5.3 Methods for Analysis and Processing of Discrete Biosignals

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nals applies and furthermore the assumption applies that the system properties (or the

system parameters or system coefficients) do not change with time. The LTI systems

form the basis of the digital filters at the end of the section.

5.3.4.1 Linear time-invariant systems

In the subsection 5.3.1, reference has already been made to some simple digital sys-

tems e.g. for changing the sampling or convolution of discrete signals. In the follow-

ing, these discrete-time systems will be formally described. There are three basic forms

of description:

1.

If one represents the relations between the input and output quantities in the time

domain, one obtains the so-called difference equations:

y(n) + d0y(n −1) = c1x(n) + c0x(n −1) ,

y(n) + d1y(n −1) + d0(n −2) = c2x(n) + c1x(n −1) + c2x(n −2) .

2.

Analogous to the convolution integral for continuous-timesystems,inthediscrete-

time domain one obtains the convolution sum according to

y(n) =

∞

∑

ν=−∞

x(ν)g(n −ν) ,

g(n): impulse response .

(5.78)

3.

The transfer function describes the behaviour of the discrete-time system N-th or-

der with respect to its input and output quantities in the image domain. For linear

systems it is a fractional rational function in z:

G(jω) = G(z = e^jωTa) = ^F{y(n)}

F{x(n)} ^{= Z{y(n)}}

Z{x(n)} ^{= c0 + c1z + ⋅⋅⋅+ cN−1zN−1 + cNzN}

d0 + d1z + ⋅⋅⋅+ dN−1z^N−1 + z^N

.

(5.79)

The three forms of representation for describing an LTI system are completely equival-

ent and can be transformed into each other.

5.3.4.2 Digital filter

In general, a digital filter is a linear algorithm that converts a sequence of numbers into

another sequence of numbers using linear mathematical operations (cf. Figure 5.31).

As a result, a digital filter also uses only the linear operators adder, subtractor,

constant multiplier and delay element (cf. Figure 5.32), i.e. . not the saturation ele-

ment known from control engineering. Discrete-time networks can be realised from

these operators such that the coefficients ci and di of the general transfer function

(Equation 5.79) occur directly in the values of the multipliers, e.g. in networks of the

1st or 2nd canonical direct form (cf. Figure 5.33 and Figure 5.34). If the coefficients di

in the denominator are non-zero, i.e. the network contains feedback, the impulse re-

sponse is infinite (IIR filter), otherwise it is finite and ends after N output values (FIR

filter, cf. Figure 5.35).