## 3.7. Frequential characterization of discrete-time system

### 3.7.1. *Amplitude and phase frequential diagrams*

The frequential characterization of a filter is obtained from the Fourier transform of the impulse response.

According to section 3.3.2, the frequency response of the system can be obtained by calculating the transfer function of the system *H*(*z*) then by being placed on the unity circle *z*, i.e., by taking as the expression of the transfer function, on condition that |*z*| = 1 is in the convergence domain of *H*(*z*).

From here, we can trace the amplitude response represented in the logarithmic scale by:

We can also trace the phase response from the z-transform of the impulse response according to the normalized frequency.

### 3.7.2. *Application*

Let us consider the system characterized by its impulse response, shown by:

We take *N* equal to 1.

If the input *x*(*k*) is the impulse δ(*k*), we have:

*y*(*k*) = *h*(*k*)*δ(*k*) = *h*(*k*)

that is:

The transfer function linked to the system equals:

The system is of finite impulse response; it is stable since

We take *N* equal to 2.

If the input *x*(*k*) is the impulse δ(*k*), we have:

that is:

The transfer function of the system equals:

The system is of finite impulse response; it is stable since

When *N* tends towards infinity.

We then have . The filter is of infinite impulse response. The filter is stable because We can also justify the stability of this system by analyzing the position of the pole of the transfer function of the system . This last, represented by is situated well inside the unity circle in the z-plane.

^{1} See equation (2.1) in Chapter 2.