Desired Impulse Response

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Fitting Filters to Measured Amplitude Responses

Measured Amplitude Response

Converting the Desired Amplitude Response to Minimum Phase

Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

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Matlab is a high-level scripting language for scientific computing — Click for https://ccrma.stanford.edu/~jos/matlab/

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

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The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.html

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Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

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The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Fitting Filters to Measured Amplitude Responses

Measured Amplitude Response

Converting the Desired Amplitude Response to Minimum Phase

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 $\ensuremath{L_2}$ norm of the outermost 20% of the impulse response divided by its total $\ensuremath{L_2}$ norm--this measure was reported to be % for this example.

**Figure 8.5:** 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 [452]. This is of course expected because the desired frequency response was real (and nonnegative).

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');

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Desired Impulse Response

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University