Raj Rao Nadakuditi

Sample Eigenvalue Based Detection of High-Dimensional Signals in White Noise Using Relatively Few Samples

The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a mathematically justifiable, computationally simple, sample-eigenvalue-based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples. The main motivation for considering a sample-eigenvalue-based scheme is the computational simplicity and the robustness to eigenvector modelling errors which can adversely impact the performance of estimators that exploit information in the sample eigenvectors. There is, however, a price we pay by discarding the information in the sample eigenvectors; we highlight a fundamental asymptotic limit of sample-eigenvalue-based detection of weak or closely spaced high-dimensional signals from a limited sample size. This motivates our heuristic definition of the effective number of identifiable signals which is equal to the number of ldquosignalrdquo eigenvalues of the population covariance matrix which exceed the noise variance by a factor strictly greater than . The fundamental asymptotic limit brings into sharp focus why, when there are too few samples available so that the effective number of signals is less than the actual number of signals, underestimation of the model order is unavoidable (in an asymptotic sense) when using any sample-eigenvalue-based detection scheme, including the one proposed herein. The analysis reveals why adding more sensors can only exacerbate the situation. Numerical simulations are used to demonstrate that the proposed estimator, like Wax and Kailaths MDL-based estimator, consistently estimates the true number of signals in the dimension fixed, large sample size limit and the effective number of identifiable signals, unlike Wax and Kailaths MDL-based estimator, in the large dimension, (relatively) large sample size limit.

Asymptotic Mean Squared Error Performance of Diagonally Loaded Capon-MVDR Processor

The asymptotic local mean squared error (MSE) performance of the Capon algorithm, a.k.a the minimum variance distortionless response (MVDR) spectral estimator, has been studied extensively by several authors. Stoica et al. [21], Vaidyanathan and Buckley [23], and Hawkes and Nehorai [11] have exploited Taylor’s theorem and complex gradient methods to provide accurate prediction of the Capon algorithm signal parameter estimate MSE performance. These predictions are valid (i) above the estimation threshold signal-to-noise ratio (SNR) and (ii) provided a sufficient number of training samples is available for covariance estimation. The goal of this present analysis is to extend these results to the case in which the sample covariance matrix is diagonally loaded, as is often done in practice for regularization, stabilizing matrix inversion, and white noise gain control [7]. Recent advances in the theory of random matrices with large dimensions facilitate simple calculation of the required moments of the eigenvalues of several modified complex Wishart matrices including the inverse of the diagonally loaded case [14, 16]. This initial work focuses on the MSE prediction of angle estimates derived for the canonical case of single and multiple planewave signals in white noise.

The bias of the MVDR beamformer outputs under diagonal loading

The MVDR beamformer is the most extensively used array processing algorithm and involves inverting the sample covariance matrix. In the snapshot deficient scenario, when the number of sensors is greater than or approximately equal to the number of snapshots, the eigenvalues of the resulting sample covariance matrix are poorly conditioned. Diagonal loading is then applied to the sample covariance matrix. Expressions for the bias of the resulting MVDR beamformer outputs in the sidelobe region are presented that are exact for asymptotically large arrays. Numerical simulations confirm the accuracy of these asymptotic expressions when predicting the bias of the outputs of moderately large arrays.

Robust adaptive vector sensor processing in the presence of mismatch and finite sample support

We present analytical results which quantify the effect of system mismatch and finite sample support on acoustic vector sensor array performance. One noteworthy result is that the vector aspect of the array ldquodampensrdquo the effect of array mismatch, enabling deeper true nulls. This is accomplished because the variance of the vector sensor array spatial response (due to rotational, positional and filter gain/phase perturbations) decreases in the sidelobes, unlike arrays of omnidirectional hydrophones. When sensor orientation is measured within a reasonable tolerance, the beampattern variance dominates the average sidelobe power response. Our analysis also suggests that vector sensor array gain performance is less sensitive to rotational than to positional perturbations in the regions of interest. We analytically characterize the eigen-SNR threshold, which depends on the signal and noise covariance and the number of noise-only and signal-plus-noise snapshots, below which (asymptotically speaking) reliable detection using sample eigenvalue based techniques is not possible. Thus for a given number of snapshots, since the dimensionality of the snapshot in a vector sensor array is larger than that of a hydrophone-only array, the eigen-SNR detection threshold will be greater whenever the eigenvector information is discarded. We present processing techniques customized to the unique characteristics of vector sensors, which exploit information encoded in the sample eigenvectors and are robust to the mismatch and finite sample support issues. These methods include adaptive processing techniques with multiple white noise constraints.

Fundamental Limit of Sample Eigenvalue based Detection of Signals in Colored Noise using Relatively Few Samples

Sample eigenvalue based estimators are often used for estimating the number of high-dimensional signals in colored noise when an independent estimate of the noise covariance matrix is available. We highlight a fundamental asymptotic limit of sample eigenvalue based detection that brings into sharp focus why in the large system, relatively large sample size limit, underestimation of the model order may be unavoidable for weak/closely spaced signals. We discuss the implication of these results for the detection of two weak, closely spaced signals.

A channel subspace filtering approach to adaptive equalization of highly dynamic realistic channels

A major challenge while communicating in dynamic channels, such as the underwater acoustic channel, is the large amount of time-varying ISI due to multipath. In realistic channels, the fluctuations between different taps of the sampled channel impulse response are correlated. Traditional least squares algorithms used for adapting channel equalizers do not exploit this correlation structure. A channel subspace post-filtering algorithm is presented that treats the least squares channel estimate as a noisy time series and exploits the channel correlation structure to reduce the channel estimation error. A channel estimate based equalizer structure that uses this improved channel estimate to determine the equalizer coefficients is presented. The improvement in performance of the equalizer is predicted theoretically and demonstrated using both simulation and experimental data.

A channel subspace filtering approach to adaptive equalization of realistic acoustic channels

One of the major problems in underwater acoustic communications is compensating for the large amount of time varying intersymbol interference (ISI) due to multipath. Associated with each of the deterministic propagation paths are macro‐multipath fluctuations which depend on large scale environmental features and geometry and micro‐multipath fluctuations which are dependent on small scale environmental inhomogeneities. For arrivals which are unsaturated or partially saturated, the fluctuations in the ISI are dominated by the macro‐multipath fluctuations resulting in correlated fluctuations between different taps of the sampled channel impulse response. Traditional recursive least squares (RLS) algorithms used for adapting channel equalizers do not exploit this structure. A post‐filtering algorithm is presented to exploit this channel correlation structure in least squares based channel estimators and equalizers. The improvement in the performance of the algorithm with respect to the traditional least squar...

Random matrix theory enabled performance analysis and algorithms for underwater signal processing

Random matrices arise naturally in many undersea signal processing applications such as sonar and underwater acoustic communications. For example, the matrix formed by stacking a noisy time series of observations collected at a sensor array alongside each other is a random matrix. Random matrix theory provides a mathematical framework for reasoning about and understanding the structure in such noisy matrix-valued signals in an analogous manner to how Fourier analysis provides us a mathematical framework for reasoning about and understanding the structure in noisy vector valued signals. We highlight some recent breakthroughs in random matrix theory that have allowed us to predict the fundamental performance limits of weak signal detection, estimation and classification and discuss some recent successes where the theory has led to the development of powerful new algorithms for better estimating weaker signals than previously thought possible.

Random matrix theory and performance prediction of subspace methods

Subspace methods constitute a powerful class of techniques for detection, estimation and classification of signals buried in noise. Recent results in random matrix theory precisely quantify the accuracy of subspaces estimates from finite, noisy data in both the white noise and colored noise setting. This advance facilitates unified performance analysis of signal processing methods that rely on these empirical subspaces. We discuss the pertinent theory and its application to the characterization of the performance of direction-of-arrival estimation, matched subspace detection and subspace clustering for large arrays in the sample-starved setting for both white and colored noise.

Analytical prediction of sample eigenvector quality deterioration in large arrays due to SNR or sample size constraints

It is well-known that subspace-based estimation methods in adaptive array processing suer a rapid degradation in performance as either the signal-to-noise ratio (SNR) or the number of available snapshots drops below a certain threshold value. In the large system, relative large sample size limit, one can use random matrix theory to analytically predict this threshold and the degradation in the “quality” of the corresponding subspace estimates. In certain settings, one observes a ”phase transition” phenonemon so that if the signals are too weak or there are insucient number of snapshots or both, the subspace estimates are, statistically speaking, noise-like. We discuss the implication of these results for the subspace based detection of signals in white and colored noise using large arrays and illustrate the accuracy of the predictions with numerical simulations.