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Signals and Systems
Lecture 12
Correlation, Energy Spectral
Density and Power
Spectral Density
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Correlation
Positively Correlated
Random CT Signals
with Zero Mean
Uncorrelated Random
CT Signals with
Zero Mean
Negatively Correlated
Random CT Signals
with Zero Mean
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Correlation
Positively Correlated
CT Sinusoids with
Non-zero Mean
Uncorrelated CT
Sinusoids with
Non-zero Mean
Negatively Correlated
CT Sinusoids with
Non-zero Mean
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Correlation
Relationships between signals can be just
as important as characteristics of
individual signals
Parseval’s Equation
Rayleigh’s energy theorem
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Correlation
Correlation function between two energy
signals:
is the area under their product as a
function of how much y is shifted relative
to x !
Rxy x t y* t dt
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Correlation
The correlation function for two energy
signals is very similar to the convolution of
two energy signals:
using convolution:
x t y t x t y d
Rxy x y
F *
xyR X Yj j
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Correlation
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Correlation
The correlation function between two
power signals, x and y, is the average
value of their product as a function of how
much y is shifted relative to x.
Rxy limT
1
Tx t y* t dt
T
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Correlation
If the two signals are both periodic and
their fundamental periods have a finite
least common period:
Rxy 1
Tx t y t dt
T
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Correlation
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Autocorrelation
A very important special case of
correlation is autocorrelation.
Autocorrelation is the correlation of a
function with a shifted version of itself.
For energy signals:
Rxx x t x t dt
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Autocorrelation
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Autocorrelation
At a shift ;
For power signals;
Rxx 0 x2t dt
energy of the signal
xx
1R x
Tt x t dt
T
2
xx
1R 0 x x
Tt dt P
T
Relative power (power on 1ohm resistor – [Volt^2])
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Autocorrelation
Power Spectral Density:
0
10
2
X x
x X x x
G F R
P G d R R
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Properties of Autocorrelation
Autocorrelation is an even function
Autocorrelation magnitude can never be
larger than it is at zero shift.
Rxx Rxx
Rxx 0 Rxx
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Properties of Autocorrelation
where is energy spectral density. XS
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Properties of Autocorrelation
If a signal is time shifted its autocorrelation
does not change.
The autocorrelation of a sum of sinusoids
of different frequencies is the sum of the
autocorrelations of the individual
sinusoids.
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Autocorrelation Example
Show that the energy spectral densities of x(t)
and x(t±to) are the same.
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Autocorrelation Example
Autocorrelations for a cosine “burst” and a sine
“burst”. Notice that they are almost (but not quite)
identical.
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Autocorrelation Example
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Autocorrelation Example
Three random power signals with different
frequency content and their autocorrelations.
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Autocorrelation Example
Autocorrelation functions for a cosine and a
sine. Notice that the autocorrelation functions
are identical even though the signals are
different.
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Autocorrelation Example
Four different random signals with identical
autocorrelations:
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Autocorrelation Example
Four different random signals with identical
autocorrelations:
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Autocorrelation Example
Four different random signals with identical
autocorrelations:
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Autocorrelation Example
Four different random signals with identical
autocorrelations:
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Correlation Example
Matched Filters:
Technique for detecting the presence of a
signal of a certain shape in the presence of
noise.
Uses correlation to detect the signal
It is often used to detect 1’s and 0’s in a
binary data stream so this filter sometimes
called a correlation filter
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Correlation Example
the optimal filter to
detect a noisy
signal is one whose
impulse response is
proportional to the
time inverse of the
signal.
some examples of
waveshapes
encoding 1’s and
0’s and the impulse
responses of
matched filters.
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Correlation Example
Noiseless Bits Noisy Bits
