J.S. Goldstein

Insights from the relationship between the multistage Wiener filter and the method of conjugate gradients

This paper demonstrates that, under certain conditions, the method of conjugate gradients (CG) and the multistage Wiener filter (MWF) produce equivalent solutions. Equivalence follows from the fact that both algorithms minimize the same cost function in the same subspace, namely a Krylov subspace. Motivation for Krylov subspaces and their properties are developed herein. New insights into both algorithms follow from the equivalence including previously unpublished results on the convergence of the multistage Wiener filter as a function of rank. Furthermore a new perspective on CG is developed where CG can now be viewed as a reduced rank algorithm for the solution of the Wiener-Hopf equations.

Subspace Expansion and the Equivalence of Conjugate Direction and Multistage Wiener Filters

We consider iterative subspace Wiener filters for solving minimum mean-squared error (MMSE) and minimum variance unbiased estimation problems in low-dimensional subspaces. In this class of subspace filters, the conjugate direction and multistage Wiener filters comprise two large subclasses, and within these the conjugate gradient and orthogonal multistage Wiener filters are the most prominent. We establish very general equivalences between conjugate direction and multistage Wiener filters, wherein the direction vectors of a conjugate direction filter and the stagewise vectors of a multistage filter are related through a one-term autoregressive recursion. By virtue of this recursion, the expanding subspaces of the two filters are identical, even though their bases for them are different. As a consequence, their subspace filters, gradients, and MSEs are identical at each stage of the subspace iteration. If the conjugate direction filter is a conjugate gradient filter, then the equivalent stagewise filter is an orthogonal multistage filter, and vice-versa. If either the conjugate gradient filter or the orthogonal multistage filter is initialized at the cross-covariance vector between the signal and the measurement, then each of the subspace filters iteratively turns out a basis for a Krylov subspace.

Application of the L-curve technique to loading level determination in adaptive beamforming

Diagonal loading is a well-known technique in adaptive beamforming for adding robustness to mismatch. Effective use of this technique, however, requires that the loading level be properly set for the given environment. A technique called the L-curve, has been developed in the numerical linear algebra community for selecting the regularization parameter in least squares problems. Motivation for the L-curve is developed herein. This technique is applied to adaptive beamforming and its performance is evaluated through Monte Carlo simulation. Initial results indicate that the L-curve predicts the optimal loading level very well.

MMSE correlator based RAKE receiver for DS-CDMA

This paper investigates the performance of a reduced rank MMSE correlator for a RAKE receiver in the context of CDMA in frequency selective multipath. The MMSE correlator based on the correlations subtractive architecture (CSA) is derived for frequency selective multipath. It is demonstrated that the standard multiuser limited RAKE receiver can achieve BER performance close to the MMSE receiver spanning multiple symbols by replacing its conventional correlator with an MMSE correlator. The CSA generates an MMSE correlator without requiring matrix inversion (thereby reducing computational complexity) and facilitates direct replacement of the standard RAKE correlator.

A low complexity MMSE multiuser detector for DS-CDMA

A novel, low complexity, reduced rank multiuser detector (MUD) for DS-CDMA is presented. The solution is derived for the general case of an asynchronous DS-CDMA system in frequency selective multipath. The joint MUD solution is shown to be equivalent to a bank of parallel, single user detectors, previously shown only for synchronous users. The algorithm is implemented in low complexity form, using the efficient correlations subtractive architecture of the reduced rank multistage Wiener filter (CSA-MWF). The reduced rank algorithm is not based on an eigen-decomposition, which requires the signal subspace rank to be greater than or equal to the number of signals present in the system, and is shown to be nearly independent of the number of signals. It is shown that the joint and parallel CSA-MWF implementations perform identically to the full rank MMSE MUD and approaches single user, full rank performance at significantly reduced ranks. Bit error rate (BER) performance, as a function of rank, signal-to-noise ratio, number of users, code type and synchronism among the codes, is analyzed.

Performance Metric Issues for Space Time Adaptive Processing Methods

Space time adaptive processing was introduced to the radar community in the 1970s as a multi-dimensional filtering method to discriminate moving targets from clutter when observed from an air-borne/space-borne multi-channel phased array radar. Numerous variants of this filtering technique have been developed and are often compared to some baseline performance with respect to output signal to interference plus noise ratio or minimum mean square error. Under clairvoyant conditions these metrics have a direct relation to probability of detection. However under non-clairvoyant conditions, such as with recorded data, probability of detection is not measured directly and secondary metrics, such as signal to interference plus noise ratio and minimum mean square error, are used to measure performance. This paper attempts to show a case where using a metric such as minimum mean square error can indicate erroneous favorable performance.

Robust steering vector mismatch technique for the multistage Wiener filter

This paper examines the impact of steering vector mismatch on the multistage Wiener filter (MWF). Since the NWF centrally features the steering vector in its formulation it is important to assess the impact of steering vector mismatch. Furthermore, since the MWF converges to the full rank MVDR solution, we would expect that steering vector mismatch would lead to signal cancellation (as with full rank MVDR) when the signal is present in the training data. We examine several standard techniques for increasing robustness and show how they apply to the MWE These include derivative constraints, quiescent pattern control (QPC) techniques, and covariance matrix tapers (CMT). We show that a combination of CMT and QPC, denoted CMTQ, is very effective at mitigating the impact of steering vector mismatch. Use of the CMTQ augmentation provides the steering vector mismatch robustness that we desire while retaining the reduced rank and reduced sample support characteristics of the MWE.

Tri-diagonalization of the covariance matrix generated by the multistage Wiener filter for whitening blind signals

The multistage Wiener filter (MWF) has been shown to provide performance benefits that meet or exceed MMSE performance. The principle behind the MWF is a whitening operation naturally created by a tri-diagonalization of the covariance matrix at each stage. In blind mode, tri-diagonalization of the implicit covariance matrix structure no longer occurs, resulting in sub-optimal performance. In this paper, we show, using the CDMA system as an example, that tri-diagonalization can occur for blind mode using larger sample support or spatial diversity at the transmit or receive antenna.

Performance analysis of a reduced rank MMSE MUD for DS-CDMA

A new reduced rank multiuser detector (MUD) for DS-CDMA is presented. The algorithm is derived for the general case of an asynchronous DS-CDMA system in frequency selective multipath. The MUD solution is shown to be equivalent to a bank of parallel, single user detectors. The algorithm is implemented in reduced rank form using the correlations subtractive algorithm of the multistage Wiener filter (CSA-MWF). It is shown that the CSA-MWF implementation performs identically to the full rank solutions of MMSE MUD and meets single user, full rank performance at a substantially reduced rank for all levels of system load.

Reduced-rank automatic target detection and recognition

The linear feature mapping detector (LFMD) developed by Yu and Reed (1995) yields excellent results for detecting a 2-D signal with limited prior information about the signal waveform and the statistical properties of the clutter. However, a direct implementation of the original version of the LFMD criterion in real-time for high resolution data may not be practical at the present time. In this paper, rank-reduction techniques for signal processing, were studied in both theory and practice in order to improve the LFMD for real-time target detection in X-band SAR imagery. It is demonstrated that the proposed reduced-rank detector can lower the computational complexity and decrease the amount of sample support for parameter estimation while providing excellent performance.