A Class of Quadratic FIR Filters with Applications to Spectral Shaping and Narrow Band Generation

In signal processing problems involving quadratic performance functional and stationary signals, solutions often involve a first stage of filtering called whitening. The whitening filter is a characterization of the signal power spectrum. For this reason, whitening and spectral estimation problems are closely related. The whitening filter obtained by Yule-Walker equations uncorrelates N samples of the output with the past input samples (beyond the zero lag) as seen by the orthogonality principle. It does not constrain the output samples to be uncorrelated with each others or to have pre-defined autocorrelation values for finite number of samples. The latter is typically achieved by additional filtering process approximately. In this paper we consider finding the unique minimum order, minimum phase FIR filter F_M (z) from the input-output autocorrelation equations directly such that for the colored input process u(n) the output y(n) has N pre-defined zero or nonzero autocorrelation samples. The input-output correlation equation is quadratic in the coefficients of F_M (z) but is solved linearly. Existence and derivation of the F_M (z) are addressed. Application and performance of such filters for spectral shaping and narrow band signal generation are shown and compared with the traditional whitening filters using Yule-Walker equations.