Showing papers by "Louis L. Scharf published in 2011"

Sensitivity to Basis Mismatch in Compressed Sensing

Abstract: The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.

Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety

Abstract: Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane. While these assumptions are convenient because they simplify computations, there are many cases where proper and circular random signals are very poor models of the underlying physics. When taking impropriety and noncircularity into account, the right type of processing can provide significant performance gains. There are two key ingredients in the statistical signal processing of complex-valued data: 1) utilizing the complete statistical characterization of complex-valued random signals; and 2) the optimization of real-valued cost functions with respect to complex parameters. In this overview article, we review the necessary tools, among which are widely linear transformations, augmented statistical descriptions, and Wirtinger calculus. We also present some selected recent developments in the field of complex-valued signal processing, addressing the topics of model selection, filtering, and source separation.

Phasor state estimation from PMU measurements with bad data

Abstract: The phasor measurement units (PMU) are expected to enhance state estimation in the power grid by providing accurate and timely measurements. However, due to communication errors and equipment failures, some detrimental data can occur among the measurements. The largest residual removal (LRR) algorithm is commonly used for phasor state estimation with bad data. Here, we show that this method cannot guarantee correctness unless data redundancy is very abundant. We then establish the equivalence between the approaches of bad data removal and bad data estimation and subtraction. In addition, we propose two new algorithms by exploiting the sparsity of the bad data. All algorithms are tested by simulations and our projection and minimization (PM) algorithm provides the best performance.

Sensitivity considerations in compressed sensing

Abstract: In [1]–[4], we considered the question of basis mismatch in compressive sensing Our motivation was to study the effect of mismatch between the mathematical basis (or frame) in which a signal was assumed to be sparse and the physical basis in which the signal was actually sparse We were motivated by the problem of inverting a complex space-time radar image for the field of complex scatterers that produced the image In this case there is no apriori known basis in which the image is actually sparse, as radar scatterers do not usually agree to place their ranges and Dopplers on any apriori agreed sampling grid The consequence is that sparsity in the physical basis is not maintained in the mathematical basis, and a sparse inversion in the mathematical basis or frame does not match up with an inversion for the field in the physical basis In [1]–[3], this effect was quantified with theorem statements about sensitivity to basis mismatch and with numerical examples for inverting time series records for their sparse set of damped complex exponential modes These inversions were compared unfavorably to inversions using fancy linear prediction In [4] and this paper, we continue these investigations by comparing the performance of sparse inversions of sparse images, using apriori selected frames that are mismatched to the physical basis, and by computing the Fisher information matrix for compressions of images that are sparse in a physical basis

Optimization of exponential error rates for a suboptimum fusion rule in wireless sensor networks

Abstract: A simple fusion rule with multiple thresholds is presented for a distributed system to resolve multiple hypotheses In contrast to the common assumption that the data is conditionally independent and identically distributed, only conditional independence under each hypothesis is assumed here This allows the modeling of situations in which different sensors have different local detection probabilities as well as situations in which different sensors have communication links of different qualities or signal-to-noise ratios A theorem is proved showing that one can select the thresholds independently in a manner that maximizes the asymptotic decay rate of the average probability of error Furthermore, it is easy to compute these individual thresholds numerically This is illustrated with an example

Linear precoding for time-varying MIMO channels with low-complexity receivers

Abstract: This paper considers linear precoding for time-varying multiple-input multiple-output (MIMO) channels. We show that linear minimum mean-squared error (LMMSE) equalization based on the conjugate gradient (CG) method can result in significantly reduced complexity compared with conventional approaches. This reduction is achieved by incorporating a condition number constraint into the precoder optimization framework, which leads to clustered eigen-values of the measurement covariance matrix. The cost is a small increase in MSE compared to the optimal precoder.

Multi-sensor beamsteering based on the asymptotic likelihood for colored signals

Abstract: In this work, we derive a maximum likelihood formula for beamsteering in a multi-sensor array. The novelty of the work is that the impinging signal and noises are wide sense stationary (WSS) time series with unknown power spectral densities, unlike in previous work that typically considers white signals. Our approach naturally provides a way of fusing frequency-dependent information to obtain a broadband beamformer. In order to obtain the compressed likelihood, it is necessary to find the maximum likelihood estimates of the unknown parameters. However, this problem turns out to be an ML estimation of a block-Toeplitz matrix, which does not have a closed-form solution. To overcome this problem, we derive the asymptotic likelihood, which is given in the frequency domain. Finally, some simulation results are presented to illustrate the performance of the proposed technique. In these simulations, it is shown that our approach presents the best results.