Correlated and Uncorrelated Signals

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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
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Example Radar Return
Since we know what we are looking for, we keep
comparing what we receive with what we sent.

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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

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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
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How we measure similarity (correlation
coefficient)
Assume zero mean
Compute
Check the value
x,y strongly correlated
x,y uncorrelated
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Back to the Example with no return
NO Correlation!
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Back to the Example with return
Good Correlation!
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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
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Example
Take two signals
Then
Compute these
x,y are not correlated
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Example
Take two signals
Compute these
Then
x,y are strongly correlated
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Example
Take two signals
Then
Compute these
x,y are strongly correlated
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Typical Application Radar
Send a Pulse
and receive it back with noise, distortion
Problem estimate the time delay , ie detect
when we receive it.
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Use Inner Product
Slide the pulse sn over the received signal
and see when the inner product is maximum
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Use Inner Product
Slide the pulse xn over the received signal
and see when the inner product is maximum
if
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Matched Filter
Take the expression
Compare this, with the output of the following
FIR Filter
Then
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Matched Filter
This Filter is called a Matched Filter
The output is maximum when
i.e.
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Example
We transmit the pulse
shown below, with length
Max at n119
Received signal
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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 sn.
This characterizes the pulse.
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Example a square pulse
See a few values
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Compute it in Matlab
N20 data length sones(1,N) square
pulse rssxcorr(s) autocorr n-N1N-1
indices for plot stem(n,rss) plot
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Example Sinusoid
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Example Chirp
schirp(049,0,49,0.1)
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Example Pseudo Noise
srandn(1,50)
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Compare them
chirp
pseudonoise
cos
Two best!
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Detection with Noise
Now see with added noise
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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
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White Noise
The autocorrelation of a white noise signal tends
to be a delta function, ie it is always zero,
apart from when n0.
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White Noise and Filters
The output of a Filter
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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
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Back to the Match Filter
At the peak
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Match Filter and SNR
At the peak
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Example
Transmit a Chirp of length N50 samples, with
SNR0dB
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Example
Transmit a Chirp of length N100 samples, with
SNR0dB
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Example
Transmit a Chirp of length N300 samples, with
SNR0dB