Filtering and Windowing the Ideal
Hilbert-Transform Impulse Response

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Let denote the convolution kernel of the continuous-time Hilbert transform from (4.17) above:

Convolving a real signal with this kernel produces the imaginary part of the corresponding analytic signal. The way the ``window method'' for digital filter design is classically done is to simply sample the ideal impulse response to obtain and then window it to give . However, we know from above (e.g., §4.5.2) that we need to provide transition bands in order to obtain a reasonable design. A single-sideband filter needs a transition band between dc and , or higher, where denotes the main-lobe width (in rad/sample) of the window we choose, and a second transition band is needed between and . Note that we cannot allow a time-domain sample at time 0 in (4.22) because it would be infinity. Instead, time 0 should be taken to lie between two samples, thereby introducing a small non-integer advance or delay. We'll choose a half-sample delay. As a result, we'll need to delay the real-part filter by half a sample as well when we make a complete single-sideband filter. The matlab below illustrates the design of an FIR Hilbert-transform filter by the window method using a Kaiser window. For a more practical illustration, the sampling-rate assumed is set to Hz instead of being normalized to 1 as usual. The Kaiser-window parameter is set to , which normally gives ``pretty good'' audio performance (cf. Fig.3.28). From Fig.3.28, we see that we can expect a stop-band attenuation better than dB. The choice of , in setting the time-bandwidth product of the Kaiser window, determines both the stop-band rejection and the transition bandwidths required by our FIR frequency response.

  M = 257;    % window length = FIR filter length (Window Method)
  fs = 22050; % sampling rate assumed (Hz)
  f1 = 530;   % lower pass-band limit = transition bandwidth (Hz)
  beta = 8;   % beta for Kaiser window for decent side-lobe rejection

Recall that, for a rectangular window, our minimum transition bandwidth would be Hz, and for a Hamming window, Hz. In this example, using a Kaiser window with ( ), the main-lobe width is on the order of Hz, so we expect transition bandwidths of this width. The choice above should therefore be sufficient, but not ``tight''.^5.8 For each doubling of the filter length (or each halving of the sampling rate), we may cut in half.

Matlab, Continued

Given the above design parameters, we compute some derived parameters as follows:

  fn = fs/2;             % Nyquist limit (Hz)
  f2 = fn - f1;          % upper pass-band limit
  N = 2^(nextpow2(8*M)); % large FFT for interpolated display
  k1 = round(N*f1/fs);   % lower band edge in bins
  if k1<2, k1=2; end;    % cannot have dc or fn response
  kn = N/2 + 1;          % bin index at Nyquist limit (1-based)
  k2 = kn-k1+1;          % high-frequency band edge
  f1 = k1*fs/N           % quantized band-edge frequencies
  f2 = k2*fs/N

Setting the upper transition band the same as the low-frequency band ( ) provides an additional benefit: the symmetry of the desired response about cuts the computational expense of the filter in half, because it forces every other sample in the impulse response to be zero [224, p. 172].^5.9

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Kaiser Window

With the filter length and Kaiser window as given above, we may compute the Kaiser window itself in matlab via

  w = kaiser(M,beta)'; % Kaiser window in "linear phase form"

The spectrum of this window (zero-padded by more than a factor of 8) is shown in Fig.4.9 (full magnitude spectrum) and Fig.4.10 (zoom-in on the main lobe).

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Windowing a Desired Impulse Response Computed by the
Frequency Sampling Method

The next step is to apply our Kaiser window to the ``desired'' impulse response, where ``desired'' means a time-shifted (by 1/2 sample) and bandlimited (to introduce transition bands) version of the ``ideal'' impulse response in (4.22). In principle, we are using the frequency-sampling method (§4.4) to prepare a desired FIR filter of length as the inverse FFT of a desired frequency response prepared by direct Fourier intuition. This long FIR filter is then ``windowed'' down to length to give us our final FIR filter designed by the window method. If the smallest transition bandwidth is Hz, then the FFT size should satisfy . Otherwise, there may be too much time aliasing in the desired impulse response.^5.10 The only non-obvious part in the matlab below is ``.^8'' which smooths 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 an exponential taper 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)];

Figure 4.11 shows our desired amplitude response so constructed. Now we inverse-FFT the desired frequency response to obtain the desired impulse response:

h = ifft(H); % desired impulse response
hodd = imag(h(1:2:N));     % This should be zero
ierr = norm(hodd)/norm(h); % Look at numerical round-off error
% Typical value: ierr = 4.1958e-15
% Also look at time aliasing:
aerr = norm(h(N/2-N/32:N/2+N/32))/norm(h);
% Typical value: 4.8300e-04

The real part of the desired impulse response is shown in Fig.4.12, and the imaginary part in Fig.4.13. Now use the Kaiser window to time-limit the desired impulse response:

% put window in zero-phase form:
wzp = [w((M+1)/2:M), zeros(1,N-M), w(1:(M-1)/2)];
hw = wzp .* h; % single-sideband FIR filter, zero-centered
Hw = fft(hw);  % for results display: plot(db(Hw));
hh = [hw(N-(M-1)/2+1:N),hw(1:(M+1)/2)]; % caual FIR
% plot(db(fft([hh,zeros(1,N-M)]))); % freq resp plot

Figure 4.14 and Fig.4.15 show the normalized dB magnitude frequency response of our final FIR filter consisting of the nonzero samples of hw.

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Next Section:
More General FIR Filter Design
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Primer on Hilbert Transform Theory

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