Fft

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Fft

  1. 1. Fourier Transforms of Discrete Signals Microlink IT College ambawb
  2. 2. Sampling • Continuous signals are digitized using digital computers • When we sample, we calculate the value of the continuous signal at discrete points – How fast do we sample – What is the value of each point • Quantization determines the value of each samples value ambawb
  3. 3. Sampling Periodic Functions - Note that wb = Bandwidth, thus if then aliasing occurs (signal overlaps) -To avoid aliasing -According sampling theory: To hear music up to 20KHz a CD should sample at the rate of 44.1 KHz ambawb
  4. 4. Discrete Time Fourier Transform • In likely we only have access to finite amount of data sequences (after sampling) • Recall for continuous time Fourier transform, when the signal is sampled: • Assuming • Discrete-Time Fourier Transform (DTFT): ambawb
  5. 5. Discrete Time Fourier Transform • Discrete-Time Fourier Transform (DTFT): • A few points – DTFT is periodic in frequency with period of 2π – X[n] is a discrete signal – DTFT allows us to find the spectrum of the discrete signal as viewed from a window ambawb
  6. 6. Example D See map! ambawb
  7. 7. Example of Convolution • Convolution – We can write x[n] (a periodic function) as an infinite sum of the function xo[n] (a non-periodic function) shifted N units at a time – This will result – Thus See map! ambawb
  8. 8. Finding DTFT For periodic signals • Starting with xo[n] • DTFT of xo[n] Example ambawb
  9. 9. Example A & B notes notes X[n]=a|n| , 0 < a < 1. ambawb
  10. 10. DT Fourier Transforms 1. Ω is in radian and it is between 0 and 2π in each discrete time interval 2. This is different from ω where it was between – INF and + INF 3. Note that X(Ω) is periodic ambawb
  11. 11. Properties of DTFT • Remember: • For time scaling note that m>1  Signal spreading ambawb
  12. 12. Fourier Transform of Periodic Sequences • Check the map~~~~~ See map! ambawb
  13. 13. Discrete Fourier Transform • We often do not have an infinite amount of data which is required by DTFT – For example in a computer we cannot calculate uncountable infinite (continuum) of frequencies as required by DTFT • Thus, we use DTF to look at finite segment of data – We only observe the data through a window – In this case the xo[n] is just a sampled data between n=0, n=N-1 (N points) ambawb
  14. 14. Discrete Fourier Transform • It turns out that DFT can be defined as • Note that in this case the points are spaced 2pi/N; thus the resolution of the samples of the frequency spectrum is 2pi/N. • We can think of DFT as one period of discrete Fourier series ambawb
  15. 15. A short hand notation remember: ambawb
  16. 16. Inverse of DFT • We can obtain the inverse of DFT • Note that ambawb
  17. 17. Using MATLAB to Calculate DFT • Example: – Assume N=4 – x[n]=[1,2,3,4] – n=0,…,3 – Find X[k]; k=0,…,3 or ambawb
  18. 18. Example of DFT • Find X[k] – We know k=1,.., 7; N=8 ambawb
  19. 19. Example of DFT ambawb
  20. 20. Example of DFT Time shift Property of DFT Polar plot for ambawb
  21. 21. Example of DFT ambawb
  22. 22. Example of DFT Summation for X[k] Using the shift property! ambawb
  23. 23. Example of IDFT Remember: ambawb
  24. 24. Example of IDFT Remember: ambawb
  25. 25. Fast Fourier Transform Algorithms • Consider DTFT • Basic idea is to split the sum into 2 subsequences of length N/2 and continue all the way down until you have N/2 subsequences of length 2 Log2(8) N ambawb
  26. 26. Radix-2 FFT Algorithms - Two point FFT • We assume N=2^m – This is called Radix-2 FFT Algorithms • Let’s take a simple example where only two points are given n=0, n=1; N=2 y0 y1 y0 Butterfly FFT Advantage: Less computationally intensive: N/2.log(N) Advantage: Less computationally intensive: N/2.log(N) ambawb
  27. 27. General FFT Algorithm • First break x[n] into even and odd • Let n=2m for even and n=2m+1 for odd • Even and odd parts are both DFT of a N/2 point sequence • Break up the size N/2 subsequent in half by letting 2mm • The first subsequence here is the term x[0], x[4], … • The second subsequent is x[2], x[6], … 1 1)2sin()2cos( )]12[(]2[ 2/ 2 2/ 2/ 2/2/ 2/ 2/ 2/ 2 12/ 0 2/ 12/ 0 2/ −= =−−−== == = ++ − + − = − = ∑∑ N N jN N m N N N m N Nm N mk N mk N N m mk N k N N m mk N W jeW WWWW WW mxWWmxW πππ ambawb
  28. 28. Example 1 1)2sin()2cos( )]12[(]2[][ 2/ 2 2/ 2/ 2/2/ 2/ 2/ 2/ 2 12/ 0 2/ 12/ 0 2/ −= =−−−== == = ++= − + − = − = ∑∑ N N jN N m N N N m N Nm N mk N mk N N m mk N k N N m mk N W jeW WWWW WW mxWWmxWkX πππ Let’s take a simple example where only two points are given n=0, n=1; N=2 ]1[]0[]1[]0[)]1[(]0[]1[ ]1[]0[)]1[(]0[]0[ 1 1 0 0 1.0 1 1 1 0 0 1.0 1 0 0 0.0 1 0 1 0 0 0.0 1 xxxWxxWWxWkX xxxWWxWkX mm mm −=+=+== +=+== ∑∑ ∑∑ == == Same result ambawb
  29. 29. FFT Algorithms - Four point FFT First find even and odd parts and then combine them: The general form: ambawb
  30. 30. FFT Algorithms - 8 point FFT http://www.engineeringproductivitytools.com/stuff/T0001/PT07.HTM Applet: http://www.falstad.com/fourier/directions.html ambawb
  31. 31. A Simple Application for FFT t = 0:0.001:0.6; % 600 points x = sin(2*pi*50*t)+sin(2*pi*120*t); y = x + 2*randn(size(t)); plot(1000*t(1:50),y(1:50)) title('Signal Corrupted with Zero-Mean Random Noise') xlabel('time (milliseconds)') Taking the 512-point fast Fourier transform (FFT): Y = fft(y,512) The power spectrum, a measurement of the power at various frequencies, is Pyy = Y.* conj(Y) / 512; Graph the first 257 points (the other 255 points are redundant) on a meaningful frequency axis: f = 1000*(0:256)/512; plot(f,Pyy(1:257)) title('Frequency content of y') xlabel('frequency (Hz)') This helps us to design an effective filter! ML Help! ambawb
  32. 32. Example • Express the DFT of the 9-point {x[0], …,x[9]} in terms of the DFTs of 3-point sequences {x[0],x[3],x[6]}, {x[1],x[4],x[7]}, and {x[2],x[5],x[8]}. ambawb
  33. 33. References • Read Schaum’s Outlines: Chapter 6 • Do Chapter 12 problems: 19, 20, 26, 5, 7 ambawb

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