does freqz() implement DTFT or DFT?

The freqz() function typically implements the discrete-time Fourier transform (DTFT) rather than the discrete Fourier transform (DFT).

The discrete-time Fourier transform (DTFT) is a mathematical transformation that represents a discrete-time signal in the frequency domain. It provides a continuous spectrum that describes the signal's frequency content over the entire range of frequencies. The DTFT is defined for both finite-duration and infinite-duration signals.

On the other hand, the discrete Fourier transform (DFT) is a finite-length approximation of the DTFT. It computes the spectrum of a discrete-time sequence at a finite number of equally spaced frequency points. The DFT is primarily used for analyzing finite-length signals and is commonly implemented using fast Fourier transform (FFT) algorithms.

The freqz() function in MATLAB computes and returns the frequency response of a FIR or IIR digital filter using the DTFT. It accepts the filter coefficients as input and returns the complex frequency response for a range of frequencies.

Example of freqz() in Matlab

% Create a simple low-pass filter
order = 4;                    % Filter order
cutoff_freq = 0.4;            % Normalized cutoff frequency

b = fir1(order, cutoff_freq); % Generate filter coefficients

% Compute the frequency response using freqz
fs = 1000;                    % Sample rate (Hz)
freq_range = 0:0.01:1;        % Frequency range (normalized from 0 to 1)

[h, w] = freqz(b, 1, freq_range * pi);  % Compute the frequency response

% Plot the magnitude response
plot(w/pi, abs(h));
title('Magnitude Response');
xlabel('Normalized Frequency (\times\pi rad/sample)');

% Plot the phase response
plot(w/pi, angle(h));
title('Phase Response');
xlabel('Normalized Frequency (\times\pi rad/sample)');
ylabel('Phase (radians)');

In this example, we first create a simple low-pass filter using the fir1() function to generate the filter coefficients. We specify the filter order and the normalized cutoff frequency.

Then, we use the freqz() function to compute the frequency response of the filter. We pass the filter coefficients (b) and the denominator coefficients (1 since it's an FIR filter) to freqz(). The freq_range * pi argument specifies the frequency range in radians.

We store the magnitude and phase response of the filter in variables h and w, respectively. Finally, we plot the magnitude and phase responses using the plot() function.

magnitude response

phase response
Note that the frequency range is normalized from 0 to 1, where 1 corresponds to the Nyquist frequency (half the sample rate).

In summary, freqz() is commonly used to compute the frequency response using the DTFT, which provides a continuous spectrum rather than the limited set of frequency points obtained from the DFT.

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