5.3 Methods for Analysis and Processing of Discrete Biosignals

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Fig. 5.41: Graph and transfer function G(f) of the digital low-pass generated by the bilinear trans-

form method from the analogue low-pass according to Figure 5.39: A comparison with the pulse

invariance method shows that at fa/2 = 500 Hz nothing more is transmitted.

However, in a slightly different representation, this can also be written as a function

of z⁻¹, i.e..

G(z) = cN + cN−1z⁻¹ + ⋅⋅⋅+ c1z^−(N−1) + c0z^−N .

(5.102)

The element z⁻¹ represents the transfer function of a delay element according to Fig-

ure 5.32. This means that a pulse applied to this element is delayed by one clock. For

the element z⁻² it is two clocks, for z⁻³ three clocks and so on. Such a digital filter thus

generates several pulses weighted with ci, i = 0, . . . , N at its output when an impulse

is applied to its input. Therefore, the corresponding impulse response is:

g(n) = cNδ(n −0) + cN−1δ(n −1) + ⋅⋅⋅+ c1δ(n −[N −1]) + c0δ(n −N) ,

(5.103)

where δ(i) is the discrete Dirac momentum.

The filter coefficients ci thus represent the values of the impulse response in a

non-recursive filter. Conversely, however, this means that the impulse response can

be specified, and one thereby also directly obtains the filter coefficients ci. This is very

practical for filter synthesis. However, it must be noted that an FIR filter always has a

finite impulse response that ends after N clocks. Therefore, if a filter is to be realised

that has an infinite impulse response, its impulse response can only be approximated

by the impulse response of an FIR filter. In the simplest case, the given infinite impulse

response is truncated after N clocks. This corresponds to multiplying the infinite im-

pulse response ̃g(n) by a rectangular window function w(t), which is constant for the

first 0 to N beats and then drops to zero, i.e.:

g(n) = ̃g(n) ⋅w(n) ,

(5.104)

with w(n) = 1 from n = 0 to N, otherwise 0.