Digital Filters Design for Signal and Image Processing

4.2. Different types of filters and filter specifications

Let us consider the example of an ideal low-pass filter of normalized gain and whose frequency is in relation to cut-off frequency (see Figure 4.1).

With an ideal filter, transmission is total in the passband and the stopband.

We write x as the normalized frequency in relation to the cut-off frequency:

NOTE.– x is also called the normalized angular frequency in relation to the cut-off angular frequency:

Figure 4.1. Ideal low-pass filter

Figure 4.2. Low-pass filter corresponding to Figure 4.1, normalized in frequency and amplitude

In general, we will deduce normalized high-pass, band-pass and band-stop filters from normalized low-pass filters by applying frequency variable change formulae (see Figures 4.3, 4.4 and 4.5).

Obtaining the transfer function of the filter H(j2πf) from the transfer function of the normalized low-pass filter H(jx) follows the frequency transformation summarized in Table 4.1.

Obtaining a filter	Transformation carried out from the transfer function of the normalized low-pass filter
High-pass with ...