Principles of Digital Multichannel Filtering
Sven Treitel · Geophysics · 1970
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.