Paul L. Feintuch

A frequency domain model for 'filtered' LMS algorithms-stability analysis, design, and elimination of the training mode

A frequency domain model of the filtered LMS algorithm is presented for analyzing the behavior of the weights during adaptation. In particular, expressions for stable operation of the algorithm are derived as a function of the algorithm step size, the input signal power, and the transfer functions of the linear filters. The expressions show that algorithm stability can be achieved over a frequency band of interest by inserting an appropriately chosen delay in the reference input to the LMS algorithm weight update equation. This result implies that it is not necessary to use a training mode to estimate the loop transfer functions before or during adaptation if the input is limited to a band of frequencies. It is only necessary to know the approximate delay introduced by the transfer functions in the band. The single delay parameter can be estimated much more easily than the entire transfer function. Simulations of the time domain algorithm are presented to support the theoretical predictions of the frequency domain model. >

A normalized frequency domain LMS adaptive algorithm

A scheme is presented for obtaining an input power estimate for setting the algorithm gain parameter μ separately in each frequency bin in the frequency domain LMS adaptive algorithm. This is particularly important if the input has large spectral variations, and a single feedback parameter, set on the broad-band power, could result in instability in the adaptive filters in some frequency bins. The estimate is incorporated directly into the algorithm as a data dependent time-varying stochastic μ(n). Using a Gaussian data model and sample-to-sample data independence, first-order linear difference equations are derived and solved for the mean and misadjustment errors. The performance of the scheme is compared to the case for which the input power level is known a priori. For the same transient response, only about ten samples need be averaged to yield the same misadjustment error.

Statistical analysis of the single-layer backpropagation algorithm. II. MSE and classification performance

For pt.I see ibid., p.583-91, 1993. The analysis of pt.I is extended to the evaluation of the mean-square error and the probability of correct classification. It is shown that the mean-square error and the corresponding performance surface are such that the single-layer perceptron is prevented from correctly classifying with probability one until the weights converge at infinity. >

Statistical analysis of the single-layer backpropagation algorithm. I. mean weight behavior

The single-layer backpropagation algorithm is a gradient-descent method that adjusts the connection weights of a single-layer perceptron to minimize the mean-square error at the output. It is similar to the standard least mean square al- gorithm, except the output of the linear combiner contains a differentiable nonlinearity. In this paper, we present a statis- tical analysis of the mean weight behavior of the single-layer backpropagation algorithm for Gaussian input signals. It is based on a nonlinear system identification model of the desired response which is capable of generating an arbitrary hyper- plane decision boundary. It is demonstrated that, although the weights grow unbounded, the algorithm, on average, quickly learns the correct hyperplane associated with the system iden- tification model.

Non-Wiener solutions for the LMS algorithm-a time domain approach

A time domain analysis of the LMS algorithm is presented for a sinusoidal deterministic reference input. For the sinusoidal reference input only, the N-dimensional time-varying linear matrix recursion for the weight vector is solved using a 2-D orthogonal subspace decomposition. Using this weight vector solution, it is shown that there exists a linear time-invariant relationship between the desired input and the filter output. >

The LMS cancellation of narrow, extended interferences in sonar

The spatial least mean-square sense cancellation of acoustic interferences, with finite but narrow angular extent, from the output of a primary sensor using reference hydrophones spatially separated from the primary, is considered. This is done by replacing adaptive LMS filters in the canceller structure with continuous Wiener filters. A far-field model for a narrow extended source impinging on a line array of hydrophones is developed and used to determine the canceller output spectrum. Lower bounds on the canceller output spectrum are developed for arbitrary but narrow source distribution. For the special case of a spatially uncorrelated, uniformly distributed narrow interference, the cancellation is evaluated as a function of the number of reference hydrophones, the position of the reference relative to the primary sensor, and the angular extent of the interference. Explicit approximations for the cancellation achieved with such a source are developed and design guidelines described for the selectcion of reference hydrophone position and the number of references. Several significant differences between cancellation of single plane wave and narrow extended sources are demonstrated. Most notably, it is shown that the spatial rejection of an extended source may be bounded significantly above the ambient noise floor if the references are not sufficiently close to the primary sensor, regardless of the number of references.

Sonar Array Detection of Gaussian Signals in Gaussian Noise of Unknown Power

A statistical test is postulated for detecting, with an M-element hydrophone array, a Gaussian signal in spatially independent Gaussian noise of unknown power. The test is an extension of the uniformly-most-powerful (UMP) unbiased test for a two-element array. The output signal-to-noise ratio of the test is calculated and, for a large number of independent space-time samples, is shown to be no better than a mean-level detector (MLD). Receiver operating characteristic curves (ROC) for the MLD are computed and compared to the ROC curves for the optimum (Bayes) parametric detector. The input signal-to-noise power ratios required to provide a detection probability of 0.5 differ by less than 0.2 dB for a fifty-element array with wide variation in false-alarm probability and time-bandwidth product. This result suggests that both the extended bivariate UMP unbiased test and the MLD perform close to the unknown UMP unbiased test for independence of a multivariate Gaussian distribution.

Specification and Performance of Passive Sonar Spectral Estimators

The application of existing estimation theory to the problem of specification and performance of passive sonar spectral estimators is considered. The classification function is addressed, so that the signal is assumed to be present, and so that the energy arrival angle is known. The spatial filter considered is a line array of M equally spaced omnidirectional hydrophones. Signal and ambient noise are both zero-mean, wide-sense, stationary Gaussian random processes that differ in their spatial correlation across the face of the array. The signal is a plane wave that can be made totally spacially corrected between array elements by inserting delays between sensors to invert the signal propagation delay. The noise correlation is a function of frequency, bandwidth, element separation, and the relative time delay between sensors. Under these assumptions, the Cramer-Rao lower bound is derived for the class of unbiased estimates of signal power in a narrow frequency band at the hydrophone in the presence of correlated ambient noise of known power. The bound is examined numerically, resulting in a threshold phenomenon with M that constitutes a new design consideration. In addition, there is a striking insensitivity to realistic values of ambient noise correlation, and there are ranges in signal-to-noise ratio for which one gains more by increasing M than by increasing the bandwidth-time product. Specific processors, including a new unbiased estimator when noise power is unknown, are developed.

Matched-Filter Envelope Detector Deflection Performance for a Correlated Phase Channel

In this paper, the effect of channel phase coherence upon a matched filter envelope detector output is investigated for a pulsed radar or active sonar. A novel model for the correlated channel phases allows the explicit calculation of the loss in detection performance using the deflection criteria. The theoretical model yields good agreement with simulations when the phase correlation coefficients between the first and last pulses are between 0.1 and 1.0. It is shown that a 3-dB loss in performance, as compared to the optimum detector for perfect coherence, requires phase correlation between adjacent pulses of \rho_{i,i+1} = 0.91 , 0.96, and 0.96 for 10, 20, and 30 pulses, respectively. On the other hand, the same performance is obtained with a noncoherent combiner of the matched filter pulse returns when correlation between adjacent pulses, \rho_{i,i+1} = 0.8 , 0.835, and 0.84 for 10, 20, and 30 pulses, respectively. If \rho_{i,i+1} is smaller than these quantities, one is better off performing noncoherent detection.