5.3 Methods for Analysis and Processing of Discrete Biosignals

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the most common selective filters (low pass, high pass, band pass and band stop) the

following transfer functions for a frequency transformation for the case that the origin

low pass has a normalised cut-off frequency of fa/4 (see e.g. [9, 41]):

1.

normalised low-pass-low-pass and low-pass-high-pass-transformation:

zntp = ± ^{z + a0}

a0z + 1 ^,

with

|a0| < 1

a0 = tan (πfgTa −π/4) .

(5.82)

zntp is the system variable in the normalised low-pass range. The positive sign ap-

plies to the low-pass-low-pass transformation, the negative sign to the "low-pass-

high-pass-transformation. As can be seen from the equation, the coefficient a0 is

only dependent on the new cut-off frequency fg, i.e. with a0 the cut-off frequency

can be varied.

2.

normalised lowpass-bandpass and lowpass bandstop-transformation:

zntp = ∓^{z2 + a1z + a0}

a0 + a1z+1

mit

|a0| < 1 ∧|a1| < 1 + a0

a0 = tan (π/4 −π[fo −fu]Ta)

a1 =

−2 sin(2π[fo −fu]Ta)

sin(2πfuTa) + sin(2πfoTa) + cos(2πfuTa) −cos(2πfoTa) ^.

(5.83)

The negative sign applies to the lowpass-bandpass-transformation, the positive

sign to the lowpass-bandpass-transformation. The coefficient a0 depends here

only on the new bandwidth fo −fu, i.e. with a0 the bandwidth of the bandpass

or the bandstop can be varied. With the coefficient a1 the centre frequency is then

influenced; because a1 does not only depend on the difference of the upper cut-off

frequency fo and the lower fu.

Explanatory example

A discrete-time normalised low-pass filter with sampling frequency fa = 200 Hz, equi-

valent to sampling period of Ta = 1/fa = 5 ms has a cut-off frequency of fa/4 = 50 Hz.

Using an all-pass transformation, a low-pass filter with a cut-off frequency of 30 Hz

and 70 Hz is to be designed. Such a normalised low-pass with the transfer function

Gntp(zntp) can be realised by a simple averaging of two consecutive samples, i.e.:

Gntp(zntp) = ¹

2^{(1 + z−1}

ntp^{) .}

(5.84)

After applying the low-pass-low-pass-transformation according to Equation 5.82, it

then follows in general for the frequency-transformed low-pass with transfer function