2.5 Biosignal Processing and the Derivation of Diagnostic Information

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45

An overview of the common convolution theorems can be found in Table 2.6. Applic-

ations of integral transformations are given in connection with discrete transforma-

tions in section 5.2 and in practical applications in chapter 6.

Tab. 2.6: Convolution theorems.

Theorem

Relation

Commutative law

a(t) ∗b(t) = b(t) ∗a(t)

Associative law

a(t) ∗b(t) ∗c(t) = [a(t) ∗b(t)] ∗c(t) = a(t) ∗[b(t) ∗c(t)]

Distributive law

a(t) ∗(b(t) + c(t)) = a(t) ∗b(t) + a(t) ∗c(t)

Identity

a(t) ∗δ(t) = a(t)

Differentiation

a(t) ∗δ^󸀠(t) = a^󸀠(t)

Integration

a(t) ∗u(t) = ∫a(τ)dτ

Displacement

a(t −t0) = a(t) ∗δ(t −t0)

Dilation

a(t) ∗δ(bt) =

1

|b| ^a(t)

2.5 Biosignal Processing and the Derivation of

Diagnostic Information

The definition of important properties and transformations of signal processing in the

previous sections allow a mathematical description of the analysis of biosignals. A

fundamentally important idea in the analysis of biosignals, as with signals in gen-

eral, is the direct connection between the signal generator, in our case a physiological

system, and the signal itself, already discussed in section 2.2. Ideally, this direct con-

nection allows the system state (state of health) to be determined from the signal in the

form of a system diagnosis. The basic prerequisite for this is that the signal measurand

has a sufficiently high sensitivity to the physiological variable under consideration and

is also not obscured by other artefacts such as movements or noise. In most cases, for

this reason, the primary task is to separate the important from the unimportant signal

components and to generate a suitable measurement and signal processing chain for

the signal to be acquired.

The signal processing chain results from the measurement task and, as shown

in Figure 2.24, always contains a physiological process (and a physiological signal)

which takes place in the patient and which is measured (in the form of the signal

measurand) with the aid of suitable sensor technology and measuring equipment and

is digitised, stored and evaluated in the computer or microcontroller for further pro-

cessing. It should be noted that in most cases the physiological signal does not cor-

respond exactly to the signal measurand, either because of the measurement method

used or the type of signal processing that follows.