Dominant Mode Rejection Beamformer Research Articles

An asymptotically exact estimate of the median noise eigenvalue of sample covariance matrices

The Dominant Mode Rejection beamformer [Abraham & Owsley (1990)] averages the noise subspace eigenvalues of the sample covariance matrix (SCM) to estimate the background noise power. This noise power estimate from snapshot deficient SCMs can be unreliable for large arrays in time-varying environments with limited snapshots. Median filtering offers robustness to outliers and subspace mismatch for these challenging problems. Recent numerical experiments [Campos Anchieta & Buck (2022)] identified a simple regression relating the median sample eigenvalue to the true background power for snapshot deficient SCMs. However, the complicated expression of the Marchenko-Pastur (MP) distribution for the noise eigenvalues of the SCM thwarted prior attempts to find a closed form estimator for the MP distribution median. This talk exploits a coordinate transform to obtain a closed-form median of the MP distribution that is exact for the leading term in the power series expansion of the distribution. The new median estimator is more accurate than the previous numerical regression when the ratio of sensors to snapshots is 2 or more. Given the central role of SCM eigenvalues in principal component analysis and constant false alarm rate detectors, the new median expression should find application in other data science algorithms for underwater acoustics. [Work supported by ONR Code 321US.]

Improving the Robustness of the Dominant Mode Rejection Beamformer With Median Filtering

Improving the Robustness of the Dominant Mode Rejection Beamformer With Median Filtering

Random matrix theory analysis of the dominant mode rejection beamformer white noise gain

The dominant mode rejection (DMR) beamformer [Abraham and Owsley, IEEE Oceans (1990)] is an adaptive spatial processor that has been effectively used to null interference sources in scenarios where the number of training samples to estimate the sample covariance matrix (SCM) is limited. In these scenarios the ratio of the SCM dimension (array size) to the number of snapshots (training samples) is close to or even greater than one. Whereas classical asymptotic SCM results require the number of samples grow to infinity while keeping the dimension fixed, random matrix theory (RMT) provides a robust mathematical framework to analyze the SCM spectrum in snapshot-limited scenarios. RMT predictions of the SCM eigenvectors have already been applied to predict DMR beamformer notch depth in a single interferer environment (Buck and Wage, IEEE SSP (2012)]. This talk presents an RMT-based asymptotic approximation of the DMR SCM inverse in a loud multi-interferer environment. The RMT SCM model is used to predict the white noise gain (WNG) of the DMR beamformer, accounting for the impact when bulk spectral components are included in the dominant subspace. The performance of the RMT-based WNG prediction is compared to the sample mean using Monte Carlo simulations. [Funded by ONR 321US.]

Performance of the median dominant mode rejection beamformer against array element perturbations

Performance of the median dominant mode rejection beamformer against array element perturbations

Frequency-wavenumber spectrum estimation using blended dominant mode rejection beamforming

Capon [Proc. IEEE (1969)] designed the minimum variance distortionless response (MVDR) beamformer to obtain spectral estimates with better resolution than the conventional averaged-periodogram estimator. The MVDR spectrum is a function of the inverse of the sample covariance matrix (SCM), which often must be regularized prior to inversion. To address conditioning problems, Abraham and Owsley [IEEE Oceans (1990)] developed a modified MVDR approach called dominant mode rejection (DMR). The DMR beamformer defines its weights using a structured covariance consisting of a low-rank interference subspace plus an orthogonal noise subspace. It assumes the rank of the interference is known. Recently, Buck and Singer [IEEE SAM (2018)] proposed the blended DMR beamformer that eliminates the need for rank estimation by defining a weight vector that is an affine combination of fixed-rank DMR beamformers. This talk investigates frequency-wavenumber estimation using blended DMR, focusing particularly on efficient implementations for large linear arrays. Adapting the approach described by Therrien [Prentice Hall (1992)] for MVDR, the blended DMR spectrum for an equally spaced array can be computed using fast Fourier transforms of the sample eigenvectors. Results will be illustrated using experimental data from underwater vertical arrays. [Work supported by ONR.]

Improving the dominant mode rejection beamformer with median filtering

Adaptive beamformers (ABFs) outperform conventional beamformers in detecting weak sources while attenuating background noise and strong interferers. Changing environments limit the number of snapshots that an ABF can average to estimate the sample covariance matrix (SCM). The dominant mode rejection (DMR) beamformer [Abraham and Owsley, Oceans (1990)] overcomes rank-deficient SCMs by imposing a structured covariance matrix which replaces the noise subspace eigenvalues by their average. However, the DMR beamformer often overestimates the dominant subspace dimension to avoid interferers contaminating the beamformer output. Overestimating the dominant subspace dimension introduces a bias to the background noise power estimate. This bias impairs the array gain and output signal to interferer and noise ratio (SINR). This talk proposes a modification to the DMR, replacing the average of the noise eigenvalues with an estimate of background noise power derived from the median of all eigenvalues. Simulations demonstrated that this new median-DMR beamformer improves the output SINR by up to 0.9dB compared to the standard DMR beamformer in scenarios which conservatively overestimated the dominant subspace dimension and suffered from a rank-deficient SCM. [Research supported by ONR 321US.]

Performance weighted blending of nested array processors

Nested arrays consist of two uniform subarrays: a short aperture array with half-wavelength spacing and a long aperture array with greater than half-wavelength spacing. Two approaches for estimating the scanned response require computing either the product or the minimum of the subarray responses in each look direction. In multi-source environments, the multiplicative and min scanned responses may be corrupted by cross terms [Wage, Acoustics Today (2018)]. When the sources are uncorrelated, snapshot averaging is often used to mitigate cross term interference. The min processor's response can contain cross terms that do not decay with snapshot averaging. Alternatively, the multiplicative processor’s response contains cross terms that decay, though large numbers of snapshots may be required for the cross terms to fall below the level of those in the min processor. This talk proposes a performance weighted combination of the two processors based on Buck and Singer's (IEEE, 2018) blended dominant mode rejection beamformer. The same performance weighting function can be used to combine the nested multiplicative and min processors. The resulting universal processor adjusts the weighting in each look direction as the number of snapshots increases to favor the multiplicative processor as its cross terms average out. [Work supported by ONR]

The blended dominant mode rejection adaptive beamformer

Adaptive beamformers balance competing demands of enhancing desired signals, suppressing discrete interferers and attenuating background noise. This requires an accurate estimate of the spatial covariance matrix that is challenging to obtain in nonstationary environments. The dominant mode rejection (DMR) beamformer [Abraham & Owsley, Oceans, 1990] addresses this challenge by replacing the sample covariance matrix (SCM) eigenvalues of the noise subspace with the average of these eigenvalues to improve its conditioning. The DMR beamformer requires an estimate of the interferer subspace dimension, and the DMR beamformer's performance degrades when this estimated dimension is inaccurate. We propose the Blended DMR beamformer, whose array weights are formed as a mixture of fixed-dimension DMR beamformer array weights across a range of interferer dimensions. The contribution of each fixed-dimension DMR beamformer is a function of its recent performance in suppressing interferers and attenuating noise. The perf...

Snapshot Performance of the Dominant Mode Rejection Beamformer

The dominant mode rejection (DMR) beamformer constructs its weight vector using a structured covariance estimate derived from the eigendecomposition of the sample covariance matrix (SCM). Like all adaptive beamformers (ABFs), the DMR ABF places notches in the direction of loud interferers to facilitate the detection of quiet targets. This paper investigates how DMR performs as a function of the number of snapshots used to estimate the SCM. The analysis focuses on the fundamental case of a single interferer in white noise. Theoretical calculations for the ensemble case reveal the relationships among notch depth, white noise gain, and SINR. The centerpiece of the paper is a detailed empirical study of the single-interferer case, which includes snapshot-deficient scenarios often ignored in previous work. Empirical data demonstrate that the sample eigenvectors determine the mean performance of the DMR ABF. On a log-log plot the mean notch depth is a piecewise linear function of the number of snapshots and the interference-to-noise ratio. The paper interprets the behavior of the DMR ABF using recent results on sample eigenvectors derived from random matrix theory.

Random matrix theory analysis of the dominant mode rejection beamformer

The Dominant Mode Rejection (DMR) beamformer developed by Abraham and Owsley [Proc. Oceans, 1990] determines the beamformer weights from the sensor covariance matrix eigendecomposition. The weights are designed to reject signals contained in the dominant subspace, which is defined by the eigenvectors associated with the largest eigenvalues. In previous work, we developed a model for the mean notch depth (ND) of the DMR beamformer from random matrix theory (RMT) results on the sample eigenvector fidelity [IEEE Stat. Sig. Proc. workshop, 2012]. While ND is useful, other metrics such as white noise gain (WNG) and signal to interference and noise ratio (SINR) are of great interest. WNG characterizes the beamformer robustness to mismatch, and SINR quantifies overall performance. SINR loss is defined as the ratio of the SINR for a beamformer designed using sample statistics to the SINR for the optimal beamformer designed using ensemble statistics. This talk extends our previous work by considering the relationship among ND, WNG, and SINR for the DMR beamformer. A surprising result obtained from RMT is that for a single loud interferer and twice as many snapshots as sensors, the expected SINR loss depends only on the number of snapshots. [Work supported by ONR.]

Modeling dominant mode rejection beamformer notch depth using random matrix theory

Any practical tracking algorithm must mitigate the effect of both nonstationary environments and loud interfering sources. The dominant mode rejection (DMR) adaptive beamformer (ABF) is often used to address these challenges. The DMR beamformer periodically updates the sample covariance matrix (SCM) employed to compute the ABF weights. A tension arises between the large number of snapshots required for each SCM update to null loud interferers effectively and the small number of snapshots required for each SCM update to track nonstationarities. In practice, observed notch depths for DMR fall dramatically short of those predicted theoretically from ensemble statistics. This talk bridges the gap between theoretical and observed performance by presenting a simple linear asymptotic model for the DMR notch depth derived from random matrix theory results on the accuracy of the SCM eigenvectors. The model predicts the mean DMR notch depth as a function of the number of snapshots given the interferer-to-noise rati...

Initial evaluation of the dominant mode rejection beamformer

Long integration times are needed to give accurate cross-spectral matrix (CSM) estimates; however, integration times must be short enough so that the dynamic behavior of the noise described by the CSM is captured. The dominant mode rejection (DMR) beamformer, described by Owsley and Abraham, calculates adaptive weights based on a reduced rank (CSM) estimate, where the CSM estimate is formed with a subset of the eigenvalues containing the largest eigenvalues and their eigenvectors. Thus the integration time required to obtain this CSM estimate is reduced, as the largest eigenvalue/eigenvector pairs are estimated much more rapidly than the smaller ones. The purpose of this study was to determine whether performance comparable to a fully adaptive beamformer could be achieved with the DMR beamformer, without the potential penalties associated with requiring the increased integration time for the CSM. Data acquired with bottom-mounted horizontal line arrays, in both deep- and shallow-water environments, were used to test the dominant mode rejection beamformer performance. Appropriate DMR parameters were found using the deep-water data so that the DMR performance emulated the performance of a fully adaptive beamformer with a white-noise gain constraint. Additional processing performed on the shallow-water data showed that the DMR beamformer had equivalent performance to a robust fully adaptive beamformer, but with much less output power bias at small integration times.

Evaluation of the dominant mode rejection beamformer using reduced integration times

Increasing the number of hydrophones in an array should increase beamformer performance. However, when the number of hydrophones is large, integration times must be long enough to give accurate cross-spectral matrix (CSM) estimates, but short enough so that the dynamic behavior of the noise described by the CSM is captured. The dominant mode rejection (DMR) beamformer calculates adaptive weights based on a reduced rank CSM estimate, where the CSM estimate is formed with a subset of the largest eigenvalues and their eigenvectors. Since the largest eigenvalue/eigenvector pairs are estimated rapidly, the integration time required is reduced. The purpose of this study was to examine the DMR beamformer performance using a bottom-mounted horizontal line array in a shallow-water environment. The data were processed with a fully adaptive beamformer and the DMR beamformer. The DMR beamformer showed better performance than the fully adaptive beamformer when using arrays with larger numbers of hydrophones. Thus, in highly dynamic noise environments, the DMR beamformer may be a more appropriate implementation to use for passive sonar detection systems.