4.2. Different types of filters and filter specifications – 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 cut-off frequency fc
Passband characterized by low and high cutoff frequencies fc1 and fc2
Stopband characterized by low and high cutoff angular frequencies of fc1 and fc2

Table 4.1. Frequency transformation to obtain the transfer function of a filter H(j2πf) from a normalized transfer function of a low-pass filter

Figure 4.3. Ideal high-pass filter

Figure 4.4. Ideal stopband filter

Figure 4.5. Ideal passband filter

NOTE 4.1.– in practice, according to Paley Wiener's theorem, it is impossible to obtain ideal filters that completely reject the frequential components of a signal on a finite band of frequencies. For this reason, we define a specification as a device on which we inscribe the filtering curve of the real filter. From here, we will no longer use the term stopband, but rather attenuated band. Moreover, unlike an ideal specification, a real filter contains a transition band (see Figure 4.6).

Figure 4.6. Low-pass filter specification

The response curve can then be approximated in several ways. In this chapter, we present different approximation approaches that lead to filters whose squared transfer function module is a rational fraction. Since the module |H(j2πf)| and the phase (f) of the transfer function are, respectively, even and odd functions of the frequency f, the expression of the squared transfer function module |H (j2πf)|2 is expressed as:

If we introduce x, the normalized frequency in relation to the cut-off frequency, we have:

where b0 = 1 and a0 = 1.

If the degree of the denominator is superior to that of the numerator, we know that and that the filter is of the low-pass type. From there, we take the series:

We then introduce the concept of attenuation A(jx) of a filter, satisfying the relation:

No matter which filter being considered, we can demonstrate that the attenuation will appear in the following form:

According to the nature of ø(jx), which is a polynomial or a rational fraction, we then speak of polynomial or elliptic filters.