4.5 Design of Analogue Filters

|

143

at even filter order n.

A0 =

1

√^{1 +}

ϵ²

κ^n/2 ∏^n/2

j=1 ^Ω2

0j

,

(4.85)

at odd filtering order n.

A0 = ¹

ϵ ^(√κ)n−2

(n−1)/2

∏

j=1

Ω²

0j ^.

(4.86)

The further filter design is also done here as for the Chebyshev-filter, i.e. application of

suitable frequency transformation, circuit selection, determination of the components

and final analysis by simulation.

4.5.2 Selective filters with Group Delay Optimisation

In the previous section, the filters were optimised according to specifications for the

magnitude frequency response. However, for some applications, e.g. ECG measure-

ment, it is also important that the waveform of the measurement signal is preserved.

To achieve this, all frequency components of the wanted signal must be passed on

from the input to the output with the same speed. In this case, the group delay of the

filter must have as constant value as possible in the passband. This is possible both

by non-recursive digital filtering, which will be discussed in a later section, and by

an analogue filter before the analogue-to-digital converter. The Bessel filter will be

presented as an example.

4.5.2.1 Bessel Filter

In the Bessel filter, an ideal delay element is first approximated with the normal-

ised delay time T and the transfer function of the normalised low-pass according to

AnTP(Ω) = e^−PT = 1/e^PT and then the cut-off frequency is adjusted according to the at-

tenuation specifications. However, since this ideal transfer function is not fractionally

rational in P, it cannot be realised directly by analogue components. For this reason,

the e- function is approximated in a series with fractional rational members in P, e.g.

by a Taylor series n-th order:

e^PT ≈1 + ^P

1! ^{+ P2}

2! ^{+ ⋅⋅⋅+ Pn}

n! ^.

(4.87)

The disadvantage is that the resulting polynomial does not always have zeros in the

left P- half plane. The system can therefore become unstable. The remedy is a Hurwitz-

polynomial. Its zeros lie always in the left P- half plane [51]. This Hurwitz polynomial

must be a Bessel function Bn(P) (hence its name), i.e.

AnTP(P) ≈^Bn(0)

Bn(PT) ^,

n: polynomial order

(4.88)