4.5 Design of Analogue Filters

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135

Fig. 4.43: Characteristic functions K(jΩ) = cos(n ⋅arccos(Ω)) of a Chebyshev-filter (left) and mag-

nitudes of the transfer functions of the associated normalised low-pass filters up to the 5th order for

ϵ = 1 (right).

The advantage of this choice of approximation of the ideal normalised low-pass filter

is that the transition between passband and stopband becomes steeper than with a

power filter, allowing a wider bandwidth for the useful signal at a prescribed stop-

band frequency, as with an antialiasing-filter to satisfy the sampling theorem (cf. Fig-

ure 4.43). If one now extends the normalised frequency Ωto the complex frequency

P := Σ + jΩagain one obtains, starting from Σ = 0 for the product

GnTP(P) := AnTP(P) ⋅AnTP(−P) :

GnTP(P) =

1

1 + K(P/j)^{2 =}

1

1 + ϵ²T²n(P/j)

.

(4.47)

Zeros of GnTP(P) are again none. The polar places Pk of GnTP(P) lie at the zeros of the

denominator and can be determined by Equation 4.24:

K(P±k/j) = ϵ ⋅Tn(P±k/j) = ±j .

(4.48)

For the derivation of the equation for the pole points, it should first be assumed that

the amounts of the pole points are smaller than 1 (which must be checked afterwards).

In this case |Ω| < 1 can also be assumed, and one obtains from Equation 4.48 and

Equation 4.45:

Tn(P±k/j) = cos(n ⋅arccos(P±k/j)) = ±j/ϵ .

(4.49)

Furthermore, in order to improve clarity, the parameterrepresentationoftheChebyshev-

polynomials is given with

t±k = arccos(P±k/j) orP±k = j ⋅cos t±k

(4.50)