Degenerate Solutions and an Algebraic Approach to the Multiple-Input Linear Filter Design Problem

This paper describes three common types of multiple-input filter design problems; in each case Wiener's general optimum multiple-input solution is reduced to a simpler equivalent single-input solution. The first type of situation discussed is the "distortionless dual-input system." Conditions under which this additional criterion of fidelity may be advisable are discussed, two general distortionless systems are shown, and it is demonstrated that Wiener's single-input solution may be used in designing such a system. The second semigeneral solution obtained (or semispecial, as the case may be) pertains to a situation in which interference is present in one channel which is statistically similar to, but still independent of, the actual signal of interest. The third case is probably the most interesting of all in that when several signal sources are available, such as in a diversity receiving system, each corrupted by additive independent noises, the optimum system is one in which all channels are added together directly (after possible simple gain adjustments) and then passed through a single filter. It is demonstrated that this result is true even though the signal-to-noise ratios are quite different for the various input channels.