Filtering and Windowing the Ideal
Hilbert-Transform Impulse Response

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

$\displaystyle h_i(t) = \frac{1}{\pi t} \protect$ (5.22)

Convolving a real signal $ x(t)$ 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 $ h_i(nT)$ and then window it to give $ \hat{h}_i(nT)=w(n)h_i(nT)$ . 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 $ \Omega_w$ , or higher, where $ \Omega_w$ denotes the main-lobe width (in rad/sample) of the window $ w$ we choose, and a second transition band is needed between $ \pi-\Omega_w$ and $ \pi$ .

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 $ f_s=22,050$ Hz instead of being normalized to 1 as usual. The Kaiser-window $ \beta $ parameter is set to $ 8$ , 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 $ 50$ dB. The choice of $ \beta $ , 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 $ 2f_s/M \approx 172$ Hz, and for a Hamming window, $ 4f_s/M \approx 343$ Hz. In this example, using a Kaiser window with $ \beta=8$ ( $ \alpha=\beta/\pi\approx 2.5$ ), the main-lobe width is on the order of $ 5f_s/M \approx 430$ Hz, so we expect transition bandwidths of this width. The choice $ f_1=530$ 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 $ f_1$ 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 ( $ f_2 = f_s/2 - f_1$ ) provides an additional benefit: the symmetry of the desired response about $ f_s/4$ 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


Kaiser Window

With the filter length $ M=257$ and Kaiser window $ \beta=8$ 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).

Figure 4.9: The Kaiser window magnitude spectrum.
\includegraphics[width=0.8\twidth]{eps/KaiserFR}

Figure 4.10: Kaiser window magnitude spectrum near dc.
\includegraphics[width=0.8\twidth]{eps/KaiserZoomFR}


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 method4.4) to prepare a desired FIR filter of length $ N=4096$ as the inverse FFT of a desired frequency response prepared by direct Fourier intuition. This long FIR filter is then ``windowed'' down to length $ M=257$ to give us our final FIR filter designed by the window method.

If the smallest transition bandwidth is $ f_1$ Hz, then the FFT size $ N$ should satisfy $ N\gg f_s/f_1$ . 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.

Figure 4.11: Desired single-sideband-filter frequency response.
\includegraphics[width=0.8\twidth]{eps/hilbertFRIdeal}

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.

Figure 4.12: Real part of the desired single-sideband-filter impulse response.
\includegraphics[width=0.8\twidth]{eps/hilbertIRRealIdeal}

Figure 4.13: Desired Hilbert-transform-filter impulse response.
\includegraphics[width=0.8\twidth]{eps/hilbertIRImagIdeal}

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 $ M$ nonzero samples of hw.

Figure 4.14: Frequency response of the Kaiser-windowed impulse response.
\includegraphics[width=0.8\twidth]{eps/KaiserHilbertFR}

Figure 4.15: Zoomed frequency response of the Kaiser-windowed impulse response.
\includegraphics[width=0.8\twidth]{eps/KaiserHilbertZoomedFR}


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More General FIR Filter Design
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Primer on Hilbert Transform Theory