Correlated and Uncorrelated Signals

1. Transmit Receive Correlated and Uncorrelated Signals Problem: we have two signals and . How “close” are they to each other? Example:in a radar (or sonar) we transmit a pulse and we expect a return

2. Receive Similar? NO! Think so! Example: Radar Return Since we know what we are looking for, we keep comparing what we receive with what we sent.

3. Inner Product between two Signals • We need a “measure” of how close two signals are to each other. • This leads to the concepts of • Inner Product • Correlation Coefficient

4. Inner Product Problem: we have two signals and . How “close” are they to each other? Define: Inner Product between two signals of the same length Properties: for some constant C if and only if

5. How we measure similarity (correlation coefficient) Assume: zero mean Compute: Check the value: x,y uncorrelated x,y strongly correlated

6. Back to the Example: with no return NO Correlation!

7. Back to the Example: with return Good Correlation!

8. Inner Product in Matlab Take two signals of the same length. Each one is a vector: Row vector Row vector Define: Inner Product between two vectors conjugate, transpose

9. Example Take two signals: Then: Compute these: x,y are not correlated

10. Example Take two signals: Compute these: Then: x,y are strongly correlated

11. Example Take two signals: Then: Compute these: x,y are strongly correlated

12. Typical Application: Radar Send a Pulse… … and receive it back with noise, distortion … Problem: estimate the time delay , ie detect when we receive it.

13. Use Inner Product “Slide” the pulse s[n] over the received signal and see when the inner product is maximum:

14. Use Inner Product “Slide” the pulse x[n] over the received signal and see when the inner product is maximum: if

15. Matched Filter Take the expression Compare this, with the output of the following FIR Filter Then

16. Matched Filter This Filter is called a Matched Filter The output is maximum when i.e.

17. Example We transmit the pulse shown below, with length Max at n=119 Received signal:

18. How do we choose a “good pulse” We transmit the pulse and we receive (ignore the noise for the time being) where The term is called the “autocorrelation of s[n]”. This characterizes the pulse.

19. Example: a square pulse See a few values:

20. Compute it in Matlab N=20; % data length s=ones(1,N); % square pulse rss=xcorr(s); % autocorr n=-N+1:N-1; % indices for plot stem(n,rss) % plot

21. Example: Sinusoid

22. Example: Chirp s=chirp(0:49,0,49,0.1)

23. Example: Pseudo Noise s=randn(1,50)

24. Compare them chirp pseudonoise cos Two best!

25. Detection with Noise Now see with added noise

26. White Noise A first approximation of a disturbance is by “White Noise”. White noise is such that any two different samples are uncorrelated with each other:

27. White Noise The autocorrelation of a white noise signal tends to be a “delta” function, ie it is always zero, apart from when n=0.

28. White Noise and Filters The output of a Filter

29. White Noise The output of a Filter In other words the Power of the Noise at the ouput is related to the Power of the Noise at the input as

30. Back to the Match Filter At the peak:

31. Match Filter and SNR At the peak:

32. Transmitted Detected with Matched Filter Example Transmit a Chirp of length N=50 samples, with SNR=0dB

33. Detected with Matched Filter Transmitted Example Transmit a Chirp of length N=100 samples, with SNR=0dB

34. Transmitted Detected with Matched Filter Example Transmit a Chirp of length N=300 samples, with SNR=0dB