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2 Fundamentals of Information, Signal and System Theory

2.3 Definition and Classification of Signals

The methods of biosignal processing refer more often to different classes and defin-

itions of signals. These are also of great importance in continuous signal processing

as special test functions in the proof. In the following, we will mathematically define

important signals for the course of the book and classify them according to their prop-

erties.

2.3.1 Univariate and Multivariate Signals

A common mathematical representation of signals can be achieved through the no-

tions of dependent and independent quantities. According to this definition, a signal

is a physical quantity that depends on one or several independent quantities. If there

is dependence on only one variable, signals are referred to in statistics as univariate

signals, whereas if there is dependence on several independent variables, signals are

referred to as multivariate signals. Another common definition of multivariate signals

concerns the measurement of several signals (e.g. for an EEG) in a measurement ar-

rangement with several sensors. If there is a common dependence on an independent

quantity, such as time, when measuring M univariate signals x1(t), x2(t), . . . , xM(t),

the following definition applies to the resulting M-dimensional multivariate signal:

Xm = {x1(t), x2(t), . . . , xM(t)}

∀m ∈ℕ.

(2.10)

In most cases, however, a biosignal is simply the temporal course (time is thus the

independent quantity) of a physical (dependent) quantity such as the electrical

voltage U – the associated signal thus becomes U(t) or Ut for short. In addition to

time as an independent quantity, the location in its three spatial directions (x, y, z)

is used in the representation of image signals as two- or three-dimensional spatially

resolved intensity signals I(x, y) and I(x, y, z), respectively. If these image signals

also have a time dependence, an image signal sequence I(t, x, y, z) is obtained as a

function of four independent variables. In principle, the methods of signal processing

can also be applied to image or video signals, but in this book we restrict ourselves to

signal sequences that depend on only one independent variable, i.e. to univariate or

scalar signals. The special features of the evaluation of multivariate signals are only

dealt with in part in this book in section 6.1.

Figure 2.9 shows on the left a time-dependent raw ECG signal U(t) acquired over

a time range of t = 0, . . . , 10 s, and on the right a two-dimensional MRI image with a

resolution of x = y = 512 pixels.

In principle, all physical quantities such as pressure, temperature, voltage, cur-

rent, etc. can be considered as dependent quantities of a signal. Normally, however, it

is an indirect electrical, easily measurable quantity that is assigned to the direct phys-