Digital Filters Design for Signal and Image Processing

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).

Thus, we write:

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( ...