Search

In this work, we show that the incoherence alone is sufficient to establish the statistical restricted isometry property (StRIP) and statistical incoherence property (SInCoP) for unit norm tight frames (UNTFs). Further, we derive three simple properties that binary matrices need to satisfy, in order to produce incoherent UNTFs (IUNTFs) with high redundancy (ratio of the number of columns to the number of rows) via an existing embedding operation. We show that biadjacency matrices corresponding to biregular graphs satisfy the required properties. Thereby, we provide a connection between graph theory and the construction of IUNTFs. We also provide a bouquet of constructions of IUNTFs from finite fields and combinatorial designs. Another important aspect of our construction is that the sparse recovery guarantees for the embedded IUNTFs can in fact be translated to the constituent binary matrix. We show that if the constituent $m\times M$ binary matrix has constant row and column weight, it can support sparse recovery through $\ell_{1}$ -minimization for all but an $\epsilon$ -fraction of $t$ -sparse signals chosen from a random signal model, provided $m=O(t(\log(\frac{M}{\epsilon}))^{3}),$ which is a significant improvement over the existing $m=O(t^{2})$ bound, where $m$ denotes the number of measurements. Also, the StRIP and SInCoP based approach results in matrices whose column size is exponential in the fourth root of the row size. To the best of our knowledge, this is the first construction of deterministic matrices satisfying StRIP and SInCoP with such high redundancy.

The temporal variations in the wireless propagation channel, referred to as channel aging , cause a mismatch between the estimated channel and the channel state at the time of data detection. This mismatch has been shown to severely impair the performance of massive MIMO systems. In this paper, we present data aided MSE-optimal channel tracking algorithms to decode the received symbols and update the channel estimates at the base station (BS) and the UEs. These algorithms combine ideas from Kalman filtering for channel tracking and deterministic equivalent analysis for symbol estimation. In the uplink case, we first develop a minimum mean squared error (MMSE) data estimator and the MSE-optimal channel predictor based on the Kalman filtering algorithm. We analytically show that the updated channel estimate obtained from the estimates of the data symbols leads to significantly larger signal to interference-plus-noise ratio (SINR), and hence achievable rate, compared to that obtained from the channel estimate based on pilot symbols. Following this, in the downlink case, we develop an algorithm to track the effective channel at the UEs and analyze its MSE, SINR and achievable rate performance. We show that tracking the effective downlink channel mitigates the effects of channel aging and leads to improved performance. However, since the beamforming matrices at the BS are not updated, downlink channel tracking is not as effective as uplink channel tracking. Finally, via Monte Carlo simulations, we validate our derived results, and demonstrate the gains achievable by tracking the time-varying channels in massive MIMO systems.

In this work, we address the problem of multiple-input multiple-output mmWave channel estimation in a hybrid analog-digital architecture, by exploiting both the underlying spatial sparsity as well as the spatial correlation in the channel. We accomplish this via compressive covariance estimation, where we estimate the channel covariance matrix from noisy low dimensional projections of the channel obtained in the pilot transmission phase. We use the estimated covariance matrix as a plug-in to the linear minimum mean square estimator to obtain the channel estimate. We present a new Gaussian prior model, inspired by sparse Bayesian learning (SBL), which incorporates parameters to capture the channel correlation in addition to sparsity. Based on this prior, we develop the Corr-SBL algorithm, which uses an expectation maximization procedure to learn the parameters of the prior and update the posterior channel estimates. A closed form solution is obtained for the maximization step based on fixed-point iterations. To facilitate practical implementation, an online version of the algorithm is developed which significantly reduces the latency at a marginal loss in performance. The efficacy of the prior model is studied by analyzing the normalized mean squared error in the channel estimate. Our results show that, when compared to a genie-aided estimator and other existing sparse recovery algorithms, exploiting both sparsity and correlation results in significant performance gains, even under imperfect covariance estimates obtained using a limited number of samples.

Sweep spread carrier (S2C) based underwater acoustic (UWA) communications is a practically attractive but less explored modulation scheme in the published literature. In this paper, we present a rigorous treatment of the S2C communication receiver design and propose a data detection scheme that can handle challenging UWA channels. State-of-the-art S2C receivers based on the gradient heterodyne processing are only effective when the path delay and Doppler spread are moderate. We develop a new variational soft symbol decoding (VSSD) algorithm based on the principle of variational Bayes’ inference for a general linear channel model. In channels with moderate delay and Doppler spreads, we show that the VSSD algorithm is equivalent to the existing gradient heterodyne receivers for S2C communications. We apply the VSSD algorithm to the i.i.d. Gaussian multiple-input multiple-output channel and show, through numerical simulations, that it far outperforms the minimum mean squared error (MMSE) data detection. We illustrate the dramatic improvement in the performance of the VSSD based S2C receiver in two different models of simulated UWA channels and two contrasting measured UWA environments publicly available in the Watermark channel dataset. The proposed VSSD algorithm recovers data symbols at a signal-to-noise ratio (SNR) which is at least 10 dB (8 dB) lower than the MMSE decoder for uncoded (rate 2/3 LDPC coded) communications over UWA channels where the existing receivers either fail completely or must compromise on the data rate to maintain the bit error rate (BER) performance.

We consider the performance of time-division duplex (TDD) massive multiple-input multiple-output (MIMO) with imperfect calibration of the transmit and receive radio frequency chains. By deriving the achievable signal-to-interference-plus-noise ratio (SINR) and the per-user bit error rate (BER) for constant modulus constellations, we establish that, under linear precoding, reciprocity imperfections can result in substantial reduction of the array gain. To mitigate this loss, we propose an algorithm for blind estimation of the effective channel gain at each user. We show that, with sufficiently many downlink data symbols, our blind channel estimation algorithm restores the array gain. In addition, the proposed blind gain estimation algorithm can improve performance compared to standard hardening-based receivers even under perfect reciprocity. Following this, we derive the BERs for non-constant modulus constellations, viz. $M$ -PAM and $M$ -QAM. We corroborate all our derived results using numerical simulations.

Dictionary learning (DL) is a well-researched problem, where the goal is to learn a dictionary from a finite set of noisy training signals, such that the training data admits a sparse representation over the dictionary. While several solutions are available in the literature, relatively little is known about their convergence and optimality properties. In this paper, we make progress on this problem by analyzing a Bayesian algorithm for DL. Specifically, we cast the DL problem into the sparse Bayesian learning (SBL) framework by imposing a hierarchical Gaussian prior on the sparse vectors. This allows us to simultaneously learn the dictionary as well as the parameters of the prior on the sparse vectors using the expectation-maximization algorithm. The dictionary update step turns out to be a non-convex optimization problem, and we present two solutions, namely, an alternating minimization (AM) procedure and an Armijo line search (ALS) method. We analytically show that the ALS procedure is globally convergent, and establish the stability of the solution by characterizing its limit points. Further, we prove the convergence and stability of the overall DL-SBL algorithm, and show that the minima of the cost function of the overall algorithm are achieved at sparse solutions. As a concrete example, we consider the application of the SBL-based DL algorithm to image denoising, and demonstrate the efficacy of the algorithm relative to existing DL algorithms.

The columnwise Khatri-Rao product of two matrices is an important matrix type, reprising its role as a structured sensing matrix in many fundamental linear inverse problems. Robust signal recovery in such inverse problems is often contingent on proving the restricted isometry property (RIP) of a certain system matrix expressible as a Khatri-Rao product of two matrices. In this work, we analyze the RIP of a generic columnwise Khatri-Rao product matrix by deriving two upper bounds for its $k^{\text{th}}$ order Restricted Isometry Constant ($k$-RIC) for different values of $k$. The first RIC bound is computed in terms of the individual RICs of the input matrices participating in the Khatri-Rao product. The second RIC bound is probabilistic, and is specified in terms of the input matrix dimensions. We show that the Khatri-Rao product of a pair of $m \times n$ sized random matrices comprising independent and identically distributed subgaussian entries satisfies $k$-RIP with arbitrarily high probability, provided $m$ exceeds $O(k \log n)$. Our RIC bounds confirm that the Khatri-Rao product exhibits stronger restricted isometry compared to its constituent matrices for the same RIP order. The proposed RIC bounds are potentially useful in the sample complexity analysis of several sparse recovery problems., The probabilistic RIC bounds presented as Theorems 2 and 3 in the earlier version (v1 and v2) could be potentially incorrect and have been restated with new proofs in this revised manuscript

The impulse response of wireless channels between the N-t transmit and N-r receive antennas of a MIMO-OFDM system are group approximately sparse (ga-sparse), i.e., NtNt the channels have a small number of significant paths relative to the channel delay spread and the time-lags of the significant paths between transmit and receive antenna pairs coincide. Often, wireless channels are also group approximately cluster-sparse (gac-sparse), i.e., every ga-sparse channel consists of clusters, where a few clusters have all strong components while most clusters have all weak components. In this paper, we cast the problem of estimating the ga-sparse and gac-sparse block-fading and time-varying channels in the sparse Bayesian learning (SBL) framework and propose a bouquet of novel algorithms for pilot-based channel estimation, and joint channel estimation and data detection, in MIMO-OFDM systems. The proposed algorithms are capable of estimating the sparse wireless channels even when the measurement matrix is only partially known. Further, we employ a first-order autoregressive modeling of the temporal variation of the ga-sparse and gac-sparse channels and propose a recursive Kalman filtering and smoothing (KFS) technique for joint channel estimation, tracking, and data detection. We also propose novel, parallel-implementation based, low-complexity techniques for estimating gac-sparse channels. Monte Carlo simulations illustrate the benefit of exploiting the gac-sparse structure in the wireless channel in terms of the mean square error (MSE) and coded bit error rate (BER) performance.

Presents corrections to the paper, “On the restricted isometry of the columnwise Khatri-Rao product,” (Khanna, S. and Murthy, C.R.), IEEE Trans. Signal Process., vol. 66, no. 5, pp. 1170– 1183, Mar. 2018.

In this paper, we derive Hybrid, Bayesian and Marginalized Cram\'{e}r-Rao lower bounds (HCRB, BCRB and MCRB) for the single and multiple measurement vector Sparse Bayesian Learning (SBL) problem of estimating compressible vectors and their prior distribution parameters. We assume the unknown vector to be drawn from a compressible Student-t prior distribution. We derive CRBs that encompass the deterministic or random nature of the unknown parameters of the prior distribution and the regression noise variance. We extend the MCRB to the case where the compressible vector is distributed according to a general compressible prior distribution, of which the generalized Pareto distribution is a special case. We use the derived bounds to uncover the relationship between the compressibility and Mean Square Error (MSE) in the estimates. Further, we illustrate the tightness and utility of the bounds through simulations, by comparing them with the MSE performance of two popular SBL-based estimators. It is found that the MCRB is generally the tightest among the bounds derived and that the MSE performance of the Expectation-Maximization (EM) algorithm coincides with the MCRB for the compressible vector. Through simulations, we demonstrate the dependence of the MSE performance of SBL based estimators on the compressibility of the vector for several values of the number of observations and at different signal powers., Comment: Accepted for publication in the IEEE Transactions on Signal Processing, 11 pages, 10 figures

This paper considers the high-rate performance of source coding for noisy discrete symmetric channels with random index assignment (IA). Accurate analytical models are developed to characterize the expected distortion performance of vector quantization (VQ) for a large class of distortion measures. It is shown that when the point density is continuous, the distortion can be approximated as the sum of the source quantization distortion and the channel-error induced distortion. Expressions are also derived for the continuous point density that minimizes the expected distortion. Next, for the case of mean squared error distortion, a more accurate analytical model for the distortion is derived by allowing the point density to have a singular component. The extent of the singularity is also characterized. These results provide analytical models for the expected distortion performance of both conventional VQ as well as for channel-optimized VQ. As a practical example, compression of the linear predictive coding parameters in the wideband speech spectrum is considered, with the log spectral distortion as performance metric. The theory is able to correctly predict the channel error rate that is permissible for operation at a particular level of distortion.

We consider the design and analysis of quantizers for equal gain transmission (EGT) systems with finite rate feedback-based communication in flat-fading multiple input single output (MISO) systems. EGT is a beamforming technique that maximizes the MISO channel capacity when there is an equal power-per-antenna constraint at the transmitter, and requires the feedback of t-1 phase angles, when there are t antennas at the transmitter. In this paper, we contrast two popular approaches for quantizing the phase angles: vector quantization (VQ) and scalar quantization (SQ). On the VQ side, using the capacity loss with respect to EGT with perfect channel information at transmitter as performance metric, we develop a criterion for designing the beamforming codebook for quantized EGT (Q-EGT). We also propose an iterative algorithm based on the well-known generalized Lloyd algorithm, for computing the beamforming vector codebook. On the analytical side, we study the performance of Q-EGT and derive closed-form expressions for the performance in terms of capacity loss and outage probability in the case of i.i.d. Rayleigh flat-fading channels. On the SQ side, assuming uniform scalar quantization and i.i.d. Rayleigh flat-fading channels, we derive the high-resolution performance of quantized EGT and contrast the performance with that of VQ. We find that although both VQ and SQ achieve the same rate of convergence (to the capacity with perfect feedback) as the number of feedback bits B increases, there exists a fixed gap between the two

This paper is a comparative study of training-based and semiblind multiple-input multiple-output (MIMO) flat-fading channel estimation schemes when the transmitter employs maximum ratio transmission (MRT). We present two competing schemes for estimating the transmit and receive beamforming vectors of the channel matrix: a training-based conventional least-squares estimation (CLSE) scheme and a closed-form semiblind (CFSB) scheme that employs training followed by information-bearing spectrally white data symbols. Employing matrix perturbation theory, we develop expressions for the mean-square error (MSE) in the beamforming vector, the average received signal-to-noise ratio (SNR) and the symbol error rate (SER) performance of both the semiblind and the conventional schemes. Finally, we describe a weighted linear combiner of the CFSB and CLSE estimates for additional improvement in performance. The analytical results are verified through Monte Carlo simulations