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
[452]. 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}

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']); |

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