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2 Fundamentals of Information, Signal and System Theory

where A is the amplitude, φ0 is the zero phase angle and ω0 is the angular frequency

of the harmonic function. The zero phase φ0 can alternatively be written down as the

time shift t0 = φ/ω0. The angular frequency is defined as

ω0 = ^2π

T0

= 2πf0 ,

(2.12)

i.e. the frequency f0 of the periodic process multiplied by 2π, where f0 is equal to the

reciprocal of the period T0 of the function y(t):

f0 = ¹

T0

= ^ω0

2π ^.

(2.13)

Harmonic signals with zero phase angle can be represented as the sum of sine and

cosine functions with the help of the addition theorems. A detailed description can be

found in [90]. In signal processing, the pointer representation in the complex plane has

become established for the calculation of harmonically excited systems. Between the

two representations can be converted with the help of the Eulerian relation as follows:

e^jφ = cos φ + j sin φ .

(2.14)

Thus a cosine and sine function can be understood as the real and imaginary part of

a complex exponential function, respectively:

z(t) = Ae^j(ω0t+φ) = Ae^jφe^j(ω0t) = A (cos(ω0t + φ) + j sin(ω0t + φ)) .

(2.15)

This mathematical representation can be represented by a pointer of length A, which

rotates in the complex plane around the coordinate origin. The time for a full rotation

is the period duration T0. In this representation, the cosine function x(t) results from

the projection of the pointer onto the real axis ℜ{z}, while the sine function y(t) rep-

resents the projection onto the imaginary axis ℑ{z}. To illustrate this common form of

representation, the projection of the complex pointer onto the imaginary axis in the

complex plane is shown in Figure 2.10.

Mathematically, one defines periodic signals by requiring a constant periodic dur-

ation T0 = const.∀t ∈ℝ. The function sper(t) repeats exactly after all k multiples of T0,

with k ∈ℤ. This class of signals also includes arbitrary superpositions of harmonic

signals as expressed in the real form of the Fourier series:

sper(t) = ^a0

2 ^{+ ∑ak cos(kω0t) + bk sin(kω0t) ,}

ω0 = ^2π

T0

.

(2.16)

The Fourier coefficients ak, bk of the equation indicate the real amplitudes of the sine

and cosine oscillations (cf. section 2.4):

a0 = ²

T0

∫sper(t)dt ,

ak = ²

T0

∫sper(t) cos(kω0t)dt ,

bk = ²

T0

∫sper(t) sin(kω0t)dt ,

k = 1, 2, 3, . . . .

(2.17)