D.1 The Fourier Series

Periodic signals may be expanded into a Fourier series [1 3]. A signal for which

(D.1)

is valid is called a periodic signal with the period T ₀. The fundamental frequency f ₀ is given by f ₀ = 1/ T ₀ and the corresponding angular frequency is ? ₀ = 2 ?f ₀. A signal

(D.2)

which is a superposition of harmonic signals at the fundamental frequency f and the harmonics nf is periodic with T ₀. The series according to (D.2) is called a Fourier series. The expansion coefficients a _n are the complex amplitudes at the frequencies n ? ₀. The signal s( t) is real if the condition is fulfilled. For real signals s( t) it follows from (D.2) and that

(D.3)

Decomposing a _n in magnitude a _n and phase ? _n by

(D.4)

we obtain from (D.3)

(D.5)

We introduce the real amplitudes

(D.6)

(D.7)

and obtain from (D.3)

(D.8)

Magnitude a _n and phase ? _n are obtained from the real amplitudes by b _n and c _n by

(D.9)

(D.10)

(D.11)

The functions form a complete orthogonal system of basis functions in the interval . Applying

(D.12)

to (D.2) we obtain

(D.13)

It can be shown that arbitrary periodic functions s( t) with period T