Fourier Series | examples- sawtooth (triangular) and square wave | Formula

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  b_n=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 = a_n = 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= a_n = b_n = 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

Related :-
      (1) Fourier Transform - Formula - Properties