Introduction of Fourier Series

Introduction of Fourier Series

A Fourier series is called as a linear combination of harmonically related complex exponentials. The Fourier series is used for the spectrum analysis of periodic signals i.e. to get frequency spectrum of a time-domain signal, when the signal is a periodic function of time. The Fourier series is important because it allows one to model periodic signals as a sum of distinct harmonic components. In other words, representing signals in this way allows one to see the harmonics in a signal distinctly, which makes it easy to see what frequencies the signal contains in order to filter/manipulate particular frequency components.

Fourier series is very useful in Electronics and Electrical Engineering or Theory of wave.

Fourier series and Fourier Transform are used in applications like file compression (JPEG image format), signal processing in communications and astronomy, acoustics, optics and cryptography.

Harmonically related signals, fundamental and harmonic components of signal x(t).

Harmonically related signals are the sets of periodic complex exponential/sinusoidal signals with fundamental frequencies that are multiples of a single positive frequency. The basic signals for continuous time, harmonically related exponentials are,

Here, each value of k, S_k(t) is periodic with fundamental period 1/kF_o = T_o/k or fundamental frequency kF_o. Since a signal that is periodic with period T_p/k is also periodic with period k (T_p/k) = T_p for any positive integer k, we see that all of the S_k(t) have a common period of T_p.

Let us consider the signal x(t) = Ce^at  …………………………………………………....................….(2)

where, C is a real number

            a is a pure imaginary number.

Assuming C = 1 and a = jω, Eqⁿ (2) becomes,      

                                                           

This signal is known as complex exponential signal having periodic property.

Now,    

                                      

where, T_o is the fundamental time period.

For signal to be periodic,  which implies that is a multiple of 2π, i.e.

Comparing Eqⁿ (3) and (4), we get

                                     ω  = kω_o ……………………………………………………....................…. (5)

 

where, ω_o is known as fundamental frequency. And equation (6) represents harmonically related complex exponentials. Thus the linear combination of harmonically related complex exponentials of the form, 

 

is also periodic with period T.