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:
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 f_{c} | |
Passband characterized by low and high cutoff frequencies f_{c1} and f_{c2} | |
Stopband characterized by low and high cutoff angular frequencies of f_{c1} and f_{c2} |
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).
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 b_{0} = 1 and a_{0} = 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.