Adaptive frequency response identification

Given a stable, discrete time, single input single output system G(z), but with only the input signal and the noise corrupted output signal available for measurement, we seek to find an approximation G(z) - a finite impulse response (FIR) filter - with ||G - ∞|| = sup |G(ejθ) - Ĝ(ejθ)| θE(-Π,Π] bounded and small. The infinity norm in (1) has application in control theory and signal processing; furthermore, it is a measure of the deviation in frequency response between G and Ĝ. Several previous papers, attempt to identify G(z) in the frequency domain; these papers fail to bound G-Ĝ in any norm. Central to our method of identification is interpolation. First, one estimates accurately G(z) at n equally spaced frequencies. Here, n is a design parameter one may freely choose. This estimation relies on filtering the input and output signals appropriately. Then estimates of G(eJ2Πk/n) come from a bank of n/2 decoupled least mean squares algorithms, each of two parameters; Ĝ(z) is then the unique FIR filter of degree n-1 with transfer function interpolating to these estimates. Ĝ(z) is computationally easy to evaluate. The resulting error bound has the form ||G - Ĝ||∞ < MRn + K(1 + log2n) Here M and R are constants, dependent on G(z), with R<1; the accuracy of estimating G(z) at the interpolation points determines K.