5.3 Methods for Analysis and Processing of Discrete Biosignals

|

199

about half. After that, the amplification increases again, since the spectrum repeats

according to the sampling theorem (cf. Figure 5.5). This effect is due to the fact that

the analogue RC low-pass filter used does not have an infinitely large attenuation at

the cut-off frequency fg like an ideal low-pass filter, but only an attenuation of 3 dB,

which is equivalent to a drop in the magnitude square of the transfer function by

1/^√2 ≈0.707. At higher frequencies, the attenuation increases but is non-zero at half

the sampling frequency of fab/2 = 500 Hz. This means that there is overlapping of the

spectrum after sampling. This cannot result in an ideal low-pass filter that has an in-

finitely large attenuation at its cut-off frequency. Furthermore, the property of the cor-

responding analogue filter according to Figure 5.39, which also has an infinitely large

attenuation at a higher frequency. A remedy is achieved by not requiring the same val-

ues as those of the impulse response of the analogue filter for the sampling times in

the filter design, but only striving for the closest possible approximation of the trans-

fer function. However, because of the basically periodic course of the transfer function

of the digital system, this means that the transfer function of the analogue system in

the range from 0 to ∞corresponds to that of the digital system in the range from 0

to half the sampling frequency. The frequency axis of the analogue system is virtually

mapped onto the frequency axis of the digital system from 0 to half the sampling fre-

quency. This can be done according to the figure after the bilinear transformation with

the following procedure:

1.

The transfer function ^—G(z = e^jωTa) shall match that of the associated analogue

filter G(p = jω up to half the sampling frequency, i.e..

—G(z = e^jωTa) = G(p = jω) .

(5.93)

2.

substituting the inverse function of z, i.e..

p = ^ln(z)

Ta

,

(5.94)

into the transfer function G(p) of the analogue filter does not lead to a fractional

rational function in z, and the digital filter is thereby not realisable with adders,

delayers and multipliers (as, for example, in the canonical circuits).

3.

The ideal transformation between p- and z-range, z = e^pTa or p = ^ln(z)

Ta ^{, is therefore}

approximated in such a way that a realisable transfer function in the z- range is

obtained.

4.

To do this, ln(z) is generated by a series expansion with fractional functions in

z and broken off after the 1st member. This is then an approximation to be used

called bilinear transformation; bilinear here means that there are linear functions

in p or z in both the numerator and denominator:

ln z = 2 { ^{z −1}

z + 1 ^{+ 1}

3 ^{( z −1}

z + 1⁾

3

+ ¹

5 ^{( z −1}

z + 1⁾

5

+ ⋅⋅⋅} ≈2 ^{z −1}

z + 1 ^.

(5.95)