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

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.