Chapter 12

1. Chapter 12 Fourier Transforms of Discrete Signals

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

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

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):

5. Discrete Time Fourier Transform • Discrete-Time Fourier Transform (DTFT): • A few points • DTFT is periodic in frequency with period of 2p • X[n] is a discrete signal • DTFT allows us to find the spectrum of the discrete signal as viewed from a window

6. Example D See map!

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!

8. Finding DTFT For periodic signals • Starting with xo[n] • DTFT of xo[n] Example

9. Example A & B notes X[n]=a|n|, 0 < a < 1. notes

10. DT Fourier Transforms • W is in radian and it is between 0 and 2p in each discrete time interval • This is different from w where it was between – INF and + INF • Note that X(W) is periodic

11. Properties of DTFT • Remember: • For time scaling note that m>1  Signal spreading

12. Fourier Transform of Periodic Sequences • Check the map~~~~~ See map!

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)

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

15. A short hand notation remember:

16. Inverse of DFT • We can obtain the inverse of DFT • Note that

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

18. Example of DFT • Find X[k] • We know k=1,.., 7; N=8

19. Example of DFT

20. Example of DFT Polar plot for Time shift Property of DFT Other DFT properties: http://cnx.org/content/m12019/latest/

21. Example of DFT

22. Example of DFT Summation for X[k] Using the shift property!

23. Example of IDFT Remember:

24. Example of IDFT Remember:

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

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 Butterfly FFT y0 y0 y1 Advantage: Less computationally intensive: N/2.log(N) http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fft.html

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], …

28. Example Let’s take a simple example where only two points are given n=0, n=1; N=2 Same result

29. FFT Algorithms - Four point FFT First find even and odd parts and then combine them: The general form:

30. FFT Algorithms - 8 point FFT Applet: http://www.falstad.com/fourier/directions.html http://www.engineeringproductivitytools.com/stuff/T0001/PT07.HTM

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)') ML Help! This helps us to design an effective filter!

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]}. Later

33. References • Read Schaum’s Outlines: Chapter 6 • Do Chapter 12 problems: 19, 20, 26, 5, 7