140

|

4 Measurement of Biosignals and Analog Signal Processing

must be (cf. Figure 4.46). The pole positions Pk of GnTP(P) lie at the zeros of the de-

nominator and can be determined with Equation 4.24:

K(P±k/j) = ϵ ⋅

1

Tn ( ^ΩS

Ω⁾

= ±j .

(4.69)

These are not to be derived this time, but as in in [70] only to be stated:

P±k = Σ±k + jΩ±k

with

Σ±k = ± ^ΩS

Ne ^{sin(π(2k + 1)/2n) ⋅sinh(̃tI±k) ,}

Ω±k = ^ΩS

Ne ^{cos(π(2k + 1)/2n) ⋅cosh(̃tI±k) ,}

̃tI±k = ¹

n ^{arcsinh(ϵ) ,}

Ne = ^󵄨󵄨󵄨󵄨cos(π(2k + 1)/2n) + j^̃tI±k^󵄨󵄨󵄨󵄨

2 .

(4.70)

These poles also lie on an ellipse. Zeros, unlike the power and Chebyshev-filter, are

present this time and are imaginary. According to [70] they lie at

P0l = j

ΩS

cos ( ^2l−1

2n ⁾

.

(4.71)

However, to calculate the constant factor A0 of the transfer function AnTP(jΩ) to Equa-

tion 4.28, we now have to consider the zeros and obtain

|A0| =

1

√1 + ϵ^{2 ⋅}

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

∏ⁿ⁻¹

l=0 ^{(j −Ppl)}

∏^m−1

l=0 ^{(j −Pnl)}

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

.

(4.72)

If these are now substituted into this equation for the poles and zeros (cf. Equation 4.70

and Equation 4.71), it follows:

at even filter order n:

A0 =

1

√1 + ϵ^{2 ,}

(4.73)

at odd filter order n:

A0 = (−1)^(n−1)/2 ⋅^{n ⋅ΩS}

ϵ

.

(4.74)

The rest of the filter design is the same as for the Chebyshev-filter. This means the

application of suitable frequency transformations, circuit selection (passive or active

realisation), determination of the components and a final analysis by simulation.

4.5.1.5 Cauer Filter

With the Cauer filter, both the passband and thestopband are optimallyapproximated.

The transition between passband and stopband is steepest here and thus the distance

between passband and stopband cut-off frequency is smallest. This approximation