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2 Fundamentals of Information, Signal and System Theory

the even part is always described by the cosine and the odd part by the sine of the

integral core. The largest values in S(jω) are obtained when s(t) itself is a periodic

function that can be represented by a Fourier series according to Equation 2.16.¹³

The Fourier-transformion has a special significance in the physical-technical con-

text, where the magnitude of S(jω) is interpreted as the frequency spectrum of s(t).

The frequency spectrum provides an indication of which frequency components are

contained in the signal s(t). The analysis in the frequency domain is explained in

detail for discrete-time signals in subsection 5.3.2. The frequency spectrum further

forms the basis for understanding the transmission behaviour of filters in terms of

magnitude and phase frequency response, the application of which is discussed in

section 4.4 and subsubsection 5.3.4.2.

In system theory, the transformation into the frequency domain offers an altern-

ative to the consideration in the time domain. As already explained in Figure 2.20,

in the time domain the result y(t) of the transmission of a signal through a linear,

time-invariant system h(t) is obtained from the mathematical convolution of the in-

put quantity s(t) with the impulse response of the system: y(t) = s(t) ∗h(t). Trans-

formation into the frequency domain, the convolution is represented by an algebraic

multiplication (cf. last row in Table 2.3).

Tab. 2.3: Fourier transform theorems.

Theorem

Time domain s(t)

Frequency domain S(jω)

Linearity

a1s1(t) + a2s2(t)

a1S1(jω) + a2S2(jω)

Similarity

s(bt)

1

|b| ^{S( jω}

b ⁾

Time shift

s(t −t0)

S(jω)e^−jωt0

Frequency shift

s(t)e^jω0t

S(j(ω −ω0))

Differentiation

∂ⁿs(t)

∂tⁿ

(jω)ⁿS(jω)

Integration

∫s(t)dt

S(jω)

jω

+ ¹

2 ^S(0)δ(jω)

Multiplication

s(t) h(t)

S(jω) ∗H(jω)

Convolution

s(t) ∗h(t)

S(jω) H(jω)

Thus, the spectrum of the system output Y(jω) is obtained from the multiplication of

the Fourier-transformation of input S(jω) and the Fourier-transformation of the im-

pulse response H(jω):

Y(jω) = S(jω) H(jω) .

(2.52)

This relationship is illustrated by Figure 2.21. The spectrum of the impulse term H(jω)

is called the transfer function of the system. In principle, neither of the two ways of

looking at linear time-invariant systems (LTI systems for short) is to be preferred to the

13 Every periodic function sper(t) with the period T0 can be represented in the form of a Fourier series.