Waheed U. Bajwa

Compressed Channel Sensing: A New Approach to Estimating Sparse Multipath Channels

High-rate data communication over a multipath wireless channel often requires that the channel response be known at the receiver. Training-based methods, which probe the channel in time, frequency, and space with known signals and reconstruct the channel response from the output signals, are most commonly used to accomplish this task. Traditional training-based channel estimation methods, typically comprising linear reconstruction techniques, are known to be optimal for rich multipath channels. However, physical arguments and growing experimental evidence suggest that many wireless channels encountered in practice tend to exhibit a sparse multipath structure that gets pronounced as the signal space dimension gets large (e.g., due to large bandwidth or large number of antennas). In this paper, we formalize the notion of multipath sparsity and present a new approach to estimating sparse (or effectively sparse) multipath channels that is based on some of the recent advances in the theory of compressed sensing. In particular, it is shown in the paper that the proposed approach, which is termed as compressed channel sensing (CCS), can potentially achieve a target reconstruction error using far less energy and, in many instances, latency and bandwidth than that dictated by the traditional least-squares-based training methods.

Compressed Sensing for Networked Data

This article describes a very different approach to the decentralized compression of networked data. Considering a particularly salient aspect of this struggle that revolves around large-scale distributed sources of data and their storage, transmission, and retrieval. The task of transmitting information from one point to another is a common and well-understood exercise. But the problem of efficiently transmitting or sharing information from and among a vast number of distributed nodes remains a great challenge, primarily because we do not yet have well developed theories and tools for distributed signal processing, communications, and information theory in large-scale networked systems.

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.

Compressive wireless sensing

Compressive sampling is an emerging theory that is based on the fact that a relatively small number of random projections of a signal can contain most of its salient information. In this paper, we introduce the concept of compressive wireless sensing for sensor networks in which a fusion center retrieves signal field information from an ensemble of spatially distributed sensor nodes. Energy and bandwidth are scarce resources in sensor networks and the relevant metrics of interest in our context are 1) the latency involved in information retrieval; and 2) the associated power-distortion trade-off. It is generally recognized that given sufficient prior knowledge about the sensed data (e.g., statistical characterization, homogeneity etc.), there exist schemes that have very favorable power-distortion-latency trade-offs. We propose a distributed matched source-channel communication scheme, based in part on recent results in compressive sampling theory, for estimation of sensed data at the fusion center and analyze, as a function of number of sensor nodes, the trade-offs between power, distortion and latency. Compressive wireless sensing is a universal scheme in the sense that it requires no prior knowledge about the sensed data. This universality, however, comes at the cost of optimality (in terms of a less favorable power-distortion-latency trade-off) and we quantify this cost relative to the case when sufficient prior information about the sensed data is assumed

Toeplitz-Structured Compressed Sensing Matrices

The problem of recovering a sparse signal x Rn from a relatively small number of its observations of the form y = Ax Rk, where A is a known matrix and k « n, has recently received a lot of attention under the rubric of compressed sensing (CS) and has applications in many areas of signal processing such as data cmpression, image processing, dimensionality reduction, etc. Recent work has established that if A is a random matrix with entries drawn independently from certain probability distributions then exact recovery of x from these observations can be guaranteed with high probability. In this paper, we show that Toeplitz-structured matrices with entries drawn independently from the same distributions are also sufficient to recover x from y with high probability, and we compare the performance of such matrices with that of fully independent and identically distributed ones. The use of Toeplitz matrices in CS applications has several potential advantages: (i) they require the generation of only O(n) independent random variables; (ii) multiplication with Toeplitz matrices can be efficiently implemented using fast Fourier transform, resulting in faster acquisition and reconstruction algorithms; and (iii) Toeplitz-structured matrices arise naturally in certain application areas such as system identification.

Compressed channel sensing

Reliable wireless communications often requires accurate knowledge of the underlying multipath channel. This typically involves probing of the channel with a known training waveform and linear processing of the input probe and channel output to estimate the impulse response. Many real-world channels of practical interest tend to exhibit impulse responses characterized by a relatively small number of nonzero channel coefficients. Conventional linear channel estimation strategies, such as the least squares, are ill-suited to fully exploiting the inherent low-dimensionality of these sparse channels. In contrast, this paper proposes sparse channel estimation methods based on convex/linear programming. Quantitative error bounds for the proposed schemes are derived by adapting recent advances from the theory of compressed sensing. The bounds come within a logarithmic factor of the performance of an ideal channel estimator and reveal significant advantages of the proposed methods over the conventional channel estimation schemes.

Joint Source–Channel Communication for Distributed Estimation in Sensor Networks

Power and bandwidth are scarce resources in dense wireless sensor networks and it is widely recognized that joint optimization of the operations of sensing, processing and communication can result in significant savings in the use of network resources. In this paper, a distributed joint source-channel communication architecture is proposed for energy-efficient estimation of sensor field data at a distant destination and the corresponding relationships between power, distortion, and latency are analyzed as a function of number of sensor nodes. The approach is applicable to a broad class of sensed signal fields and is based on distributed computation of appropriately chosen projections of sensor data at the destination - phase-coherent transmissions from the sensor nodes enable exploitation of the distributed beamforming gain for energy efficiency. Random projections are used when little or no prior knowledge is available about the signal field. Distinct features of the proposed scheme include: (1) processing and communication are combined into one distributed projection operation; (2) it virtually eliminates the need for in-network processing and communication; (3) given sufficient prior knowledge about the sensed data, consistent estimation is possible with increasing sensor density even with vanishing total network power; and (4) consistent signal estimation is possible with power and latency requirements growing at most sublinearly with the number of sensor nodes even when little or no prior knowledge about the sensed data is assumed at the sensor nodes.

Why Gabor frames? Two fundamental measures of coherence and their role in model selection

The problem of model selection arises in a number of contexts, such as subset selection in linear regression, estimation of structures in graphical models, and signal denoising. This paper studies non-asymptotic model selection for the general case of arbitrary (random or deterministic) design matrices and arbitrary nonzero entries of the signal. In this regard, it generalizes the notion of incoherence in the existing literature on model selection and introduces two fundamental measures of coherence — termed as the worst-case coherence and the average coherence — among the columns of a design matrix. It utilizes these two measures of coherence to provide an in-depth analysis of a simple, model-order agnostic one-step thresholding (OST) algorithm for model selection and proves that OST is feasible for exact as well as partial model selection as long as the design matrix obeys an easily verifiable property, which is termed as the coherence property. One of the key insights offered by the ensuing analysis in this regard is that OST can successfully carry out model selection even when methods based on convex optimization such as the lasso fail due to the rank deficiency of the submatrices of the design matrix. In addition, the paper establishes that if the design matrix has reasonably small worst-case and average coherence then OST performs near-optimally when either (i) the energy of any nonzero entry of the signal is close to the average signal energy per nonzero entry or (ii) the signal-to-noise ratio in the measurement system is not too high. Finally, two other key contributions of the paper are that (i) it provides bounds on the average coherence of Gaussian matrices and Gabor frames, and (ii) it extends the results on model selection using OST to low-complexity, model-order agnostic recovery of sparse signals with arbitrary nonzero entries. In particular, this part of the analysis in the paper implies that an Alltop Gabor frame together with OST can successfully carry out model selection and recovery of sparse signals irrespective of the phases of the nonzero entries even if the number of nonzero entries scales almost linearly with the number of rows of the Alltop Gabor frame.

Identification of Parametric Underspread Linear Systems and Super-Resolution Radar

Identification of time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), whose responses lie within a unit-area region in the delay-Doppler space, by probing them with a known input signal. It is shown that sufficiently-underspread parametric linear systems, described by a finite set of delays and Doppler-shifts, are identifiable from a single observation as long as the time-bandwidth product of the input signal is proportional to the square of the total number of delay-Doppler pairs in the system. In addition, an algorithm is developed that enables identification of parametric ULSs from an input train of pulses in polynomial time by exploiting recent results on sub-Nyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials. Finally, application of these results to super-resolution target detection using radar is discussed. Specifically, it is shown that the proposed procedure allows to distinguish between multiple targets with very close proximity in the delay-Doppler space, resulting in a resolution that substantially exceeds that of standard matched-filtering based techniques without introducing leakage effects inherent in recently proposed compressed sensing-based radar methods.

Learning sparse doubly-selective channels

Coherent data communication over doubly-selective channels requires that the channel response be known at the receiver. Training-based schemes, which involve probing of the channel with known signaling waveforms and processing of the corresponding channel output to estimate the channel parameters, are commonly employed to learn the channel response in practice. Conventional training-based methods, often comprising of linear least squares channel estimators, are known to be optimal under the assumption of rich multipath channels. Numerous measurement campaigns have shown, however, that physical multipath channels tend to exhibit a sparse structure at high signal space dimension (time-bandwidth product), and can be characterized with significantly fewer parameters compared to the maximum number dictated by the delay-Doppler spread of the channel. In this paper, it is established that traditional training-based channel learning techniques are ill-suited to fully exploiting the inherent low-dimensionality of sparse channels. In contrast, key ideas from the emerging theory of compressed sensing are leveraged to propose sparse channel learning methods for both single-carrier and multicarrier probing waveforms that employ reconstruction algorithms based on convex/linear programming. In particular, it is shown that the performance of the proposed schemes come within a logarithmic factor of that of an ideal channel estimator, leading to significant reductions in the training energy and the loss in spectral efficiency associated with conventional training-based methods.