5.3 Methods for Analysis and Processing of Discrete Biosignals

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5.3.1.3 Linear and cyclic convolution

Convolution uses the impulse response to determine the output signal of a causal lin-

ear system from the input signal, as in the analogue case. In a linear discrete-time

system, the superposition principle applies as in the analogue case, i.e., if two signals

are applied to an input of a linear discrete-time system in succession and the asso-

ciated output signals are generated and then added, the same output result is also

obtained if the two input signals are added beforehand, the sum is applied to the in-

put and the output signal is then determined. Or formulated in mathematical notation

for discrete-time systems: If

y(n) = Ta{x(n)} ,

Ta : Operator ,

(5.45)

then

Ta{

N

∑

ν=1

kν ⋅xν(n)} =

N

∑

ν=1

kν ⋅Ta {xν(n)} .

(5.46)

For time-invariant systems, their property does not change with time. It does not mat-

ter when a certain input signal is given to the system. The same signal then always

occurs at the output. This is not the case, for example, with mobile communication

using a mobile phone. In the city centre, the reception is worse than in an open field.

This transmission system is then time-variant with regard to its transmission proper-

ties. Then the spectrum is not only dependent on frequency, but also on time. To plan

network coverage and standardisation, various models for the associated transmis-

sion channels were developed for this purpose.

However, in the calculation of the convolution, the Dirac momentum δ(t) is not

needed as in the analogue case, but the unit momentum. However, the unit momentum

is not obtained from the "sampling", but has the value 1 at the point n = 0 and is

calculated according to

δ(n) =

{

{

{

0

for n

̸= 0

1

for n = 0

(5.47)

is defined. In contrast to the analogue δ(t), in this case no transition from a square

pulse to an infinite pulse of vanishing width with an area of 1 is required. The digital

unit impulse according to Equation 5.47 and Figure 5.15 thus has a value corresponding

to the area of the analogue Dirac impulse and therefore has no mathematical peculi-

arity.

The digital impulse response g(n) serves as the basis for the digital convolution.

This is the output signal of a discrete-time system if the unit impulse δ(n) is present

at its input (cf. Figure 5.16).

Because the superposition principle applies to a linear system, each discrete-time

input signal x(n) can also be expressed as a weighted sum of impulse responses. In

this case, the discrete-time convolution sum is obtained:

y(n) =

∞

∑

ν=−∞

x(ν)g(n −ν) =

∞

∑

μ=−∞

x(n −μ)g(μ) ,

μ = n −ν .

(5.48)