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4 Measurement of Biosignals and Analog Signal Processing
Bn(P) :=
n
∑
i=0
(2n −i)!Pi
2n−ii!(n −i)! ,
i = 0, 1, . . . , n .
(4.89)
Examples are:
B1(P) = 1 + P
B2(P) = 3 + 3P + P2
B3(P) = 15 + 15P + 6P2 + P3
B4(P) = 105 + 105P + 45P2 + 10P3 + P4
B5(P) = 945 + 945P + 420P2 + 105P3 + 15P4 + P5 .
Explanatory Example
For ECG filtering, design a passive Bessel filter of 2nd order using an RLC element
which has a passband cut-off frequency of 200 Hz and at this an attenuation of 3 dB.
In this case the Bessel polynomial of 2nd order is B2(P) = 3 + 3P + P2. The delay
time T is chosen such that at the normalised passband cut-off frequency ΩD = 1 the
attenuation of 3 dB can be maintained, i.e.
|AnTP(P = jΩD)| = |AnTP(P = j)| =
3
3 + j3T + (jT)2
= 1
√2
.
(4.90)
One thereby obtains T = 1.3823 s. If we now denormalise so that the passband cut-off
frequency is fD = 200 Hz and compare this with the transfer function ARLC(jω) of an
RLC element one obtains because of
A(jω)Bessel = AnTP (P = j ω
ωD
) =
3
3 + 3j ω
ωD T + (j ω
ωD T)
2
= ARLC(jω) =
1
1 + jωRC + (jω)2LC
(4.91)
the relation:
RC = T/ωD
und
LC = 1
3 (T/ωD)2 .
(4.92)
Choosing R = 100 Ω, it follows with the forward frequency fD = 200 Hz and T =
1.3823 s for the inductance L = 36.7 mH and for the capacitance C = 11 μF. The
phase response in the passband is linear (cf. Figure 4.48).
4.6 Post-Reading and Exercises
Measurement of Electrical Biosignals
1.
Why is analog signal processing needed in the digital age?
2.
What would be the skin impedance for a infinitely high frequency current?