4.5 Design of Analogue Filters

|

141

comes closest to the ideal low-pass filter in terms of its selectivity. However, such filters

are most sensitive to parameter fluctuations of the components, so that even small

changes can cause a completely different filter behaviour or instability.

For the characteristic function K(jΩ) a factor ϵ weighted rational elliptic function

Rn(κ, Ω) of order n is chosen [70]:

K(Ω) = ϵ ⋅Rn(κ, Ω) ,

κ:

module (sensitivity measure) .

(4.75)

The rational elliptic function Rn(κ, ω) has the property that at the reciprocal of the an-

gular frequency Ωthe function is also transformed into its reciprocal, i.e. Rn(κ, 1/Ω) =

1/Rn(κ, Ω). Thus, at the normalised passband frequency ΩD = 1, one achieves an op-

timal approximation of the ideal lowpass in the passband as well as in the stopband.

For even filter order n is

Rn(κ, Ω) = κ^n/2 ⋅

n/2

∏

k=1

Ω² −Ω0k

Ω²κ²Ω²

0k ⁻¹

,

(4.76)

and for odd filter order n is

Rn(κ, Ω) = (^√κ)ⁿ ⋅

(n−1)/2

∏

k=1

Ω² −Ω0k

Ω²κ²Ω²

0k ⁻¹

.

(4.77)

Where

Ω0k = sn ( ^{n −2k + 1}

n

K0) ,

k = 1 . . . [ ⁿ

2^{] ,}

(4.78)

where [ ⁿ

2^{] is said to be the smallest integer less than n}

2^{. sn is the Jacobian elliptic func-}

tion with modulus κ < 1, and K0 is the complete elliptic integral of the first kind also

with modulus κ, which is a measure of the cutoff frequency:

K0 := ∫

dx

√(1 −x²) ⋅(1 −κ²x²)

,

ΩS = ¹

κ ^.

An example of the magnitude frequency response of a Cauer filter for filter grades 1 to

5 is shown Figure 4.47. There is a ripple of 0.3 in both the passband and the stopband.

Betragsfrequenzgang-Cauer The poles of the transfer function A(P) result in (cf. [74]):

Pk = ^{σ0F0k ± jΩ0kF0}

1 + κ²σ0Ω²

0k

,

k = 1 . . . [ ^{n + 1}

2

] ,

(4.79)

σ0 = ^{sn(u0, κ󸀠)}

cn(u0, κ^󸀠) ^,

F0 = ^{dn(u0, κ󸀠)}

cn²(u0, κ^󸀠) ^{, F0k = cn(u0k, κ) ⋅dn(u0k, κ) .}

(4.80)