PowerPoint Presentation

FOURIER TRANSFORMATIONPresented by:Ashish Singh Bhandari(2012AMD2601)Saurabh Sahu(2012AMD2611)Rajanish Prasad(2013AME2035)Kartikey Mishra(2013AME8306)

Fourier TransformationAFourier transformconverts time (or space) to frequency and vice versa FT rapidly computes such transformations at very fast rateFourier Transformed are easy to integrate and Differentiate f(x) F(x) f(x) fourier Inverse fourier

This is continuous form of FT

But in actual practice we have functions of time which are discontinuous and not continuous as the value of t(Time) 0 to Infinity

Discrete Fourier TransformationDFT of x[n], for n=0,1.,N-1

for k=0,1.,N-1

Where

Why FTT(Fast Fourier Transformation)?DFT is an expensive method time wize and computationallyAssume x[n] has N data and each data is a complex number

For k=0,1,N-1For each k, it needs N complex multiplications and N-1 complex additionsIn overall , it needs N2 complex multiplications and N(N-1) complex additions

Basic Principle of FTTDIVIDE & CONQUER and Conquer To break down a big problem to a number of smaller problems and tackle them individually

Example for N=8

(A) The order of input data need to be rearranged (according to binary bit-reversed pattern. (B) Values for all k can be evaluated in place. No additional memory is needed.Bit Reversal

Example of FFTx0x1x2x3x4x5x6x7x0x4x2x6x1x5x3x7Swap data according to bit reversalSpacing =1x0x4x2+x6x2x6x1+x5x1x5x3+x7x3x7x0+x44= eik2= eik/2Spacing =2x0+x4+x2+x6x0-x4+i(x2-x6)x0+x4-(x2+x6)x0-x4-i(x2-x6)x1+x5+x3+x7x1-x5+i(x3-x7)x1+x5-(x3+x7)x1-x5-i(x3-x7)= eik/4Spacing =4x0+x4+x2+x6+x1+x5+x3+x7x0-x4+i(x2-x6)+ei/4 (x1-x5+i(x3-x7))x0+x4-x2-x6+i(x1+x5-x3-x7)x0-x4-i(x2-x6)+ ei3/4(x1-x5-i(x3-x7))x0+x4+x2+x6-(x1+x5+x3+x7)x0-x4+i(x2-x6)-ei/4 (x1-x5+i(x3-x7))x0+x4-x2-x6-i(x1+x5-x3-x7)x0-x4-i(x2-x6)- ei3/4(x1-x5-i(x3-x7))FT of x in placeF2F4F8Response function

Response functionModal superposition

FFT of force function