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4 Measurement of Biosignals and Analog Signal Processing

is chosen. Since in general the pole locations P±k are complex, t±k is also assumed to

be complex, i.e. t±k = tR±k + jtI±k. Inserted in Equation 4.50, it follows:

P±k = Σ±k + jΩ±k = j ⋅cos(tR±k + jtI±k)

= j ⋅cos tR±1 cos(jtI±k) −j ⋅sin tR±1 sin(jtI±k)

= sin tR±1 sinh tI±k + j ⋅cos tR±1 cosh(jtI±k) .

(4.51)

The pole locations P±k can thus be calculated from t±k and Equation 4.51. If the para-

meter t±k is substituted into Equation 4.49, it follows:

Tn(P±k/j) = cos(n ⋅(tR±k + jtI±k))

= cos(ntR±k) ⋅cosh(ntI±k) −j ⋅sin(ntR±k) ⋅sinh(ntI±k)

= ±j/ϵ .

(4.52)

This gives rise to two conditions:

1. cos(ntR±k) ⋅cosh(ntI±k) = 0 ,

2. sin(ntR±k) ⋅sinh(ntI±k) = ∓1/ϵ .

(4.53)

From these conditions it follows:

n ⋅tR±k = (2k + 1) ⋅π/2

and thus

sin(ntR±k) = 1

and

tI±k = −¹

n ^{arcsinh(1/ϵ) .}

(4.54)

This can now be used to calculate the pole positions from Equation 4.51 , and Equa-

tion 4.54 can be calculated:

P±k = Σ±k + jΩ±k

with

Σ±k = ± sin(π(2k + 1)/2n) ⋅sinh(tI±k)

Ω±k = cos(π(2k + 1)/2n) ⋅cosh(tI±k)

tI±k = −¹

n ^{arcsinh(1/ϵ) .}

(4.55)

If the real part is divided by sinh(tI±k) and the imaginary part of the pole by cosh(tI±k),

and if the squares of each are added, it follows:

(

Σ±k

cosh(tI±k)⁾

2

+ (

Ω±k

sinh(tI±k)⁾

2

= 1 .

(4.56)

This is the equation of an ellipse, i.e. the pole locations of a Chebyshev-filter all lie

on an ellipse, since their semi-axes sinh(tI±k) and cosh(tI±k) are independent of k and

depend only on the filter order n and the damping parameter ϵ. They are also all smal-

ler than unity in magnitude, so our assumption above about using the Chebyshev-

polynomials for Equation 4.49 was correct. Substituting the polynomials for Equa-

tion 4.55 into Equation 4.28, it follows for the constant factor A0:

A0 =

1

ϵ ⋅2^{n−1 .}

(4.57)