• Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC.

• The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series.

• Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 and publishing Analytical theory of heat in 1822.

• The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave.

• Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine. This superposition or linear combination is called the Fourier series.

• Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems.

Determining the Fourier Series Representation of a

Continuous Time Signal

tjntjkk

k

tjn eeaetx 000)(

1)( )/2(0

tTjkk

k

tjkk

keaeatx

A periodic CT signal can be expressed as a linear combination of harmonically related complex exponentials of the form :-

Multiplying both sides with , we get :-tjne 0

Now if we integrate both sides from 0 to T, we have :-

dteeadtetxT

tjntjkk

k

Ttjn

00

000)(

Interchanging the order of integration and summation :-

2)(00

000

T

tjntjkk

k

Ttjn dteeadtetx

Applying Euler’s formula to the bracketed Integral :-

TTT

tnkj tdtnkjtdtnkdte0

0

0

0

0

)( )sin()cos(0

T

tnkj dte0

)( 0 T, if k = n

0, if k n{

In the R.H.S. side of the integral, for ‘k’ not equal to ‘n’, both the integrals will be zero.Fr k=n, the integrand equals ‘1’ and thus, the integral equals ‘T’.

T

tnkj dte0

)( 0 T, if k = n

0, if k n{ dteT

tnkj 0)(

Integrating from 0 to T is same as integrating over any interval of length T because we are only concerned with integral number of periods of cosine and sine function in the previous equation.

Now the R.H.S. of equation (2) reduces to Tan . Therefore :-

dtetxT

aT

tjnn

0

0)(1

Consequently, we can write

dtetxT

aT

tjnn 0)(

1

Hence the Fourier series of x(t) can be expressed as :-

4)(1

)(1

3)(

)/2(

)/2(

0

0

dtetxT

dtetxT

a

eaeatx

T

tTjk

T

tjkk

tTjkk

k

tjkk

k

Here equation (3) is the analysis equation and equation (4) is the synthesis equation.Coefficients ak are called Fourier Series Coefficients or Spectral Coefficients of x(t).

Here the DC or constant component of x(t) is :-

dttxT

aT

0

0 )(1

7

Example :- Periodic square wave defined over one period as

2/ t T 0

t 1

1

1

T

Ttx

1 tx

-T -T/2 –T1 0 T1 T/2 T t

Defining

101 sin

2Tkc

T

Tak

k

Tkak

10sin When k 0

At k = 0 T

Tdttx

Ta

T

T

12/

/2- 0

21

x

xxc

sinsin

Fourier Series of some Common Signals

Conditions for convergence of CT Fourier series

• Every function ƒ(x) of period 2п satisfying following conditions known as DIRICHLET’S CONDITIONS, can be expressed in the form of Fourier series.

1. Over any period, x(t) must be absolutely integrable :-

it guarantees that each coefficient ak will be finite

2. In a single period, x(t) should have finite number of max and min3. In any finite interval of time, there are only a finite number of discontinuities. Each

discontinuity should be finite.

dttxT

2)(

ka

Properties of CT Fourier Series

1. Linearityz(t) = Ax(t) + By(t) ck = Aak + Bbk

2. Time Shiftingz(t) = x(t-t0) ck =

• In time shifting magnitude of Fourier series coefficient remains the same|ck| = |ak|

3. Time Reversalz(t) = x(-t) ck = a-k

• If x(t) is even, ak = a-k

• If x(t) is odd, ak = -a-k

x(t) & y(t) are two periodic signals with period T andFourier coefficients ak & bk respectively

00tjkkea

4. Time Scalingz(t) = x(αt) ck = ak

• But, the fundamental period is (T/α)

5. Multiplicationz(t) = x(t)y(t) ck =

• (DT convolution between coefficients)

6. Conjugation & Conjugate Symmetry• Real x(t) a-k = a*k (conjugate symmetric)

• Real & Even x(t) ak = a*k (real & even ak)

• Real & Odd x(t) ak = -a*k (purely imaginary & odd ak ; a0 = 0)

• z(t) = Even part of x(t) ck = Real{ak}

• z(t) = Odd part of x(t) ck = jImaginary{ak}

llklba

7. Periodic Convolution

8. Parseval’s Relation

• Total average power = sum of average power in all harmonic components• Energy in time domain equals to energy in frequency domain

dtyxtytxT

)()()(*)( Takbk

kk

T

tjk

kk

T k

tjkk

T

adteaT

dteaT

dttxT

222

22

0

0

1

1)(

1

Determining the Fourier Series Representation of a

Discrete Time Signal

1][ )/2(0 Nk

nNjkk

Nk

njkk eaeanx

A periodic DT signal can be expressed as set of N linear equations for N unknown coeffficients ak as k ranges over a set of N successive integers :-

Nn

nNjke )/2( N, if k = 0,+N,+2N…

0, otherwise{According to the fact that :-

Nk

nNrkjk

nNjr eaenx )/2)(()/2(][

Multiplying both sides with , we get :-nNjre )/2(

Now if summing over N terms, we have :-

Nn Nk

nNrkjk

Nn

nNjr eaenx )/2)(()/2(][

Interchanging the order of summation, we have :-

Nk Nn

nNrkjk

Nn

nNjr eaenx )/2)(()/2(][

Nn

nNrkje )/2)(( N, if k = 0,+N,+2N…

0, otherwise{According to the fact stated before, we can conclude that :-

So the R.H.S. of the equation reduces to Nar

Nn

nNjrr enxN

a )/2(][1

Hence the Fourier series of x[n] can be expressed as below where equation (3) is the analysis equation and equation (4) is the synthesis equation

4][1

][1

3][

)/2(

)/2(

0

0

Nn

nNjk

Nn

njkk

Nk

nNjkk

Nk

njkk

enxN

enxN

a

eaeanx

Properties of DT Fourier Series

1. Linearityz[n] = Ax[n] + By[t] ck = Aak + Bbk

2. Time Shiftingz[t] = x[n-n0] ck =

• In time shifting magnitude of Fourier series coefficient remains the same|ck| = |ak|

3. Time Reversalz[t] = x[-n] ck = a-k

• If x[n] is even, ak = a-k

• If x[n] is odd, ak = -a-k

x[n] & y[n] are two periodic signals with period N andFourier coefficients ak & bk respectively periodic with period N

0)/2( nNjkkea

4. Multiplicationz[n] = x[n]y[n] ck =

5. Conjugation & Conjugate Symmetry• Real x[n] a-k = a*k (conjugate symmetric)

• Real & Even x[n] ak = a*k (real & even ak)

• Real & Odd x[n] ak = -a*k (purely imaginary & odd ak )

• z[n] = Even part of x[n] ck = Real{ak}

• z[n] = Odd part of x[n] ck = jImaginary{ak}

6. Periodic Convolution

Nl

lklba

Nr

rnyrxnynx ][][][*][ Nakbk

7. Parseval’s Relation

• Total average power = sum of average power in all harmonic components• Energy in time domain equals to energy in frequency domain

22][

1

Nk

kNn

anxN

FOURIER SERIES, which is an infinite series representation of such functions in terms of ‘sine’ and ‘cosine’ terms, is useful here. Thus,

FOURIER SERIES, are in certain sense, more UNIVERSAL than TAYLOR’s SERIES as it applies to all continuous, periodic functions and also to the

functions which are discontinuous in their values and derivatives. FOURIER SERIES a very powerful method to solve ordinary and partial

differential equation..

As we know that TAYLOR SERIES representation of functions are valid only for those functions which are continuous and differentiable. But

there are many discontinuous periodic function which requires to express in terms of an infinite series containing ‘sine’ and ‘cosine’

terms.

Advantages of using Fourier Series

Consider a mass-spring system as before, where we have a mass m on a spring with springconstant k, with damping c, and a force F(t) applied to the mass.Suppose the forcing function F(t) is 2L-periodic for some L > 0.

The equation that governs this particular setup is

The general solution consists of the complementary solution xc, which solves the associatedhomogeneous equation mx” + cx’ + kx = 0, and a particular solution of (1) we call xp.

mx”(t) + cx’(t) + kx(t) = F(t)

Applications of using Fourier Series1. Forced Oscillation

For c > 0,the complementary solution xc will decay as time goes by. Therefore, we are mostly interested in aparticular solution xp that does not decay and is periodic with the same period as F(t). We call thisparticular solution the steady periodic solution and we write it as xsp as before. What will be new inthis section is that we consider an arbitrary forcing function F(t) instead of a simple cosine.For simplicity, let us suppose that c = 0. The problem with c > 0 is very similar. The equation

mx” + kx = 0has the general solution,

x(t) = A cos(ωt) + B sin(ωt);Where,

Any solution to mx”(t) + kx(t) = F(t) is of the form A cos(ωt) + B sin(ωt) + xsp.

The steady periodic solution xsp has the same period as F(t).

In the spirit of the last section and the idea of undetermined coefficients we first write,

Then we write a proposed steady periodic solution x as,

where an and bn are unknowns. We plug x into the deferential equation and solve for an and bn in terms of cn and dn.

• It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if a voice is recorded for one second saying something, I can find its Fourier series which may look something like this for example

• and this interactive module shows you how when you add sines and/or cosines the graph of cosines and sines becomes closer and closer to the original graph we are trying to approximate.

• The really cool thing about Fourier series is that first, almost any kind of a wave can be approximated. Second, when Fourier series converge, they converge very fast.

• So one of many applications is compression. Everyone's favorite MP3 format uses this for audio compression. You take a sound, expand its Fourier series. It'll most likely be an infinite series BUT it converges so fast that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be ignored because they add so little that a human ear can likely tell no difference. So I just save the first few terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less memory.

• JPEG for pictures is the same idea.

2. Speech/Music Recognition

3. Approximation Theory :- We use Fourier series to write a function as a trigonometric polynomial.

4. Control Theory :- The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution.

5. Partial Differential equation :- We use it to solve higher order partial differential equations by the method of separation of variables.