PRINCIPLES OF DIGITAL MULTICHANNEL FILTERING

Abstract

The transition from single‐channel to multichannel data processing systems requires substantial modifications of the simpler single‐channel model. While the response function of a single‐channel digital filter can be specified in terms of scalar‐valued weighting coefficients, the corresponding response function of a multichannel filter is more conveniently described by matrix‐valued weighting coefficients. Correlation coefficients, which are scalars in the single‐channel case, now become matrices. Multichannel sampled data are manipulated with greater ease by recourse to multichannel z‐transform theory. Exact inverse filters are calculable by a matrix inversion technique which is the counterpart to the computation of exact single‐channel inverse operators by polynomial division. The delay properties of the original filter govern the stability of its inverse. This inverse is expressible in the form of a two‐stage cascaded system, whose first stage is a single‐channel recursive filter. Optimum multichannel filtering systems result from a generalization of the single‐channel least squares error criterion. The corresponding correlation matrices are now functions of coefficients which are themselves matrices. The system of normal matrix‐valued equations that is obtained in this manner can be solved by means of Robinson’s generalization of the Wiener‐Levinson algorithm. Inverse multichannel filters are designed by specifying the desired output to be an identity matrix rather than a unit spike; if this matrix occurs at zero lag, the least squares filter is minimum‐delay. Simple numerical examples serve to illustrate the design principles involved and to indicate the types of problems that can be attacked with multichannel least squares processors.