Article records
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 12:14:40 +0000Blind channel identification based on second-order statistics: a frequency-domain approach
https://resolver.caltech.edu/CaltechAUTHORS:TONieeetit95
Authors: Tong, Lang; Xu, Guanghan; Hassibi, B.; Kailath, T.
Year: 1995
DOI: 10.1109/18.370088
In this communication, necessary and sufficient conditions are presented for the unique blind identification of possibly nonminimum phase channels driven by cyclostationary processes. Using a frequency domain formulation, it is first shown that a channel can be identified by the second-order statistics of the observation if and only if the channel transfer function does not have special uniformly spaced zeros. This condition leads to several necessary and sufficient conditions on the observation spectra and the channel impulse response. Based on the frequency-domain formulation, a new identification algorithm is proposed.https://authors.library.caltech.edu/records/szh0w-p7179Linear estimation in Krein spaces. Part II. Applications
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetac96b
Authors: Hassibi, Babak; Sayed, Ali H.; Kailath, Thomas
Year: 1996
DOI: 10.1109/9.481606
We have shown that several interesting problems in H∞-filtering, quadratic game theory, and risk sensitive control and estimation follow as special cases of the Krein-space linear estimation theory developed in Part I. We show that all these problems can be cast into the problem of calculating the stationary point of certain second-order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein-space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns.https://authors.library.caltech.edu/records/hch06-tpk15Linear estimation in Krein spaces. I. Theory
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetac96a
Authors: Hassibi, Babak; Sayed, Ali H.; Kailath, Thomas
Year: 1996
DOI: 10.1109/9.481605
The authors develop a self-contained theory for linear estimation in Krein spaces. The derivation is based on simple concepts such as projections and matrix factorizations and leads to an interesting connection between Krein space projection and the recursive computation of the stationary points of certain second-order (or quadratic) forms. The authors use the innovations process to obtain a general recursive linear estimation algorithm. When specialized to a state-space structure, the algorithm yields a Krein space generalization of the celebrated Kalman filter with applications in several areas such as H ∞-filtering and control, game problems, risk sensitive control, and adaptive filtering.https://authors.library.caltech.edu/records/7ay1e-0z482Inertia properties of indefinite quadratic forms
https://resolver.caltech.edu/CaltechAUTHORS:SAYieeespl96
Authors: Sayed, Ali H.; Hassibi, Babak; Kailath, Thomas
Year: 1996
DOI: 10.1109/97.484217
We study the relation between the solutions of two estimation problems with indefinite quadratic forms. We show that a complete link between both solutions can be established by invoking a fundamental set of inertia conditions. While these inertia conditions are automatically satisfied in a standard Hilbert space setting, they nevertheless turn out to mark the differences between the two estimation problems in indefinite metric spaces. They also include, as special cases, the well-known conditions for the existence of H-infinity-filters and controlers.https://authors.library.caltech.edu/records/fz0f7-v2147H∞ optimality of the LMS algorithm
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetsp96
Authors: Hassibi, Babak; Sayed, Ali H.; Kailath, Thomas
Year: 1996
DOI: 10.1109/78.485923
We show that the celebrated least-mean squares (LMS) adaptive algorithm is H∞ optimal. The LMS algorithm has been long regarded as an approximate solution to either a stochastic or a deterministic least-squares problem, and it essentially amounts to updating the weight vector estimates along the direction of the instantaneous gradient of a quadratic cost function. We show that the LMS can be regarded as the exact solution to a minimization problem in its own right. Namely, we establish that it is a minimax filter: it minimizes the maximum energy gain from the disturbances to the predicted errors, whereas the closely related so-called normalized LMS algorithm minimizes the maximum energy gain from the disturbances to the filtered errors. Moreover, since these algorithms are central H∞ filters, they minimize a certain exponential cost function and are thus also risk-sensitive optimal. We discuss the various implications of these results and show how they provide theoretical justification for the widely observed excellent robustness properties of the LMS filter.https://authors.library.caltech.edu/records/k8kpy-wcb12Array algorithms for H-infinity estimation
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetac00
Authors: Hassibi, Babak; Kailath, Thomas; Sayed, Ali H.
Year: 2000
DOI: 10.1109/9.847105
In this paper we develop array algorithms for H-infinity filtering. These algorithms can be regarded as the Krein space generalizations of H-2 array algorithms, which are currently the preferred method for implementing H-2 biters, The array algorithms considered include typo main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H-infinity filters, as these conditions are built into the algorithms themselves. However, since H-infinity square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H-2 case, further investigation is needed to determine the numerical behavior of such algorithms.https://authors.library.caltech.edu/records/7zhjb-wbm31On linear H∞ equalization of communication channels
https://resolver.caltech.edu/CaltechAUTHORS:ERDieeetsp00
Authors: Erdogan, Alper T.; Hassibi, Babak; Kailath, Thomas
Year: 2000
DOI: 10.1109/78.875478
As an alternative to existing techniques and algorithms, we investigate the merit of the H∞ approach to the linear equalization of communication channels. We first give the formulation of all causal H∞ equalizers using the results of and then look at the finite delay case. We compare the risk-sensitive H∞ equalizer with the MMSE equalizer with respect to both the average and the worst-case BER performances and illustrate the improvement due to the use of the H∞ equalizer.https://authors.library.caltech.edu/records/qp1z7-r1q78Estimation-based synthesis of H∞-optimal adaptive FIR filtersfor filtered-LMS problems
https://resolver.caltech.edu/CaltechAUTHORS:SAYieeetsp01
Authors: Sayyarrodsari, Bijan; How, Jonathan P.; Hassibi, Babak; Carrier, Alain
Year: 2001
DOI: 10.1109/78.890358
This paper presents a systematic synthesis procedure for H∞-optimal adaptive FIR filters in the context of an active noise cancellation (ANC) problem. An estimation interpretation of the adaptive control problem is introduced first. Based on this interpretation, an H∞ estimation problem is formulated, and its finite horizon prediction (filtering) solution is discussed. The solution minimizes the maximum energy gain from the disturbances to the predicted (filtered) estimation error and serves as the adaptation criterion for the weight vector in the adaptive FIR filter. We refer to this adaptation scheme as estimation-based adaptive filtering (EBAF). We show that the steady-state gain vector in the EBAF algorithm approaches that of the classical (normalized) filtered-X LMS algorithm. The error terms, however, are shown to be different. Thus, these classical algorithms can be considered to be approximations of our algorithm. We examine the performance of the proposed EBAF algorithm (both experimentally and in simulation) in an active noise cancellation problem of a one-dimensional (1-D) acoustic duct for both narrowband and broadband cases. Comparisons to the results from a conventional filtered-LMS (FxLMS) algorithm show faster convergence without compromising steady-state performance and/or robustness of the algorithm to feedback contamination of the reference signal.https://authors.library.caltech.edu/records/0tjm9-pbf77H∞ bounds for least-squares estimators
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetac01
Authors: Hassibi, Babak; Kaliath, Thomas
Year: 2001
DOI: 10.1109/9.905700
We obtain upper and lower bounds for the H∞ norm of the Kalman filter and the recursive-least-squares (RLS) algorithm, with respect to prediction and filtered errors. These bounds can be used to study the robustness properties of such estimators. One main conclusion is that, unlike H∞-optimal estimators which do not allow for any amplification of the disturbances, the least-squares estimators do allow for such amplification. This fact can be especially pronounced in the prediction error case, whereas in the filtered error case the energy amplification is at most four. Moreover, it is shown that the H∞ norm for RLS is data dependent, whereas for least-mean-squares (LMS) algorithms and normalized LMS, the H∞ norm is simply unity.https://authors.library.caltech.edu/records/ck6v7-p4m62FIR H∞ equalization
https://resolver.caltech.edu/CaltechAUTHORS:20150217-071050303
Authors: Erdogan, A. T.; Hassibi, B.; Kailath, T.
Year: 2001
DOI: 10.1016/S0165-1684(00)00253-X
We approach finite impulse response (FIR) equalization problem from an H∞ perspective. First, we formulate the calculation of the optimal H∞ performance for a given equalization setting as a semidefinite programming (SDP) problem. H∞ criterion provides a set of FIR equalizers with different optimality properties. Among these, we formulate the calculation of risk-sensitive or minimum entropy FIR filter as the constrained analytic centring problem and mixed H2/H∞ problem as another SDP. We provide examples to illustrate the procedures we described.https://authors.library.caltech.edu/records/1qb89-d7d04Representation theory for high-rate multiple-antenna code design
https://resolver.caltech.edu/CaltechAUTHORS:SHOieeetit01
Authors: Shokrollahi, Amin; Hassibi, Babak; Hochwald, Bertrand M.; Sweldens, Wim
Year: 2001
DOI: 10.1109/18.945251
Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels.https://authors.library.caltech.edu/records/993me-xy585Space-time autocoding
https://resolver.caltech.edu/CaltechAUTHORS:HOCieeetit01
Authors: Hochwald, Bertrand M.; Marzetta, Thomas L.; Hassibi, Babak
Year: 2001
DOI: 10.1109/18.959258
Prior treatments of space-time communications in Rayleigh flat fading generally assume that channel coding covers either one fading interval-in which case there is a nonzero "outage capacity"-or multiple fading intervals-in which case there is a nonzero Shannon capacity. However, we establish conditions under which channel codes span only one fading interval and yet are arbitrarily reliable. In short, space-time signals are their own channel codes. We call this phenomenon space-time autocoding, and the accompanying capacity the space-time autocapacity. Let an M-transmitter antenna, N-receiver antenna Rayleigh flat fading channel be characterized by an M×N matrix of independent propagation coefficients, distributed as zero-mean, unit-variance complex Gaussian random variables. This propagation matrix is unknown to the transmitter, it remains constant during a T-symbol coherence interval, and there is a fixed total transmit power. Let the coherence interval and number of transmitter antennas be related as T=βM for some constant β. A T×M matrix-valued signal, associated with R·T bits of information for some rate R is transmitted during the T-symbol coherence interval. Then there is a positive space-time autocapacity Ca such that for all Rhttps://authors.library.caltech.edu/records/sp4ab-cq754Maximum-Likelihood Sequence Detection of Multiple Antenna Systems over Dispersive Channels via Sphere Decoding
https://resolver.caltech.edu/CaltechAUTHORS:VIKeurasipjasp02
Authors: Vikalo, Haris; Hassibi, Babak
Year: 2002
DOI: 10.1155/S1110865702204011
Multiple antenna systems are capable of providing high data rate transmissions over wireless channels. When the channels are dispersive, the signal at each receive antenna is a combination of both the current and past symbols sent from all transmit antennas corrupted by noise. The optimal receiver is a maximum-likelihood sequence detector and is often considered to be practically infeasible due to high computational complexity (exponential in number of antennas and channel memory). Therefore, in practice, one often settles for a less complex suboptimal receiver structure, typically with an equalizer meant to suppress both the intersymbol and interuser interference, followed by the decoder. We propose a sphere decoding for the sequence detection in multiple antenna communication systems over dispersive channels. The sphere decoding provides the maximum-likelihood estimate with computational complexity comparable to the standard space-time decision-feedback equalizing (DFE) algorithms. The performance and complexity of the sphere decoding are compared with the DFE algorithm by means of simulations.https://authors.library.caltech.edu/records/a2z2j-cnc72Structured unitary space-time autocoding constellations
https://resolver.caltech.edu/CaltechAUTHORS:MARieeetit02
Authors: Marzetta, Thomas L.; Hassibi, Babak; Hochwald, Bertrand M.
Year: 2002
DOI: 10.1109/18.992790
We previously showed that arbitrarily reliable communication is possible within a single coherence interval in Rayleigh flat fading as the symbol duration of the coherence interval and the number of transmit antennas grow simultaneously. This effect, where the space-time signals act as their own channel codes, is called autocoding. For relatively short (e.g., 16-symbol) coherence intervals, a codebook of independent isotropically random unitary space-time signals theoretically supports transmission rates that are a significant fraction of autocapacity with an extremely low probability of error. The exploitation of space-time autocoding requires the creation and decoding of extraordinarily large constellations-typically L = 2^80. We make progress on the first part of the problem through a random, but highly structured, constellation that is completely specified by log2 L independent isotropically distributed unitary matrices. The distinguishing property of this construction is that any two signals in the constellation are pairwise statistically independent and isotropically distributed. Thus, the pairwise probability of error, and hence the union bound on the block probability of error, of the structured constellation is identical to that of a fully random constellation of independent signals. We establish the limitations of an earlier construction through a subsidiary result that is interesting in its own right: the square (or for that matter, any integer power greater than one) of an isotropically random unitary matrix is not isotropically random, with the sole exception of the one-by-one unitary matrix.https://authors.library.caltech.edu/records/ysax1-m3j55Cayley differential unitary space-time codes
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetit02c
Authors: Hassibi, Babak; Hochwald, Bertrand M.
Year: 2002
DOI: 10.1109/TIT.2002.1003836
One method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter, and to decode differentially without knowing the channel coefficients at the receiver. Since channel knowledge is not required at the receiver, differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Although this basic principle is well understood, it is not known how to generate good-performing constellations of unitary matrices, for any number of transmit and receive antennas and for any rate. This is especially true at high rates where the constellations must be rapidly encoded and decoded.
We propose a class of Cayley codes that works with any number of antennas, and has efficient encoding and decoding at any rate. The codes are named for their use of the Cayley transform, which maps the highly nonlinear Stiefel manifold of unitary matrices to the linear space of skew-Hermitian matrices. This transformation leads to a simple linear constellation structure in the Cayley transform domain and to an information-theoretic design criterion based on emulating a Cauchy random matrix. Moreover, the resulting Cayley codes allowpolynomial-time near-maximum-likelihood (ML) decoding based on either successive nulling/canceling or sphere decoding. Simulations show that the Cayley codes allow efficient and effective high-rate data transmission in multiantenna communication systems without knowing the channel.https://authors.library.caltech.edu/records/5q8sr-5ta06Multiple-antennas and isotropically random unitary inputs: the received signal density in closed form
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetit02b.874
Authors: Hassibi, Babak; Marzetta, Thomas L.
Year: 2002
DOI: 10.1109/TIT.2002.1003835
An important open problem in multiple-antenna communications theory is to compute the capacity of a wireless link subject to flat Rayleigh block-fading, with no channel-state information (CSI) available either to the transmitter or to the receiver. The isotropically random (i.r.) unitary matrix-having orthonormal columns, and a probability density that is invariant to premultiplication by an independent unitary matrix-plays a central role in the calculation of capacity and in some special cases happens to be capacity-achieving. We take an important step toward computing this capacity by obtaining, in closed form, the probability density of the received signal when transmitting i.r. unitary matrices. The technique is based on analytically computing the expectation of an exponential quadratic function of an i.r. unitary matrix and makes use of a Fourier integral representation of the constituent Dirac delta functions in the underlying density. Our formula for the received signal density enables us to evaluate the mutual information for any case of interest, something that could previously only be done for single transmit and receive antennas. Numerical results show that at high signal-to-noise ratio (SNR), the mutual information is maximized for M=min(N, T/2) transmit antennas, where N is the number of receive antennas and T is the length of the coherence interval, whereas at low SNR, the mutual information is maximized by allocating all transmit power to a single antenna.https://authors.library.caltech.edu/records/1esf5-r9j06High-rate codes that are linear in space and time
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetit02a
Authors: Hassibi, Babak; Hochwald, Bertrand M.
Year: 2002
DOI: 10.1109/TIT.2002.1013127
Multiple-antenna systems that operate at high rates require simple yet effective space-time transmission schemes to handle the large traffic volume in real time. At rates of tens of bits per second per hertz, Vertical Bell Labs Layered Space-Time (V-BLAST), where every antenna transmits its own independent substream of data, has been shown to have good performance and simple encoding and decoding. Yet V-BLAST suffers from its inability to work with fewer receive antennas than transmit antennas-this deficiency is especially important for modern cellular systems, where a base station typically has more antennas than the mobile handsets. Furthermore, because V-BLAST transmits independent data streams on its antennas there is no built-in spatial coding to guard against deep fades from any given transmit antenna. On the other hand, there are many previously proposed space-time codes that have good fading resistance and simple decoding, but these codes generally have poor performance at high data rates or with many antennas. We propose a high-rate coding scheme that can handle any configuration of transmit and receive antennas and that subsumes both V-BLAST and many proposed space-time block codes as special cases. The scheme transmits substreams of data in linear combinations over space and time. The codes are designed to optimize the mutual information between the transmitted and received signals. Because of their linear structure, the codes retain the decoding simplicity of V-BLAST, and because of their information-theoretic optimality, they possess many coding advantages. We give examples of the codes and show that their performance is generally superior to earlier proposed methods over a wide range of rates and signal-to-noise ratios (SNRs).https://authors.library.caltech.edu/records/81we3-0ct81How much training is needed in multiple-antenna wireless links?
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetit03
Authors: Hassibi, Babak; Hochwald, Bertrand M.
Year: 2003
DOI: 10.1109/TIT.2003.809594
Multiple-antenna wireless communication links promise very high data rates with low error probabilities, especially when the wireless channel response is known at the receiver. In practice, knowledge of the channel is often obtained by sending known training symbols to the receiver. We show how training affects the capacity of a fading channel-too little training and the channel is improperly learned, too much training and there is no time left for data transmission before the channel changes. We compute a lower bound on the capacity of a channel that is learned by training, and maximize the bound as a function of the received signal-to-noise ratio (SNR), fading coherence time, and number of transmitter antennas. When the training and data powers are allowed to vary, we show that the optimal number of training symbols is equal to the number of transmit antennas-this number is also the smallest training interval length that guarantees meaningful estimates of the channel matrix. When the training and data powers are instead required to be equal, the optimal number of symbols may be larger than the number of antennas. We show that training-based schemes can be optimal at high SNR, but suboptimal at low SNR.https://authors.library.caltech.edu/records/y6rsb-e9851The academic and industrial embrace of space-time methods
https://resolver.caltech.edu/CaltechAUTHORS:HOCieeetit03
Authors: Hochwald, Bertrand M.; Caire, Giuseppe; Hassibi, Babak; Marzetta, Thomas L.
Year: 2003
DOI: 10.1109/TIT.2003.817832
[Guest Editors introduction to: Special issue on space-time transmission, reception, coding and signal processing]
Every episode of the classic 1966–1969 television series Star Trek begins with Captain Kirk's (played by William Shatner) famous words : "Space: The final frontier…." While space may not be the final frontier for the information and communication theory community, it is proving to be an important and fruitful one.
In the information theory community, the notion of space can be broadly defined as the simultaneous use of multiple, possibly coupled, channels. The notions of space–time and multiple-input multiple-output (MIMO) channels are therefore often used interchangeably. The connection between space and MIMO is most transparent when we view the multiple channels as created by two or more spatially separated antennas at a wireless transmitter or receiver.
A large component of the current interest in space–time methods can be attributed to discoveries in the late 1980s and early 1990s that a rich wireless scattering environment can be beneficial when multiple antennas are used on a point-to-point link. We now know that adding antennas in a rich environment provides proportional increases in point-to-point data rates, without extra transmitted power or bandwidth.https://authors.library.caltech.edu/records/jze5r-14q44Unitary space-time modulation via Cayley transform
https://resolver.caltech.edu/CaltechAUTHORS:JINieeetsp03
Authors: Jing, Yindi; Hassibi, Babak
Year: 2003
DOI: 10.1109/TSP.2003.818202
A prevoiusly proposed method for communicating with multiple antennas over block fading channels is unitary space-time modulation (USTM). In this method, the signals transmitted from the antennas, viewed as a matrix with spatial and temporal dimensions, form a unitary matrix, i.e., one with orthonormal columns. Since channel knowledge is not required at the receiver, USTM schemes are suitable for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Previous results have shown that if suitably designed, USTM schemes can achieve full channel capacity at high SNR and, moreover, that all this can be done over a single coherence interval, provided the coherence interval and number of transmit antennas are sufficiently large, which is a phenomenon referred to as autocoding. While all this is well recognized, what is not clear is how to generate good performing constellations of (nonsquare) unitary matrices that lend themselves to efficient encoding/decoding. The schemes proposed so far either exhibit poor performance, especially at high rates, or have no efficient decoding algorithms. We propose to use the Cayley transform to design USTM constellations. This work can be viewed as a generalization, to the nonsquare case, of the Cayley codes that have been proposed for differential USTM. The codes are designed based on an information-theoretic criterion and lend themselves to polynomial-time (often cubic) near-maximum-likelihood decoding using a sphere decoding algorithm. Simulations suggest that the resulting codes allow for effective high-rate data transmission in multiantenna communication systems without knowing the channel. However, our preliminary results do not show a substantial advantage over training-based schemes.https://authors.library.caltech.edu/records/mtyhb-enn28On the achievable average power reduction of MSM optical signals
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeecl04
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2004
DOI: 10.1109/LCOMM.2004.823418
In this letter, we consider the achievable average power reduction of multiple subcarrier modulated optical signals by using optimized reserved carriers. Based on Nehari's result we present a lower bound for the maximum average power of the signal after adding the reserved carriers. Simulations show that the mean value of the average required power behaves very close to /spl radic/(2nloglogn) for binary phase-shift keying (BPSK) constellations where n is the number of subcarriers. We further remark on evaluating optimum values for reserved carriers using convex optimization and Nehari's result.https://authors.library.caltech.edu/records/bjdjd-76820MIMO decision feedback equalization from an H∞ perspective
https://resolver.caltech.edu/CaltechAUTHORS:ERDieeetsp04
Authors: Erdogan, Alper Tunga; Hassibi, Babak; Kailath, Thomas
Year: 2004
DOI: 10.1109/TSP.2003.822289
We approach the multiple input multiple output (MIMO) decision feedback equalization (DFE) problem in digital communications from an H∞ estimation point of view. Using the standard (and simplifying) assumption that all previous decisions are correct, we obtain an explicit parameterization of all H∞ optimal DFEs. In particular, we show that, under the above assumption, minimum mean square error (MMSE) DFEs are H∞ optimal. The H∞ approach also suggests a method for dealing with errors in previous decisions.https://authors.library.caltech.edu/records/jq31k-deb32On multicarrier signals where the PMEPR of a random codeword is asymptotically log n
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetit04
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2004
DOI: 10.1109/TIT.2004.826681
Multicarrier signals exhibit a large peak-to-mean envelope power ratio (PMEPR). In this correspondence, without using a Gaussian assumption, we derive lower and upper probability bounds for the PMEPR distribution when the number of subcarriers n is large. Even though the worst case PMEPR is of the order of n, the main result is that the PMEPR of a random codeword C=(c/sub 1/,...,c/sub n/) is logn with probability approaching one asymptotically, for the following three general cases: i) c/sub i/'s are independent and identically distributed (i.i.d.) chosen from a complex quadrature amplitude modulation (QAM) constellation in which the real and imaginary part of c/sub i/ each has i.i.d. and even distribution (not necessarily uniform), ii) c/sub i/'s are i.i.d. chosen from a phase-shift keying (PSK) constellation where the distribution over the constellation points is invariant under /spl pi//2 rotation, and iii) C is chosen uniformly from a complex sphere of dimension n. Based on this result, it is proved that asymptotically, the Varshamov-Gilbert (VG) bound remains the same for codes with PMEPR of less than logn chosen from QAM/PSK constellations.https://authors.library.caltech.edu/records/md2w5-20b56Optimal quantum detectors for unambiguous detection of mixed states
https://resolver.caltech.edu/CaltechAUTHORS:20150211-074354159
Authors: Eldar, Yonina C.; Stojnic, Mihailo; Hassibi, Babak
Year: 2004
DOI: 10.1103/PhysRevA.69.062318
We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that maximizes the probability of correct detection. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these states share the same symmetries, and can be computed very efficiently by solving a reduced size semidefinite program.https://authors.library.caltech.edu/records/ta06a-k3718On the capacity of frequency-selective channels in training-based transmission schemes
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetsp04
Authors: Vikalo, Haris; Hassibi, Babak; Hochwald, Bertrand; Kailath, Thomas
Year: 2004
DOI: 10.1109/TSP.2004.832020
Communication systems transmitting over frequency-selective channels generally employ an equalizer to recover the transmitted sequence corrupted by intersymbol interference (ISI). Most practical systems use a training sequence to learn the channel impulse response and thereby design the equalizer. An important issue is determining the optimal amount of training: too little training and the channel is not learned properly, too much training and there is not enough time available to transmit data before the channel changes and must be learned anew. We use an information-theoretic approach to find the optimal parameters in training-based transmission schemes for channels described by a block-fading model. The optimal length of the training interval is found by maximizing a lower bound on the training-based channel capacity. When the transmitter is capable of providing two distinct transmission power levels (one for training and one for data transmission), the optimal length of the training interval is shown to be equal to the length of the channel. Further, we show that at high SNR, training-based schemes achieve the capacity of block-fading frequency selective channels, whereas at low SNR, they are highly suboptimal.https://authors.library.caltech.edu/records/va3nq-acs71Analysis of multiple-antenna wireless links at low SNR
https://resolver.caltech.edu/CaltechAUTHORS:RAOieeetit04
Authors: Rao, Chaitanya; Hassibi, Babak
Year: 2004
DOI: 10.1109/TIT.2004.833369
Wireless channels with multiple transmit/receive antennas are known to provide a high spectral efficiency both when the channel is known to the receiver, and when the channel is not known to the receiver if the signal-to-noise ratio (SNR) is high. Here we analyze such systems at low SNR, which may find application in sensor networks and other low-power devices. The key point is that, since channel estimates are not reliable, it is often not reasonable to assume that the channel is known at the receiver at low SNR. In this unknown channel case, we show that for sensible input distributions, in particular all practical modulation schemes, the capacity is asymptotically quadratic in the SNR, /spl rho/, and thus much less than the known channel case where it exhibits a linear growth in /spl rho/. We show that under various signaling constraints, e.g., Gaussian modulation, unitary space-time modulation, and peak constraints, that mutual information is maximized by using a single transmit antenna. We also show that at low SNR, sending training symbols leads to a rate reduction in proportion to the fraction of training duration time so that it is best not to perform training. Furthermore, we show that the per-channel use mutual information is linear in both the number of receive antennas and the channel coherence interval.https://authors.library.caltech.edu/records/cc36x-yzm64Existence of codes with constant PMEPR and related design
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetsp04
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2004
DOI: 10.1109/TSP.2004.834343
Recently, several coding methods have been proposed to reduce the high peak-to-mean envelope ratio (PMEPR) of multicarrier signals. It has also been shown that with probability one, the PMEPR of any random codeword chosen from a symmetric quadrature amplitude modulation/phase shift keying (QAM/PSK) constellation is logn for large n, where n is the number of subcarriers. Therefore, the question is how much reduction beyond logn can one asymptotically achieve with coding, and what is the price in terms of the rate loss? In this paper, by optimally choosing the sign of each subcarrier, we prove the existence of q-ary codes of constant PMEPR for sufficiently large n and with a rate loss of at most log/sub q/2. We also obtain a Varsharmov-Gilbert-type upper bound on the rate of a code, given its minimum Hamming distance with constant PMEPR, for large n. Since ours is an existence result, we also study the problem of designing signs for PMEPR reduction. Motivated by a derandomization algorithm suggested by Spencer, we propose a deterministic and efficient algorithm to design signs such that the PMEPR of the resulting codeword is less than clogn for any n, where c is a constant independent of n. For symmetric q-ary constellations, this algorithm constructs a code with rate 1-log/sub q/2 and with PMEPR of clogn with simple encoding and decoding. Simulation results for our algorithm are presented.https://authors.library.caltech.edu/records/dbd0d-b9a64Iterative decoding for MIMO channels via modified sphere decoding
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetwc04
Authors: Vikalo, H.; Hassibi, B.; Kailath, T.
Year: 2004
DOI: 10.1109/TWC.2004.837271
In recent years, soft iterative decoding techniques have been shown to greatly improve the bit error rate performance of various communication systems. For multiantenna systems employing space-time codes, however, it is not clear what is the best way to obtain the soft information required of the iterative scheme with low complexity. In this paper, we propose a modification of the Fincke-Pohst (sphere decoding) algorithm to estimate the maximum a posteriori probability of the received symbol sequence. The new algorithm solves a nonlinear integer least squares problem and, over a wide range of rates and signal-to-noise ratios, has polynomial-time complexity. Performance of the algorithm, combined with convolutional, turbo, and low-density parity check codes, is demonstrated on several multiantenna channels. The results for systems that employ space-time modulation schemes seem to indicate that the best performing schemes are those that support the highest mutual information between the transmitted and received signals, rather than the best diversity gain.https://authors.library.caltech.edu/records/82req-ch147Design of fully diverse multiple-antenna codes based on Sp(2)
https://resolver.caltech.edu/CaltechAUTHORS:JINieeetit04
Authors: Jing, Yindi; Hassibi, Babak
Year: 2004
DOI: 10.1109/TIT.2004.836701
Fully diverse constellations, i.e., sets of unitary matrices whose pairwise differences are nonsingular, are useful in multiple-antenna communications, especially in multiple-antenna differential modulation, since they have good pairwise error properties. Recently, group theoretic ideas, especially fixed-point-free (fpf) groups, have been used to design fully diverse constellations of unitary matrices. Here we construct four-transmit-antenna constellations appropriate for differential modulation based on the symplectic group Sp(2). They can be regarded as extensions of Alamouti's celebrated two-transmit-antenna orthogonal design which can be constructed from the group Sp(1). We further show that the structure of Sp(2) codes lends itself to efficient maximum-likelihood (ML) decoding via the sphere decoding algorithm. Finally, the performance of Sp(2) codes is compared with that of other existing codes including Alamouti's orthogonal design, a 4/spl times/4 complex orthogonal design, Cayley differential unitary space-time codes and group-based codes.https://authors.library.caltech.edu/records/xb4ws-26615On robust signal reconstruction in noisy filter banks
https://resolver.caltech.edu/CaltechAUTHORS:20150210-072703948
Authors: Vikalo, Haris; Hassibi, Babak; Erdogan, Alper T.
Year: 2005
DOI: 10.1016/j.sigpro.2004.08.011
We study the design of synthesis filters in noisy filter bank systems using an H∞ estimation point of view. The H∞ approach is most promising in situations where the statistical properties of the disturbances (arising from quantization, compression, etc.) in each subband of the filter bank is unknown, or is too difficult to model and analyze. For the important special case of unitary analysis polyphase matrices we obtain an explicit expression for the minimum achievable disturbance attenuation. For arbitrary analysis polyphase matrices, standard state-space H∞ techniques can be employed to obtain numerical solutions. When the synthesis filters are restricted to being FIR, as is often the case in practice, the design can be cast as a finite-dimensional semi-definite program. In this case, we can effectively exploit the inherent non-uniqueness of the H∞ solution to optimize for an additional criteria. By optimizing for average performance in addition to the H∞ criteria, we obtain mixed H^2/H∞ optimal FIR synthesis filters. Alternatively, if the additional criteria is concerned with penalizing occasional occurrence of large values of reconstruction errors more than frequent occurrence of small to moderate ones, we obtain risk-sensitive FIR synthesis filters. Numerical examples and comparisons with existing methods are also included.https://authors.library.caltech.edu/records/0z4gt-x3w83On the capacity of MIMO broadcast channels with partial side information
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetit05
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2005
DOI: 10.1109/TIT.2004.840897
In multiple-antenna broadcast channels, unlike point-to-point multiple-antenna channels, the multiuser capacity depends heavily on whether the transmitter knows the channel coefficients to each user. For instance, in a Gaussian broadcast channel with M transmit antennas and n single-antenna users, the sum rate capacity scales like Mloglogn for large n if perfect channel state information (CSI) is available at the transmitter, yet only logarithmically with M if it is not. In systems with large n, obtaining full CSI from all users may not be feasible. Since lack of CSI does not lead to multiuser gains, it is therefore of interest to investigate transmission schemes that employ only partial CSI. We propose a scheme that constructs M random beams and that transmits information to the users with the highest signal-to-noise-plus-interference ratios (SINRs), which can be made available to the transmitter with very little feedback. For fixed M and n increasing, the throughput of our scheme scales as MloglognN, where N is the number of receive antennas of each user. This is precisely the same scaling obtained with perfect CSI using dirty paper coding. We furthermore show that a linear increase in throughput with M can be obtained provided that M does not not grow faster than logn. We also study the fairness of our scheduling in a heterogeneous network and show that, when M is large enough, the system becomes interference dominated and the probability of transmitting to any user converges to 1/n, irrespective of its path loss. In fact, using M=αlogn transmit antennas emerges as a desirable operating point, both in terms of providing linear scaling of the throughput with M as well as in guaranteeing fairness.https://authors.library.caltech.edu/records/4h51t-3kt97On the sphere-decoding algorithm II. Generalizations, second-order statistics, and applications to communications
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetsp05
Authors: Vikalo, Haris; Hassibi, Babak
Year: 2005
DOI: 10.1109/TSP.2005.850350
In Part 1, we found a closed-form expression for the expected complexity of the sphere-decoding algorithm, both for the infinite and finite lattice. We continue the discussion in this paper by generalizing the results to the complex version of the problem and using the expected complexity expressions to determine situations where sphere decoding is practically feasible. In particular, we consider applications of sphere decoding to detection in multiantenna systems. We show that, for a wide range of signal-to-noise ratios (SNRs), rates, and numbers of antennas, the expected complexity is polynomial, in fact, often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can, in fact, be implemented in real-time-a result with many practical implications. To provide complexity information beyond the mean, we derive a closed-form expression for the variance of the complexity of sphere-decoding algorithm in a finite lattice. Furthermore, we consider the expected complexity of sphere decoding for channels with memory, where the lattice-generating matrix has a special Toeplitz structure. Results indicate that the expected complexity in this case is, too, polynomial over a wide range of SNRs, rates, data blocks, and channel impulse response lengths.https://authors.library.caltech.edu/records/4b4t3-b2086On the sphere-decoding algorithm I. Expected complexity
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetsp05
Authors: Hassibi, Babak; Vikalo, Haris
Year: 2005
DOI: 10.1109/TSP.2005.850352
The problem of finding the least-squares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worst-case complexity of the integer least-squares problem, we study its expected complexity, averaged over the noise and over the lattice. For the "sphere decoding" algorithm of Fincke and Pohst, we find a closed-form expression for the expected complexity, both for the infinite and finite lattice. It is demonstrated in the second part of this paper that, for a wide range of signal-to-noise ratios (SNRs) and numbers of antennas, the expected complexity is polynomial, in fact, often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can, in fact, be implemented in real time - a result with many practical implications.https://authors.library.caltech.edu/records/qg8yc-5e185Amplitude and Sign Adjustment for Peak-to-Average-Power Reduction
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetc05
Authors: Sharif, Masoud; Florens, Cedric; Fazel, Maryam; Hassibi, Babak
Year: 2005
DOI: 10.1109/TCOMM.2005.852830
In this letter, we propose a method to reduce the peak-to-mean-envelope-power ratio (PMEPR) of multicarrier signals by modifying the constellation. For$M$-ary phase-shift keying constellations, we minimize the maximum of the multicarrier signal over the sign and amplitude of each subcarrier. In order to find an efficient solution to the aforementioned nonconvex optimization problem, we present a suboptimal solution by first optimizing over the signs, and then optimizing over the amplitudes given the signs. We prove that the minimization of the maximum of a continuous multicarrier signal over the amplitude of each subcarrier can be written as a convex optimization problem with linear matrix inequality constraints. We also generalize the idea to other constellations such as 16-quadrature amplitude modulation. Simulation results show that by an average power increase of 0.21 dB, and not sending information over the sign of each subcarrier, PMEPR can be decreased by 5.1 dB for a system with 128 subcarriers.https://authors.library.caltech.edu/records/j8nsm-qea18Three-transmit-antenna space-time codes based on SU(3)
https://resolver.caltech.edu/CaltechAUTHORS:JINieeetsp05
Authors: Jing, Yindi; Hassibi, Babak
Year: 2005
DOI: 10.1109/TSP.2005.855092
Fully diverse constellations, i.e., a set of unitary matrices whose pairwise differences are nonsingular, are useful in multiantenna communications especially in multiantenna differential modulation, since they have good pairwise error properties. Recently, group theoretic ideas, especially fixed-point-free (fpf) groups, have been used to design fully diverse constellations of unitary matrices. Here, we give systematic design methods of space-time codes which are appropriate for three-transmit-antenna differential modulation. The structures of the codes are motivated by the special unitary Lie group SU(3). One of the codes, which is called the AB code, has a fast maximum-likelihood (ML) decoding algorithm using complex sphere decoding. Diversity products of the codes can be easily calculated, and simulated performance shows that they are better than group-based codes, especially at high rates and as good as the elaborately designed nongroup code.https://authors.library.caltech.edu/records/1szab-swq82On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage
https://resolver.caltech.edu/CaltechAUTHORS:20110426-130530739
Authors: Gupta, Vijay; Chung, Timothy H.; Hassibi, Babak; Murray, Richard M.
Year: 2006
DOI: 10.1016/j.automatica.2005.09.016
In this note we consider the following problem. Suppose a set of sensors is jointly trying to estimate a process. One sensor takes a measurement at every time step and the measurements are then exchanged among all the sensors. What is the sensor schedule that results in the minimum error covariance? We describe a stochastic sensor selection strategy that is easy to implement and is computationally tractable. The problem described above comes up in many domains out of which we discuss two. In the sensor selection problem, there are multiple sensors that cannot operate simultaneously (e.g., sonars in the same frequency band). Thus measurements need to be scheduled. In the sensor coverage problem, a geographical area needs to be covered by mobile sensors each with limited range. Thus from every position, the sensors obtain a different view-point of the area and the sensors need to optimize their trajectories. The algorithm is applied to these problems and illustrated through simple examples.https://authors.library.caltech.edu/records/5j7kx-bdm52MIMO linear equalization with an H∞ criterion
https://resolver.caltech.edu/CaltechAUTHORS:HASieeetsp06
Authors: Hassibi, Babak; Erdogan, Alper T.; Kailath, Thomas
Year: 2006
DOI: 10.1109/TSP.2005.861888
In this paper, we study the problem of linearly equalizing the multiple-input multiple-output (MIMO) communications channels from an H∞ point of view. H∞ estimation theory has been recently introduced as a method for designing filters that have acceptable performance in the face of model uncertainty and lack of statistical information on the exogenous signals. In this paper, we obtain a closed-form solution to the square MIMO linear H∞ equalization problem and parameterize all possible H∞-optimal equalizers. In particular, we show that, for minimum phase channels, a scaled version of the zero-forcing equalizer is H∞-optimal. The results also indicate an interesting dichotomy between minimum phase and nonminimum phase channels: for minimum phase channels the best causal equalizer performs as well as the best noncausal equalizer, whereas for nonminimum phase channels, causal equalizers cannot reduce the estimation error bounds from their a priori values. Our analysis also suggests certain remedies in the nonminimum phase case, namely, allowing for finite delay, oversampling, or using multiple sensors. For example, we show that H∞ equalization of nonminimum phase channels requires a time delay of at least l units, where l is the number of nonminimum phase zeros of the channel.https://authors.library.caltech.edu/records/zcv78-7c540Capacity of wireless erasure networks
https://resolver.caltech.edu/CaltechAUTHORS:DANieeetit06a
Authors: Dana, Amir F.; Gowaikar, Radhika; Palanki, Ravi; Hassibi, Babak; Effros, Michelle
Year: 2006
DOI: 10.1109/TIT.2005.864424
In this paper, a special class of wireless networks, called wireless erasure networks, is considered. In these networks, each node is connected to a set of nodes by possibly correlated erasure channels. The network model incorporates the broadcast nature of the wireless environment by requiring each node to send the same signal on all outgoing channels. However, we assume there is no interference in reception. Such models are therefore appropriate for wireless networks where all information transmission is packetized and where some mechanism for interference avoidance is already built in. This paper looks at multicast problems over these networks. The capacity under the assumption that erasure locations on all the links of the network are provided to the destinations is obtained. It turns out that the capacity region has a nice max-flow min-cut interpretation. The definition of cut-capacity in these networks incorporates the broadcast property of the wireless medium. It is further shown that linear coding at nodes in the network suffices to achieve the capacity region. Finally, the performance of different coding schemes in these networks when no side information is available to the destinations is analyzed.https://authors.library.caltech.edu/records/yrg6y-jan67The p-norm generalization of the LMS algorithm for adaptive filtering
https://resolver.caltech.edu/CaltechAUTHORS:KIVieeetsp06
Authors: Kivinen, Jyrki; Warmuth, Manfred K.; Hassibi, Babak
Year: 2006
DOI: 10.1109/TSP.2006.872551
Recently much work has been done analyzing online machine learning algorithms in a worst case setting, where no probabilistic assumptions are made about the data. This is analogous to the H/sup /spl infin// setting used in adaptive linear filtering. Bregman divergences have become a standard tool for analyzing online machine learning algorithms. Using these divergences, we motivate a generalization of the least mean squared (LMS) algorithm. The loss bounds for these so-called p-norm algorithms involve other norms than the standard 2-norm. The bounds can be significantly better if a large proportion of the input variables are irrelevant, i.e., if the weight vector we are trying to learn is sparse. We also prove results for nonstationary targets. We only know how to apply kernel methods to the standard LMS algorithm (i.e., p=2). However, even in the general p-norm case, we can handle generalized linear models where the output of the system is a linear function combined with a nonlinear transfer function (e.g., the logistic sigmoid).https://authors.library.caltech.edu/records/a8h45-2zf83A statistical model for microarrays, optimal estimation algorithms, and limits of performance
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetsp06
Authors: Vikalo, Haris; Hassibi, Babak; Hassibi, Arjang
Year: 2006
DOI: 10.1109/TSP.2006.873716
DNA microarray technology relies on the hybridization process, which is stochastic in nature. Currently, probabilistic cross hybridization of nonspecific targets, as well as the shot noise (Poisson noise) originating from specific targets binding, are among the main obstacles for achieving high accuracy in DNA microarray analysis. In this paper, statistical techniques are used to model the hybridization and cross-hybridization processes and, based on the model, optimal algorithms are employed to detect the targets and to estimate their quantities. To verify the theory, two sets of microarray experiments are conducted: one with oligonucleotide targets and the other with complementary DNA (cDNA) targets in the presence of biological background. Both experiments indicate that, by appropriately modeling the cross-hybridization interference, significant improvement in the accuracy over conventional methods such as direct readout can be obtained. This substantiates the fact that the accuracy of microarrays can become exclusively noise limited, rather than interference (i.e., cross-hybridization) limited. The techniques presented in this paper potentially increase considerably the signal-to-noise ratio (SNR), dynamic range, and resolution of DNA and protein microarrays as well as other affinity-based biosensors. A preliminary study of the Cramer-Rao bound for estimating the target concentrations suggests that, in some regimes, cross hybridization may even be beneficial-a result with potential ramifications for probe design, which is currently focused on minimizing cross hybridization. Finally, in its current form, the proposed method is best suited to low-density arrays arising in diagnostics, single nucleotide polymorphism (SNP) detection, toxicology, etc. How to scale it to high-density arrays (with many thousands of spots) is an interesting challenge.https://authors.library.caltech.edu/records/ae9de-70f88Sphere-constrained ML detection for frequency-selective channels
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetc06
Authors: Vikalo, H.; Hassibi, B.; Mitra, U.
Year: 2006
DOI: 10.1109/TCOMM.2006.877946
The maximum-likelihood (ML) sequence detection problem for channels with memory is investigated. The Viterbi algorithm (VA) provides an exact solution. Its computational complexity is linear in the length of the transmitted sequence, but exponential in the channel memory length. On the other hand, the sphere decoding (SD) algorithm also solves the ML detection problem exactly, and has expected complexity which is a low-degree polynomial (often cubic) in the length of the transmitted sequence over a wide range of signal-to-noise ratios. We combine the sphere-constrained search strategy of SD with the dynamic programming principles of the VA. The resulting algorithm has the worst-case complexity determined by the VA, but often significantly lower expected complexity.https://authors.library.caltech.edu/records/jw7tg-cmj66On the Power Efficiency of Sensory and Ad Hoc Wireless Networks
https://resolver.caltech.edu/CaltechAUTHORS:DANieeetit06b
Authors: Dana, Amir F.; Hassibi, Babak
Year: 2006
DOI: 10.1109/TIT.2006.876245
We consider the power efficiency of a communications channel, i.e., the maximum bit rate that can be achieved per unit power (energy rate). For additive white Gaussian noise (AWGN) channels, it is well known that power efficiency is attained in the low signal-to-noise ratio (SNR) regime where capacity is proportional to the transmit power. In this paper, we first show that for a random sensory wireless network with n users (nodes) placed in a domain of fixed area, with probability converging to one as n grows, the power efficiency scales at least by a factor of sqrt n. In other words, each user in a wireless channel with n nodes can support the same communication rate as a single-user system, but by expending only 1/(sqrt n) times the energy. Then we look at a random ad hoc network with n relay nodes and r simultaneous transmitter/receiver pairs located in a domain of fixed area. We show that as long as r ≤ sqrt n, we can achieve a power efficiency that scales by a factor of sqrt n. We also give a description of how to achieve these gains.https://authors.library.caltech.edu/records/fqf7r-rf941High-rate codes with bounded PMEPR for BPSK and other symmetric constellations
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetc06
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2006
DOI: 10.1109/TCOMM.2006.877964
In this letter, we consider the problem of constructing high-rate codes with low peak-to-mean-envelope power ratio (PMEPR) for multicarrier signals. Assuming coefficients of the multicarrier signal are chosen from a symmetric q-ary constellation, we construct codes with rate 1-(1/r)log/sub q/2 and PMEPR of less than crlogn for any r and n, where n is the number of subcarriers and c is a constant independent of n and r. The construction is based on dividing n subcarriers into n/r groups of r subcarriers and choosing a sign for each group to minimize the PMEPR. The signs are chosen using a variation of the algorithm proposed by the authors in previous papers. For large n, we can, in fact, construct a code with a rate of 1-O(1/logn) and PMEPR of less than clog/sup 2/n. For binary phase-shift-keying-modulated signals, this partially solves the problem posed by Litsyn and implies a construction of 2/sup n/2/ codewords with PMEPR less than 2clogn.https://authors.library.caltech.edu/records/125j6-8ay42Efficient joint maximum-likelihood channel estimation and signal detection
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetwc06
Authors: Vikalo, Haris; Hassibi, Babak; Stoica, Petre
Year: 2006
In wireless communication systems, channel state information is often assumed to be available at the receiver. Traditionally, a training sequence is used to obtain the estimate of the channel. Alternatively, the channel can be identified using known properties of the transmitted signal. However, the computational effort required to find the joint ML solution to the symbol detection and channel estimation problem increases exponentially with the dimension of the problem. To significantly reduce this computational effort, we formulate the joint ML estimation and detection as an integer least-squares problem, and show that for a wide range of signal-to-noise ratios (SNR) and problem dimensions it can be solved via sphere decoding with expected complexity comparable to the complexity of heuristic
techniques.https://authors.library.caltech.edu/records/nn052-yp585Communication Over a Wireless Network With Random Connections
https://resolver.caltech.edu/CaltechAUTHORS:GOWieeetit06
Authors: Gowaikar, Radhika; Hochwald, Bertrand; Hassibi, Babak
Year: 2006
DOI: 10.1109/TIT.2006.876254
A network of nodes in which pairs communicate over a shared wireless medium is analyzed. We consider the maximum total aggregate traffic flow possible as given by the number of users multiplied by their data rate. The model in this paper differs substantially from the many existing approaches in that the channel connections in this network are entirely random: rather than being governed by geometry and a decay-versus-distance law, the strengths of the connections between nodes are drawn independently from a common distribution. Such a model is appropriate for environments where the first-order effect that governs the signal strength at a receiving node is a random event (such as the existence of an obstacle), rather than the distance from the transmitter. It is shown that the aggregate traffic flow as a function of the number of nodes n is a strong function of the channel distribution. In particular, for certain distributions the aggregate traffic flow is at least n/(log n)^d for some d≫0, which is significantly larger than the O(sqrt n) results obtained for many geometric models. The results provide guidelines for the connectivity that is needed for large aggregate traffic. The relation between the proposed model and existing distance-based models is shown in some cases.https://authors.library.caltech.edu/records/r62rc-wvz93Rate maximization in multi-antenna broadcast channels with linear preprocessing
https://resolver.caltech.edu/CaltechAUTHORS:STOieeetwc06
Authors: Stojnic, Mihailo; Vikalo, Haris; Hassibi, Babak
Year: 2006
DOI: 10.1109/TWC.2006.04475
The sum rate capacity of the multi-antenna broadcast channel has recently been computed. However, the search for efficient practical schemes that achieve it is still ongoing. In this paper, we focus on schemes with linear preprocessing of the transmitted data. We propose two criteria for the precoding matrix design: one maximizing the sum rate and the other maximizing the minimum rate among all users. The latter problem is shown to be quasiconvex and is solved exactly via a bisection method. In addition to precoding, we employ a signal scaling scheme that minimizes the average bit-error-rate (BER). The signal scaling scheme is posed as a convex optimization problem, and thus can be solved exactly via efficient interior-point methods. In terms of the achievable sum rate, the proposed technique significantly outperforms traditional channel inversion methods, while having comparable (in fact, often superior) BER performance.https://authors.library.caltech.edu/records/5y25z-y8v67On joint detection and decoding of linear block codes on Gaussian vector channels
https://resolver.caltech.edu/CaltechAUTHORS:VIKieeetsp06b
Authors: Vikalo, Haris; Hassibi, Babak
Year: 2006
DOI: 10.1109/TSP.2006.877675
Optimal receivers recovering signals transmitted across noisy communication channels employ a maximum-likelihood (ML) criterion to minimize the probability of error. The problem of finding the most likely transmitted symbol is often equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In systems that employ error-correcting coding for data protection, the symbol space forms a sparse lattice, where the sparsity structure is determined by the code. In such systems, ML data recovery may be geometrically interpreted as a search for the closest point in the sparse lattice. In this paper, motivated by the idea of the "sphere decoding" algorithm of Fincke and Pohst, we propose an algorithm that finds the closest point in the sparse lattice to the given vector. This given vector is not arbitrary, but rather is an unknown sparse lattice point that has been perturbed by an additive noise vector whose statistical properties are known. The complexity of the proposed algorithm is thus a random variable. We study its expected value, averaged over the noise and over the lattice. For binary linear block codes, we find the expected complexity in closed form. Simulation results indicate significant performance gains over systems employing separate detection and decoding, yet are obtained at a complexity that is practically feasible over a wide range of system parameters.https://authors.library.caltech.edu/records/26nqj-jd261Distributed Space-Time Coding in Wireless Relay Networks
https://resolver.caltech.edu/CaltechAUTHORS:JINieeetwc06
Authors: Jing, Yindi; Hassibi, Babak
Year: 2006
DOI: 10.1109/TWC.2006.256975
We apply the idea of space-time coding devised for multiple-antenna systems to the problem of communications over a wireless relay network with Rayleigh fading channels. We use a two-stage protocol, where in one stage the transmitter sends information and in the other, the relays encode their received signals into a "distributed" linear dispersion (LD) code, and then transmit the coded signals to the receive node. We show that for high SNR, the pairwise error probability (PEP) behaves as (log P/P)min{T,R}, with T the coherence interval, that is, the number of symbol periods during which the channels keep constant, R the number of relay nodes, and P the total transmit power. Thus, apart from the log P factor, the system has the same diversity as a multiple-antenna system with R transmit antennas, which is the same as assuming that the R relays can fully cooperate and have full knowledge of the transmitted signal. We further show that for a network with a large number of relays and a fixed total transmit power across the entire network, the optimal power allocation is for the transmitter to expend half the power and for the relays to collectively expend the other half. We also show that at low and high SNR, the coding gain is the same as that of a multiple-antenna system with R antennas. However, at intermediate SNR, it can be quite different, which has implications for the design of distributed space-time codes.https://authors.library.caltech.edu/records/s82rv-z8c96A scheme for cancelling intercarrier interference using conjugate transmission in multicarrier communication systems
https://resolver.caltech.edu/CaltechAUTHORS:YEHieeetwc07
Authors: Yeh, Hen-Geul; Chang, Yuan-Kwei; Hassibi, Babak
Year: 2007
DOI: 10.1109/TWC.2007.04541
To mitigate intercarrier interference (ICI), a two-path algorithm is developed for multicarrier communication systems, including orthogonal frequency division multiplexing (OFDM) systems. The first path employs the regular OFDM algorithm. The second path uses the conjugate transmission of the first path. The combination of both paths forms a conjugate ICI cancellation scheme at the receiver. This conjugate cancellation (CC) scheme provides (1) a high signal to interference power ratio (SIR) in the presence of small frequency offsets (50 dB and 33 dB higher than that of the regular OFDM and linear self-cancellation algorithms [1], [2], respectively, at ΔfT = 0.1% of subcarrier frequency spacing); (2) better bit error rate (BER) performance in both additive white Gaussian noise (AWGN) and fading channels; (3) backward compatibility with the existing OFDM system; (4) no channel equalization is needed for reducing ICI, a simple low cost receiver without increasing system complexity. Although the two-path transmission reduces bandwidth efficiency, the disadvantage can be balanced by increasing signal alphabet sizes.https://authors.library.caltech.edu/records/acn7e-vhr29A Comparison of Time-Sharing, DPC, and Beamforming for MIMO Broadcast Channels With Many Users
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetc07
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2007
DOI: 10.1109/TCOMM.2006.887480
In this letter, we derive the scaling laws of the sum rate for fading multiple-input multiple-output Gaussian broadcast channels using time sharing to the strongest user, dirty-paper coding (DPC), and beamforming, when the number of users (receivers) n is large. Throughout the letter, we assume a fix average transmit power and consider a block-fading Rayleigh channel. First, we show that for a system with M transmit antennas and users equipped with N antennas, the sum rate scales like M log logn N for DPC, and beamforming when M is fixed and for any N (either growing to infinity or not). On the other hand, when both M and N are fixed, the sum rate of time sharing to the strongest user scales like min(M,N)log log n. Therefore, the asymptotic gain of DPC over time sharing for the sum rate is (M/min(M,N)) when M and N are fixed. It is also shown that if M grows as logn, the sum rate of DPC and beamforming will grow linearly in M, but with different constant multiplicative factors. In this region, the sum-rate capacity of time-sharing scales like N log log n.https://authors.library.caltech.edu/records/64cdr-j9g19A Practical Scheme for Wireless Network Operation
https://resolver.caltech.edu/CaltechAUTHORS:GOWieeetc07
Authors: Gowaikar, Radhika; Dana, Amir F.; Hassibi, Babak; Effros, Michelle
Year: 2007
DOI: 10.1109/TCOMM.2007.892448
In many problems in wireline networks, it is known that achieving capacity on each link or subnetwork is optimal for the entire network operation. In this paper, we present examples of wireless networks in which decoding and achieving capacity on certain links or subnetworks gives us lower rates than other simple schemes, like forwarding. This implies that the separation of channel and network coding that holds for many classes of wireline networks does not, in general, hold for wireless networks. Next, we consider Gaussian and erasure wireless networks where nodes are permitted only two possible operations: nodes can either decode what they receive (and then re-encode and transmit the message) or simply forward it. We present a simple greedy algorithm that returns the optimal scheme from the exponential-sized set of possible schemes. This algorithm will go over each node at most once to determine its operation, and hence, is very efficient. We also present a decentralized algorithm whose performance can approach the optimum arbitrarily closely in an iterative fashion.https://authors.library.caltech.edu/records/08dkc-0re60Algebraic Cayley Differential Space–Time Codes
https://resolver.caltech.edu/CaltechAUTHORS:OGGieeetit07
Authors: Oggier, Frédérique; Hassibi, Babak
Year: 2007
DOI: 10.1109/TIT.2007.894681
Cayley space-time codes have been proposed as a solution for coding over noncoherent differential multiple-input multiple-output (MIMO) channels. Based on the Cayley transform that maps the space of Hermitian matrices to the manifold of unitary matrices, Cayley codes are particularly suitable for high data rate, since they have an easy encoding and can be decoded using a sphere-decoder algorithm. However, at high rate, the problem of evaluating if a Cayley code is fully diverse may become intractable, and previous work has focused instead on maximizing a mutual information criterion. The drawback of this approach is that it requires heavy optimization which depends on the number of antennas and rate. In this work, we study Cayley codes in the context of division algebras, an algebraic tool that allows to get fully diverse codes. We present an algebraic construction of fully diverse Cayley codes, and show that this approach naturally yields, without further optimization, codes that perform similarly or closely to previous unitary differential codes, including previous Cayley codes, and codes built from Lie groups.https://authors.library.caltech.edu/records/pe2fs-1jf50Optimal LQG control across packet-dropping links
https://resolver.caltech.edu/CaltechAUTHORS:20150205-075834386
Authors: Gupta, Vijay; Hassibi, Babak; Murray, Richard M.
Year: 2007
DOI: 10.1016/j.sysconle.2006.11.003
We examine two special cases of the problem of optimal linear quadratic Gaussian control of a system whose state is being measured by sensors that communicate with the controller over packet-dropping links. We pose the problem as an information transmission problem. Using a separation principle, we decompose the problem into a standard LQR state-feedback controller design, along with an optimal encoder–decoder design for propagating and using the information across the unreliable links. Our design is optimal among all causal algorithms for any arbitrary packet-drop pattern. Further, the solution is appealing from a practical point of view because it can be implemented as a small modification of an existing LQG control design.https://authors.library.caltech.edu/records/dyha8-v6k82Statistical Pruning for Near-Maximum Likelihood Decoding
https://resolver.caltech.edu/CaltechAUTHORS:GOWieeetsp07
Authors: Gowaikar, Radhika; Hassibi, Babak
Year: 2007
DOI: 10.1109/TSP.2006.890912
In many communications problems, maximum-likelihood (ML) decoding reduces to finding the closest (skewed) lattice point in N-dimensions to a given point xisin CN. In its full generality, this problem is known to be NP-complete. Recently, the expected complexity of the sphere decoder, a particular algorithm that solves the ML problem exactly, has been computed. An asymptotic analysis of this complexity has also been done where it is shown that the required computations grow exponentially in N for any fixed SNR. At the same time, numerical computations of the expected complexity show that there are certain ranges of rates, SNRs and dimensions N for which the expected computation (counted as the number of scalar multiplications) involves no more than N3 computations. However, when the dimension of the problem grows too large, the required computations become prohibitively large, as expected from the asymptotic exponential complexity. In this paper, we propose an algorithm that, for large N, offers substantial computational savings over the sphere decoder, while maintaining performance arbitrarily close to ML. We statistically prune the search space to a subset that, with high probability, contains the optimal solution, thereby reducing the complexity of the search. Bounds on the error performance of the new method are proposed. The complexity of the new algorithm is analyzed through an upper bound. The asymptotic behavior of the upper bound for large N is also analyzed which shows that the upper bound is also exponential but much lower than the sphere decoder. Simulation results show that the algorithm is much more efficient than the original sphere decoder for smaller dimensions as well, and does not sacrifice much in terms of performance.https://authors.library.caltech.edu/records/x121b-9dj45Fundamental Limits in MIMO Broadcast Channels
https://resolver.caltech.edu/CaltechAUTHORS:HASieeejsac07
Authors: Hassibi, Babak; Sharif, Masoud
Year: 2007
DOI: 10.1109/JSAC.2007.070907
This paper studies the fundamental limits of MIMO broadcast channels from a high level, determining the sum-rate capacity of the system as a function of system paramaters, such as the number of transmit antennas, the number of users, the number of receive antennas, and the total transmit power. The crucial role of channel state information at the transmitter is emphasized, as well as the emergence of opportunistic transmission schemes. The effects of channel estimation errors, training, and spatial correlation are studied, as well as issues related to fairness, delay and differentiated rate scheduling.https://authors.library.caltech.edu/records/gz9vm-vxa38Delay Considerations for Opportunistic Scheduling in Broadcast Fading Channels
https://resolver.caltech.edu/CaltechAUTHORS:SHAieeetwc07
Authors: Sharif, Masoud; Hassibi, Babak
Year: 2007
DOI: 10.1109/TWC.2007.06067
We consider a single-antenna broadcast block fading
channel with n users where the transmission is packetbased.
We define the (packet) delay as the minimum number of channel uses that guarantees all n users successfully receive m packets. This is a more stringent notion of delay than average delay and is the worst case (access) delay among the users. A delay optimal scheduling scheme, such as round-robin, achieves the delay of mn. For the opportunistic scheduling (which is throughput optimal) where the transmitter sends the packet to the user with the best channel conditions at each channel use, we derive the mean and variance of the delay for any m and n. For large n and in a homogeneous network, it is proved that the expected delay in receiving one packet by all the receivers scales as n log n, as opposed to n for the round-robin scheduling. We also show that when m grows faster than (log n)^r, for some r > 1, then the delay scales as mn. This roughly determines the timescale required for the system to behave fairly in a homogeneous network. We then propose a scheme to significantly reduce the delay at the expense of a small throughput hit. We further look into the advantage of multiple transmit antennas on the delay. For a system with M antennas in the transmitter where at each channel use packets are sent to M different users, we obtain the expected delay in receiving one packet by all the users.https://authors.library.caltech.edu/records/aw712-gxv91Diversity Analysis of Distributed Space-Time Codes in Relay Networks with Multiple Transmit/Receive Antennas
https://resolver.caltech.edu/CaltechAUTHORS:JINeurasipjasp08
Authors: Jing, Yindi; Hassibi, Babak
Year: 2007
DOI: 10.1155/2008/254573
The idea of space-time coding devised for multiple-antenna systems is applied to the problem of communication over a wireless relay network, a strategy called distributed space-time coding, to achieve the cooperative diversity provided by antennas of the relay nodes. In this paper, we extend the idea of distributed space-time coding to wireless relay networks with multiple-antenna nodes and fading channels. We show that for a wireless relay network with M antennas at the transmit node, N antennas at the receive node, and a total of ℛ antennas at all the relay nodes, provided that the coherence interval is long enough, the high SNR pairwise error probability (PEP) behaves as (1/P)^min{M,N}ℛ if M≠N and (log^(1/M)P/P)^ℳℛ if M=N, where P is the total power consumed by the network. Therefore, for the case of M≠N, distributed space-time coding achieves the maximal diversity. For the case of M=N, the penalty is a factor of log^(1/M)P which, compared to P, becomes negligible when P is very high.https://authors.library.caltech.edu/records/cex5v-ht929Code Design for Multihop Wireless Relay Networks
https://resolver.caltech.edu/CaltechAUTHORS:OGGeurasipjasp08
Authors: Oggier, Frédérique; Hassibi, Babak
Year: 2008
DOI: 10.1155/2008/457307
We consider a wireless relay network, where a transmitter node communicates with a receiver node with the help of relay nodes. Most coding strategies considered so far assume that the relay nodes are used for one hop. We address the problem of code design when relay nodes may be used for more than one hop. We consider as a protocol a more elaborated version of amplify-and-forward, called distributed space-time coding, where the relay nodes multiply their received signal with a unitary matrix, in such a way that the receiver senses a space-time code. We first show that in this scenario, as expected, the so-called full-diversity condition holds, namely, the codebook of distributed space-time codewords has to be designed such that the difference of any two distinct codewords is full rank. We then compute the diversity of the channel, and show that it is given by the minimum number of relay nodes among the hops. We finally give a systematic way of building fully diverse codebooks and provide simulation results for their performance.https://authors.library.caltech.edu/records/ge04x-c4s77Speeding up the Sphere Decoder With ℋ∞ and SDP Inspired Lower Bounds
https://resolver.caltech.edu/CaltechAUTHORS:STOieeetsp08
Authors: Stojnic, Mihailo; Vikalo, Haris; Hassibi, Babak
Year: 2008
DOI: 10.1109/TSP.2007.906697
It is well known that maximum-likelihood (ML) decoding in many digital communication schemes reduces to solving an integer least-squares problem, which is NP hard in the worst-case. On the other hand, it has recently been shown that, over a wide range of dimensions and signal-to-noise ratios (SNRs), the sphere decoding algorithm can be used to find the exact ML solution with an expected complexity that is often less than N^3. However, the computational complexity of sphere decoding becomes prohibitive if the SNR is too low and/or if the dimension of the problem is too large. In this paper, we target these two regimes and attempt to find faster algorithms by pruning the search tree beyond what is done in the standard sphere decoding algorithm. The search tree is pruned by computing lower bounds on the optimal value of the objective function as the algorithm proceeds to descend down the search tree. We observe a tradeoff between the computational complexity required to compute a lower bound and the size of the pruned tree: the more effort we spend in computing a tight lower bound, the more branches that can be eliminated in the tree. Using ideas from semidefinite program (SDP)-duality theory and H∞ estimation theory, we propose general frameworks for computing lower bounds on integer least-squares problems. We propose two families of algorithms, one that is appropriate for large problem dimensions and binary modulation, and the other that is appropriate for moderate-size dimensions yet high-order constellations. We then show how in each case these bounds can be efficiently incorporated in the sphere decoding algorithm, often resulting in significant improvement of the expected complexity of solving the ML decoding problem, while maintaining the exact ML-performance.https://authors.library.caltech.edu/records/477s9-6sy33Recovering Sparse Signals Using Sparse Measurement Matrices in Compressed DNA Microarrays
https://resolver.caltech.edu/CaltechAUTHORS:20090916-154606685
Authors: Parvaresh, Farzad; Vikalo, Haris; Misra, Sidhant; Hassibi, Babak
Year: 2008
DOI: 10.1109/JSTSP.2008.924384
Microarrays (DNA, protein, etc.) are massively parallel affinity-based biosensors capable of detecting and quantifying a large number of different genomic particles simultaneously. Among them, DNA microarrays comprising tens of thousands of probe spots are currently being employed to test multitude of targets in a single experiment. In conventional microarrays, each spot contains a large number of copies of a single probe designed to capture a single target, and, hence, collects only a single data point. This is a wasteful use of the sensing resources in comparative DNA microarray experiments, where a test sample is measured relative to a reference sample. Typically, only a fraction of the total number of genes represented by the two samples is differentially expressed, and, thus, a vast number of probe spots may not provide any useful information. To this end, we propose an alternative design, the so-called compressed microarrays, wherein each spot contains copies of several different probes and the total number of spots is potentially much smaller than the number of targets being tested. Fewer spots directly translates to significantly lower costs due to cheaper array manufacturing, simpler image acquisition and processing, and smaller amount of genomic material needed for experiments. To recover signals from compressed microarray measurements, we leverage ideas from compressive sampling. For sparse measurement matrices, we propose an algorithm that has significantly lower computational complexity than the widely used linear-programming-based methods, and can also recover signals with less sparsity.https://authors.library.caltech.edu/records/856jy-pc460Modeling and Estimation for Real-Time Microarrays
https://resolver.caltech.edu/CaltechAUTHORS:20090916-153943053
Authors: Vikalo, Haris; Hassibi, Babak; Hassibi, Arjang
Year: 2008
DOI: 10.1109/JSTSP.2008.924383
Microarrays are used for collecting information about a large number of different genomic particles simultaneously. Conventional fluorescent-based microarrays acquire data after the hybridization phase. During this phase, the target analytes (e.g., DNA fragments) bind to the capturing probes on the array and, by the end of it, supposedly reach a steady state. Therefore, conventional microarrays attempt to detect and quantify the targets with a single data point taken in the steady state. On the other hand, a novel technique, the so-called real-time microarray, capable of recording the kinetics of hybridization in fluorescent-based microarrays has recently been proposed. The richness of the information obtained therein promises higher signal-to-noise ratio, smaller estimation error, and broader assay detection dynamic range compared to conventional microarrays. In this paper, we study the signal processing aspects of the real-time microarray system design. In particular, we develop a probabilistic model for real-time microarrays and describe a procedure for the estimation of target amounts therein. Moreover, leveraging on system identification ideas, we propose a novel technique for the elimination of cross hybridization. These are important steps toward developing optimal detection algorithms for real-time microarrays, and to understanding their fundamental limitations.https://authors.library.caltech.edu/records/3m9hf-9zp95An Algebraic Coding Scheme for Wireless Relay Networks With Multiple-Antenna Nodes
https://resolver.caltech.edu/CaltechAUTHORS:OGGieeetsp08
Authors: Oggier, Frédérique; Hassibi, Babak
Year: 2008
DOI: 10.1109/TSP.2008.917410
We consider the problem of coding over a half-duplex wireless relay network where both the transmitter and the receiver have respectively several transmit and receive antennas, whereas each relay is a small device with only a single antenna. Since, in this scenario, requiring the relays to decode results in severe rate hits, we propose a full rate strategy where the relays do a simple operation before forwarding the signal, based on the idea of distributed space-time coding. Our scheme relies on division algebras, an algebraic object which allows the design of fully diverse matrices. The code construction is applicable to systems with any number of transmit/receive antennas and relays, and has better performance than random code constructions, with much less encoding complexity. Finally, the robustness of the proposed distributed space-time codes to node failures is considered.https://authors.library.caltech.edu/records/emats-q3e25Differentiated rate scheduling for the down-link of cellular systems
https://resolver.caltech.edu/CaltechAUTHORS:DANieeetc08
Authors: Dana, Amir F.; Sharif, Masoud; Vakili, Ali; Hassibi, Babak
Year: 2008
DOI: 10.1109/TCOMM.2008.4641899
We consider the problem of differentiated rate scheduling for the downlink (i.e., multi-antenna broadcast channel), in the sense that the rates required by different users must satisfy certain constraints on their ratios. When full channel state information (CSI) is available at the transmitter and receivers, the problem can be readily solved using dirty paper coding (DPC) and the application of convex optimization techniques on the dual problem which is the multiple access channel (MAC). Since in many practical application full CSI may not be feasible and computational complexity prohibitive when the number of users is large, we focus on other simple schemes that require very little CSI: time-division opportunistic (TO) beamforming where in different time slots (of different lengths) the transmitter performs opportunistic beamforming to the users requiring the same rate, and weighted opportunistic (WO) beamforming where the random beams are assigned to those users having the largest weighted SINR. For single antenna systems we also look at the capacity-achieving superposition coding (SC) scheme. In all cases, we determine explicit schedules to guarantee the rate constraints and show that, in the limit of large number of users, the throughput loss compared to the unconstrained throughput (sum-rate capacity) tends to zero. We further provide bounds on the rate of convergence of the sum-rates of these schemes to the sum-rate capacity. Finally, we provide simulation results of the performance of different scheduling schemes considered in the paper.https://authors.library.caltech.edu/records/2jesa-k9046Scaling laws of multiple antenna group-broadcast channels
https://resolver.caltech.edu/CaltechAUTHORS:NAFieeetwc08
Authors: Al-Naffouri, Tareq Y.; Dana, Antir F.; Hassibi, Babak
Year: 2008
DOI: 10.1109/T-WC.2008.070902
Broadcast (or point to multipoint) communication has attracted a lot of research recently. In this paper, we consider the group broadcast channel where the users' pool is divided into groups, each of which is interested in common information. Such a situation occurs for example in digital audio and video broadcast where the users are divided into various groups according to the shows they are interested in. The paper obtains upper and lower bounds for the sum rate capacity in the large number of users regime and quantifies the effect of spatial correlation on the system capacity. The paper also studies the scaling of the system capacity when the number of users and antennas grow simultaneously. It is shown that in order to achieve a constant rate per user, the number of transmit antennas should scale at least logarithmically in the number of users.https://authors.library.caltech.edu/records/e9h35-xk546Real-time DNA microarray analysis
https://resolver.caltech.edu/CaltechAUTHORS:20100108-102354698
Authors: Hassibi, Arjang; Vikalo, Haris; Riechmann, José Luis; Hassibi, Babak
Year: 2009
DOI: 10.1093/nar/gkp675
We present a quantification method for affinity-based
DNA microarrays which is based on the
real-time measurements of hybridization kinetics.
This method, i.e. real-time DNA microarrays,
enhances the detection dynamic range of conventional
systems by being impervious to probe
saturation in the capturing spots, washing
artifacts, microarray spot-to-spot variations, and
other signal amplitude-affecting non-idealities. We
demonstrate in both theory and practice that the
time-constant of target capturing in microarrays,
similar to all affinity-based biosensors, is inversely
proportional to the concentration of the target
analyte, which we subsequently use as the fundamental
parameter to estimate the concentration
of the analytes. Furthermore, to empirically
validate the capabilities of this method in practical
applications, we present a FRET-based assay which
enables the real-time detection in gene expression
DNA microarrays.https://authors.library.caltech.edu/records/fsnzp-m9w21How much does transmit correlation affect the sum-rate scaling of MIMO Gaussian broadcast channels?
https://resolver.caltech.edu/CaltechAUTHORS:20090805-113256208
Authors: Al-Naffouri, T. Y.; Sharif, M.; Hassibi, B.
Year: 2009
DOI: 10.1109/TCOMM.2009.02.060065
This paper considers the effect of spatial correlation between transmit antennas on the sum-rate capacity of the MIMO Gaussian broadcast channel (i.e., downlink of a cellular system). Specifically, for a system with a large number of users n, we analyze the scaling laws of the sum-rate for the dirty paper coding and for different types of beamforming transmission schemes. When the channel is i.i.d., it has been shown that for large n, the sum rate is equal to M log log n + M log P/M + o(1) where M is the number of transmit antennas, P is the average signal to noise ratio, and o(1) refers to terms that go to zero as n → ∞. When the channel exhibits some spatial correlation with a covariance matrix R (non-singular with tr(R) = M), we prove that the sum rate of dirty paper coding is M log log n + M log P/M + log det(R) + o(1). We further show that the sum-rate of various beamforming schemes achieves M log log n + M log P/M + M log c + o(1) where c ≤ 1 depends on the type of beamforming. We can in fact compute c for random beamforming proposed in and more generally, for random beamforming with preceding in which beams are pre-multiplied by a fixed matrix. Simulation results are presented at the end of the paper.https://authors.library.caltech.edu/records/a6j3q-1r022Peak Power Reduction of OFDM Signals with Sign Adjustment
https://resolver.caltech.edu/CaltechAUTHORS:20090824-142302617
Authors: Sharif, Masoud; Tarokh, Vahid; Hassibi, Babak
Year: 2009
DOI: 10.1109/TCOMM.2009.07.080012
It has recently been shown that significant reduction in the peak to mean envelope power (PMEPR) can be obtained by altering the sign of each subcarrier in a multicarrier system with n subcarriers. However, finding the best sign not only requires a search over 2n possible signs but also may lead to a substantial rate loss for small size constellations. In this paper, we first propose a greedy algorithm to choose the signs based on p-norm minimization and prove that the resulting PMEPR is guaranteed to be less than c log n where c is a constant independent of n for any n. This approach has lower complexity in each iteration compared to the derandomization approach of while achieving similar PMEPR reduction. We further improve the performance of the proposed algorithm by enlarging the search space using pruning. Simulation results show that PMEPR of a multicarrier signal with 128 subcarriers can be reduced to within 1.6 dB of the PMEPR of a single carrier system. In the second part of the paper, we address the rate loss by proposing a block coding scheme in which only one sign vector is chosen for K different modulating vectors. The sign vector can be computed using the greedy algorithm in n iterations. We show that the multi-symbol encoding approach can reduce the rate loss by a factor of K while achieving the PMEPR of c logKn, i.e., only logarithmic growth in K. Simulation results show that the rate loss can be made smaller than %10 at the cost of only 1 db increase in the resulting PMEPR for a system with 128 subcarriers.https://authors.library.caltech.edu/records/r9prc-sg678On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements
https://resolver.caltech.edu/CaltechAUTHORS:20091012-122739100
Authors: Stojnic, Mihailo; Parvaresh, Farzad; Hassibi, Babak
Year: 2009
DOI: 10.1109/TSP.2009.2020754
Let A be an M by N matrix (M < N) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations A x = y for a given y. In general, whenever the sparsity of x is smaller than half the dimension of y then with overwhelming probability over A the sparsest solution is unique and can be found by an exhaustive search over x with an exponential time complexity for any y. The recent work of Candes, Donoho, and Tao shows that minimization of the ℓ_1 norm of x subject to Ax = y results in the sparsest solution provided the sparsity of x, say K, is smaller than a certain threshold for a given number of measurements. Specifically, if the dimension of y approaches the dimension of x , the sparsity of x should be K < 0.239 N. Here, we consider the case where x is block sparse, i.e., x consists of n = N /d blocks where each block is of length d and is either a zero vector or a nonzero vector (under nonzero vector we consider a vector that can have both, zero and nonzero components). Instead of ℓ_1 -norm relaxation, we consider the following relaxation: times min ||X_1||_2 + ||X_2||_2 + • • • + ||X_n ||_2, subject to A x = y (*) where X_i = (x_(i-1)d+1, x)_(i-1)d+2,• • •, x_(id))^T for i = 1, 2,• • •, N. Our main result is that as n → ∞, (*) finds the sparsest solution to A=x = y, with overwhelming probability in A, for any x whose sparsity is k/n < (1/2) - O (ε), provided m/n > 1 - 1/d, and d = Ω(log(1/ε)/ε^3). The rlaxation given in (*) can be solved in polynomial time using semi-definite programming.https://authors.library.caltech.edu/records/qwq6v-1sq46Data Transmission Over Networks for Estimation and Control
https://resolver.caltech.edu/CaltechAUTHORS:20090826-112853239
Authors: Gupta, Vijay; Dana, Amir F.; Hespanha, Joao P.; Murray, Richard M.; Hassibi, Babak
Year: 2009
DOI: 10.1109/TAC.2009.2024567
We consider the problem of controlling a linear time invariant process when the controller is located at a location remote from where the sensor measurements are being generated. The communication from the sensor to the controller is supported by a communication network with arbitrary topology composed of analog erasure channels. Using a separation principle, we prove that the optimal linear-quadratic-Gaussian (LQG) controller consists of an LQ optimal regulator along with an estimator that estimates the state of the process across the communication network. We then determine the optimal information processing strategy that should be followed by each node in the network so that the estimator is able to compute the best possible estimate in the minimum mean squared error sense. The algorithm is optimal for any packet-dropping process and at every time step, even though it is recursive and hence requires a constant amount of memory, processing and transmission at every node in the network per time step. For the case when the packet drop processes are memoryless and independent across links, we analyze the stability properties and the performance of the closed loop system. The algorithm is an attempt to escape the viewpoint of treating a network of communication links as a single end-to-end link with the probability of successful transmission determined by some measure of the reliability of the network.https://authors.library.caltech.edu/records/1gnam-v4926Efficient and Robust Compressed Sensing Using Optimized Expander Graphs
https://resolver.caltech.edu/CaltechAUTHORS:20090908-083705068
Authors: Jafarpour, Sina; Xu, Weiyu; Hassibi, Babak; Calderbank, Robert
Year: 2009
DOI: 10.1109/TIT.2009.2025528
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(klog n) measurements and only O(klog n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal.https://authors.library.caltech.edu/records/dq7j6-z0w73Achievable Throughput in Two-Scale Wireless Networks
https://resolver.caltech.edu/CaltechAUTHORS:20090911-153601945
Authors: Gowaikar, Radhika; Hassibi, Babak
Year: 2009
DOI: 10.1109/JSAC.2009.090913
We propose a new model of wireless networks which we refer to as "two-scale networks." At a local scale, characterised by nodes being within a distance r, channel strengths are drawn independently and identically from a distance-independent distribution. At a global scale, characterised by nodes being further apart from each other than a distance r, channel connections are governed by a Rayleigh distribution, with the power satisfying a distance-based decay law. Thus, at a local scale, channel strengths are determined primarily by random effects such as obstacles and scatterers whereas at the global scale channel strengths depend on distance.
For such networks, we propose a hybrid communications scheme, combining elements of distance-dependent networks and random networks. For particular classes of two-scale networks with N nodes, we show that an aggregate throughput that is slightly sublinear in N, for instance, of the form N/ log^4 N is achievable. This offers a significant improvement over a throughput scaling behaviour of O(√N) that is obtained in other work.https://authors.library.caltech.edu/records/2x8rv-dxf28Performance of sphere decoding of block codes
https://resolver.caltech.edu/CaltechAUTHORS:20091106-102918514
Authors: El-Khamy, Mostafa; Vikalo, Haris; Hassibi, Babak; McEliece, Robert J.
Year: 2009
DOI: 10.1109/TCOMM.2009.10.080402
A sphere decoder searches for the closest lattice point within a certain search radius. The search radius provides a tradeoff between performance and complexity. We focus on analyzing the performance of sphere decoding of linear block codes. We analyze the performance of soft-decision sphere decoding on AWGN channels and a variety of modulation schemes. A harddecision sphere decoder is a bounded distance decoder with the corresponding decoding radius. We analyze the performance of hard-decision sphere decoding on binary and q-ary symmetric channels. An upper bound on the performance of maximumlikelihood decoding of linear codes defined over F_q (e.g. Reed- Solomon codes) and transmitted over q-ary symmetric channels is derived and used in the analysis.We then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes. The tradeoff between the performance and complexity of a sphere decoder is then discussed.https://authors.library.caltech.edu/records/vtm54-w7v65Reduced feedback and random beamforming for OFDM MIMO broadcast channels
https://resolver.caltech.edu/CaltechAUTHORS:20100126-100241646
Authors: Fakhereddin, Maralle J.; Sharif, Masoud; Hassibi, Babak
Year: 2009
DOI: 10.1109/TCOMM.2009.12.060236
It has been shown that random beamforming using partial channel state information (CSI) achieves the same throughput scaling as obtained from dirty paper coding for a broadcast (downlink) channel with M transmit antennas and K users where K is large. In this paper, we apply this scheme to wideband MIMO broadcast channels. By using OFDM, an L-tap wideband channel can be decomposed to N parallel narrowband channels (subcarriers), where N > L. Neighboring subcarriers are highly correlated. Therefore, we consider neighboring subcarriers as a cluster and find the closed form solution for the joint characteristic function of SINR values at two subcarriers in a cluster. We show numerically how the knowledge of the quality of the center subcarrier sheds light about the quality of other subcarriers in the same cluster, and address the issue of cluster size. In addition, through complex and asymptotic analysis, we show that for cluster size of order N/L√(log K) (for large K), users need only feedback the best SINR at the center subcarrier of each cluster in order for the transmitter to perform opportunistic beamforming and maintain the same throughput scaling as when full CSI is available. Using simulation results, we verify our analytical result and show that even fewer feedback can be tolerated, and larger clusters (N/2L) can be implemented for a small throughput hit.https://authors.library.caltech.edu/records/emmq2-3sr92Cyclic Distributed Space–Time Codes for Wireless Relay Networks With No Channel Information
https://resolver.caltech.edu/CaltechAUTHORS:20100120-105549993
Authors: Oggier, Frédérique; Hassibi, Babak
Year: 2010
DOI: 10.1109/TIT.2009.2034801
In this paper, we present a coding strategy for half duplex wireless relay networks, where we assume no channel knowledge at any of the transmitter, receiver, or relays. The coding scheme uses distributed space–time coding, that is, the relay nodes cooperate to encode the transmitted signal so that the receiver senses a space–time codeword. It is inspired by noncoherent differential techniques. The proposed strategy is available for any number of relays nodes. It is analyzed, and shown to yield a diversity linear in the number of relays. We also study the resistance of the scheme to relay node failures, and show that a network with R relay nodes and d of them down behaves, as far as diversity is concerned, as a network with R-d nodes. Finally, our construction can be easily generalized to the case where the transmitter and receiver nodes have several antennas.https://authors.library.caltech.edu/records/evx5c-nne46Limits of Performance of Quantitative Polymerase
Chain Reaction Systems
https://resolver.caltech.edu/CaltechAUTHORS:20110222-133846369
Authors: Vikalo, Haris; Hassibi, Babak; Hassibi, Arjang
Year: 2010
DOI: 10.1109/TIT.2009.2037088
Estimation of the DNA copy number in a given biological
sample is an important problem in genomics. Quantitative
polymerase chain reaction (qPCR) systems detect the target
DNA molecules by amplifying their number through a series of
thermal cycles and measuring the amount of created amplicons in each cycle. Ideally, the number of target molecules doubles at the end of each cycle. However, in practice, due to biochemical noise the efficiency of the qPCR reaction—defined as the fraction of the target molecules which are successfully copied during a cycle—is
always less than 1. In this paper, we formulate the problem of the joint maximum-likelihood estimation of the qPCR efficiency and the initial DNA copy number. Then, we analytically determine the limits of performance of qPCR by deriving the Cramer–Rao lower bound on the mean-square estimation error. As indicated by simulation studies, the performance of the proposed estimator is superior compared to competing statistical approaches. The proposed approach is validated using experimental data.https://authors.library.caltech.edu/records/mb7kp-w5b67Distance-Dependent Kronecker Graphs for Modeling Social Networks
https://resolver.caltech.edu/CaltechAUTHORS:20101025-094230043
Authors: Bodine-Baron, Elizabeth; Hassibi, Babak; Wierman, Adam
Year: 2010
DOI: 10.1109/JSTSP.2010.2049412
This paper focuses on a generalization of stochastic
Kronecker graphs, introducing a Kronecker-like operator and
defining a family of generator matrices H dependent on distances
between nodes in a specified graph embedding. We prove
that any lattice-based network model with sufficiently small
distance-dependent connection probability will have a Poisson
degree distribution and provide a general framework to prove
searchability for such a network. Using this framework, we focus
on a specific example of an expanding hypercube and discuss
the similarities and differences of such a model with recently
proposed network models based on a hidden metric space. We
also prove that a greedy forwarding algorithm can find very short
paths of length O((log log n)^2) on the hypercube with n nodes,
demonstrating that distance-dependent Kronecker graphs can
generate searchable network models.https://authors.library.caltech.edu/records/b6mjq-7zh42Sparse Recovery of Nonnegative Signals With Minimal Expansion
https://resolver.caltech.edu/CaltechAUTHORS:20110610-094646559
Authors: Khajehnejad, M. Amin; Dimakis, Alexandros G.; Xu, Weiyu; Hassibi, Babak
Year: 2011
DOI: 10.1109/TSP.2010.2082536
We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than l_1 minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when l_1 minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when l_1 minimization is used. We further show that the success of l_1 optimization is equivalent to the existence of a "unique" vector in the set of solutions to the linear equations, which enables alternative algorithms for l_1 minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to l_1 minimization.https://authors.library.caltech.edu/records/cvbtx-c9348Null space conditions and thresholds for rank minimization
https://resolver.caltech.edu/CaltechAUTHORS:20110401-112500555
Authors: Recht, Benjamin; Xu, Weiyu; Hassibi, Babak
Year: 2011
DOI: 10.1007/s10107-010-0422-2
Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in machine learning, control theory, and discrete geometry. This class of optimization problems, known as rank minimization, is NP-hard, and for most practical problems there are no efficient algorithms that yield exact solutions. A popular heuristic replaces the rank function with the nuclear norm—equal to the sum of the singular values—of the decision variable and has been shown to provide the optimal low rank solution in a variety of scenarios. In this paper, we assess the practical performance of this heuristic for finding the minimum rank
matrix subject to linear equality constraints. We characterize properties of the null space of the linear operator defining the constraint set that are necessary and sufficient for the heuristic to succeed. We then analyze linear constraints sampled uniformly at random, and obtain dimension-free bounds under which our null space properties hold almost surely as the matrix dimensions tend to infinity. Finally, we provide empirical evidence that these probabilistic bounds provide accurate predictions of the heuristic's performance in non-asymptotic scenarios.https://authors.library.caltech.edu/records/fxcmj-vhy70Analyzing Weighted ℓ_1 Minimization for Sparse Recovery With Nonuniform Sparse Models
https://resolver.caltech.edu/CaltechAUTHORS:20110502-112413009
Authors: Khajehnejad, M. Amin; Xu, Weiyu; Avestimehr, A. Salman; Hassibi, Babak
Year: 2011
DOI: 10.1109/TSP.2011.2107904
In this paper, we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted ℓ_1 minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted ℓ_1 minimization recovery algorithm and analyze its performance using a Grassmann angle approach. We compute explicitly the relationship between the system parameters-the weights, the number of measurements, the size of the two sets, the probabilities of being nonzero-so that when i.i.d. random Gaussian measurement matrices are used, the weighted ℓ_1 minimization recovers a randomly selected signal drawn from the considered sparsity model with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We demonstrate through rigorous analysis and simulations that for the case when the support of the signal can be divided into two different subclasses with unequal sparsity fractions, the weighted ℓ_1 minimization outperforms the regular ℓ_1 minimization substantially. We also generalize our results to signal vectors with an arbitrary number of subclasses for sparsity.https://authors.library.caltech.edu/records/bxm8q-jqv05The Secrecy Capacity of the MIMO Wiretap Channel
https://resolver.caltech.edu/CaltechAUTHORS:20111108-150023605
Authors: Oggier, Frédérique; Hassibi, Babak
Year: 2011
DOI: 10.1109/TIT.2011.2158487
We consider the MIMO wiretap channel, that is a MIMO broadcast channel where the transmitter sends some confidential information to one user which is a legitimate receiver, while the other user is an eavesdropper. Perfect secrecy is achieved when the transmitter and the legitimate receiver can communicate at some positive rate, while insuring that the eavesdropper gets zero bits of information. In this paper, we compute the perfect secrecy capacity of the multiple antenna MIMO broadcast channel, where the number of antennas is arbitrary for both the transmitter and the two receivers. Our technique involves a careful study of a Sato-like upper bound via the solution of a certain algebraic Riccati equation.https://authors.library.caltech.edu/records/s93q2-1f955Precise Stability Phase Transitions for ℓ_1 Minimization: A Unified Geometric Framework
https://resolver.caltech.edu/CaltechAUTHORS:20111101-085058069
Authors: Xu, Weiyu; Hassibi, Babak
Year: 2011
DOI: 10.1109/TIT.2011.2165825
ℓ_1 minimization is often used for recovering sparse
signals from an under-determined linear system. In this paper, we focus on finding sharp performance bounds on recovering approximately sparse signals using ℓ_1 minimization under noisy measurements. While the restricted isometry property is powerful for the analysis of recovering approximately sparse signals with noisy measurements, the known bounds on the achievable
sparsity level can be quite loose. The neighborly polytope
analysis which yields sharp bounds for perfectly sparse signals cannot be readily generalized to approximately sparse signals. We start from analyzing a necessary and sufficient condition, the "balancedness" property of linear subspaces, for achieving a certain signal recovery accuracy. Then we give a unified null space Grassmann angle-based geometric framework to give sharp bounds on this "balancedness" property of linear subspaces. By
investigating the "balancedness" property, this unified framework characterizes sharp quantitative tradeoffs between signal sparsity and the recovery accuracy of ℓ_1 minimization for approximately sparse signal. As a consequence, this generalizes the neighborly polytope result for perfectly sparse signals. Besides the robustness
in the "strong" sense for all sparse signals, we also discuss the notions of "weak" and "sectional" robustness. Our results concern fundamental properties of linear subspaces and so may be of independent mathematical interest.https://authors.library.caltech.edu/records/faejm-8dp64Divide-and-Conquer: Approaching the Capacity of the Two-Pair Bidirectional Gaussian Relay Network
https://resolver.caltech.edu/CaltechAUTHORS:20120502-151314658
Authors: Sezgin, Aydin; Avestimehr, A. Salman; Khajehnejad, M. Amin; Hassibi, Babak
Year: 2012
DOI: 10.1109/TIT.2011.2177773
The capacity region of multi-pair bidirectional relay
networks, in which a relay node facilitates the communication between
multiple pairs of users, is studied. This problem is first examined
in the context of the linear shift deterministic channel model.
The capacity region of this network when the relay is operating at
either full-duplex mode or half-duplex mode for arbitrary number
of pairs is characterized. It is shown that the cut-set upper-bound
is tight and the capacity region is achieved by a so called divide-and-
conquer relaying strategy. The insights gained from the deterministic
network are then used for the Gaussian bidirectional
relay network. The strategy in the deterministic channel translates
to a specific superposition of lattice codes and random Gaussian
codes at the source nodes and successive interference cancelation
at the receiving nodes for the Gaussian network. The achievable
rate of this scheme with two pairs is analyzed and it is shown that
for all channel gains it achieves to within 3 bits/sec/Hz per user of
the cut-set upper-bound. Hence, the capacity region of the two-pair
bidirectional Gaussian relay network to within 3 bits/sec/Hz per
user is characterized.https://authors.library.caltech.edu/records/axdfq-7f883Reweighted LP Decoding for LDPC Codes
https://resolver.caltech.edu/CaltechAUTHORS:20121008-110544402
Authors: Khajehnejad, Amin; Dimakis, Alexandros G.; Hassibi, Babak; Vigoda, Benjamin; Bradley, William
Year: 2012
DOI: 10.1109/TIT.2012.2202211
We introduce a novel algorithm for decoding binary linear codes by linear programming (LP). We build on the LP decoding algorithm of Feldman and introduce a postprocessing step that solves a second linear program that reweights the objective function based on the outcome of the original LP decoder output. Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. We also show significant empirical performance gains for the reweighted LP decoding algorithm with very small additional computational complexity.https://authors.library.caltech.edu/records/35g4p-4n578Low-Complexity Blind Equalization for OFDM Systems With General Constellations
https://resolver.caltech.edu/CaltechAUTHORS:20130116-105358832
Authors: Al Naffouri, Tareq Y.; Dahman, Ala A.; Sohail, Muhammad S.; Xu, Weiyu; Hassibi, Babak
Year: 2012
DOI: 10.1109/TSP.2012.2218808
This paper proposes a low-complexity algorithm for blind equalization of data in orthogonal frequency division multiplexing (OFDM)-based wireless systems with general constellations. The proposed algorithm is able to recover the transmitted data even when the channel changes on a symbol-by-symbol basis, making it suitable for fast fading channels. The proposed algorithm does not require any statistical information about the channel and thus does not suffer from latency normally associated with blind methods. The paper demonstrates how to reduce the complexity of the algorithm, which becomes especially low at high signal-to-noise ratio (SNR). Specifically, it is shown that in the high SNR regime, the number of operations is of the order O(LN), where L is the cyclic prefix length and N is the total number of subcarriers. Simulation results confirm the favorable performance of the proposed algorithm.https://authors.library.caltech.edu/records/wt84s-x6805The Kalman-Like Particle Filter: Optimal Estimation With
Quantized Innovations/Measurements
https://resolver.caltech.edu/CaltechAUTHORS:20130225-083341103
Authors: Sukhavasi, Ravi Teja; Hassibi, Babak
Year: 2013
DOI: 10.1109/TSP.2012.2226164
We study the problem of optimal estimation and control of linear systems using quantized measurements. We show that the state conditioned on a causal quantization of the measurements can be expressed as the sum of a Gaussian random vector and a certain truncated Gaussian vector. This structure bears close resemblance to the full information Kalman filter and so allows us to effectively combine the Kalman structure with a particle filter to recursively compute the state estimate. We call the resulting filter the Kalman-like particle filter (KLPF) and observe that it delivers close to optimal performance using far fewer particles than that of a particle filter directly applied to the original problem.https://authors.library.caltech.edu/records/hrh74-yh810Optimized Markov Chain Monte Carlo for Signal Detection in MIMO Systems: An Analysis of the Stationary Distribution and Mixing Time
https://resolver.caltech.edu/CaltechAUTHORS:20140918-142934393
Authors: Hassibi, Babak; Hansen, Morten; Dimakis, Alexandros G.; Alshamary, Haider Ali Jasim; Xu, Weiyu
Year: 2014
DOI: 10.1109/TSP.2014.2334558
We introduce an optimized Markov chain Monte Carlo (MCMC) technique for solving integer least-squares (ILS) problems, which include maximum likelihood (ML) detection in multiple-input multiple-output (MIMO) systems. Two factors contribute to its speed of finding the optimal solution: the probability of encountering the optimal solution when the Markov chain has converged to the stationary distribution, and the mixing time of the MCMC detector. First, we compute the optimal "temperature" parameter value, so that once the Markov chain has mixed to its stationary distribution, there is a polynomially small probability ( 1/poly(N), instead of exponentially small) of encountering the optimal solution, where N is the system dimension. This temperature is shown to be O(√{SNR}/ln(N)), where SNR > 2ln(N) is the SNR. Second, we study the mixing time of the underlying Markov chain of the MCMC detector. We find that, the mixing time is closely related to whether there is a local minimum in the ILS problem's lattice structure. For some lattices without local minima, the mixing time is independent of SNR, and grows polynomially in N. Conventional wisdom proposed to set temperature as the noise standard deviation, but our results show that, under such a temperature, the mixing time grows unbounded with SNR if the lattice has local minima. Our results suggest that, very often the temperature should instead be scaling at least as Ω(√{SNR}). Simulation results show that the optimized MCMC detector efficiently achieves approximately ML detection in MIMO systems having a huge number of transmit and receive dimensions.https://authors.library.caltech.edu/records/cxycy-aav38Equivalent relaxations of optimal power flow
https://resolver.caltech.edu/CaltechAUTHORS:20150112-101936328
Authors: Bose, Subhonmesh; Low, Steven H.; Teeraratkul, Thanchanok; Hassibi, Babak
Year: 2015
DOI: 10.1109/TAC.2014.2357112
Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper, we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxation based on a chordal extension of the network graph, and a second-order cone relaxation that computes the smallest partial matrix. We prove a bijection between the feasible sets of the OPF in the bus injection model and the branch flow model, establishing the equivalence of these two models and their second-order cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the second-order cone relaxation. For mesh networks, the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second-order cone relaxation. Therefore, for mesh networks, one should either solve the chordal relaxation or the SOCP relaxation, trading off tightness and the required computational effort. Simulations are used to illustrate these results.https://authors.library.caltech.edu/records/cm86v-dwe23Simultaneously Structured Models with Application to Sparse and Low-rank Matrices
https://resolver.caltech.edu/CaltechAUTHORS:20150126-073012139
Authors: Oymak, Samet; Jalali, Amin; Fazel, Maryam; Eldar, Yonina C.; Hassibi, Babak
Year: 2015
DOI: 10.1109/TIT.2015.2401574
Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that promote the individual structures are known, and allow for recovery using an order-wise optimal number of measurements (e.g., ℓ_1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, using multiobjective optimization with these norms can do no better, orderwise, than exploiting only one of the structures, thus revealing a fundamental limitation in sample complexity. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation. Further, specializing our results to the case of sparse and low-rank matrices, we show that a nonconvex formulation recovers the model from very few measurements (on the order of the degrees of freedom), whereas the convex problem combining the ℓ_1 and nuclear norms requires many more measurements, illustrating a gap between the performance of the convex and nonconvex recovery problems. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give sample complexity bounds for problems such as sparse phase retrieval and low-rank tensor completion.https://authors.library.caltech.edu/records/tyzyf-bk365Regularized Linear Regression: A Precise Analysis of the Estimation Error
https://resolver.caltech.edu/CaltechAUTHORS:20200221-130234534
Authors: Thrampoulidis, Christos; Oymak, Samet; Hassibi, Babak
Year: 2015
Non-smooth regularized convex optimization procedures have emerged as a powerful tool to recover structured signals (sparse, low-rank, etc.) from (possibly compressed) noisy linear measurements. We focus on the problem of linear regression and consider a general class of optimization methods that minimize a loss function measuring the misfit of the model to the observations with an added structured-inducing regularization term. Celebrated instances include the LASSO, Group-LASSO, Least-Absolute Deviations method, etc.. We develop a quite general framework for how to determine precise prediction performance guaranties (e.g. mean-square-error) of such methods for the case of Gaussian measurement ensemble. The machinery builds upon Gordon's Gaussian min-max theorem under additional convexity assumptions that arise in many practical applications. This theorem associates with a primary optimization (PO) problem a simplified auxiliary optimization (AO) problem from which we can tightly infer properties of the original (PO), such as the optimal cost, the norm of the optimal solution, etc. Our theory applies to general loss functions and regularization and provides guidelines on how to optimally tune the regularizer coefficient when certain structural properties (such as sparsity level, rank, etc.) are known.https://authors.library.caltech.edu/records/gm0zm-6jw70Improving the Thresholds of Sparse Recovery: An Analysis of a Two-Step Reweighted Basis Pursuit Algorithm
https://resolver.caltech.edu/CaltechAUTHORS:20150127-072715568
Authors: Khajehnejad, M. Amin; Xu, Weiyu; Avestimehr, A. Salman; Hassibi, Babak
Year: 2015
DOI: 10.1109/TIT.2015.2448690
It is well known that ℓ_1 minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. Exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from independent identically distributed (i.i.d.) Gaussian measurements, have been computed and are referred to as weak thresholds. In this paper, we introduce a reweighted ℓ_1 recovery algorithm composed of two steps: 1) a standard ℓ_1 minimization step to identify a set of entries where the signal is likely to reside and 2) a weighted ℓ_1 minimization step where entries outside this set are penalized. For signals where the non-sparse component entries are independent and identically drawn from certain classes of distributions, (including most well-known continuous distributions), we prove a strict improvement in the weak recovery threshold. Our analysis suggests that the level of improvement in the weak threshold depends on the behavior of the distribution at the origin. Numerical simulations verify the distribution dependence of the threshold improvement very well, and suggest that in the case of i.i.d. Gaussian nonzero entries, the improvement can be quite impressive—over 20% in the example we consider.https://authors.library.caltech.edu/records/937tq-bqj71A Perspective on the MIMO Wiretap Channel
https://resolver.caltech.edu/CaltechAUTHORS:20151016-081115003
Authors: Oggier, Frédérique; Hassibi, Babak
Year: 2015
DOI: 10.1109/JPROC.2015.2468077
A wiretap channel is a communication channel between a transmitter Alice and a legitimate receiver Bob, in the presence of an eavesdropper Eve. The goal of communication is to achieve reliability between Alice and Bob, but also confidentiality despite Eve's presence. Wiretap channels are declined in all kinds of flavors, depending on the underlying channels used by the three players: discrete memoryless channels, additive Gaussian noise channels, or fading channels, to name a few. In this survey, we focus on the case where the three players use multiple-antenna channels with Gaussian noise to communicate. After summarizing known results for multiple-input-multiple-output (MIMO) channels, both in terms of achievable reliable data rate (capacity) and code design, we introduce the MIMO wiretap channel. We then state the MIMO wiretap capacity, summarize the idea of the proof(s) behind this result, and comment on the insights given by the proofs on the physical meaning of the secrecy capacity. We finally discuss design criteria for MIMO wiretap codes.https://authors.library.caltech.edu/records/5jpmm-tzr27Group Frames With Few Distinct Inner Products and Low Coherence
https://resolver.caltech.edu/CaltechAUTHORS:20150911-153303704
Authors: Thill, Matthew; Hassibi, Babak
Year: 2015
DOI: 10.1109/TSP.2015.2450195
Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original "group frame" design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with n elements from n(n-1)/2 to n-1. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When n is prime, for example, we use cyclic groups to construct m-dimensional frame vectors with at most n-1/m distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance.https://authors.library.caltech.edu/records/rsgbw-wgz62On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables
https://resolver.caltech.edu/CaltechAUTHORS:20160211-110501483
Authors: Al-Naffouri, Tareq Y.; Moinuddin, Muhammed; Ajeeb, Nizar; Hassibi, Babak; Moustakas, Aris L.
Year: 2016
DOI: 10.1109/TCOMM.2015.2496592
In this work, we propose a unified approach to evaluating the CDF and PDF of indefinite quadratic forms in Gaussian random variables. Such a quantity appears in many applications in communications, signal processing, information theory, and adaptive filtering. For example, this quantity appears in the mean-square-error (MSE) analysis of the normalized least-mean-square (NLMS) adaptive algorithm, and SINR associated with each beam in beam forming applications. The trick of the proposed approach is to replace inequalities that appear in the CDF calculation with unit step functions and to use complex integral representation of the the unit step function. Complex integration allows us then to evaluate the CDF in closed form for the zero mean case and as a single dimensional integral for the non-zero mean case. Utilizing the saddle point technique allows us to closely approximate such integrals in non zero mean case. We demonstrate how our approach can be extended to other scenarios such as the joint distribution of quadratic forms and ratios of such forms, and to characterize quadratic forms in isotropic distributed random variables. We also evaluate the outage probability in multiuser beamforming using our approach to provide an application of indefinite forms in communications.https://authors.library.caltech.edu/records/mrbkc-qve57Optimal Placement of Distributed Energy Storage in Power Networks
https://resolver.caltech.edu/CaltechAUTHORS:20150126-071149016
Authors: Thrampoulidis, Christos; Bose, Subhonmesh; Hassibi, Babak
Year: 2016
DOI: 10.1109/TAC.2015.2437527
We formulate the optimal placement, sizing and control of storage devices in a power network to minimize generation costs with the intent of load shifting. We assume deterministic demand, a linearized DC approximated power flow model and a fixed available storage budget. Our main result proves that when the generation costs are convex and nondecreasing, there always exists an optimal storage capacity allocation that places zero storage at generation-only buses that connect to the rest of the network via single links. This holds regardless of the demand profiles, generation capacities, line-flow limits and characteristics of the storage technologies. Through a counterexample, we illustrate that this result is not generally true for generation buses with multiple connections. For specific network topologies, we also characterize the dependence of the optimal generation cost on the available storage budget, generation capacities and flow constraints.https://authors.library.caltech.edu/records/dt8cw-pj310STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20160520-110029620
Authors: Jaganathan, Kishore; Eldar, Yonina C.; Hassibi, Babak
Year: 2016
DOI: 10.1109/JSTSP.2016.2549507
The problem of recovering a signal from its Fourier magnitude is of paramount importance in various fields of engineering and applied physics. Due to the absence of Fourier phase information, some form of additional information is required in order to be able to uniquely, efficiently, and robustly identify the underlying signal. Inspired by practical methods in optical imaging, we consider the problem of signal reconstruction from the short-time Fourier transform (STFT) magnitude. We first develop conditions under, which the STFT magnitude is an almost surely unique signal representation. We then consider a semidefinite relaxation-based algorithm (STliFT) and provide recovery guarantees. Numerical simulations complement our theoretical analysis and provide directions for future work.https://authors.library.caltech.edu/records/96aa3-evk67Sharp MSE Bounds for Proximal Denoising
https://resolver.caltech.edu/CaltechAUTHORS:20160825-102038882
Authors: Oymak, Samet; Hassibi, Babak
Year: 2016
DOI: 10.1007/s10208-015-9278-4
Denoising has to do with estimating a signal x_0 from its noisy observations y = x_0 + z. In this paper, we focus on the "structured denoising problem," where the signal x_0 possesses a certain structure and z has independent normally distributed entries with mean zero and variance σ^2. We employ a structure-inducing convex function f(⋅) and solve min_x {1/2 ∥y−x∥^2_2 +σλf(x)}to estimate x_0, for some λ>0. Common choices for f(⋅) include the ℓ_1 norm for sparse vectors, the ℓ_1 −ℓ_2 norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate x∗ is the normalized mean-squared error NMSE(σ)=E∥x∗ − x_0∥^2_2/σ^2. We show that NMSE is maximized as σ→0 and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the λ-scaled subdifferential λ∂f(x_0). When λ is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem min_(f(x)≤f(x_0)){∥y−x∥2}. The paper also connects these results to the generalized LASSO problem, in which one solves min_(f(x)≤f(x_0)){∥y−Ax∥2} to estimate x_0 from noisy linear observations y=Ax_0 + z. We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a "phase transition" as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.https://authors.library.caltech.edu/records/fbe5r-xn288Linear Time-Invariant Anytime Codes for Control Over Noisy Channels
https://resolver.caltech.edu/CaltechAUTHORS:20160524-085405487
Authors: Sukhavasi, Ravi Teja; Hassibi, Babak
Year: 2016
DOI: 10.1109/TAC.2016.2529960
The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in problems of distributed control and networked control systems. Although the work of Schulman, and Sahai and Mitter over the past two decades, and their development of the notions of "tree codes" and "anytime capacity" respectively, provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one often needs real-time encoding and real-time decoding and a reliability which increases exponentially with delay, which is what tree codes guarantee. We propose an ensemble of random causal linear codes with a time invariant structure and show that they are tree codes with probability one. For erasure channels, we show that the average complexity of maximum likelihood decoding is bounded by a constant for all time if the code rate is smaller than the computational cutoff rate. For rates larger than the computational cutoff rate, we present an alternate way to perform maximum likelihood decoding with a complexity that grows linearly with time. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight.https://authors.library.caltech.edu/records/nj3jg-ww782On the Ingleton-Violating Finite Groups
https://resolver.caltech.edu/CaltechAUTHORS:20150126-075202822
Authors: Mao, Wei; Thill, Matthew; Hassibi, Babak
Year: 2017
DOI: 10.1109/TIT.2016.2627530
Given n discrete random variables, its entropy vector is the 2^n - 1-dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a close relation between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups; indeed, roughly speaking, knowing the region of all such group-characterizable vectors is equivalent to knowing the region of all entropy vectors. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al. that linear network codes cannot achieve capacity in general network coding problems (since linear network codes come from abelian groups). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. General entropy vectors, however, do not necessarily have this property. It is, therefore, of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors, which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S_5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family PGL(2,q) with a prime power q ≥ 5 , i.e., the projective group of 2×2 nonsingular matrices with entries in F_q . We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as PGL(n,q) and GL(n,q) . The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes.https://authors.library.caltech.edu/records/xtfq4-bcp29Training Signal Design for Correlated Massive MIMO Channel Estimation
https://resolver.caltech.edu/CaltechAUTHORS:20170216-151436862
Authors: Soltanalian, Mojtaba; Naghsh, Mohammad Mahdi; Shariati, Nafiseh; Stoica, Petre; Hassibi, Babak
Year: 2017
DOI: 10.1109/TWC.2016.2639485
In this paper, we propose a new approach to the design of training sequences that can be used for an accurate estimation of multi-input multi-output channels. The proposed method is particularly instrumental in training sequence designs that deal with three key challenges: 1) arbitrary channel and noise statistics that do not follow specific models, 2) limitations on the properties of the transmit signals, including total power, per-antenna power, having a constant-modulus, discrete-phase, or low peak-to-average-power ratio, and 3) signal design for large-scale or massive antenna arrays. Several numerical examples are provided to examine the proposed method.https://authors.library.caltech.edu/records/r5ckw-88y65Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20150121-073440098
Authors: Jaganathan, Kishore; Oymak, Samet; Hassibi, Babak
Year: 2017
DOI: 10.1109/TSP.2017.2656844
The problem of signal recovery from its Fourier
transform magnitude is of paramount importance in various
fields of engineering and has been around for over 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this work, we focus our attention on discrete-time sparse signals (of length n). We first show that, if the DFT dimension is greater than or equal to 2n, then almost all signals with aperiodic support can be uniquely identified by
their Fourier transform magnitude (up to time-shift, conjugate-flip and global phase).
Then, we develop an efficient Two-stage Sparse Phase Retrieval algorithm (TSPR), which involves: (i) identifying the support, i.e., the locations of the non-zero components, of the signal using a combinatorial algorithm (ii) identifying the signal values in the support using a convex algorithm. We show that TSPR can provably recover most O(n^(1/2-ϵ)-sparse signals (up to a timeshift,
conjugate-flip and global phase). We also show that, for
most O(n^(1/4-ϵ)-sparse signals, the recovery is robust in the presence of measurement noise. These recovery guarantees are asymptotic in nature. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR.https://authors.library.caltech.edu/records/m1zhe-6q443Low-Coherence Frames from Group Fourier Matrices
https://resolver.caltech.edu/CaltechAUTHORS:20170322-160021533
Authors: Thill, Matthew; Hassibi, Babak
Year: 2017
DOI: 10.1109/TIT.2017.2686420
Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of such frames can in turn be used to build high-performance spherical codes, quantum measurement operators, and compressive sensing measurement matrices, to name a few applications.
In this work, we allude to the group-frame construction first described by Slepian and further explored in the works of Vale and Waldron. We present a method for selecting representations of a nite group to construct a group frame that achieves low coherence. Our technique produces a tight frame with a small number of distinct inner product values between the frame elements, in a sense approximating a Grassmannian frame. We identify special cases in which our construction yields some previously-known frames with optimal coherence meeting the Welch lower bound, and other cases in which the entries of our frame vectors come from small alphabets. In particular, we apply our technique to the problem choosing a subset of rows of a Hadamard matrix so that the resulting columns form a low-coherence frame. Finally, we give an explicit calculation of the average coherence of our frames, and nd regimes in which they satisfy the Strong Coherence Property described by Mixon, Bajwa, and Calderbank.https://authors.library.caltech.edu/records/pq22v-39a56Capacity Analysis of Discrete Energy Harvesting Channels
https://resolver.caltech.edu/CaltechAUTHORS:20170712-150437770
Authors: Mao, Wei; Hassibi, Babak
Year: 2017
DOI: 10.1109/TIT.2017.2726070
We study the channel capacity of a general discrete energy harvesting channel with a finite battery. Contrary to traditional communication systems, the transmitter of such a channel is powered by a device that harvests energy from a random exogenous energy source and has a finite-sized battery. As a consequence, at each transmission opportunity, the system can only transmit a symbol whose energy is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is simultaneously random, instantaneous, and influenced by the full history of the inputs and the energy harvesting process. Furthermore, naturally, in such a channel, the energy information is observed causally at the transmitter. Both of these characteristics pose great challenges for the analysis of the channel capacity. In this paper, we use techniques developed for channels with side information and finite-state channels, to obtain lower and upper bounds on the capacity of energy harvesting channels. In particular, in a general case with Markov energy harvesting processes, we use stationarity and ergodicity theory to compute and optimize the achievable rates for the channels, and derive a series of computable capacity upper and lower bounds.https://authors.library.caltech.edu/records/rqhs1-39n10Low-Rank Riemannian Optimization on Positive Semidefinite Stochastic Matrices with Applications to Graph Clustering
https://resolver.caltech.edu/CaltechAUTHORS:20210511-092854252
Authors: Douik, Ahmed; Hassibi, Babak
Year: 2018
This paper develops a Riemannian optimization framework for solving optimization problems on the set of symmetric positive semidefinite stochastic matrices. The paper first reformulates the problem by factorizing the optimization variable as X=YY^T and deriving conditions on p, i.e., the number of columns of Y, under which the factorization yields a satisfactory solution. The reparameterization of the problem allows its formulation as an optimization over either an embedded or quotient Riemannian manifold whose geometries are investigated. In particular, the paper explicitly derives the tangent space, Riemannian gradients and retraction operator that allow the design of efficient optimization methods on the proposed manifolds. The numerical results reveal that, when the optimal solution has a known low-rank, the resulting algorithms present a clear complexity advantage when compared with state-of-the-art Euclidean and Riemannian approaches for graph clustering applications.https://authors.library.caltech.edu/records/5gpd3-v5r21Symbol Error Rate Performance of Box-relaxation Decoders in Massive MIMO
https://resolver.caltech.edu/CaltechAUTHORS:20180503-105428947
Authors: Thrampoulidis, Christos; Xu, Weiyu; Hassibi, Babak
Year: 2018
DOI: 10.1109/TSP.2018.2831622
The maximum-likelihood (ML) decoder for symbol detection in large multiple-input multiple-output wireless communication systems is typically computationally prohibitive. In this paper, we study a popular and practical alternative, namely the box-relaxation optimization (BRO) decoder, which is a natural convex relaxation of the ML. For independent identically distributed real Gaussian channels with additive Gaussian noise, we obtain exact asymptotic expressions for the symbol error rate (SER) of the BRO. The formulas are particularly simple, they yield useful insights, and they allow accurate comparisons to the matched-filter bound (MFB) and to linear decoders, such as zero-forcing and linear minimum mean square error. For binary phase-shift keying signals, the SER performance of the BRO is within 3 dB of the MFB for square systems, and it approaches the MFB as the number of receive antennas grows large compared to the number of transmit antennas. Our analysis further characterizes the empirical density function of the solution of the BRO, and shows that error events for any fixed number of symbols are asymptotically independent. The fundamental tool behind the analysis is the convex Gaussian min–max theorem.https://authors.library.caltech.edu/records/fftkp-83563Precise Error Analysis of Regularized M-estimators in High-dimensions
https://resolver.caltech.edu/CaltechAUTHORS:20180601-084821157
Authors: Thrampoulidis, Christos; Abbasi, Ehsan; Hassibi, Babak
Year: 2018
DOI: 10.1109/TIT.2018.2840720
A popular approach for estimating an unknown signal x_0 ∈ ℝ^n from noisy, linear measurements y = Ax_0 + z ∈ ℝ^m is via solving a so called regularized M-estimator: x:= arg min_x L(y – Ax) + λf(x). Here, L is a convex loss function, f is a convex (typically, non-smooth) regularizer, and, λ > 0 is a regularizer parameter. We analyze the squared error performance ‖x – x_0‖^2_2 of such estimators in the high-dimensional proportional regime where m,n → ∞ and m/n → δ. The design matrix is assumed to have entries iid Gaussian; only minimal and rather mild regularity conditions are imposed on the loss function, the regularizer, and on the noise and signal distributions. We show that the squared error converges in probability to a nontrivial limit that is given as the solution to a minimax convex-concave optimization problem on four scalar optimization variables. We identify a new summary parameter, termed the expected Moreau envelope to play a central role in the error characterization. The precise nature of the results permits an accurate performance comparison between different instances of regularized M-estimators and allows to optimally tune the involved parameters (such as the regularizer parameter, the number of measurements, etc.). The key ingredient of our proof is the convex Gaussian min-max theorem (CGMT) which is a tight and strengthened version of a classical Gaussian comparison inequality that was proved by Gordon in 1988.https://authors.library.caltech.edu/records/df7qx-jfb18Multiplexed identification, quantification and genotyping of infectious agents using a semiconductor biochip
https://resolver.caltech.edu/CaltechAUTHORS:20180723-095009655
Authors: Hassibi, Arjang; Manickam, Arun; Singh, Rituraj; Bolouki, Sara; Sinha, Ruma; Jirage, Kshama B.; McDermott, Mark W.; Hassibi, Babak; Vikalo, Haris; Mazarei, Gelareh; Pei, Lei; Bousse, Luc; Miller, Mark; Heshami, Mehrdad; Savage, Michael P.; Taylor, Michael T.; Gamini, Nader; Wood, Nicholas; Mantina, Pallavi; Grogan, Patrick; Kuimelis, Peter; Savalia, Piyush; Conradson, Scott; Li, Yuan; Meyer, Rich B.; Ku, Edmond; Ebert, Jessica; Pinsky, Benjamin A.; Dolganov, Gregory; Van, Tran; Johnson, Kirsten A.; Naraghi-Arani, Pejman; Kuimelis, Robert G.; Schoolnik, Gary
Year: 2018
DOI: 10.1038/nbt.4179
The emergence of pathogens resistant to existing antimicrobial drugs is a growing worldwide health crisis that threatens a return to the pre-antibiotic era. To decrease the overuse of antibiotics, molecular diagnostics systems are needed that can rapidly identify pathogens in a clinical sample and determine the presence of mutations that confer drug resistance at the point of care. We developed a fully integrated, miniaturized semiconductor biochip and closed-tube detection chemistry that performs multiplex nucleic acid amplification and sequence analysis. The approach had a high dynamic range of quantification of microbial load and was able to perform comprehensive mutation analysis on up to 1,000 sequences or strands simultaneously in <2 h. We detected and quantified multiple DNA and RNA respiratory viruses in clinical samples with complete concordance to a commercially available test. We also identified 54 drug-resistance-associated mutations that were present in six genes of Mycobacterium tuberculosis, all of which were confirmed by next-generation sequencing.https://authors.library.caltech.edu/records/dfxsd-tv133Sparse and Balanced Reed–Solomon and Tamo–Barg Codes
https://resolver.caltech.edu/CaltechAUTHORS:20181005-093939651
Authors: Halbawi, Wael; Liu, Zihan; Duursma, Iwan M.; Dau, Hoang; Hassibi, Babak
Year: 2019
DOI: 10.1109/tit.2018.2873128
We study the problem of constructing balanced generator matrices for Reed–Solomon and Tamo–Barg codes. More specifically, we are interested in realizing generator matrices, for the full-length cyclic versions of these codes, where all rows have the same weight and the difference in weight between any columns is at most one. The results presented in this paper translate to computationally balanced encoding schemes, which can be appealing in distributed storage applications. Indeed, the balancedness of these generator matrices guarantees that the computation effort exerted by any storage node is essentially the same. In general, the framework presented can accommodate various values for the required row weight. We emphasize the possibility of constructing sparsest and balanced generator matrices for Reed–Solomon codes, i.e., each row is a minimum distance codeword. The number of storage nodes contacted once a message symbol is updated decreases with the row weight, so sparse constructions are appealing in that context. Results of similar flavor are presented for cyclic Tamo–Barg codes. In particular, we show that for a code with minimum distance d and locality r, a construction in which every row is of weight d+r−1 is possible. The constructions presented are deterministic and operate over the codes' original underlying finite field. As a result, efficient decoding from both errors and erasures is possible thanks to the plethora of efficient decoders available for the codes considered.https://authors.library.caltech.edu/records/qfwv6-c2z59Tracking and Control of Gauss-Markov Processes over Packet-Drop Channels with Acknowledgments
https://resolver.caltech.edu/CaltechAUTHORS:20180629-113827049
Authors: Khina, Anatoly; Kostina, Victoria; Khisti, Ashish; Hassibi, Babak
Year: 2019
DOI: 10.1109/TCNS.2018.2850225
We consider the problem of tracking the state of Gauss–Markov processes over rate-limited erasure-prone links. We concentrate first on the scenario in which several independent processes are seen by a single observer. The observer maps the processes into finite-rate packets that are sent over the erasure-prone links to a state estimator, and are acknowledged upon packet arrivals. The aim of the state estimator is to track the processes with zero delay and with minimum mean square error (MMSE). We show that, in the limit of many processes, greedy quantization with respect to the squared error distortion is optimal. That is, there is no tension between optimizing the MMSE of the process in the current time instant and that of future times. For the case of packet erasures with delayed acknowledgments, we connect the problem to that of compression with side information that is known at the observer and may be known at the state estimator—where the most recent packets serve as side information that may have been erased, and demonstrate that the loss due to a delay by one time unit is rather small. For the scenario where only one process is tracked by the observer–state estimator system, we further show that variable-length coding techniques are within a small gap of the many-process outer bound. We demonstrate the usefulness of the proposed approach for the simple setting of discrete-time scalar linear quadratic Gaussian control with a limited data-rate feedback that is susceptible to packet erasures.https://authors.library.caltech.edu/records/9gce2-ngy16Reconstruction of signals from their autocorrelation and cross-correlation vectors, with applications to phase retrieval and blind channel estimation
https://resolver.caltech.edu/CaltechAUTHORS:20190425-103104881
Authors: Jaganathan, Kishore; Hassibi, Babak
Year: 2019
DOI: 10.1109/TSP.2019.2911254
We consider the problem of reconstructing two signals from the autocorrelation and cross-correlation measurements. This inverse problem is a fundamental one in signal processing, and arises in many applications, including phase retrieval and blind channel estimation. In a typical phase retrieval setup, only the autocorrelation measurements are obtainable. We show that, when the measurements are obtained using three simple "masks", phase retrieval reduces to the aforementioned reconstruction problem. The classic solution to this problem is based on finding common factors between the z-transforms of the autocorrelation and cross-correlation vectors. This solution has enjoyed limited practical success, mainly due to the fact that it is not sufficiently stable in the noisy setting. In this paper, inspired by the success of convex programming in provably and stably solving various quadratic constrained problems, we develop a semidefinite programming-based algorithm and provide theoretical guarantees. In particular, we show that almost all signals can be uniquely recovered by this algorithm (up to a global phase). Comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting.https://authors.library.caltech.edu/records/ngff9-35390Distributed Solution of Large-Scale Linear Systems via Accelerated Projection-Based Consensus
https://resolver.caltech.edu/CaltechAUTHORS:20190610-092256233
Authors: Azizan-Ruhi, Navid; Lahouti, Farshad; Avestimehr, Salman; Hassibi, Babak
Year: 2019
DOI: 10.1109/TSP.2019.2917855
Solving a large-scale system of linear equations is a key step at the heart of many algorithms in scientific computing, machine learning, and beyond. When the problem dimension is large, computational and/or memory constraints make it desirable, or even necessary, to perform the task in a distributed fashion. In this paper, we consider a common scenario in which a taskmaster intends to solve a large-scale system of linear equations by distributing subsets of the equations among a number of computing machines/cores. We propose a new algorithm called Accelerated Projection-based Consensus , in which at each iteration every machine updates its solution by adding a scaled version of the projection of an error signal onto the nullspace of its system of equations, and the taskmaster conducts an averaging over the solutions with momentum. The convergence behavior of the proposed algorithm is analyzed in detail and analytically shown to compare favorably with the convergence rate of alternative distributed methods, namely distributed gradient descent, distributed versions of Nesterov's accelerated gradient descent and heavy-ball method, the block Cimmino method, and Alternating Direction Method of Multipliers. On randomly chosen linear systems, as well as on real-world data sets, the proposed method offers significant speed-up relative to all the aforementioned methods. Finally, our analysis suggests a novel variation of the distributed heavy-ball method, which employs a particular distributed preconditioning and achieves the same theoretical convergence rate as that in the proposed consensus-based method.https://authors.library.caltech.edu/records/210vm-r1p04Control over Gaussian Channels With and Without Source-Channel Separation
https://resolver.caltech.edu/CaltechAUTHORS:20190425-110709737
Authors: Khina, Anatoly; Riedel Gårding, Elias; Pettersson, Gustav M.; Kostina, Victoria; Hassibi, Babak
Year: 2019
DOI: 10.1109/TAC.2019.2912255
We consider the problem of controlling an unstable linear plant with Gaussian disturbances over an additive white Gaussian noise channel with an average transmit power constraint, where the signaling rate of communication may be different from the sampling rate of the underlying plant. Such a situation is quite common since sampling is done at a rate that captures the dynamics of the plant and that is often lower than the signaling rate of the communication channel. This rate mismatch offers the opportunity of improving the system performance by using coding over multiple channel uses to convey a single control action. In a traditional, separation-based approach to source and channel coding, the analog message is first quantized down to a few bits and then mapped to a channel codeword whose length is commensurate with the number of channel uses per sampled message. Applying the separation-based approach to control meets its challenges: first, the quantizer needs to be capable of zooming in and out to be able to track unbounded system disturbances, and second, the channel code must be capable of improving its estimates of the past transmissions exponentially with time, a characteristic known as anytime reliability. We implement a separated scheme by leveraging recently developed techniques for control over quantized-feedback channels and for efficient decoding of anytime-reliable codes. We further propose an alternative, namely, to perform analog joint source–channel coding, by this avoiding the digital domain altogether. For the case where the communication signaling rate is twice the sampling rate, we employ analog linear repetition as well as Shannon–Kotel'nikov maps to show a significant improvement in stability margins and linear-quadratic costs over separation-based schemes. We conclude that such analog coding performs better than separation, and can stabilize all moments as well as guarantee almost-sure stability.https://authors.library.caltech.edu/records/amxh8-wgg97Rate-Cost Tradeoffs in Control
https://resolver.caltech.edu/CaltechAUTHORS:20190425-103602105
Authors: Kostina, Victoria; Hassibi, Babak
Year: 2019
DOI: 10.1109/TAC.2019.2912256
Consider a control problem with a communication channel connecting the observer of a linear stochastic system to the controller. The goal of the controller is to minimize a quadratic cost function in the state variables and control signal, known as the linear quadratic regulator (LQR). We study the fundamental tradeoff between the communication rate r bits/sec and the expected cost b. We obtain a lower bound on a certain rate-cost function, which quantifies the minimum directed mutual information between the channel input and output that is compatible with a target LQR cost. The rate-cost function has operational significance in multiple scenarios of interest: among others, it allows us to lower-bound the minimum communication rate for fixed and variable length quantization, and for control over noisy channels. We derive an explicit lower bound to the rate-cost function, which applies to the vector, non-Gaussian, and partially observed systems, thereby extending and generalizing an earlier explicit expression for the scalar Gaussian system, due to Tatikonda el al. [2]. The bound applies as long as the differential entropy of the system noise is not −∞ . It can be closely approached by a simple lattice quantization scheme that only quantizes the innovation, that is, the difference between the controller's belief about the current state and the true state. Via a separation principle between control and communication, similar results hold for causal lossy compression of additive noise Markov sources. Apart from standard dynamic programming arguments, our technical approach leverages the Shannon lower bound, develops new estimates for data compression with coding memory, and uses some recent results on high resolution variablelength vector quantization to prove that the new converse bounds are tight.https://authors.library.caltech.edu/records/j04q3-ds323MOCZ for Blind Short-Packet Communication: Basic Principles
https://resolver.caltech.edu/CaltechAUTHORS:20190816-081650823
Authors: Walk, Philipp; Jung, Peter; Hassibi, Babak
Year: 2019
DOI: 10.1109/TWC.2019.2932668
We introduce a novel blind (noncoherent) communication scheme, called modulation on conjugate-reciprocal zeros (MOCZ), pronounced as "Moxie," to reliably transmit sporadic short-packets over unknown wireless multipath channels. In MOCZ, the information is modulated onto the zeros of the transmitted discrete-time baseband signal's z-transform, which yields to a codebook of non-orthogonal signals. In the absence of additive noise, the zero structure of the signal is perfectly preserved at the receiver, no matter what the channel impulse response (CIR) is. Furthermore, by a proper selection of the zeros, we show that MOCZ is not only invariant to the CIR but also robust against additive noise. Starting with the maximum-likelihood estimator, we define a low complexity and reliable decoder and compare it to various state-of-the-art noncoherent multipath schemes, such as OFDM index-modulation (IM), OFDM pilot-aided, OFDM differential-modulation, and pulse-position-modulation. Our scheme outperforms all schemes and maintains its performance even if the length becomes shorter than the CIR.https://authors.library.caltech.edu/records/n9rd7-x5s76Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry
https://resolver.caltech.edu/CaltechAUTHORS:20191010-144909166
Authors: Douik, Ahmed; Hassibi, Babak
Year: 2019
DOI: 10.1109/tsp.2019.2946024
Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a more general set of convex programs known as semi-definite and second-order cone programs. However, these methods are excessively slow for high dimensions. On the other hand, optimization algorithms on manifolds have shown great abilities in finding solutions to non-convex problems in a reasonable time. This paper suggests using a Riemannian optimization approach to solve a subset of convex optimization problems wherein the optimization variable is a doubly stochastic matrix. Optimization over the set of doubly stochastic matrices is crucial for multiple communications and signal processing applications, especially graph-based clustering. The paper introduces and investigates the geometries of three convex manifolds, namely the doubly stochastic, the symmetric, and the definite multinomial manifolds which generalize the simplex, also known as the multinomial manifold. Theoretical complexity analysis and numerical simulation results testify the efficiency of the proposed method over state-of-the-art algorithms. In particular, they reveal that the proposed framework outperforms conventional generic and specialized approaches, especially in high dimensions.https://authors.library.caltech.edu/records/21fp2-75v15Optimum Linear Codes with Support-Constrained Generator Matrices over Small Fields
https://resolver.caltech.edu/CaltechAUTHORS:20190814-151834005
Authors: Yildiz, Hikmet; Hassibi, Babak
Year: 2019
DOI: 10.1109/tit.2019.2932663
We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed–Solomon code of small field size and with the same minimum distance. In particular, if the code has length n, and maximum minimum distance d (over all generator matrices with the given support), then an optimal code exists for any field size q ≥ 2n-d. As a by-product of this result, we settle the GM–MDS conjecture in the affirmative.https://authors.library.caltech.edu/records/ay6q2-m4645Precise 3-D GNSS Attitude Determination Based on Riemannian Manifold Optimization Algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20191216-132407663
Authors: Douik, Ahmed; Liu, Xing; Ballal, Tarig; Al-Naffouri, Tareq Y.; Hassibi, Babak
Year: 2020
DOI: 10.1109/tsp.2019.2959226
In the past few years, Global Navigation Satellite Systems (GNSS) based attitude determination has been widely used thanks to its high accuracy, low cost, and real-time performance. This paper presents a novel 3-D GNSS attitude determination method based on Riemannian optimization techniques. The paper first exploits the antenna geometry and baseline lengths to reformulate the 3-D GNSS attitude determination problem as an optimization over a non-convex set. Since the solution set is a manifold, in this manuscript we formulate the problem as an optimization over a Riemannian manifold. The study of the geometry of the manifold allows the design of efficient first and second order Riemannian algorithms to solve the 3-D GNSS attitude determination problem. Despite the non-convexity of the problem, the proposed algorithms are guaranteed to globally converge to a critical point of the optimization problem. To assess the performance of the proposed framework, numerical simulations are provided for the most challenging attitude determination cases: the unaided, single-epoch, and single-frequency scenarios. Numerical results reveal that the proposed algorithms largely outperform state-of-the-art methods for various system configurations with lower complexity than generic non-convex solvers, e.g., interior point methods.https://authors.library.caltech.edu/records/sk71d-gh394A Silicon Photonics Computational Lensless Active-Flat-Optics Imaging System
https://resolver.caltech.edu/CaltechAUTHORS:20200212-111715644
Authors: White, Alexander; Khial, Parham; Salehi, Fariborz; Hassibi, Babak; Hajimiri, Ali
Year: 2020
DOI: 10.1038/s41598-020-58027-1
PMCID: PMC6997425
The need for lightweight, miniature imaging systems is becoming increasingly prevalent in light of the development of wearable electronics, IoT devices, and drones. Computational imaging enables new types of imaging systems that replace standard optical components like lenses with cleverly designed computational processes. Traditionally, many of these types of systems use conventional complementary metal oxide semiconductor (CMOS) or charge coupled device (CCD) sensors for data collection. While this allows for rapid development of large-scale systems, the lack of system-sensor co-design limits the compactness and performance. Here we propose integrated photonics as a candidate platform for the implementation of such co-integrated systems. Using grating couplers and co-designed computational processing in lieu of a lens, we demonstrate the use of silicon photonics as a viable platform for computational imaging with a prototype lensless imaging device. The proof-of-concept device has 20 sensors and a 45-degree field of view, and its optics and sensors are contained within a 2,000 μm × 200 μm × 20 μm volume.https://authors.library.caltech.edu/records/9tjrk-xgy56Gabidulin Codes with Support Constrained Generator Matrices
https://resolver.caltech.edu/CaltechAUTHORS:20190401-161416001
Authors: Yildiz, Hikmet; Hassibi, Babak
Year: 2020
DOI: 10.1109/tit.2019.2955106
Gabidulin codes are the first general construction of linear codes that are maximum rank distant (MRD). They have found applications in linear network coding, for example, when the transmitter and receiver are oblivious to the inner workings and topology of the network (the so-called incoherent regime). The reason is that Gabidulin codes can be used to map information to linear subspaces, which in the absence of errors cannot be altered by linear operations, and in the presence of errors can be corrected if the subspace is perturbed by a small rank. Furthermore, in distributed coding and distributed systems, one is led to the design of error correcting codes whose generator matrix must satisfy a given support constraint. In this paper, we give necessary and sufficient conditions on the support of the generator matrix that guarantees the existence of Gabidulin codes and general MRD codes. When the rate of the code is not very high, this is achieved with the same field size necessary for Gabidulin codes with no support constraint. When these conditions are not satisfied, we characterize the largest possible rank distance under the support constraints and show that they can be achieved by subcodes of Gabidulin codes. The necessary and sufficient conditions are identical to those that appear for MDS codes which were recently proven by Yildiz et al. and Lovett in the context of settling the GM-MDS conjecture.https://authors.library.caltech.edu/records/p2s6k-czs94MOCZ for Blind Short-Packet Communication: Practical Aspects
https://resolver.caltech.edu/CaltechAUTHORS:20190402-083621511
Authors: Walk, Philipp; Jung, Peter; Hassibi, Babak; Jafarkhani, Hamid
Year: 2020
DOI: 10.1109/TWC.2020.3004588
We investigate practical aspects of a recently introduced blind (noncoherent) communication scheme, called modulation on conjugate-reciprocal zeros (MOCZ). MOCZ is suitable for a reliable transmission of sporadic and short-packets at ultra-low latency and high spectral efficiency via unknown multipath channels, which are assumed to be static over the receive duration of one packet. The information is modulated on the zeros of the transmitted discrete-time baseband signal's z− transform. Because of ubiquitous impairments between the transmitter and receiver clocks, a carrier frequency offset occurs after down-conversion to the baseband. This results in a common rotation of the zeros. To identify fractional rotations of the base angle in the zero-pattern, we propose an oversampled direct zero-testing decoder to identify the most likely one. Integer rotations correspond to cyclic shifts of the binary message, which we determine by cyclically permutable codes (CPC). Additionally, the embedding of CPCs into cyclic codes, enables additive error-correction which reduces the bit-error-rate tremendously. Furthermore, we exploit the trident structure in the signal's autocorrelation for an energy based detector to estimate timing offsets and the effective channel delay spread. We finally demonstrate how this joint data and channel estimation can be largely improved by receive antenna diversity at low SNR.https://authors.library.caltech.edu/records/8da53-4by32Regret-Optimal Filtering
https://resolver.caltech.edu/CaltechAUTHORS:20210225-132748732
Authors: Sabag, Oron; Hassibi, Babak
Year: 2021
DOI: 10.48550/arXiv.2101.10357
We consider the problem of filtering in linear state-space models (e.g., the Kalman filter setting) through the lens of regret optimization. Specifically, we study the problem of causally estimating a desired signal, generated by a linear state-space model driven by process noise, based on noisy observations of a related observation process. We define a novel regret criterion for estimator design as the difference of the estimation error energies between a clairvoyant estimator that has access to all future observations (a so-called smoother) and a causal one that only has access to current and past observations. The regret-optimal estimator is the causal estimator that minimizes the worst-case regret across all bounded-energy noise sequences. We provide a solution for the regret filtering problem at two levels. First, an horizon-independent solution at the operator level is obtained by reducing the regret to the well-known Nehari problem. Secondly, our main result for state-space models is an explicit estimator that achieves the optimal regret. The regret-optimal estimator is represented as a finite-dimensional state-space whose parameters can be computed by solving three Riccati equations and a single Lyapunov equation. We demonstrate the applicability and efficacy of the estimator in a variety of problems and observe that the estimator has average and worst-case performances that are simultaneously close to their optimal values.https://authors.library.caltech.edu/records/4kpts-tg502Stability and Identification of Random Asynchronous Linear Time-Invariant Systems
https://resolver.caltech.edu/CaltechAUTHORS:20210225-132728423
Authors: Lale, Sahin; Teke, Oguzhan; Hassibi, Babak; Anandkumar, Anima
Year: 2021
DOI: 10.48550/arXiv.2012.04160
In many computational tasks and dynamical systems, asynchrony and randomization are naturally present and have been considered as ways to increase the speed and reduce the cost of computation while compromising the accuracy and convergence rate. In this work, we show the additional benefits of randomization and asynchrony on the stability of linear dynamical systems. We introduce a natural model for random asynchronous linear time-invariant (LTI) systems which generalizes the standard (synchronous) LTI systems. In this model, each state variable is updated randomly and asynchronously with some probability according to the underlying system dynamics. We examine how the mean-square stability of random asynchronous LTI systems vary with respect to randomization and asynchrony. Surprisingly, we show that the stability of random asynchronous LTI systems does not imply or is not implied by the stability of the synchronous variant of the system and an unstable synchronous system can be stabilized via randomization and/or asynchrony. We further study a special case of the introduced model, namely randomized LTI systems, where each state element is updated randomly with some fixed but unknown probability. We consider the problem of system identification of unknown randomized LTI systems using the precise characterization of mean-square stability via extended Lyapunov equation. For unknown randomized LTI systems, we propose a systematic identification method to recover the underlying dynamics. Given a single input/output trajectory, our method estimates the model parameters that govern the system dynamics, the update probability of state variables, and the noise covariance using the correlation matrices of collected data and the extended Lyapunov equation. Finally, we empirically demonstrate that the proposed method consistently recovers the underlying system dynamics with optimal rate.https://authors.library.caltech.edu/records/12k1k-p2p46Regret-optimal measurement-feedback control
https://resolver.caltech.edu/CaltechAUTHORS:20210719-210216911
Authors: Goel, Gautam; Hassibi, Babak
Year: 2021
DOI: 10.48550/arXiv.2011.12785
We consider measurement-feedback control in linear dynamical systems from the perspective of regret minimization. Unlike most prior work in this area, we focus on the problem of designing an online controller which competes with the optimal dynamic sequence of control actions selected in hindsight, instead of the best controller in some specific class of controllers. This formulation of regret is attractive when the environment changes over time and no single controller achieves good performance over the entire time horizon. We show that in the measurement-feedback setting, unlike in the full-information setting, there is no single online controller which outperforms every other online controller on every disturbance, and propose a new H₂-optimal online controller as a benchmark for the online controller to compete against. We show that the corresponding regret-optimal online controller can be found via a novel reduction to the classical Nehari problem from robust control and present a tight data-dependent bound on its regret.https://authors.library.caltech.edu/records/bbmm2-tp094Finite-time System Identification and Adaptive Control in Autoregressive Exogenous Systems
https://resolver.caltech.edu/CaltechAUTHORS:20210727-162630002
Authors: Lale, Sahin; Azizzadenesheli, Kamyar; Hassibi, Babak; Anandkumar, Animashree
Year: 2021
Autoregressive exogenous (ARX) systems are the general class of input-output dynamical system used for modeling stochastic linear dynamical system (LDS) including partially observable LDS such as LQG systems. In this work, we study the problem of system identification and adaptive control of unknown ARX systems. We provide finite-time learning guarantees for the ARX systems under both open-loop and closed-loop data collection. Using these guarantees, we design adaptive control algorithms for unknown ARX systems with arbitrary strongly convex or non-strongly convex quadratic regulating costs. Under strongly convex cost functions, we design an adaptive control algorithm based on online gradient descent to design and update the controllers that are constructed via a convex controller reparametrization. We show that our algorithm has Õ(√T) regret via explore and commit approach and if the model estimates are updated in epochs using closed-loop data collection, it attains the optimal regret of polylog(T) after T time-steps of interaction. For the case of non-strongly convex quadratic cost functions, we propose an adaptive control algorithm that deploys the optimism in the face of uncertainty principle to design the controller. In this setting, we show that the explore and commit approach has a regret upper bound of Õ(√T^(2/3)), and the adaptive control with continuous model estimate updates attains Õ(√T) regret after T time-steps.https://authors.library.caltech.edu/records/rszc7-7g943Manifold Optimization for High-Accuracy Spatial Location Estimation Using Ultrasound Waves
https://resolver.caltech.edu/CaltechAUTHORS:20210414-080100426
Authors: AlSharif, Mohammed H.; Douik, Ahmed; Ahmed, Mohanad; Al-Naffouri, Tareq Y.; Hassibi, Babak
Year: 2021
DOI: 10.1109/TSP.2021.3109792
This paper reports the design of a high-accuracy spatial location estimation method using ultrasound waves by exploiting the fixed geometry of the transmitters. Assuming an isosceles triangle antenna configuration, where three antennas are placed as the vertices of an isosceles triangle, the spatial location problem can be formulated as a non-convex optimization problem whose interior is shown to admit a Riemannian manifold structure. Our investigation of the geometry of the newly introduced manifold (i.e., the manifold of all isosceles triangles in R³) enables the design of highly efficient optimization algorithms. Simulations are presented to compare the performance of the proposed approach with popular methods from the literature. The results suggest that the proposed Riemannian-based methods outperform the state-of-the-art methods. Furthermore, the proposed Riemannian methods require much less computation time compared to popular generic non-convex approaches.https://authors.library.caltech.edu/records/gsxyg-y9782The CEO Problem With Inter-Block Memory
https://resolver.caltech.edu/CaltechAUTHORS:20211222-495165300
Authors: Kostina, Victoria; Hassibi, Babak
Year: 2021
DOI: 10.1109/tit.2021.3111658
An n-dimensional source with memory is observed by K isolated encoders via parallel channels, who compress their observations to transmit to the decoder via noiseless rate-constrained links while leveraging their memory of the past. At each time instant, the decoder receives K new codewords from the observers, combines them with the past received codewords, and produces a minimum-distortion estimate of the latest block of n source symbols. This scenario extends the classical one-shot CEO problem to multiple rounds of communication with communicators maintaining the memory of the past. We extend the Berger-Tung inner and outer bounds to the scenario with inter-block memory, showing that the minimum asymptotically (as n→∞) achievable sum rate required to achieve a target distortion is bounded by minimal directed mutual information problems. For the Gauss-Markov source observed via K parallel AWGN channels, we show that the inner bound is tight and solve the corresponding minimal directed mutual information problem, thereby establishing the minimum asymptotically achievable sum rate. Finally, we explicitly bound the rate loss due to a lack of communication among the observers; that bound is attained with equality in the case of identical observation channels. The general coding theorem is proved via a new nonasymptotic bound that uses stochastic likelihood coders and whose asymptotic analysis yields an extension of the Berger-Tung inner bound to the causal setting. The analysis of the Gaussian case is facilitated by reversing the channels of the observers.https://authors.library.caltech.edu/records/38jst-wdx28Low-Rank Riemannian Optimization for Graph-Based Clustering Applications
https://resolver.caltech.edu/CaltechAUTHORS:20210503-115705141
Authors: Douik, Ahmed; Hassibi, Babak
Year: 2022
DOI: 10.1109/tpami.2021.3074467
With the abundance of data, machine learning applications engaged increased attention in the last decade. An attractive approach to robustify the statistical analysis is to preprocess the data through clustering. This paper develops a low-complexity Riemannian optimization framework for solving optimization problems on the set of positive semidefinite stochastic matrices. The low-complexity feature of the proposed algorithms stems from the factorization of the optimization variable X = YY^T and deriving conditions on the number of columns of Y under which the factorization yields a satisfactory solution. The paper further investigates the embedded and quotient geometries of the resulting Riemannian manifolds. In particular, the paper explicitly derives the tangent space, Riemannian gradients and Hessians, and a retraction operator allowing the design of efficient first and second-order optimization methods for the graph-based clustering applications of interest. The numerical results reveal that the resulting algorithms present a clear complexity advantage as compared with state-of-the-art Euclidean and Riemannian approaches for graph clustering applications.https://authors.library.caltech.edu/records/xyckf-8xv90Thompson Sampling Achieves Õ(√T) Regret in Linear Quadratic Control
https://resolver.caltech.edu/CaltechAUTHORS:20220714-212445251
Authors: Kargin, Taylan; Lale, Sahin; Azizzadenesheli, Kamyar; Anandkumar, Anima; Hassibi, Babak
Year: 2022
DOI: 10.48550/arXiv.2206.08520
Thompson Sampling (TS) is an efficient method for decision-making under uncertainty, where an action is sampled from a carefully prescribed distribution which is updated based on the observed data. In this work, we study the problem of adaptive control of stabilizable linear-quadratic regulators (LQRs) using TS, where the system dynamics are unknown. Previous works have established that Õ(√T) frequentist regret is optimal for the adaptive control of LQRs. However, the existing methods either work only in restrictive settings, require a priori known stabilizing controllers, or utilize computationally intractable approaches. We propose an efficient TS algorithm for the adaptive control of LQRs, TS-based Adaptive Control, TSAC, that attains Õ(√T)regret, even for multidimensional systems, thereby solving the open problem posed in Abeille and Lazaric (2018). TSAC does not require a priori known stabilizing controller and achieves fast stabilization of the underlying system by effectively exploring the environment in the early stages. Our result hinges on developing a novel lower bound on the probability that the TS provides an optimistic sample. By carefully prescribing an early exploration strategy and a policy update rule, we show that TS achieves order-optimal regret in adaptive control of multidimensional stabilizable LQRs. We empirically demonstrate the performance and the efficiency of TSAC in several adaptive control tasks.https://authors.library.caltech.edu/records/87tww-zn973Differentially Quantized Gradient Methods
https://resolver.caltech.edu/CaltechAUTHORS:20220909-232702000
Authors: Lin, Chung-Yi; Kostina, Victoria; Hassibi, Babak
Year: 2022
DOI: 10.1109/tit.2022.3171173
Consider the following distributed optimization scenario. A worker has access to training data that it uses to compute the gradients while a server decides when to stop iterative computation based on its target accuracy or delay constraints. The server receives all its information about the problem instance from the worker via a rate-limited noiseless communication channel. We introduce the principle we call differential quantization (DQ) that prescribes compensating the past quantization errors to direct the descent trajectory of a quantized algorithm towards that of its unquantized counterpart. Assuming that the objective function is smooth and strongly convex, we prove that differentially quantized gradient descent (DQ-GD) attains a linear contraction factor of $\max \{\sigma _{\mathrm {GD}}, \rho _{n} 2^{-R}\}$ , where $\sigma _{\mathrm {GD}}$ is the contraction factor of unquantized gradient descent (GD), $\rho _{n} \geq 1$ is the covering efficiency of the quantizer, and $R$ is the bitrate per problem dimension $n$ . Thus at any $R\geq \log _{2} \rho _{n} /\sigma _{\mathrm {GD}}$ bits, the contraction factor of DQ-GD is the same as that of unquantized GD, i.e., there is no loss due to quantization. We show a converse demonstrating that no algorithm within a certain class can converge faster than $\max \{\sigma _{\mathrm {GD}}, 2^{-R}\}$ . Since quantizers exist with $\rho _{n} \to 1$ as $n \to \infty $ (Rogers, 1963), this means that DQ-GD is asymptotically optimal. In contrast, naively quantized GD where the worker directly quantizes the gradient barely attains $\sigma _{\mathrm {GD}} + \rho _{n}2^{-R}$ . The principle of differential quantization continues to apply to gradient methods with momentum such as Nesterov's accelerated gradient descent, and Polyak's heavy ball method. For these algorithms as well, if the rate is above a certain threshold, there is no loss in contraction factor obtained by the differentially quantized algorithm compared to its unquantized counterpart, and furthermore, the differentially quantized heavy ball method attains the optimal contraction achievable among all (even unquantized) gradient methods. Experimental results on least-squares problems validate our theoretical analysis.https://authors.library.caltech.edu/records/thnm4-0jq77Regret-Optimal Filtering for Prediction and Estimation
https://resolver.caltech.edu/CaltechAUTHORS:20221114-805047300.12
Authors: Sabag, Oron; Hassibi, Babak
Year: 2022
DOI: 10.1109/tsp.2022.3212153
We study the filtering problem of causally estimating a desired signal from a related observation signal, through the lens of regret optimization. Classical filter designs, such as ℋ₂ (i.e., Kalman) and ℋ_∞, minimize the average and worst-case estimation errors, respectively. As a result ℋ₂ filters are sensitive to inaccuracies in the underlying statistical model, and ℋ_∞ filters are overly conservative since they safeguard against the worst-case scenario. In order to design filters that perform well in different noise regimes, we propose instead to minimize the regret by comparing the performance of the designed filter with that of a clairvoyant filter. More explicitly, we minimize the largest deviation of the squared estimation error of a causal filter from that of a non-causal filter that also has access to future observations. For the important case of signals that can be described with a linear state-space, we provide an explicit solution for the regret optimal filter in the estimation (causal) and the prediction (strictly-causal) regimes. These solutions are obtained by reducing the regret filtering problem to a Nehari problem, i.e., approximating a non-causal operator by a causal one in spectral norm. The regret-optimal filters bear some resemblance to Kalman and ℋ_∞ filters: they are expressed as state-space models, inherit the finite dimension of the original state-space, and their solutions require solving algebraic Riccati equations. Numerical simulations demonstrate that regret minimization inherently interpolates between the performances of the ℋ₂ and ℋ_∞ filters and is thus a viable approach for filter design.https://authors.library.caltech.edu/records/qfna3-wqz15How to Query an Oracle? Efficient Strategies to Label Data
https://resolver.caltech.edu/CaltechAUTHORS:20221031-572094900.1
Authors: Lahouti, Farshad; Kostina, Victoria; Hassibi, Babak
Year: 2022
DOI: 10.1109/tpami.2021.3118644
We consider the basic problem of querying an expert oracle for labeling a dataset in machine learning. This is typically an expensive and time consuming process and therefore, we seek ways to do so efficiently. The conventional approach involves comparing each sample with (the representative of) each class to find a match. In a setting with N equally likely classes, this involves N/2 pairwise comparisons (queries per sample) on average. We consider a k-ary query scheme with k ≥ 2 samples in a query that identifies (dis)similar items in the set while effectively exploiting the associated transitive relations. We present a randomized batch algorithm that operates on a round-by-round basis to label the samples and achieves a query rate of O(N/k²). In addition, we present an adaptive greedy query scheme, which achieves an average rate of ≈0.2N queries per sample with triplet queries. For the proposed algorithms, we investigate the query rate performance analytically and with simulations. Empirical studies suggest that each triplet query takes an expert at most 50% more time compared with a pairwise query, indicating the effectiveness of the proposed k-ary query schemes. We generalize the analyses to nonuniform class distributions when possible.https://authors.library.caltech.edu/records/476ee-hxx78Stochastic Mirror Descent on Overparameterized Nonlinear Models
https://resolver.caltech.edu/CaltechAUTHORS:20190628-084821622
Authors: Azizan, Navid; Lale, Sahin; Hassibi, Babak
Year: 2022
DOI: 10.1109/TNNLS.2021.3087480
Most modern learning problems are highly overparameterized, i.e., have many more model parameters than the number of training data points. As a result, the training loss may have infinitely many global minima (parameter vectors that perfectly "interpolate" the training data). It is therefore imperative to understand which interpolating solutions we converge to, how they depend on the initialization and learning algorithm, and whether they yield different test errors. In this article, we study these questions for the family of stochastic mirror descent (SMD) algorithms, of which stochastic gradient descent (SGD) is a special case. Recently, it has been shown that for overparameterized linear models, SMD converges to the closest global minimum to the initialization point, where closeness is in terms of the Bregman divergence corresponding to the potential function of the mirror descent. With appropriate initialization, this yields convergence to the minimum-potential interpolating solution, a phenomenon referred to as implicit regularization. On the theory side, we show that for sufficiently- overparameterized nonlinear models, SMD with a (small enough) fixed step size converges to a global minimum that is "very close" (in Bregman divergence) to the minimum-potential interpolating solution, thus attaining approximate implicit regularization. On the empirical side, our experiments on the MNIST and CIFAR-10 datasets consistently confirm that the above phenomenon occurs in practical scenarios. They further indicate a clear difference in the generalization performances of different SMD algorithms: experiments on the CIFAR-10 dataset with different regularizers, ℓ₁ to encourage sparsity, ℓ₂ (SGD) to encourage small Euclidean norm, and ℓ∞ to discourage large components, surprisingly show that the ℓ∞ norm consistently yields better generalization performance than SGD, which in turn generalizes better than the ℓ₁ norm.https://authors.library.caltech.edu/records/6rrav-h4569Regret-Optimal Estimation and Control
https://resolver.caltech.edu/CaltechAUTHORS:20230705-538377800.2
Authors: Goel, Gautam; Hassibi, Babak
Year: 2023
DOI: 10.1109/tac.2023.3253304
In this article, we consider estimation and control in linear dynamical systems from the perspective of regret minimization. Unlike most prior work in this area, we focus on the problem of designing causal state estimators and causal controllers, which compete against a clairvoyant noncausal policy, instead of the best policy selected in hindsight from some fixed parametric class. We show that regret-optimal filters and regret-optimal controllers can be derived in state space form using operator-theoretic techniques from robust control. Our results can be viewed as extending traditional robust estimation and control, which focuses on minimizing worst-case cost, to minimizing worst-case regret. We propose regret-optimal analogs of model-predictive control and the extended Kalman filter for systems with nonlinear dynamics and present numerical experiments which show that these algorithms can significantly outperform standard approaches to estimation and control.https://authors.library.caltech.edu/records/y9nd3-x3w31