In this chapter, we provide an overview of existing algorithms for blind source separation of convolutive audio mixtures. We provide a taxonomy, wherein many of the existing algorithms can be organized, and we present published results from those algorithms that have been applied to real-world audio separation tasks.

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A survey of convolutive blind source separation methods

An analytical performance evaluation of the errors of the direction-of-arrival estimates obtained by the MUSIC algorithm for uncorrelated sources is provided. Explicit asymptotic formulas are derived for the means and the covariance of the estimates. The covariances are then compared to the Cramer-Rao lower bound. It is shown that for a single course, the MUSIC algorithm is asymptotically efficient. For multiple sources, the algorithm is not efficient in general. However, it approaches asymptotic efficiency when the SNRs (signal-to-noise ratios) of all sources tend to infinity. It is illustrated by several test cases that the relative efficiency of the MUSIC algorithm is nearly one under a wide range of parameter variations. The analytic performance evaluation thus confirms empirical evidence to the excellent performance of the MUSIC algorithm for narrowband signals. >

Analysis of the asymptotic relative efficiency of the MUSIC algorithm

A first-order analysis of the sensitivity of the MUSIC algorithm to system errors which cause differences between the array manifold used by MUSIC and the true array manifold is presented. The effect of such errors on the direction-of-arrival estimates is investigated. The ability of the MUSIC algorithm to resolve two closely spaced sources in the presence of phase and gain errors in the array elements or the receivers, or errors in the element locations, is analyzed. A formula for computing the failure threshold of MUSIC as a function of source separation and other system parameters is derived and tested by simulation. The analysis assumes that the exact covariance matrix of array element outputs is known (the infinite data case). >

A sensitivity analysis of the MUSIC algorithm

A method and apparatus for processing cochannel signals received at a sensor array (110) in a cumulant-based signal processing and separation engine to obtain a desired set of output signals (38) or parameters. For use in a signal recovery system, the output signals are recovered and separated versions of the originally transmitted cochannel signals. An important feature that distinguishes the cumulant-based system from other signal separation and recovery systems is that it generates an estimated generalized steering vector associated with each signal source, and representative of all received coherent signal components attributable to the source. This feature enables the invention to perform well in multipath conditions, by combining all coherent multipath components from the same source. In a receiver/transmitter system (316), the estimated generalized steering vectors associated with each source are used to generate transmit beamformer weight vectors that permit cochannel transmission to multiple user stations (310). The basic cumulant-based processing and separation engine can also be used in a variety of applications, such as high density recording, complex phase angle equalization, receiving systems with enhanced effective dynamic range, and signal separation in the presence of strong interference. Various embodiments and extensions of the basic cumulant-based system are disclosed.

Cochannel signal processing system

The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization problem. Some classical optimization techniques on Euclidean space are generalized to Riemannian manifolds. Several algorithms are presented and their convergence properties are analyzed employing the Riemannian structure of the manifold. Specifically, two new algorithms, which can be thought of as Newton's method and the conjugate gradient method on Riemannian manifolds, are presented and shown to possess quadratic and superlinear convergence, respectively. These methods are applied to several eigenvalue and singular value problems, which are posed as constrained optimization problems. New efficient algorithms for the eigenvalue problem are obtained by exploiting the special homogeneous space structure of the constraint manifold. It is shown that Newton's method applied to the Rayleigh quotient on a sphere converges cubically, and that the Rayleigh quotient iteration is an efficient approximation of Newton's method. The Riemannian version of the conjugate gradient method applied to this function gives a new algorithm for finding the eigenvectors corresponding to the extreme eigenvalues of a symmetric matrix. The Riemannian version of the conjugate gradient method applied to a generalized Rayleigh quotient yields a superlinearly convergent algorithm for computing the k eigenvectors corresponding to the extreme eigenvalues of an n-by-n matrix. This algorithm requires $O(nk\sp2)$ operations and O(k) matrix-vector multiplications per step. Several gradient flows are analyzed that solve eigenvalue and singular value problems. The new optimization algorithms are applied to the subspace tracking problem of adaptive signal processing. A new algorithm for subspace tracking is given, which is based upon the conjugate gradient method applied to the generalized Rayleigh quotient. The results of several numerical experiments demonstrating the convergence properties of the new algorithms are given.

Geometric optimization methods for adaptive filtering

The multiple signal characterization (MUSIC) algorithm is an implementation of the signal subspace approach to compute parameter estimates of multiple point-source signals from the observed voltages received on an array of M antennas. In it, the solution to the multiple source direction finding (DF) problem is provided by the intersection of the signal subspace (obtained from the received data) and the array manifold (obtained via array calibration or prior knowledge of array directional characteristics). The MUSIC algorithm was implemented to experimentally verify the performance of the signal subspace approach to DF under very general scenarios and conditions which are regarded as difficult to impossible in traditional systems. The results of those experiments are described herein. The experimental system consisted of an eight-element antenna array 13 wavelengths in diameter, an eight-channel receiver and digitizer, and a minicomputer with disk storage to process the digitized data. With ideal instrumentation, the MUSIC algorithm provides performance that, as the amount of data collected increases without limit, is asymptotically ideal. However, with finite precision and finite data collection, the performance of even an ideal system can be a sensitive function of source and scenario parameters. Tests demonstrated the resolution of three sources all within one beamwidth (5 \deg ), even when the closer two were spaced less than 0.2 beamwidths. Sources that were polarized differently could be resolved at closer spacings. Experimental DF accuracy was limited by the oncalibrated scattering of source energy from the test range support tower and from the ground. The measured direction of arrival of one source changed by less than 0.01 beamwidths as the other two sources were switched on and off in all combinations. In general, results indicated that all parameters of a source can be measured and the signal waveform can be recovered as well in the presence of other sources less than a beamwidth away.

R. Schmidt

Papers

Multiple source DF signal processing: An experimental system