4.5 Design of Analogue Filters

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Fig. 4.37: Example of pole selection when the magnitude frequency response of a 4th order select-

ive filter is specified: the pole locations of the extended magnitude frequency response function

AnTP(P) ⋅A^∗

nTP^{(−P) (left) and the pole locations of the transfer function AnTP(P) (right).}

Fig. 4.38: Tolerance scheme of the characteristic function K(Ω) of a normalised low-pass filter.

case – i.e. all zeros lie in the left P- half plane. An example of the distribution of the

pole positions for a 4th order filter is shown in Figure 4.37.

In the further synthesis of normalised low-pass filters, apart from normalisation, it

is easier to express the magnitude of the transfer function |A(jΩ)| by the characteristic

function K(Ω) defined by the following relation:

|A(jΩ)|² =

1

1 + K(Ω)^{2 .}

(4.19)

This makes it easier to describe the standard filters, since for these the characteristic

function can be described by simple polynomials and the associated tolerance scheme

can be simplified (cf. Figure 4.38).

4.5.1.2 Butterworth or Power Filter

In a Butterworth or power filter, only a factor ϵ weighted power of the normalised fre-

quency Ωis chosen for the characteristic function K(Ω), i.e..

K(jΩ) = ϵ ⋅Ωⁿ ,

n: filtering order , ϵ: attenuation constant

(4.20)