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Fitting Filters to Measured Amplitude Responses

The preceding filter-design example digitized an ideal differentiator, which is an example of converting an LTI lumped modeling element into a digital filter while maximally preserving its frequency response over the audio band. Another situation that commonly arises is the need for a digital filter that matches a measured frequency response over the audio band.

Measured Amplitude Response

Figure 8.3 shows a plot of simulated amplitude-response measurements at 10 frequencies equally spread out between 100 Hz and 3 kHz on a log frequency scale. The ``measurements'' are indicated by circles. Each circle plots, for example, the output amplitude divided by the input amplitude for a sinusoidal input signal at that frequency [449]. These ten data points are then extended to dc and half the sampling rate, interpolated, and resampled to a uniform frequency grid (solid line in Fig.8.3), as needed for FFT processing. The details of these computations are listed in Fig.8.4. We will fit a four-pole, one-zero, digital-filter frequency-response to these data.9.14

Figure 8.3: Example measured amplitude-response samples at 10 exponentially spaced frequencies. Circles: Measured amplitude-response points. Solid: Extrapolated, spline-interpolated, and uniformly resampled amplitude response, ready for ifft.
\includegraphics[width=\twidth]{eps/tmps2-G}

Figure: Script (matlab) for simulating a measured amplitude response at 10 exponentially spaced frequencies and extrapolating/interpolating/resampling to obtain a complete, nonparametric amplitude response, uniformly sampled at FFT frequencies. This script generated Fig.8.3.
  
NZ = 1;      % number of ZEROS in the filter to be designed
NP = 4;      % number of POLES in the filter to be designed
NG = 10;     % number of gain measurements
fmin = 100;  % lowest measurement frequency (Hz)
fmax = 3000; % highest measurement frequency (Hz)
fs = 10000;  % discrete-time sampling rate
Nfft = 512;  % FFT size to use
df = (fmax/fmin)^(1/(NG-1)); % uniform log-freq spacing
f = fmin * df .^ (0:NG-1);   % measurement frequency axis

% Gain measurements (synthetic example = triangular amp response):
Gdb = 10*[1:NG/2,NG/2:-1:1]/(NG/2); % between 0 and 10 dB gain

% Must decide on a dc value.
% Either use what is known to be true or pick something "maximally
% smooth".  Here we do a simple linear extrapolation:
dc_amp = Gdb(1) - f(1)*(Gdb(2)-Gdb(1))/(f(2)-f(1));

% Must also decide on a value at half the sampling rate.
% Use either a realistic estimate or something "maximally smooth".
% Here we do a simple linear extrapolation. While zeroing it
% is appealing, we do not want any zeros on the unit circle here.
Gdb_last_slope = (Gdb(NG) - Gdb(NG-1)) / (f(NG) - f(NG-1));
nyq_amp = Gdb(NG) + Gdb_last_slope * (fs/2 - f(NG));

Gdbe = [dc_amp, Gdb, nyq_amp];
fe = [0,f,fs/2];
NGe = NG+2;

% Resample to a uniform frequency grid, as required by ifft.
% We do this by fitting cubic splines evaluated on the fft grid:
Gdbei = spline(fe,Gdbe); % say `help spline'
fk = fs*[0:Nfft/2]/Nfft; % fft frequency grid (nonneg freqs)
Gdbfk = ppval(Gdbei,fk); % Uniformly resampled amp-resp

figure(1);
semilogx(fk(2:end-1),Gdbfk(2:end-1),'-k'); grid('on');
axis([fmin/2 fmax*2 -3 11]);
hold('on'); semilogx(f,Gdb,'ok');
xlabel('Frequency (Hz)');   ylabel('Magnitude (dB)');
title(['Measured and Extrapolated/Interpolated/Resampled ',...
       'Amplitude Response']);

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 $ L2$ norm of the outermost 20% of the impulse response divided by its total $ L2$ norm--this measure was reported to be $ 0.02$% for this example.

Figure: Desired impulse response obtained from a length 512 inverse FFT of the interpolated measured amplitude response in Fig.8.3.
\includegraphics[width=\twidth]{eps/tmps2-ir}

Note also that the desired impulse response is noncausal. In fact, it is zero phase [449]. 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');

Converting the Desired Amplitude Response to Minimum Phase

Phase-sensitive filter-design methods such as the equation-error method implemented in invfreqz are normally constrained to produce filters with causal impulse responses.9.15 In cases such as this (phase-sensitive filter design when we don't care about phase--or don't have it), it is best to compute the minimum phase corresponding to the desired amplitude response [449].

As detailed in Fig.8.8, the minimum phase is constructed by the cepstral method [449].9.16

The four-pole, one-zero filter fit using invfreqz is shown in Fig.8.7.

Figure 8.7: Overlay of desired amplitude response (solid) and that of a fourth-order filter fit (dashed) using invfreqz.
\includegraphics[width=\twidth]{eps/tmps2-Hh}

Figure: Script (matlab) for converting the (real) desired amplitude response to minimum-phase form for invfreqz. This script generated Fig.8.7.
  
c = ifft(Sdb); % compute real cepstrum from log magnitude spectrum
% Check aliasing of cepstrum (in theory there is always some):
caliaserr = 100*norm(c(round(Ns*0.9:Ns*1.1)))/norm(c);
disp(sprintf(['Cepstral time-aliasing check: Outer 20%% of ' ...
    'cepstrum holds %0.2f %% of total rms'],caliaserr));
% = 0.09 percent
if caliaserr>1.0 % arbitrary limit
  error('Increase Nfft and/or smooth Sdb to shorten cepstrum');
end
% Fold cepstrum to reflect non-min-phase zeros inside unit circle:
% If complex:
% cf=[c(1),c(2:Ns-1)+conj(c(Nfft:-1:Ns+1)),c(Ns),zeros(1,Nfft-Ns)];
cf = [c(1), c(2:Ns-1)+c(Nfft:-1:Ns+1), c(Ns), zeros(1,Nfft-Ns)];
Cf = fft(cf); % = dB_magnitude + j * minimum_phase
Smp = 10 .^ (Cf/20); % minimum-phase spectrum

Smpp = Smp(1:Ns); % nonnegative-frequency portion
wt = 1 ./ (fk+1); % typical weight fn for audio
wk = 2*pi*fk/fs;
[B,A] = invfreqz(Smpp,wk,NZ,NP,wt);
Hh = freqz(B,A,Ns);

figure(3);
plot(fk,db([Smpp(:),Hh(:)])); grid('on');
xlabel('Frequency (Hz)');
ylabel('Magnitude (dB)');
title('Magnitude Frequency Response');
% legend('Desired','Filter');

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Digital Differentiator Design