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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.36: Transmission chain when an analogue signal is processed by a digital filter.

Fig. 5.37: Periodic magnitude spectra of an ideal discrete-time high-pass (top) and low-pass (bot-

tom).

second value of the discrete impulse response of the low-pass its sign is changed. This

is a very practical property when the cut-off frequency of the low-pass is fa/4, i.e. in

half of the total usable frequency range up to fa/2. This fact is used, for example, in

the half-band decomposition of the discrete wavelet-transformation.

However, it is often desirable not only to realise a high pass from a low pass, but

also to design other selective filters if a suitable low pass is already available. This can

be achieved by a frequency transformation. Unlike analogue filters, where the imagin-

ary axis of the complex frequency plane p = σ + jω is mapped onto itself in a suitable

way by a reactance function, now in the discrete-time domain the corresponding fre-

quency line, which now describes a circle around the origin of the coordinate system

because of z = e^jωTa, must be mapped onto itself again as a circle, ie, a circle is mapped

onto a circle, but in such a way that the scaling changes and, for example, a low-pass

becomes a high-pass or band-pass. The magnitude of z always remains unchanged at

one with this transformation. Transfer functions that leave the magnitude of the input

spectrum unchanged are allpasses. These only change the phase (or group delay) and

can thus influence the circular frequency line in the z-range. One therefore obtains for