2.3 Definition and Classification of Signals

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2.3.7 Continuous and Discrete Signals

Due to their deterministic nature, purely analytical signals are used for proof in the

fundamentals of theoretical signal processing. This class of continuous signals is

called value and time continuous because both their dependent and independent

quantities can take all values of a continuum. This means that the value set of a time-

/value interval is in principle infinite, as in the case of the representatives of the real

numbers ℝ. Due to their analogue character, these continuous signals are also found

in analogue signal processing.

In contrast, a digital signal is always discrete in value and time, and the informa-

tion contained therein consists of a limited number of possible symbols, i.e. the time-

value interval of these signals is limited to a countably large number of different val-

ues. The number of possible values M is called the interval number. Digital signals are

named according to the number of intervals, such as binary signal (M = 2) or tern-

ary signal (M = 3), etc. Only the finite range of values or information content of these

signals makes it possible to store them on data carriers for further processing.

Whereas with continuous signals the signal value was defined at any time t, with

discrete-time signals this is only the case at discrete times t(n) = tn. These time points

are usually chosen equidistant, i.e. as a whole multiple n ∈ℕof a discrete time interval

tn = nTs. In practice, these discrete signals are created by sampling the continuous-

time signal. Therefore n is called the sample index, Ts the sampling index or sample

interval and its reciprocal fs = 1/Ts the sample or sample frequency. A discrete-time

signal x(ts) is thus completely determined by the time sequence of its samples:

xs(ts) = {x(n1Ts), x(n2Ts), x(n3Ts), . . . , x(nNTs)} ,

n ∈ℕ.

(2.47)

These discrete signal values can be both continuous and discrete. With complete di-

gitisation, however, the instantaneous continuous signal value x(t) at the time ts is

rounded to a discrete signal value. This is then held as a signal value in the digital

signal for the duration of a sampling interval.

The signals shown in Figure 2.19 were generated with the programming language

Matlab, i.e. due to the numerical calculation method, the signals are all discrete in

nature. However, the form of representation can also be influenced there by selecting

the appropriate function. For the representation of continuous signals, high time res-

olutions and the command plot() are used (top), whereas for the representation of

discrete signals low time resolutions and the command stem() (bottom) is applicable.

In section 5.1, the digitisation of continuous signals by time discretisation and value

quantisation and the associated laws are discussed in detail.