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5 Methods for Discrete Processing and Analysis of Biosignals

In the frequency range, however, larger deviations can occur, which are particu-

larly noticeable with steep filter edges. If this is undesirable, the rectangular window

can be replaced by another window that does not have such a steep edge, depending

on the application. In this case, the original values of the given impulse response are

no longer obtained in these windows, but they have a more favourable behaviour in

the frequency range, e.g. no overshoots of the magnitude frequency response at a filter

edge.

If the filter coefficients are symmetrical, selective filters can be realised with a non-

recursive filter, which have no phase distortions or a constant group delay, which is

not possible with analogue filters. Four cases can be distinguished:

1.

filter order N even, coefficients mirrorsymmetric (ci = cN−i).

G(z = e^jωTa) = {cN/2 + 2

N

2 ⁻¹

∑

i=0

ci cos[( ^N

2 ^{−i)ωTa]}e−jωNTa/2 ,}

(5.105)

2.

filter order N even, coefficients pointsymmetric (ci = −cN−i)

G(jω) = −j2{

N

2 ⁻¹

∑

i=0

ci sin[( ^N

2 ^{−i)ωTa]}e−jωNTa/2 ,}

(5.106)

3.

filter order N odd, coefficients mirrorsymmetric (ci = cN−i)

G(jω) = 2{

N−1

2

∑

i=0

ci cos[( ^N

2 ^{−i)ωTa]}e−jωNTa/2 ,}

(5.107)

4.

filter order N odd, coefficients pointsymmetric (ci = −cN−i)

G(jω) = −j2{

N−1

2

∑

i=0

ci sin[( ^N

2 ^{−i)ωTa]}e−jωNTa/2 .}

(5.108)

In all four cases, the phase of the filter is linearly dependent on the angular fre-

quency ω, and it has a constant group delay of t0 = ^NTa

2 ^.

Explanatory Example

An ideal digital low-pass filter with a constant magnitude frequency response from 0

to the cut-off frequency fg = 200 Hz and a sampling frequency of fa = 1 kHz is to be

realised by an 8th order FIR filter. With symmetrical filter coefficients, this filter has a

group delay of t0 = ^NT

2 ^{= 8⋅1}

2 ^{ms = 4 ms.}

An ideal analogue low pass with a constant magnitude frequency response of 1 in

the passband would have the impulse response

gan(t) = ^{sin(ωg(t −t0))}

π(t −t0)