Frequency Response in Matlab

In practice, we usually work with a *sampled* frequency axis. That
is, instead of evaluating the transfer function
at
to obtain the frequency response
, where
is *continuous* radian frequency, we compute instead

where is the desired number of spectral samples around the unit circle in the plane. From Eq.(7.5) we have that this is the same thing as the

The uniformly sampled DTFT has its own name: the

where

DFT

and , where denotes the sampling rate in Hz.

To avoid undersampling
, we must have
, and to
avoid undersampling
, we must have
. In general,
*will be undersampled* (when
), because it is the
quotient of
over
. This means, for example, that
computing the impulse response
from the sampled frequency
response
will be *time aliased* in general. *I.e.*,

IDFT

will be time-aliased in the IIR case. In other words, an infinitely long impulse response cannot be Fourier transformed using a finite-length DFT, and this corresponds to not being able to sample the frequency response of an IIR filter without some loss of information. In practice, we simply choose sufficiently large so that the sampled frequency response is accurate enough for our needs. A conservative practical rule of thumb when analyzing stable digital filters is to choose , where denotes the maximum pole magnitude. This choice provides more than 60 dB of decay in the impulse response over a duration of samples, which is the time-aliasing block size. (The time to decay 60 dB, or `` '', is a little less than time constants [84], and the time-constant of decay for a single pole at radius can be approximated by samples, when is close to 1, as derived in §8.6.)

As is well known, when the DFT length
is a power of 2, *e.g.*,
, the DFT can be computed extremely efficiently using
the *Fast Fourier Transform (FFT)*. Figure 7.1 gives an
example matlab script for computing the frequency response of an IIR
digital filter using two FFTs. The Matlab function `freqz`
also uses this method when possible (*e.g.*, when
is a power of 2).

function [H,w] = myfreqz(B,A,N,whole,fs) %MYFREQZ Frequency response of IIR filter B(z)/A(z). % N = number of uniform frequency-samples desired % H = returned frequency-response samples (length N) % w = frequency axis for H (length N) in radians/sec % Compatible with simple usages of FREQZ in Matlab. % FREQZ(B,A,N,whole) uses N points around the whole % unit circle, where 'whole' is any nonzero value. % If whole=0, points go from theta=0 to pi*(N-1)/N. % FREQZ(B,A,N,whole,fs) sets the assumed sampling % rate to fs Hz instead of the default value of 1. % If there are no output arguments, the amplitude and % phase responses are displayed. Poles cannot be % on the unit circle. A = A(:).'; na = length(A); % normalize to row vectors B = B(:).'; nb = length(B); if nargin < 3, N = 1024; end if nargin < 4, if isreal(b) & isreal(a), whole=0; else whole=1; end; end if nargin < 5, fs = 1; end Nf = 2*N; if whole, Nf = N; end w = (2*pi*fs*(0:Nf-1)/Nf)'; H = fft([B zeros(1,Nf-nb)]) ./ fft([A zeros(1,Nf-na)]); if whole==0, w = w(1:N); H = H(1:N); end if nargout==0 % Display frequency response if fs==1, flab = 'Frequency (cyles/sample)'; else, flab = 'Frequency (Hz)'; end subplot(2,1,1); % In octave, labels go before plot: plot([0:N-1]*fs/N,20*log10(abs(H)),'-k'); grid('on'); xlabel(flab'); ylabel('Magnitude (dB)'); subplot(2,1,2); plot([0:N-1]*fs/N,angle(H),'-k'); grid('on'); xlabel(flab); ylabel('Phase'); end |

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