IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 43. NO. 1, JANUARY 1995

233

Low-Rank Detection of Multichannel Gaussian Signals Using Block Matrix Approximation Peter Strobach, Senior Member, IEEE

A6stracf- The exact design of an m-channel matched filter with L taps requires the solution of an m L x m L block system of linear equations with Toeplitz entries. Practical cases where m > 50 and L > 100 are not uncommon. When the individual sensors of an array are closely spaced, the temporal and spatial correlation of the resulting vector noise process may be modeled in a separable fashion. In this case, the noise covariance block matrix attains a special structure where all block entries are just weighted versions of each other. It is shown that in this case, the complexity of the detector design can be reduced drastically by a factor of mL compared to a conventional multichannel matched filter design procedure. It is further shown that the separable vector noise model facilitates a complete exploitation of the rank properties of noise and data matrices. A constructive procedure for the design of “low-rank” detectors in the multichannel case is derived. These detectors consist of two consecutive blocks: a data and noise dependent “compression” stage, which maps the significant signal energy into a subspace of minimal dimension, followed by a minimal set of independent matched filters, which point in the subspace directions in which the signal is much stronger than the noise. This low-rank detector concept enables discrimination with little or no performance penalty at a minimal computational cost.

I.

INTRODUCTION

T

HE USUAL approach of detecting a known (desired) transient signal of duration L samples in an m-channel sensor process is to apply matched filters in each channel whose outputs are summed up. The exact design of such an w-channel matched filter of length L taps requires the solution of an 7nL x 711L block system of linear equations with L x L Toeplitz entries. This unpractical size of the multichannel Gaussian matched filter design problem often prevents the application of exact matched filter concepts in the area of array or multichannel signal detection where dimensions of m. > 50 and L > 100 are not uncommon. The problem is important because a broad class of modem noninvasive investigation systems in acoustics and in medicine, especially in the area of biomagnetism [ 11, [ 2 ] and multisensorial electrophysiology [ 3 ] , as well as in multichannel sonar, radar, and geophysical systems are based on the application of sensor arrays of very many closely spaced sensors. In this application area, the sensors of an array are usually operated in a “homogeneous” noise environment where all sensors are exposed to noise of the same or almost the same spectral characteristics. It has been observed experimentally Manuscript received February 9, 1993; revised July I. 1994. The associate editor coordinating the review of this paper and approving it for publication was Prof. John Goutcias. The author is with Fachhochschule Furtwangen, Furtwangen. Germany. IEEE Log Number 95069 I 1 ,

that in this case, the L x m snapshot matrix N of the array vector noise can be whitened sufficiently using a two*

-112

sided transformation of the type 8 - 1 1 2 N G - ’ / 2where , 4p is the inverse symmetric matrix square-root of an L x L real symmetric Toeplitz matrix @, and G-’12 is the inverse symmetric matrix square-root of an rri x m real symmetric (but not necessarily Toeplitz) matrix G . A Gaussian vector noise process that can be whitened perfectly by this kind of transformation was named a “separable” vector noise process. It is shown in this paper that in all practical cases where a separable vector noise assumption is justified, the complexity of the multichannel matched filter design problem can be reduced drastically by a factor of 71iL over a conventional exact multichannel matched filter design procedure. Thus, it is one purpose of this paper to demonstrate that closed-form designs of multichannel matched filters become feasible even for very large V L and L , provided only that the observed vector noise process is approximately “separable” vector noise. In a separable vector noise model, 4 has the meaning of a temporal noise covariance matrix that describes the “spectral shape” of the noise in which the sensor array is operated. The matrix G describes the mutual “coupling” of the noise processes of all rrL sensor channels and can hence be interpreted as a spatial correlation matrix. We show that the noise covariance block matrix of a separable vector noise process is just the Kronecker product of G and 6. Hence, the noise covariance block matrix attains a special structure using identical “prototype” block entries 4 weighted by the elements of G . This special block structure provides a constructive basis for a drastic reduction of the overall complexity of the detector design procedure because the inverse of a separable vector noise covariance block matrix is just the Kronecker product of G-l and 4-l. A second important consequence of the separable vector noise assumption is that the multichannel matched filter can be transformed from a concatenated form into a state-spacelike representation. In this representation, the multichannel matched filter becomes amenable to the application of subspace rotation and rank reduction techniques [4]. This is important because in an array of many closely spaced sensors, a desired L x V L template or signal X will usually exhibit a considerable spatial correlation. Consequently, such a multichannel signal may be represented in a subspace of smaller or much smaller dimension compared with the dimension Tri of the raw data subspace.

10.53-.587X/95$04.000 199.5 IEEE

IEEE TRANSACTIONS ON SIGNAL. PROCESSING, VOL. 43, NO. I , JANUARY 1995

234

This raises the concept of “low-rank” detection. A procedure for the “rotation” or compression of the data sequence into a subspace of minimal dimension is derived. Note, however, that such a compression stage is not just “ordinary” data compression. Both multichannel signal and noise correlation properties must be taken into account in the design of the optimal “rotor” used in the compression stage. The conipression stage transforms the vi-channel input process into a &channel ( I ; 5 711.) output process with the property that the entire signal energy or information of interest is mapped into these I; output channels. These output channels are further processed by k subspace matched filters. Thus, the classical search in an in-dimensional subspace is replaced by a search in a I;-dimensional subspace of lower dimension. The computational savings achieved by this concept of lowrank detection can be dramatic when the template or desired signal is very “compressible” ( X :

233

Low-Rank Detection of Multichannel Gaussian Signals Using Block Matrix Approximation Peter Strobach, Senior Member, IEEE

A6stracf- The exact design of an m-channel matched filter with L taps requires the solution of an m L x m L block system of linear equations with Toeplitz entries. Practical cases where m > 50 and L > 100 are not uncommon. When the individual sensors of an array are closely spaced, the temporal and spatial correlation of the resulting vector noise process may be modeled in a separable fashion. In this case, the noise covariance block matrix attains a special structure where all block entries are just weighted versions of each other. It is shown that in this case, the complexity of the detector design can be reduced drastically by a factor of mL compared to a conventional multichannel matched filter design procedure. It is further shown that the separable vector noise model facilitates a complete exploitation of the rank properties of noise and data matrices. A constructive procedure for the design of “low-rank” detectors in the multichannel case is derived. These detectors consist of two consecutive blocks: a data and noise dependent “compression” stage, which maps the significant signal energy into a subspace of minimal dimension, followed by a minimal set of independent matched filters, which point in the subspace directions in which the signal is much stronger than the noise. This low-rank detector concept enables discrimination with little or no performance penalty at a minimal computational cost.

I.

INTRODUCTION

T

HE USUAL approach of detecting a known (desired) transient signal of duration L samples in an m-channel sensor process is to apply matched filters in each channel whose outputs are summed up. The exact design of such an w-channel matched filter of length L taps requires the solution of an 7nL x 711L block system of linear equations with L x L Toeplitz entries. This unpractical size of the multichannel Gaussian matched filter design problem often prevents the application of exact matched filter concepts in the area of array or multichannel signal detection where dimensions of m. > 50 and L > 100 are not uncommon. The problem is important because a broad class of modem noninvasive investigation systems in acoustics and in medicine, especially in the area of biomagnetism [ 11, [ 2 ] and multisensorial electrophysiology [ 3 ] , as well as in multichannel sonar, radar, and geophysical systems are based on the application of sensor arrays of very many closely spaced sensors. In this application area, the sensors of an array are usually operated in a “homogeneous” noise environment where all sensors are exposed to noise of the same or almost the same spectral characteristics. It has been observed experimentally Manuscript received February 9, 1993; revised July I. 1994. The associate editor coordinating the review of this paper and approving it for publication was Prof. John Goutcias. The author is with Fachhochschule Furtwangen, Furtwangen. Germany. IEEE Log Number 95069 I 1 ,

that in this case, the L x m snapshot matrix N of the array vector noise can be whitened sufficiently using a two*

-112

sided transformation of the type 8 - 1 1 2 N G - ’ / 2where , 4p is the inverse symmetric matrix square-root of an L x L real symmetric Toeplitz matrix @, and G-’12 is the inverse symmetric matrix square-root of an rri x m real symmetric (but not necessarily Toeplitz) matrix G . A Gaussian vector noise process that can be whitened perfectly by this kind of transformation was named a “separable” vector noise process. It is shown in this paper that in all practical cases where a separable vector noise assumption is justified, the complexity of the multichannel matched filter design problem can be reduced drastically by a factor of 71iL over a conventional exact multichannel matched filter design procedure. Thus, it is one purpose of this paper to demonstrate that closed-form designs of multichannel matched filters become feasible even for very large V L and L , provided only that the observed vector noise process is approximately “separable” vector noise. In a separable vector noise model, 4 has the meaning of a temporal noise covariance matrix that describes the “spectral shape” of the noise in which the sensor array is operated. The matrix G describes the mutual “coupling” of the noise processes of all rrL sensor channels and can hence be interpreted as a spatial correlation matrix. We show that the noise covariance block matrix of a separable vector noise process is just the Kronecker product of G and 6. Hence, the noise covariance block matrix attains a special structure using identical “prototype” block entries 4 weighted by the elements of G . This special block structure provides a constructive basis for a drastic reduction of the overall complexity of the detector design procedure because the inverse of a separable vector noise covariance block matrix is just the Kronecker product of G-l and 4-l. A second important consequence of the separable vector noise assumption is that the multichannel matched filter can be transformed from a concatenated form into a state-spacelike representation. In this representation, the multichannel matched filter becomes amenable to the application of subspace rotation and rank reduction techniques [4]. This is important because in an array of many closely spaced sensors, a desired L x V L template or signal X will usually exhibit a considerable spatial correlation. Consequently, such a multichannel signal may be represented in a subspace of smaller or much smaller dimension compared with the dimension Tri of the raw data subspace.

10.53-.587X/95$04.000 199.5 IEEE

IEEE TRANSACTIONS ON SIGNAL. PROCESSING, VOL. 43, NO. I , JANUARY 1995

234

This raises the concept of “low-rank” detection. A procedure for the “rotation” or compression of the data sequence into a subspace of minimal dimension is derived. Note, however, that such a compression stage is not just “ordinary” data compression. Both multichannel signal and noise correlation properties must be taken into account in the design of the optimal “rotor” used in the compression stage. The conipression stage transforms the vi-channel input process into a &channel ( I ; 5 711.) output process with the property that the entire signal energy or information of interest is mapped into these I; output channels. These output channels are further processed by k subspace matched filters. Thus, the classical search in an in-dimensional subspace is replaced by a search in a I;-dimensional subspace of lower dimension. The computational savings achieved by this concept of lowrank detection can be dramatic when the template or desired signal is very “compressible” ( X :