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

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

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

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: if and only if for some constant C

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

Back to the Example: with no return NO Correlation!

Back to the Example: with return Good Correlation!

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

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

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

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

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

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

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

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

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

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

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.

Example: a square pulse See a few values:

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

Example: Sinusoid

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

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

Compare them cos chirppseudonoise Two best!

Detection with Noise Now see with added noise

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:

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.

White Noise and Filters The output of a Filter

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

Back to the Match Filter At the peak:

Match Filter and SNR At the peak:

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

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

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