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4 Measurement of Biosignals and Analog Signal Processing

Fig. 4.39: Characteristic functions K(Ω) = ϵΩⁿ (left) and magnitudes of the transfer functions of the

associated normalised low-pass filters up to 4th order for ϵ = 1 (right).

and after Equation 4.19 results in:

|A(jΩ)|² =

1

1 + ϵ² ⋅Ω^{2n .}

(4.21)

At the normalised passband cut-off frequency Ω= 1, the characteristic function K(Ω)

has the value ϵ and is otherwise always smaller in the passband. The tolerance scheme

must therefore have this maximum value. At ϵ = 1 the magnitude of the transfer

function |AnTP(Ω= 1)| = 1/^√2 ≈0.707. This corresponds to an attenuation a =

−20⋅log(|AnTP|) of 3 dB (cf. Figure 4.39). If one now extends according to Equation 4.72

the normalised frequency Ωto the complex frequency P := Σ + jΩ, then starting from

Σ = 0 for the product GnTP(P) := AnTP(P) ⋅AnTP(−P) we obtain:

GnTP(P) =

1

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

1

1 + ϵ²( ^P

j ⁾

2n ⁼

1

1 + ϵ² (e^−jπ/2P)^{2n .}

(4.22)

Zeros of GnTP(P) are not present. The pole places Pk of GnTP(P) are at the zeros of the

denominator, i.e..

1 + K(P±k/j)² = 0

(4.23)

respespectively

K(P±k/j) = ±j

(4.24)

and have the values (cf. also [68]):

P±k = e^jπ/2 (⁻¹

ϵ^{2 )}

1/2n

= ^{ej(π/2±(π+k⋅2π)/2n)}

n√ϵ

,

k = 0, . . . , n −1 .

(4.25)

Since the amount of these poles always have the same value of 1/

n√ϵ, the poles all

lie on a circle (cf. Figure 4.37, for ϵ = 1). Now that the poles and zeros of GnTP(P) are