Desired Impulse Response

It is good to check that the desired impulse response is not overly aliased in the time domain. The impulse-response for this example is plotted in Fig.8.5. We see that it appears quite short compared with the inverse FFT used to compute it. The script in Fig.8.6 gives the details of this computation, and also prints out a measure of ``time-limitedness'' defined as the $\ensuremath{L_2}$ norm of the outermost 20% of the impulse response divided by its total $\ensuremath{L_2}$ norm--this measure was reported to be $0.02$ % for this example.

Figure 8.5: Desired impulse response obtained from a length 512 inverse FFT of the interpolated measured amplitude response in Fig.8.3.

Note also that the desired impulse response is noncausal. In fact, it is zero phase [452]. This is of course expected because the desired frequency response was real (and nonnegative).

Figure 8.6: Script (matlab) for checking the desired impulse-response for time aliasing. If it wraps around in the time buffer, one can increase the inverse FFT length (Nfft) and/or smooth the desired amplitude response (Sdb).

 

Ns = length(Gdbfk); if Ns~=Nfft/2+1, error("confusion"); end
Sdb = [Gdbfk,Gdbfk(Ns-1:-1:2)]; % install negative-frequencies

S = 10 .^ (Sdb/20); % convert to linear magnitude
s = ifft(S); % desired impulse response
s = real(s); % any imaginary part is quantization noise
tlerr = 100*norm(s(round(0.9*Ns:1.1*Ns)))/norm(s);
disp(sprintf(['Time-limitedness check: Outer 20%% of impulse ' ...
              'response is %0.2f %% of total rms'],tlerr));
% = 0.02 percent
if tlerr>1.0 % arbitrarily set 1% as the upper limit allowed
  error('Increase Nfft and/or smooth Sdb');
end

figure(2);
plot(s,'-k'); grid('on');   title('Impulse Response');
xlabel('Time (samples)');   ylabel('Amplitude');