## Phase Response

The phase of the frequency response is called the *phase
response*. Like the phase of any complex number, it is given by the
arctangent of the imaginary part of
divided by its
real part, and it specifies the *delay* of the filter at each
frequency. The phase response is a good way to look at short filter
delays which are not directly perceivable as causing an
``echo''.^{4.4} For longer delays in audio, it is
usually best to study the filter *impulse response*, which is
output of the filter when its input is
(an
``impulse''). We will show later that the impulse response is also
given by the inverse
*z* transform of the filter transfer function (or, equivalently, the inverse
Fourier transform of the filter frequency response).

In this example, the phase response is

`plotfr`is given in §J.4.) In Octave or the Matlab Signal Processing Toolbox, a figure similar to Fig.3.10 can be produced by typing simply

`freqz(B,A,Nspec)`

.

% efr.m - frequency response computation in Matlab/Octave % Example filter: g1 = 0.5^3; B = [1 0 0 g1]; % Feedforward coeffs g2 = 0.9^5; A = [1 0 0 0 0 g2]; % Feedback coefficients Nfft = 1024; % FFT size Nspec = 1+Nfft/2; % Show only positive frequencies f=[0:Nspec-1]/Nfft; % Frequency axis Xnum = fft(B,Nfft); % Frequency response of FIR part Xden = fft(A,Nfft); % Frequency response, feedback part X = Xnum ./ Xden; % Should check for divide by zero! clf; figure(1); % Matlab-compatible plot plotfr(X(1:Nspec),f);% Plot frequency response cmd = 'print -deps ../eps/efr.eps'; disp(cmd); eval(cmd); |

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Pole-Zero Analysis

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Amplitude Response