1. 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
Transmit
Receive

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

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

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

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