Fourier
series is almost always used in harmonic analysis of a
waveform.
Fourier series is applicable to periodic signals only.
Using fourier series, a periodic signal can be
expressed as a sum of a dc signal , sine function and cosine function. The
frequencies of sine and cosine functions are integral multiples of a frequency
called fundamental frequency.
Different
Forms of Fourier Series
Fourier series can have three forms :-
(1) Trigonometric fourier series
(2) Polar form
(3) Exponential fourier series
Trigonometric
Fourier Series
A periodic signal x(t) is expressed as the sum of a dc
signal, sine functions and cosine functions. Shown below is the fourier
series formula.
Fourier Series Solved Examples
(1) Fourier series for square wave
(2) Fourier
series for sawtooth (triangular) wave
Symmetry conditions in Fourier series
The
knowledge of symmetry conditions results in reduced calculations.
(1) Even
symmetry :-
A signal
x(t) has even symmetry if
x(t) =
x(-t)
For such
even symmetric signals, there will be no sine terms in fourier series i.e bn
=0
(2) Odd
symmetry :-
A signal
x(t) has even symmetry if
x(t) = -
x(-t)
The fourier
expansion of odd symmetric signals contains sine terms only i.e a0 = an = 0
(3) Half
wave symmetry :-
A signal
x(t) has half wave symmetry if
x(t) = - x[t±(T/2)]
The half
wave symmetric signal x(t) has
a0
= an = bn = 0 ,
if n is even
a0
= 0 , if n is odd
Polar Form of Fourier Series
The fourier
series can be expressed in polar form as
Exponential Fourier Series
A periodic
function x(t) can be expressed in exponential form of fourier series as
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