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2 Fundamentals of Information, Signal and System Theory

time t/s

time t/s

discrete time ts /nTs

discrete time ts /nTs

Fig. 2.19: Examples for a continuous-time and continuous-value signal szkwk (top left), for a

continuous-time and discrete-value signal szkwd (top right), for a discrete-time and discrete-value

signal szdwk (bottom left) and a discrete-time and discrete-value signal szdwd (bottom right).

2.4 Signal Processing Transformations

In the previous sections, a signal was described by the time-dependent quantity s(t)

and a system by the time-dependent impulse response h(t). However, in some cases

it is useful to represent the signal and system in a different function space called im-

age domain (or reciprocal space, frequency domain, Fourier domain). For example,

transforming the signal s(t) into the image domain can reveal signal information that

is hidden in the original function space. The transformation into the image domain

is done by means of a mathematical operation T, which transforms s(t) into a new

quantity S(ξ) with the new variable of the image domain ξ. In signal processing, how-

ever, it also happens that the analysis is performed a priori in the image domain. For

example, the effect of filters is usually considered in the frequency domain. Therefore,

the mathematical operation T must be designed in such a way that a back transform-

ation T⁻¹ exists which uniquely returns the transformed quantity S(ξ) from the image

domain back into s(t) of the original function space.

s(t)

T

S(ξ)

S(ξ)

T ⁻¹

s(t)

Fig. 2.20: By means of the mathematical operation T the time-dependent quantity s(t) is trans-

formed into S(ξ) (image domain). In signal processing, the requirement for T is that there exists a

unique back transformation T ⁻¹ from S(ξ) back to s(t).