Fourier Series

Fourier Series

I found that  almost all the books of DSP have a chapter dedicated to Fourier theory, all describing some crazy looking weird formulas and one can be easily lost in  the mathematics behind that. But basic idea behind all the horrible formulas is very simple & even  fascinating :-Any Periodic function can be represented as sum of sine and cosine wave this representation is known as Fourier series named in the honor of Jean Baptiste Joseph Baron Fourier.

Fourier series is given by :-

where,

n = 1 , 2 , 3 , … and T is the period of function f(t). 

a_n and b_n are called Fourier coefficients and are given by

 

now lets we want to represent Square wave in the form of sine & cosine wave.

periodic square wave function f(t) defined by
 

here suppose T = 6 (approx)

now lets find coefficient a₀ which is  DC value 

 

 

 

a₀ = 0  

 

i.e our waveform will oscillation either side of 0 = horizontal axis X line. 

 

now  lets find coefficient a_n which is Even Part 

 

 

 

 

 

 

 

 

 

 

a_n = 0 

 

i.e our waveform will not contain any of cosine Part (even) 

 

now, lets find coefficient b_n which is Odd Part 

Note that cos (n pi) may be written ascos (n pi) = (-1)ⁿ

and that b_n = 0 whenever n is even.

The given function f(t) has the following Fourier series
 

 

 

so, our waveform contains only of sum of sine function i.e odd part.

 

now put N=1 and observer the waveform, 

 

we got 1st harmonic of sine wave also called fundamental harmonic. 

now put n=2  

at 2nd harmonic we observe no change in the waveform because our function is zero when ( i.e bn = 0) whenever n is even 

 

now put n=3

 

its a addition of 1st harmonic + 3rd harmonic(shown in black at centre) = resultant red waveform

 

n=4 which is even so there is no change in waveform

 

 

n=5 note we are closer to approximation of square wave

 

 

n=29 we have almost approximated the square wave

 

 

Note :- fourier series is used when function is continuous and periodic.