1. Final Project Part II MATLAB Session ES 156 Signals and Systems 2007 SEAS Prepared by Frank Tompkins

2. Outline freqz() command Step by step through the communication system –Explanation of new concepts and new MATLAB functions –High-level view of flow through the system –Next week we talk in more detail about implementation Eye diagrams

3. Start Early! Project is complex Not something you can do in one day Less hand-holding than MATLAB exercises in homework –More like real-life projects Extra office hours possible –Email us if you have questions/problems

4. freqz() Same inputs as filter() Plots frequency response Outputs frequency response channelFilterTaps = [1 0 -1/2 3/8 zeros(1,28)]; freqz(channelFilterTaps, 1); channelFilterTaps = [1 0 -1/2 3/8 zeros(1,28)]; H = freqz(channelFilterTaps, 1, 256); % evaluate H(e j  ) at 256 points

5. Overall Idea Want to transmit a digital image from point A to point B using radio waves, etc. We won’t actually build antennas/wires –Simulate the whole thing in MATLAB Have to convert digital image to a wave that can travel through the air, a wire, etc. at point A Then convert wave back to a digital image at point B We’ll transmit DCTs instead of actual image pixels, as is often done in real life applications

6. Image Pre-Processing –Break image into 8 pixel by 8 pixel blocks and take DCT of each block –Quantize DCT coefficients into 256 levels by representing them as 8-bit unsigned numbers

7. blkproc() MATLAB command for applying a function in blocks to a matrix Example: apply DCT in 8 by 8 blocks I = imread(‘myimage.tif'); fun = @dct2; J = blkproc(I, [8 8], fun);

8. Quantization Approximate a continuous range of values by a set of discrete values

9. Quantization In MATLAB uint8() For images, we have to scale to [0,1] before quantizing with im2uint8() x = 5.7; % x is double precision (32-bit floating point) xq = uint8(x); % xq is an 8-bit unsigned integer xscaled = (1.4 - x) / (1.4 - 6.3); % x is an image matrix xq = im2uint8(xscaled);

10. Conversion to a bit stream –Using reshape() and permute() Arrange 8 x 8 DCT blocks into groups of N blocks each Reshape each block into a vector (length 8*8*N) to be transmitted later Convert each pixel in vector to a binary number

11. Conversion

12. reshape() >> x = [1 2 3; 4 5 6; 7 8 9]' x = 1 4 7 2 5 8 3 6 9 >> reshape(x,1,9) ans = 1 2 3 4 5 6 7 8 9 Takes elements columnwise

13. permute() >> x = rand(1,2,2) x(:,:,1) = 0.4565 0.0185 x(:,:,2) = 0.8214 0.4447 >> permute(x,[2 1 3]) ans(:,:,1) = 0.4565 0.0185 ans(:,:,2) = 0.8214 0.4447

14. de2bi() >> x = [4; 212; 19] x = 4 212 19 >> de2bi(x) ans = 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 0 0 1 0 0 0

15. Modulation –Modulate each bit by a sine wave and put into the channel –Implementation details are up to you What is written in the PDF file is a suggestion –We will however take off points if you use more than one for loop in your code That one should loop over the N-sized block groups to send each one through the channel in turn

16. Pulse Amplitude Modulation PAM for short We will use a specific simple type: half-sine pulse To send a bit –For a 1 send sin(t) –For a 0 send –sin(t)

17. Modulated [1 0 0 1]

18. Channel –Atmosphere, telephone wire, coaxial cable –We model it as an LTI system Impulse response h(t) –In using MATLAB, must approximate by discrete time system h[n]

19. Noise –We will use zero mean AWGN Additive White Gaussian Noise –Add an independent Gaussian random variable to each sample passed through the channel

20. AWGN noisePower = 2; % variance of Gaussian random variable result = signalFromChannel + sqrt(noisePower) * randn(rows, cols); >> randn(2,3) ans = -0.4326 0.1253 -1.1465 -1.6656 0.2877 1.1909 y[n] = h[n] * x[n] + noise[n]

21. [1 0 0 1] After Channel and Noise

22. Receiver Equalization –Attempt to undo distortion of modulated signal (sine wave) due to channel and noise –We will try two equalizing filters Zero Forcing (ZF) Minimum Mean Square Error (MMSE)

23. Zero Forcing (ZF) Filter Just the inverse of the channel response If H(e j  ) is the response of the channel, then the ZF filter has response 1 / H(e j  ) Clearly if there were no noise, ZF filter would perfectly recover modulated signal But since ZF doesn’t take noise into account at all, it will perform very badly if noise is strong

24. MMSE Filter Takes noise into account Derived by minimizing the average error –We’ll just take it on faith If H(e j  ) is the response of the channel, then the MMSE filter has response

25. Detection –Examine equalized signal to determine whether a 1 or a 0 was sent over channel –Optimal detector is a thresholder

26. Threshold Detector Integrate (sum) the transmitted and equalized half-sine pulse If integral (sum) < 0, decide a 0 was sent Else decide a 1 was sent

27. Conversion to an Image –After receiving all block groups, use reshape() and permute() to rebuild the block DCT image

28. Conversion to an Image

29. Image Post-Processing –Use blkproc() to compute inverse block DCT –That’s it!

30. Things To Play With Noise Power –Increasing noise power will cause more distortion in received image ZF/MMSE Equalizers –Since MMSE handles noise, it should perform better than ZF Eye Diagrams –Coming up next Different Channels –Optional

31. Eye Diagrams Used to visualize how waveforms used to send (modulate) multiple bits of data can lead to detection errors The more “open” the eye, the lower the probability of error Consider modulated half-sine pulses for four subsequent transmitted bits

32. Modulated [1 0 0 1]

33. [1 0 0 1] After Channel and Noise

34. [1 0 0 1] Eye Diagrams Before Channel After Channel