142

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4 Measurement of Biosignals and Analog Signal Processing

Fig. 4.47: Amount-frequency response of a Cauer filter for filter grades 1 to 5 with a normalised stop-

band frequency ΩD = 1 and a ripple of 0.3.

The still unknown quantities κ^󸀠, cn, dn, u0 and u0k in Equation 4.80 have the following

meaning:

κ^󸀠= ^√1 −κ² ,

cn = ^√1 −sn² ,

dn = ^√1 −κ²sn² ,

u0k = ^{n −2k + 1}

n

K0 ,

u0 =

K0

n ⋅K∆

sn⁻¹ (

1

√1 + ϵ²∆^{2 ,k∆) ,}

k∆= ^√1 −∆.

(4.81)

K∆is the complete elliptic integral with modulus ∆. The maximum change of the char-

acteristic function K(jΩ) describes ϵ ⋅∆. ∆can now be further determined by the fol-

lowing relations: For even filter order n is

∆= κ^n/2

n/2

∏

j=1

Ω²

0j

(4.82)

and for odd filter order n is

∆=

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

(^√κ)ⁿ

(n−1)/2

∏

j=1

1 −Ω²

0j

κ²Ω²

0j ⁻¹

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

.

(4.83)

There are also zeros in the transfer function AnTP(P) for the Cauer filters. As with the

inverse Chebyshev-filters, they all lie on the imaginary axis at

P0k = ± j

1

κΩ0k

,

k = 1, 2, . . . , [ ⁿ

2^{] .}

(4.84)

To calculate the constant factor A0 of the transfer function AnTP(jΩ) to Equation 4.28,

the zeros must now be considered again. From this follows (without derivation):