196

|

5 Methods for Discrete Processing and Analysis of Biosignals

fg=30Hz

fg=50Hz (norm. LP)

fg=70Hz

Fig. 5.38: Example of the magnitude transfer functions for low-pass-low-pass-transform using a

digital all-pass according to Equation 5.82 with fa = 200 Hz.

Gtp(z):

Gtp(z) = Gntp (zntp = ^{z + a0}

a0z + 1^{) = 1 + a0}

2

z + 1

z + a0

,

a0 = tan(πfgTa −π/4)

with

|a0| < 1 .

(5.85)

The coefficient a0 then has the value for the transformed low-pass with the cut-off

frequency of 30 Hz.

a0 = tan(π ⋅30 Hz ⋅5 ms −π/4) = −0.325

(5.86)

and analogously for the low pass with cut-off frequency of 70 Hz

a0 = tan(π ⋅70 Hz ⋅5 ms −π/4) = 0.325 .

(5.87)

Figure 5.38 shows the associated magnitude transfer functions |Gtp(f)| in the context

of the normalised low-pass with cut-off frequency of 50 Hz.

Another possibility to change the selectivity behaviour of a given filter with a cer-

tain property (e.g. normalised low pass) by frequency transformation is to apply the

frequency transformation to an analogue filter and then to realise a digital filter with

the same selectivity behaviour by transferring it to the discrete-time range. This does

not apply to the frequency range from 0 to ∞as with the analogue filter, but only from

0 to half the sampling frequency; because after that the spectrum repeats itself, since

this is generally always periodic in discrete-time systems (cf. Figure 5.5). For this pur-

pose, there is the transformation with the impulse invariance method and with the

help of the bilinear transformation, which will be explained in the following.

The impulse invariance method

In this method, the digital filter is realised in such a way that the values of its impulse

response gdi(n) are the same as the impulse response of the analogue filter gan(t) at