Hilbert Transform Design Example
We will now use the window method to design a complex bandpass filter which passes positive frequencies and rejects negative frequencies.
Since every real signal possesses a Hermitian spectrum , i.e., , it follows that, if we filter out the negative frequencies, we will destroy this spectral symmetry, and the output signal will be complex for every nonzero real input signal (excluding dc and half the sampling rate). In other terms, we want a filter which produces a ``single sideband'' (SSB) output signal in response to any real input signal. The Hermitian spectrum of a real signal can be viewed as two sidebands about dc (with one sideband being the ``conjugate flip'' of the other). See §2.3.3 for a review of Fourier symmetry-properties for real signals.
An ``analytic signal'' in signal processing is defined as any signal having only positive or only negative frequencies, but not both (typically only positive frequencies). In principle, the imaginary part of an analytic signal is computed from its real part by the Hilbert transform (defined and discussed below). In other words, one can ``filter out'' the negative-frequency components of a signal by taking its Hilbert transform and forming the analytic signal . Thus, an alternative problem specification is to ask for a (real) filter which approximates the Hilbert transform as closely as possible for a given filter order.
Primer on Hilbert Transform Theory
We need a Hilbert-transform filter to compute the imaginary part of the analytic signal given its real part . That is,
where . In the frequency domain, we have
where denotes the frequency response of the Hilbert transform . Since by definition we have for , we must have for , so that for negative frequencies (an allpass response with phase-shift degrees). To pass the positive-frequency components unchanged, we would most naturally define for . However, conventionally, the positive-frequency Hilbert-transform frequency response is defined more symmetrically as for , which gives and , i.e., the positive-frequency components of are multiplied by .
In view of the foregoing, the frequency response of the ideal Hilbert-transform filter may be defined as follows:
Note that the point at can be defined arbitrarily since the inverse-Fourier transform integral is not affected by a single finite point (being a ``set of measure zero'').
The ideal filter impulse response is obtained by finding the inverse Fourier transform of (4.16). For discrete time, we may take the inverse DTFT of (4.16) to obtain the ideal discrete-time Hilbert-transform impulse response, as pursued in Problem 10. We will work with the usual continuous-time limit in the next section.
The Hilbert transform of a real, continuous-time signal may be expressed as the convolution of with the Hilbert transform kernel:
That is, the Hilbert transform of is given by
Thus, the Hilbert transform is a non-causal linear time-invariant filter.
The complex analytic signal
corresponding to the real signal
then given by
To show this last equality (note the lower limit of 0
instead of the
), it is easiest to apply (4.16) in the frequency
Thus, the negative-frequency components of are canceled, while the positive-frequency components are doubled. This occurs because, as discussed above, the Hilbert transform is an allpass filter that provides a degree phase shift at all negative frequencies, and a degree phase shift at all positive frequencies, as indicated in (4.16). The use of the Hilbert transform to create an analytic signal from a real signal is one of its main applications. However, as the preceding sections make clear, a Hilbert transform in practice is far from ideal because it must be made finite-duration in some way.
Filtering and Windowing the Ideal
Hilbert-Transform Impulse Response
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 rejectionRecall 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.
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/NSetting 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
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).
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
. 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-04The 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.
More General FIR Filter Design
We have looked at more than just FIR filter design by the window method and frequency-sampling technique. The general steps were
- Prepare a desired frequency-response that ``seems achievable''
- Inverse FFT
- Window the result (time-limit it)
- FFT that to see how it looks
Comparison to Optimal Chebyshev FIR Filter
Let's now compare the window-method design using the Kaiser window to the optimal equiripple FIR filter design given by the Remez multiple exchange algorithm.
Note, by the way, that many filter-design software functions, such as firpm have special modes for designing Hilbert-transform filters .
It turns out that the Remez exchange algorithm has convergence problems for filters larger than a few hundred taps. Therefore, the FIR filter length was chosen above to be small enough to work out in this comparison. However, keep in mind that for very large filter orders, the Remez exchange method may not be an option. There are more recently developed methods for optimal Chebyshev FIR filter design, using ``convex optimization'' techniques, that may continue to work at very high orders [218,22,153]. The fast nonparametric methods discussed above (frequency sampling, window method) will work fine at extremely high orders.
The following Matlab command will try to design the FIR Hilbert-transform filter of the desired length with the desired transition bands:
hri = firpm(M-1, [f1,f2]/fn, [1,1], , 'Hilbert');Instead, however, we will use a more robust method  which uses the Remez exchange algorithm to design a lowpass filter, followed by modulation of the lowpass impulse-response by a complex sinusoid at frequency in order to frequency-shift the lowpass to the single-sideband filter we seek:
tic; % remember the current time hrm = firpm(M-1, [0,(f2-fs/4)/fn,0.5,1], [1,1,0,0], [1,10]); dt = toc; % design time dt can be minutes hr = hrm .* j .^ [0:M-1]; % modulate lowpass to single-sidebandThe weighting [1,10] in the call to firpm above says ``make the pass-band ripple times that of the stop-band.'' For steady-state audio spectra, pass-band ripple can be as high as dB or more without audible consequences.5.11 The result is shown in Fig.4.16 (full amplitude response) and Fig.4.17 (zoom-in on the dc transition band). By symmetry the high-frequency transition region is identical (but flipped):
In this case we did not normalize the peak amplitude response to 0 dB because it has a ripple peak of about 1 dB in the pass-band. Figure 4.18 shows a zoom-in on the pass-band ripple.
We can note the following points regarding our single-sideband FIR filter design by means of direct Fourier intuition, frequency-sampling, and the window-method:
- The pass-band ripple is much smaller than 0.1 dB, which is
``over designed'' and therefore wasting of taps.
- The stop-band response ``droops'' which ``wastes'' filter taps
when stop-band attenuation is the only stop-band specification. In
other words, the first stop-band ripple drives the spec (
while all higher-frequency ripples are over-designed. On the other
hand, a high-frequency ``roll-off'' of this nature is quite natural
in the frequency domain, and it corresponds to a ``smoother pulse''
in the time domain. Sometimes making the stop-band attenuation
uniform will cause small impulses at the beginning and end of
the impulse response in the time domain. (The pass-band and
stop-band ripple can ``add up'' under the inverse Fourier transform
integral.) Recall this impulsive endpoint phenomenon for the
Chebyshev window shown in Fig.3.33.
- The pass-band is degraded by early roll-off. The pass-band edge
is not exactly in the desired place.
- The filter length can be thousands of taps long without running
into numerical failure. Filters this long are actually needed for
sampling rate conversion
We can also note some observations regarding the optimal Chebyshev version designed by the Remez multiple exchange algorithm:
- The stop-band is ideal, equiripple.
- The transition bandwidth is close to half that of the
window method. (We already knew our chosen transition bandwidth was
not ``tight'', but our rule-of-thumb based on the Kaiser-window
main-lobe width predicted only about
% excess width.)
- The pass-band is ideal, though over-designed for static audio spectra.
- The computational design time is orders of magnitude larger
than that for window method.
- The design fails to converge for filters much longer than 256
taps. (Need to increase working precision or use a different
method to get longer optimal Chebyshev FIR filters.)
Generalized Window Method
Window Method for FIR Filter Design