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  • 1. Fourier Transforms of Discrete Signals Microlink IT College ambawb
  • 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. 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. 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. 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. Example D See map! ambawb
  • 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. Finding DTFT For periodic signals • Starting with xo[n] • DTFT of xo[n] Example ambawb
  • 9. Example A & B notes notes X[n]=a|n| , 0 < a < 1. ambawb
  • 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. Properties of DTFT • Remember: • For time scaling note that m>1  Signal spreading ambawb
  • 12. Fourier Transform of Periodic Sequences • Check the map~~~~~ See map! ambawb
  • 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. 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. A short hand notation remember: ambawb
  • 16. Inverse of DFT • We can obtain the inverse of DFT • Note that ambawb
  • 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. Example of DFT • Find X[k] – We know k=1,.., 7; N=8 ambawb
  • 19. Example of DFT ambawb
  • 20. Example of DFT Time shift Property of DFT Polar plot for ambawb
  • 21. Example of DFT ambawb
  • 22. Example of DFT Summation for X[k] Using the shift property! ambawb
  • 23. Example of IDFT Remember: ambawb
  • 24. Example of IDFT Remember: ambawb
  • 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. 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. 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. 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. FFT Algorithms - Four point FFT First find even and odd parts and then combine them: The general form: ambawb
  • 30. FFT Algorithms - 8 point FFT http://www.engineeringproductivitytools.com/stuff/T0001/PT07.HTM Applet: http://www.falstad.com/fourier/directions.html ambawb
  • 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. 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. References • Read Schaum’s Outlines: Chapter 6 • Do Chapter 12 problems: 19, 20, 26, 5, 7 ambawb