Windowing an ``Ideal'' Impulse Response computed by the Frequency Sampling Method

We will now create an ``ideal'' Hilbert transformer frequency response with transition bands using the ``frequency sampling'' method. As we saw in the previous section, we must provide transition bands in our ``ideal'' frequency response so that the corresponding ``ideal'' impulse response is not so long that windowing it will give poor results.

If the smallest transition band is Hz, then the FFT size should satisfy . Otherwise, there may be too much time aliasing in the ``ideal'' impulse response. It is good to try doubling the FFT size to see if it changes the designed filter. The only non-obvious part below is ``.^8'' which smoothes the taper to zero and looks better on a log magnitude scale. It would also make sense to do a linear taper on a dB scale which corresponds to exponential tapers to zero.

H = [ ([0:k1-2]/(k1-1)).^8,ones(1,k2-k1+1),...
      ([k1-2:-1:0]/(k1-1)).^8, zeros(1,N/2-1)];

% rotate buffer so negative frequencies are on the left 
% and DC is at N/2+1, and plot:
Hp = [H(N/2+2:N), H(1:N/2+1)];
f = [-0.5 + 1/N : 1/N : 0.5]; % normalized frequency axis
omega = f*(2*pi); % normalized radian frequency axis
plot(f,Hp); grid;
s = sprintf('Desired Frequency Response, FFT size = %d', N);
title(s); 
xlabel('Normalized Frequency (cycles/sample)'); 
ylabel('Gain')
if saveplots, saveplot('hilbertFRIdeal.eps'); end;
if dopause, disp 'Pausing... RETURN to continue'; pause; end;

% Compute impulse response of "ideal" 
%   bandlimited negative-frequency filter:
h = ifft(H);
hodd = imag(h(1:2:N));     % This should be zero
ierr = norm(hodd)/norm(h); % Look at numerical round-off error

ierr =

   4.1958e-15

% Compute a rough time-aliasing error estimate:
aerr = norm(h(N/2-N/32:N/2+N/32))/norm(h) 

aerr =

   4.8300e-04

hp = [h(N/2+2:N), h(1:N/2+1)]; % Plot negative times on the left

nzero = N/2; % index of time 0
nshow = (M-1)/2;
n = -nshow : nshow;
plot(n,real(hp(nzero-nshow:nzero+nshow))); grid;
s = sprintf(['Ideal SSB Filter Impulse Response,',...
             ' Real Part, FFT size = %d'], N);
title(s); xlabel('Time (samples)'); ylabel('Amplitude')
if saveplots, saveplot('hilbertIRRealIdeal.eps'); end;
if dopause, disp 'Pausing... RETURN to continue'; pause; end;

plot(n,imag(hp(nzero-nshow:nzero+nshow))); grid;
s = sprintf(['Ideal SSB Filter Impulse Response,',...
             ' Imaginary Part, FFT size = %d'], N);
title(s); xlabel('Time (samples)'); ylabel('Amplitude')
if saveplots, saveplot('hilbertIRImagIdeal.eps'); end;
if dopause, disp 'Pausing... RETURN to continue'; pause; end;

hw = wzp .* h; % Apply window to ideal impulse response
Hw = fft(hw);  % "DTFT" of windowed impulse response

% Compute total stopband attenuation:
ierr = norm(Hw(N/2+2:N))/norm(Hw)
     = 3.2350e-06
disp(sprintf(['Analytic-derived stop-band energy is',...
             ' %g percent of total spectral energy'], 100*ierr));

  Stop-band energy is 0.000324 percent of total spectral energy

Hwp = [Hw(N/2+2:N), Hw(1:N/2+1)]; % plot (-) freqs on the left
Hwpn = abs(Hwp); Hwpn = Hwpn/max(Hwpn);
plot(f,20*log10(Hwpn)); grid;
s = sprintf(...
  'Length %d FIR filter Frequency Response, FFT size = %d',M,N);
title(s); 
xlabel('Normalized Frequency (cycles/sample)'); 
ylabel('Gain (dB)')
if saveplots, saveplot('KaiserHilbertFR.eps'); end;
if dopause, disp 'Pausing... RETURN to continue'; pause; end;

% Zoom in on low-frequency transition band 
%    (same as HF band, by symmetry):
kshow = 2*k1;
krange = kzero-kshow : kzero+kshow;
frange = (krange - kzero)/N;
plot(frange,20*log10(Hwpn(krange))); grid; 
hold on; plot((k1-1)/N,1,'+'); hold off;
s = sprintf(
  'Low-Frequency Transition Band, f1/fs=%f (marked by "+")',f1/fs);
title(s); 
xlabel('Normalized Frequency (cycles/sample)'); 
ylabel('Gain (dB)')
if saveplots, saveplot('KaiserHilbertZoomedFR.eps'); end;
if dopause, disp 'Pausing... RETURN to continue'; pause; end;

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Order a Hardcopy of Spectral Audio Signal Processing

Previous: Windowing an ``Ideal'' Impulse Response
Next: Comparison to Optimal Chebyshev FIR Filter

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.