SUB: ADVANCEDE EN GINEERING MATHEMATICS ( )

Fourier Transform PEN NO.

Introduction Jean Baptiste Joseph Fourier (Mar21st 1768 –May16th 1830) French mathematician, physicist Main Work: Théorie analytique de la chaleur (The Analytic Theory of Heat) Any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable (Incorrect) The concept of dimensional homogeneity in equations Proposal of his partial differential equation for conductive diffusion of heat Discovery of the "greenhouse effect"

Fourier’s “Controversy” Work  Fourier did his important mathematical work on the theory of heat (highly regarded memoir On the Propagation of Heat in Solid Bodies ) from 1804 to 1807  This memoir received objection from Fourier’s mentors (Laplace and Lagrange) and not able to be published until 1815 Napoleon awarded him a pension of 6000 francs, payable from 1 July, However Napoleon was defeated on 1 July and Fourier did not receive any money

Expansion of a Function Example of Taylor Series: constant first-order term second-order term ……………………

Fourier Series:

Examples:

Fourier Transform:

Fourier transform:

Derive the result: Solution: According to the definition Then (a>0)

Properties and applications:

Solve the wave equation and Take the Fourier Transform of both equations. The initial condition gives And the PDE gives Which is basically an ODE in t, we can write it as

Which has the solution, and derivative So the first initial condition gives and the second gives and make the solution

Let’s first look at Then The second piece And now the first factor looks like an integral, as a derivative with respect By fundamental theorem of calculus to x would cancel the iw in bottom. Define:

So Putting both piece together we get the solution

Solve heat transfer equation B.C: (1) u(0,t)=0 (2)u(x,0)=P(x) or P(x)=1, Solution with Fourier Sine Transform: According to the B.C, we can get

Then Inverse Gives the complete solution

Passage from Fourier integral to Laplace transform

Applications of Fourier Transform Physics Solve linear PDEs (heat conduction, Laplace, wave propagation) Antenna design Seismic arrays, side scan sonar, GPS, SAR Signal processing 1D: speech analysis, enhancement … 2D: image restoration, enhancement …

Just like Calculus invented by Newton, Fourier analysis is another mathematical tool BIOM: fake iris detection CS: anti-aliasing in computer graphics CpE: hardware and software systems

FT in Biometrics: naturalfake

Can be represented by: When you let these three waves interfere with each other you get your original wave function! Let’s start with an example…in 1-D Notice that it is symmetric around the central point and that the amount of points radiating outward correspond to the distinct frequencies used in creating the image. Since this object can be made up of 3 fundamental frequencies an ideal Fourier Transform would look something like this: Increasing Frequency

This image exclusively has 32 cycles in the vertical direction. This image exclusively has 8 cycles in the horizontal direction. You will notice that the second example is a little more smeared out. This is because the lines are more blurred so more sine waves are required to build it. The transform is weighted so brighter spots indicate sine waves more frequently used. Let’s Try it with Two-Dimensions!

So what is going on here? The u axis runs from left to right and it represents the horizontal component of the frequency. The v axis runs up and down and it corresponds to vertical components of the frequency. The central dot is an average of all the sine waves so it is usually the brightest dot and used as a point of reference for the rest of the points. x-y coordinate system u-v coordinate system Fourier Transform

FT in CS Anti-aliasing in 3D graphic display

Brightness Image Fourier Transform Inverse Transformed

FT in CSE

Frequency-Domain Analysis of Interpolation Step-I: Up sampling Step-II: Low-pass filtering Different interpolation schemes correspond to different low-pass filters

Frequency Domain Representation of Up sampling

Frequency Domain Representation of Interpolation

Source Dr. K. r. kachot ( Mahajan publication) Google links