Statistical design of nonrecursive digital filters

The problem of designing a finite duration impulse response (FIR) digital filter to approximate a desired spectral response is treated in this paper. The philosophy adopted is that for a given FIR filter structure, the filter coefficients can be designed to provide a minimum mean-squared error (MMSE) estimate of a random signal sequence (the design-signal) imbedded in a random noise sequence. By treating the signal and noise covariance functions as design parameters, one can design FIR filters with spectral responses that approximate the power spectral density of the design-signal. For signal processing applications that require some attention to signal fidelity, as well as noise rejection, the MMSE philosophy seems appropriate (as opposed to a maximum signal-to-noise ratio philosophy, for example). Several practical designs are presented that emphasize the simplicity of the design technique and illustrate the selection of design parameters. The designs show quite dramatically that the MMSE design technique can be competitive with existing low-pass and bandpass design techniques. Finally, considerable attention is given to an efficient Toeplitz matrix inversion algorithm that permits rapid inversion of the covariance matrices that arise in the MMSE design. The resulting computation times for the design of high-order filters (N = 128, e.g.) appear to be shorter than computation times for competing algorithms.