Low-rank detection of multichannel Gaussian signals using block matrix approximation

The exact design of an m-channel matched filter with L taps requires the solution of an mL×mL 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