Estimating an impulse response from input-output measurements is called system identification, and a large literature exists on this topic (e.g., ).
Cross-correlation can be used to compute the impulse response of a filter from the cross-correlation of its input and output signals and , respectively. To see this, note that, by the correlation theorem,
Therefore, the frequency response equals the input-output cross-spectrum divided by the input power spectrum:
where multiplication and division of spectra are defined pointwise, i.e., . A Matlab program illustrating these relationships is listed in Fig.8.13. (A Python version8.12is also available.)
% sidex.m - Demonstration of the use of FFT cross- % correlation to compute the impulse response % of a filter given its input and output. % This is called "FIR system identification". Nx = 32; % input signal length Nh = 10; % filter length Ny = Nx+Nh-1; % max output signal length % FFT size to accommodate cross-correlation: Nfft = 2^nextpow2(Ny); % FFT wants power of 2 x = rand(1,Nx); % input signal = noise %x = 1:Nx; % input signal = ramp h = [1:Nh]; % the filter xzp = [x,zeros(1,Nfft-Nx)]; % zero-padded input yzp = filter(h,1,xzp); % apply the filter X = fft(xzp); % input spectrum Y = fft(yzp); % output spectrum Rxx = conj(X) .* X; % energy spectrum of x Rxy = conj(X) .* Y; % cross-energy spectrum Hxy = Rxy ./ Rxx; % should be the freq. response hxy = ifft(Hxy); % should be the imp. response hxy(1:Nh) % print estimated impulse response freqz(hxy,1,Nfft); % plot estimated freq response err = norm(hxy - [h,zeros(1,Nfft-Nh)])/norm(h); disp(sprintf(['Impulse Response Error = ',... '%0.14f%%'],100*err)); err = norm(Hxy-fft([h,zeros(1,Nfft-Nh)]))/norm(h); disp(sprintf(['Frequency Response Error = ',... '%0.14f%%'],100*err));