FSK demodulation with an adaptive direct form II filter

An adaptive second-order direct-form II digital filter is used for frequency-shift keying (FSK) demodulation. The adaptive filter is a bandpass filter, whose filter coefficients are self-adjusted to minimize the mean square of an error signal, which is formed by subtracting the band-pass output from the input signal. The gradient of adaptation can be found by correlating the error signal with the output of a sensitivity filter. After adaptation the resonant frequency of the adaptive filter equals the frequency of the sinusoidal input signal, which is corrupted with noise. Explicit formulas are derived for the noise variance in the filter coefficients and the frequency detector. Optimum transmitter carrier frequencies are calculated for an M-ary FSK, as well as the optimum threshold frequencies, which are used in the detector to determine the transmitted value. A formula for the number of samples needed for convergence of the adaptive filter has been derived. Using this formula one can calculate the optimum value of the adaptation constant mu for an adaptive direct-form II digital filter with a given bandwidth and a given adaptation time.<>