4.4 Interference Suppression and Analog Filtering

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magnitude / dB

phase

phase / °

Fig. 4.28: Magnitude and phase frequency response of 4th order Bessel, Butterworth and Cheby-

shev low-pass filters: For this purpose, the circuit in Figure 4.27 was duplicated and connected in

series. The filter type was determined by choosing a after Table 4.4.

C1

47n

C2

47n

AC 1

R1

68k

R2

68k

R3

10k

R4

10k

5V

-5V

Fig. 4.29: Second order notch filter for 50 Hz suppression.

biosignal had a linear phase frequency response. As the phase responses of the dif-

ferent filter types in Figure 4.28 show, this is generally not the case. There, the phase

response at 200 Hz, well into the passband, is −40^∘, while at 10 Hz it still has 0^∘. In

other words, this means that high-frequency signal components receive a different

phase rotation due to filtering than low-frequency signal components. The associated

effect on the signal is made even clearer by the introduction of the concept of group

delay. This is defined as

Tg(ω) = −^∂φ

∂ω ^,

(4.15)

i.e. the negative derivative of the phase φ with respect to the frequency ω. The group

delay can be understood as the propagation time for infinitesimally narrowband sig-

nal components. In Figure 4.32 the group delay times for a 2nd, 8th and 12th order

Bessel filter are shown.

In the example, the group delay times above approx. 100 Hz are no longer con-

stant. Signal components in this frequency range are subject to a different delay dur-