文档介绍：Fundamentals of Measurement Technology(2)
Prof. Wang Boxiong
In a finite interval of time, a periodic signal x(t) can be represented by its Fourier series when plies with the Dirichlet conditions:
where
n=0,1,2,3,……
T= the period
ω0= the angular frequency or circular frequency
ω0= 2π/T
an(including a0 and bn) are called Fourier coefficients.
Frequency representation of periodic signals
()
()
()
Fourier coefficients an and bn (functions of nω0):
an: even function of n or nω0, a-n = an.
bn: odd function of n or nω0, b-n = -bn.
Dirichlet conditions:
x(t) must be absolutely integrable,
x(t) possesses a finite number of maxima and minima and finite number of discontinuities in any finite interval.
Frequency representation of periodic signals
Rewrite Eq. ():
where
An: amplitude of signal’s ponent
φn: phase-shift
Frequency representation of periodic signals
()
()
()
a0/2 is the constant-value or the . component of a periodic signal.
The term for n=1 is referred to as the fundamental (component), or as the first ponent.
ponent for n=N is referred to as the Nth ponent.
The representation of a periodic signal in the form of Eq. () is referred to as the Fourier series representation:
An: amplitude of the nth ponent
φn: phase shift of the nth ponent
Frequency representation of periodic signals
The plots of the amplitude An and the phase φn versus signal’s angular frequency ω0 are called amplitude spectrum plot and phase spectrum plot respectively.
The frequency spectrum is displayed graphically by a number of discrete vertical lines representing the amplitude An and the phase φn of the analyzed signal respectively.
The frequency spectrum of a periodic signal is a discrete one.
Frequency representation of periodic signals
Example 1.
Find the Fourier series of the periodic square wave signal x(t) shown in Fig. .
Frequency representation of periodic signals
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