TERM PAPER

TOPIC: Application of fourier transform

SUBMITTED BY: submitted to:

Rohit lakra mr.Rohit Gandhi

C6803 A 12 {lec:mth 201}

10806972

B.tech[ece-mba]intg.

ACKNOWLEDGEMENT

I would like to express my gratitude to all faculty of Lovely who in spite of their busy schedule took personal interest to ensure that this term paper is a through learning process for me.

I would like to give thanks to Mr.ROHIT GANDHI under whose guidance this term paper was being carried out, for providing the requisite platform and support throughout the term paper.

I am grateful to all our friends for providing critical feedback & support whenever required. Finally I would like to be grateful to all those who directly or indirectly have been of great help and obliged me with their support and have helped me in converting my collection of data and information into a finely polished term paper.

TABLE OF CONTENTS:-

1) INTRODUCTION

2) DEFINITION

3) EXPONENTIAL FOURIER SERIES

4) FOURIER TRANSFORM

integral transform

fourier sine and cosine integral

complex form of integral5) APPLICATIONS

a) Solving differential equation

b) To boundary condition

c) Disctrete fourier transform

d) fourier transform parsvals theorem

6) PROPERTIES OF FOURIER TRANSFORM

LINEARITY

TIME SHIFT

MULTIPLICATION THEOREM

PARSEVALS EQUATION

MODULATION

7) CONCLUSION

8) REFRENCES

INTRODUCTION

In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely Sine and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.

The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, 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. These simple solutions are now sometimes called Eigen solutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding Eigen solutions. 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.

The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics etc.

Definition

In this section, ƒ(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2π–periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−π, π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

Fourier's formula for 2π-periodic functions using sine and cosines

For a periodic function ƒ(x) that is integral on [−π, π], the numbers

and

are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by

The partial sums for ƒ are trigonometric polynomials. One expects that the functions SN ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum

is called the Fourier series of ƒ.

The Fourier series does not always converge, and even when it does converge for a specific value x0 of x, the sum of the series at x0 may differ from the value ƒ(x0) of the function. It is one of the main questions in harmonic analysis to decide when Fourier series converge, and when the sum is equal to the original function. If a function is square-integralable on the interval [−π, π], then the Fourier series converges to the function at almost every point. In engineering applications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counter-examples to this presumption. In particular, the Fourier series converges absolutely and uniformly to ƒ(x) whenever the derivative of ƒ(x) (which may not exist everywhere) is square integrable.

Exponential Fourier series

We can use Euler's formula,

Where i is the imaginary unit, to give a more concise formula:

The Fourier coefficients are then given by:

The Fourier coefficients an, bn, cn are related via

and

The notation cn is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of ƒ (in this case), such as F or   and functional notation often replaces subscripting.  Thus:

In engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Fourier Transform

Integral transform of function f(x) is equal to

I{f(x)} = k(s,x) dx

K(s,x) = kernel depending on s & x.

K(s,k) =

=

Fourier integral:

cos (t-x)dtd

Fourier sine and cosine integral:

F(x) =

F(x) =

Complex form of Fourier integral

F(x) = dtd

APPLICATION OF FOURIER TRANSFORM SERIES :

a) TO DIFFERENTIAL EQUATIONS

Since the beginning Fourier himself was interested to find a powerful tool to be used in solving differential equations. Therefore, it is of no surprise that we discuss in this page, the application of Fourier series differential equations. We will only discuss the equations of the form

y(n) + an-1y(n-1) + ........+ a1y' + a0 y = f(x),

where f(x) is a -periodic function.

Note that we will need the complex form of Fourier series of a periodic function. Let us define this object first:

Definition. Let f(x) be -periodic. The complex Fourier series of f(x) is

where

We will use the notation

If you wonder about the existence of a relationship between the real Fourier coefficients and the complex ones, the next theorem answers that worry.

Theoreme. Let f(x) be -periodic. Consider the real Fourier coefficients and

of f(x), as well as the complex Fourier coefficients . We have

The proof is based on Euler's formula for the complex exponential function.

Remark. When f(x) is 2L-periodic, then the complex Fourier series will be defined as before where

for any .

Example. Let f(x) = x, for and f(x+2) = f(x). Find its complex Fourier

coefficients . Answer. We have d0 = 0 and

Easy calculations give

Since , we get . Consequently

Back to our original problem. In order to apply the Fourier technique to differential equations, we will need to have a result linking the complex coefficients of a function with its derivative. We have:

Theorem. Let f(x) be 2L-periodic. Assume that f(x) is differentiable. If

then

Example. Find the periodic solutions of the differential equation

y' + 2y = f(x),

where f(x) is a -periodic function. Answer. Set

Let y be any -periodic solution of the differential equation. Assume

Then, from the differential equation, we get

Hence

Therefore, we have

Example. Find the periodic solutions of the differential equation

Answer. Because

we get with

Let y be a periodic solution of the differential equation. If

then . Hence

Therefore, the differential equation has only one periodic solution

The most important result may be stated as:

Theoreme. Consider the differential equation

where f(x) is a 2c-periodic function. Assume

for . Then the differential equations has one 2c-periodic solution given by

where

b) TO BOUNDARY CONDITION

In one dimensional boundary value problems,the partial differential equation can easily be transformed into a ordinary differential equation by applying a suitable transform. The required solution is then obtained by solving this equation and inverting by means of complex inversion formula. In two dimensional ,it is required to apply transform twice and the desired solution is obtained by double inversion.

1) If in a problem u(x,t)x=0 is giventhen we use infinite Sine transform to remove

from the differential equation.

In case [ ]x=0 is given then we employ infinite cosine transform to

remove .

2) If in a problem u(o,t) and u(l,t) are given,then we use finite Sine transform to remove from the differential eqution.

In case ( )x=0 and ( ) are given,thenwe employ finite cosine transform

to remove .

The method of solution is best explained through the examples.

EXAMPLE: Determine the distribution of temperature in the semi infinite medium x when the end x=0 is maintained at zero temperature and the initial

distribution of temperature is f(x).

SOLUTION: let u(x,t) be the temperature at any point x and at any time t. We have to solve the heat –flow equation

= (x>0,t>0)..............(1)

Subject to the initial condition u(x,0)=f(x)...............(2)

And the boundary condition u(0,t)=0.................(3)

Taking fourier Sine transform of (1) and denoting Fs[u(x,t)]by s have

dus/dt= [su (0,t)- s]

d /dt + =0.................(4)

or also the fourier Sine transform of (2) is = (s) at t = 0. ..........(5)

Solving (4) and (5) ,we get = (s)

Hence taking its inverse Fourier Sine transform, we obtain

U(x,t) = Sine xs ds

c) DISCRETE FOURIER TRANSFORM

Working with the Fourier transform on a computer usually involves a form of the transform known as the discrete Fourier transform (DFT). A discrete transform is a transform whose input and output values are discrete samples, making it convenient for computer manipulation. There are two principal reasons for using this form of the transform:

The input and output of the DFT are both discrete, which makes it convenient for computer manipulations.

There is a fast algorithm for computing the DFT known as the fast Fourier transform (FFT).

The DFT is usually defined for a discrete function f(m,n) that is nonzero only over the finite region   and  . The two-dimensional M-by-N DFT and inverse M-by-N DFT relationships are given by

and

The values F(p,q) are the DFT coefficients of f(m,n). The zero-frequency coefficient, F(0,0), is often called the "DC component." DC is an electrical engineering term that stands for direct current.

Discrete Fourier Transform Computed Without Padding

This plot differs from the Fourier transform displayed Visualising the fourier transform.

First, the sampling of the Fourier transform is much coarser. Second, the zero-frequency coefficient is displayed in the upper left corner instead of the traditional location in the center.

To obtain a finer sampling of the Fourier transform, add zero padding to f when computing its DFT. The zero padding and DFT computation can be performed in a single step with this command.

F = fft2(f,256,256);This command zero-pads f to be 256-by-256 before computing the DFT.

imshow(log(abs(F)),[-1 5]); colormap (jet); colorbar

Discrete Fourier Transform Computed with Padding

d) FOURIER TRANSFORM PARSEVAL'S THEOREM

If the signal is assumed to be a voltage and it is applied across a 1 ohm resistor, then the ENERGY dissipated in the resistor is given by the left hand side of the equation below.

Parseval's theorem states that this is also equal to integral of the square of the spectrum on the right.

Parseval's theorem is similar to the expression below for periodic signals. Since the periodic signal goes from -inf to +inf the the energy dissipated is infinite. To get a finite result, an average must be taken resulting in a POWER relationship.

PROPERTIES OF FOURIER TRANSFORM

The properties of the Fourier transform are :

1) Linearity 

2) Time shift 

Proof:  Let  , i.e.,  , we have

 

 

3) Multiplication theorem

Proof: 

     

   

4) Parseval's equation

In the special case when  , the above becomes the Parseval's equation : 

where 

is the energy density function representing how the signal's energy is distributed along the frequency axes. The total energy contained in the signal is obtained by

integrating   over the entire frequency axes.

The Parseval's equation indicates that the energy or information contained in the signal is reserved, i.e., the signal is represented equivalently in either the time or frequency domain with no energy gained or lost.

5) Modulation

CONCLUSION:-

At last l concluded that the finite fourier transform is useful for problem solving boundary

conditions of heat distribution on two parallel boundries,while the cosine fourier transform is

useful for problems in which velocities narmal to two parallel boundries are among the

boundries conditions. The properties of fourier transform are important in radio and

television where the harmonic carrier wave is modulated by an envelope. The Fourier

transform is also used in nuclear magnetic resonance (NMR) and in other kinds

of spectroscopy, e.g. infrared (FT-IR). In NMR an exponentially-shaped free induction decay

(FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-

shape in the frequency domain. The Fourier transform is also used in magnetic resonance

imaging (MRI) and mass spectrometry.

REFRENCES:-

1) Higher Engineering of mathematics by Dr.B.S.Grewal (text book)

2) Engineering of mathematics by N.P.BALI

3) http://topex.ucsd.edu/geodynamics/01fourier.pdf