# Fourier series

** Jean Baptiste Joseph Fourier, ** a French mathematician and physicist; was born in Auxerre, France. He initialized the Fourier series, the Fourier transforms and their applications to the problems of heat transfer and vibrations. The Fourier series, the Fourier transforms and the Fourier law are named in his honor.

Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) ## Fourier series

To represent any periodic signal x ( t), Fourier developed an expression called the Fourier series. It is in terms of an infinite sum of sines and cosines or exponentials. The Fourier series uses the condition of orthoganality.

### Fourier series representation of periodic signals in continuous time

A signal is said to be periodic if it satisfies the condition x (t) = x (t + T) or x (n) = x (n + N).

Where T = fundamental time period,

There are two basic periodic signals:

$ x (t) = cos omega_0t $ (sinusoidal) &

$ x (t) = e ^ {j omega_0 t} $ (complex exponential)

These two signals are periodic with period $ T = 2 pi / omega_0 $.

A set of harmonically related complex exponentials can be represented by {$ phi_k (t) $}

$$ {phi_k ( t)} = {e ^ {jk omega_0t}} = {e ^ {jk ({2 pi on T}) t}} text {where}, k = 0 pm 1, pm 2 ..n,,, .. ... (1) $$

All of this signals are periodic of period T

According to the orthogonal approximation of the signal space of a function x (t) with n, mutually orthogonal functions are given by

$$ x (t) = sum_ {k = - infty} ^ {infty} a_k e ^ {jk omega_0t} ..... (2) $$

$$ = sum_ {k = - infty} ^ {infty} a_kk e ^ { jk omega_0 t} $$

Where $ a_k $ = Fourier coefficient= coefficient of approximation.

This signal x (t) is also periodic with period T.

Equation 2 represents the Fourier series representation of the periodic signal x (t).

The term k = 0 is constant.

The term $ k = pm1 $ having the fundamental frequency $ omega_0 $, is called as 1 ^{ st } harmonic.

The term $ k = pm2 $ having a fundamental frequency $ 2 omega_0 $, is called as 2 ^{ nd } harmonics, and so on ...

The term $ k = ± n $ having a fundamental frequency $ n omega0 $, is called n ^{ e } harmonics.

### Derivation of the Fourier coefficient

We know that $ x (t) = Sigma_ {k = - infty} ^ {infty} a_k e ^ {jk omega_0 t} ... ... (1) $

Multiply $ e ^ {- jn omega_0 t} $ on both sides. So

$$ x (t) e ^ {- jn omega_0 t} = sum_ {k = - infty} ^ {infty} a_k e ^ {jk omega_0 t}. e ^ {- jn omega_0 t} $$

Considerthe integral on both sides.

$$ int_ {0} ^ {T} x (t) e ^ {jk omega_0 t} dt = int_ {0} ^ {T} sum_ { k = - infty} ^ {infty} a_k e ^ {jk omega_0 t}. e ^ {- jn omega_0 t} dt $$

$$ quad quad quad quad,, = int_ {0} ^ {T} sum_ {k = - infty} ^ {infty} a_k e ^ {j (kn) omega_0 t}. dt $$

$$ int_ {0} ^ {T} x (t) e ^ {jk omega_0 t} dt = sum_ {k = - infty} ^ {infty} a_k int_ {0} ^ {T} e ^ {j (kn) omega_0 t} dt. ,, ..... (2) $$

by Euler 's formula,

$$ int_ {0} ^ {T} e ^ {j (kn) omega_0 t} dt. = int_ {0} ^ {T} cos (kn) omega_0 dt + j int_ {0} ^ {T} sin (kn) omega_0t, dt $$

$$ int_ {0} ^ {T} e ^ {j (kn) omega_0 t} dt. = left {begin {array} {l l} T & quad k = n 0 & quad k neq n end {array} right. $$

So in Equation 2, the integral is zero for all values of k except k = n. Put k = n in l 'equation 2.

$$ Rightarrow int_ {0} ^ {T} x (t) e ^ {- jn omega_0 t} dt = a_n T $ $

$$ Rightarrow a_n = {1 over T} int_ {0} ^ {T} e ^ {- jn omega_0 t} dt $$

Replace n with k.

$$ Rightarrow a_k = {1 over T} int_ {0} ^ {T} e ^ {- jk omega_0 t} dt $$

$$ so x (t) = sum_ {k = - infty} ^ {infty} a_k e ^ {j (kn) omega_0 t} $$

$$ text {where} a_k = {1 over T} int_ {0} ^ {T} e ^ {- jk omega_0 t} dt $$