Chapter 6: Correlation Functions
6-1
Introduction
6-2
Example: Autocorrelation Function of a Binary Process
6-3
Properties of Autocorrelation Functions
6-4
Measurement of Autocorrelation Functions
6-5
Examples of Autocorrelation Functions
6-6
Crosscorrelation Functions
6-7
Properties of Crosscorrelation Functions
6-8
Example and Applications of Crosscorrelation Functions
6-9
Correlation Matrices for Sampled Functions
Concepts









Autocorrelation
o of a Binary/Bipolar Process
o of a Sinusoidal Process
Linear Transformations
Properties of Autocorrelation Functions
Measurement of Autocorrelation Functions
An Anthology of Signals
Crosscorrelation
Properties of Crosscorrelation
Examples and Applications of Crosscorrelation
Correlation Matrices For Sampled Functions
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
5-5.3 A random process has sample functions of the form

 A  f t  nT  t0 
X t  
n  
where A and T are constants and t0 is a random variable that is uniformly distributed
between 0 and T. The function f(t) is defined by
f t   1
0t T
2
and zero elsewhere.
pdf t 0  
Also Note:
a.
1
T
0  t0  T
Find the mean and second moment.
 

E X t   E   A  f t  nT  t 0   E A  E  f t  nT  t 0 
 n  

E X t   A  E  f t  nT  t 0 
For any time t, there is a probability that f(t) is present or absent, based on the random variable
t0. As t0 is uniformly distributed across the bit period T and f(t) is present for exactly ½ of the
period,
T
E  f t  nT  t 0    f t 0  
0
1
1 T 1
 dt 0   
T
T 2 2
and
E X t   A 

E X t 
2

1
2
2
 
 
 

 E   A  f t  nT  t 0    E   A 2  f t  nT  t 0 
 
n  

 n  


E X t   A 2  E f t  nT  t 0 
2
As previously stated


E X t   A 2 
2
1
2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
2 of 27
ECE 3800
b.
Find the time average and squared average.
1
X t   
T
T  t0

t0
n  
  A  f t  nT  t   dt
0
T
A 2
A T A
1
X t     A  f t   dt    1  dt   
T 0
T 0
T 2 2
T
and
X t 
X t 
c.
2
2
1
 
T
T  t0

t0
2
 

  A  f t  nT  t 0   dt
 n  

T
A2 2
A2 T A2
1
   A 2  f t   dt 
  1  dt 
 
T 0
T 0
T 2
2
T
Can this process be stationary?
Yes, the mean and variance are not functions of time. Therefore, we expect it to be wide-sense
stationary.
d.
Can this process be ergodic?
Based on the information provide, there is no specific reason to suggest that the process is not
ergodic.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Chapter 6: Correlation Functions
The Autocorrelation Function
For a sample function defined by samples in time of a random process, how alike are the
different samples?
Define:
X 1  X t1  and X 2  X t 2 
The autocorrelation is defined as:
R XX t1 , t 2   EX t1   X t 2 
The above function is valid for all processes, stationary and non-stationary.
For WSS processes:
R XX t1 , t 2   E X t X t     R XX  
If the process is ergodic, the time average is equivalent to the probabilistic expectation, or
1
 XX    lim
T   2T
T
 xt   xt     dt 
xt   xt   
T
and
 XX    R XX  
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
4 of 27
ECE 3800
Example: xt   A  sin2  f  t  for A a uniformly distributed random variable A   2,2
R XX t1 , t 2   E X t1   X t 2   EA  sin2  f  t1   A  sin2  f  t 2 
1


R XX t1 , t 2   E X t1   X t 2   E  A 2   cos2  f  t1  t 2   cos2  f  t1  t 2 
2


 
1
 E A 2  cos2  f  t1  t 2   cos2  f  t1  t 2 
2
R XX t1 , t 2  
for   t 2  t1
R XX t1 , t 2  
1 2
  cos2  f     cos2  f  t1  t 2 
2 12
R XX t1 , t 2  
16
 cos2  f     cos2  f  t1  t 2 
24
A non-stationary process
The time based formulation:
1
 XX    lim
T   2T
1
T   2T
1
 XX    A  lim
T   2T
 XX   
 xt   xt     dt 
T
 A  sin 2  f  t   A  sin 2  f  t     dt
T
T
1
 2  cos2  f      cos2  f  2  t    dt
T
A2
A2
1
 cos2  f     
 lim


T
2
2
2T
 XX   
xt   xt   
T
 XX    lim
2
T
T
 cos2  f  2  t     dt
T
A2
A2
 cos2  f     
 cos2  f   
2
2
Acceptable, but the R.V. is still present?!
 A2 
16
E  XX    E    cos2  f    
 cos2  f   
24
 2 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
5 of 27
ECE 3800
Example: xt   A  sin2  f  t    for  a uniformly distributed random variable   0,2 
R XX t1 , t 2   EX t1   X t 2   EA  sin2  f  t1     A  sin2  f  t 2   
1

R XX t1 , t 2   E X t1   X t 2   A 2  E   cos2  f  t1  t 2   cos2  f  t1  t 2   2   
2

2
2
A
A
R XX t1 , t 2   E  X t1   X t 2  
 cos2  f  t1  t 2  
 E cos2  f  t1  t 2   2   
2
2
Of note is that the phase need only be uniformly distributed over 0 to π in the previous step!
R XX t1 , t 2   E X t1   X t 2  
A2
 cos2  f  t1  t 2 
2
for   t 2  t1
A2
R XX   
 cos2  f    
2
but
R XX    R XX    
A2
 cos2  f   
2
Assuming a uniformly distributed random phase “simplifies the problem” …
Also of note, if the amplitude is an independent random variable, then
R XX   
 
E A2
 cos2    f   
2
The time based formulation:
1
 XX    lim
T   2T
1
T   2T
 XX    lim
T
 xt   xt     dt 
xt   xt   
T
T
 A  sin 2  f  t     A  sin 2  f  t        dt
T
A2
1
 XX   
 lim
2 T  2T
T
 cos2  f     cos2  f  2t     2  dt
T
 XX   
A2
 cos2  f   
2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
6 of 27
ECE 3800
 t  t0 
Example: xt   B  rect 
 for B =+/-A with probability p and (1-p) and t0 a uniformly
 T 
 T T
distributed random variable t 0   ,  . Assume B and t0 are independent.
 2 2

t t 
t t
R XX t1 , t 2   E X t1   X t 2   E  B  rect  1 0   B  rect  1 0
 T 
 T


t t 
t t
R XX t1 , t 2   E X t1   X t 2   E  B 2  rect  1 0   rect  1 0
 T 
 T







As the RV are independent

t t 
t t
R XX t1 , t 2   E X t1   X t 2   E B 2  E rect  1 0   rect  2 0
 T
 T 

 









t t 
t t
2
R XX t1 , t 2   A 2  p   A  1  p   E rect  1 0   rect  2 0
 T
 T 

T
R XX t1 , t 2   A 
2
2

T
2
t t 
t t  1
rect  1 0   rect  2 0    dt 0
 T 
 T  T
For t1  0 and t 2  
T
R XX 0,   A 
2
2
   t0  1
   dt 0
T  T
 1  rect 
T
2
The integral can be recognized as being a triangle, extending from –T to T and zero everywhere
else.
 
R XX    A 2  tri  
T 
0,

 A 2  1  T   ,

T
R XX    
 A 2  1  T   ,

T
0,

  T
T    0
0  T
T 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
7 of 27
ECE 3800
The time based formulation:
1
 XX    lim
T  2T
T
 xt   xt     dt  xt   xt   
T
1
 t    t0 
 t  t0 
 XX    lim
B  rect 
  B  rect 
  dt
C   2C 
T
 T 


C
C
A change in variable for the integral t  t  t 0 . And only integrate over the finite interval T.
T
1 2
 t  
 XX    B 2   1  rect 
  dt
T T
 T 
2
 t  
 XX    B 2  tri 

 T 
 t  
E  XX    E B 2  tri

 T 
 


 t  
2
E  XX    A 2  p   A  1  p   tri 

 T 
 t  
E  XX    A 2  tri 

 T 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
8 of 27
ECE 3800
Properties of Autocorrelation Functions
1)
 
R XX 0   E X 2  X 2 or  XX 0   xt 2
the mean squared value of the random process can be obtained by observing the zeroth lag of the
autocorrelation function.
2)
R XX    R XX   
The autocorrelation function is an even function in time. Only positive (or negative) needs to be
computed for an ergodic, WSS random process.
3)
R XX    R XX 0
The autocorrelation function is a maximum at 0. For periodic functions, other values may equal
the zeroth lag, but never be larger.
4)
If X has a DC component, then Rxx has a constant factor.
X t   X  N t 
R XX    X 2  R NN  
Note that the mean value can be computed from the autocorrelation function constants!
5)
If X has a periodic component, then Rxx will also have a periodic component of the same
period.
Think of:
X t   A  cosw  t   ,
0    2 
where A and w are known constants and theta is a uniform random variable.
A2
R XX    E  X t X t    
 cosw   
2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
9 of 27
ECE 3800
5b)
For signals that are the sum of independent random variable, the autocorrelation is the
sum of the individual autocorrelation functions.
W t   X t   Y t 
RWW    R XX    RYY    2   X  Y
For non-zero mean functions, (let w, x, y be zero mean and W, X, Y have a mean)
RWW    R XX    RYY    2   X  Y
RWW    Rww    W 2  R xx     X 2  R yy    Y 2  2   X   Y
RWW    Rww    W 2  R xx    R yy     X 2  2   X   Y   Y 2
RWW    Rww    W 2  R xx    R yy     X   Y 2
Then we have
W 2   X  Y 2
Rww    R xx    R yy  
6)
If X is ergodic and zero mean and has no periodic component, then
lim R XX    0
 
No good proof provided.
The logical argument is that eventually new samples of a non-deterministic random process will
become uncorrelated. Once the samples are uncorrelated, the autocorrelation goes to zero.
7)
Autocorrelation functions can not have an arbitrary shape. One way of specifying shapes
permissible is in terms of the Fourier transform of the autocorrelation function. That is, if
R XX   

 R XX    exp jwt   dt

then the restriction states that
R XX    0 for all w
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Additional concepts that are useful … dealing with constant:
X t   a  N t 
R XX    EX t X t     Ea  N t   a  N t   
R XX    a 2  E N t   N t     a 2  R NN  
X t   a  Y t 
R XX    Ea  Y t   a  Y t   


R XX    E a 2  a  Y t   a  Y t     Y t   Y t   
R XX    a 2  a  E Y t   a  E Y t     E Y t   Y t   
R XX    a 2  2  a  E Y t   RYY  
Exercise 6-1.1
A random process has a sample function of the form
0  t 1
else
 A,
X t   
0,
where A is a random variable that is uniformly distributed from 0 to 10.
Find the autocorrelation of the process.
f a  
1
,
10
for 0  a  10
Using
R XX t1 , t 2   EX t1   X t 2 
 
R XX t1 , t 2   E A 2 ,
R XX t1 , t 2  
10
1
 10  a
2
 da,
for 0  t1 , t 2  1
for 0  t1 , t 2  1
0
10
a3
R XX t1 , t 2  
,
30 0
R XX t1 , t 2  
1000 100

,
30
3
for 0  t1 , t 2  1
for 0  t1 , t 2  1
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
11 of 27
ECE 3800
Exercise 6-1.2
Define a random variable Z(t) as
Z t   X t   X t   1 
where X(t) is a sample function from a stationary random process whose autocorrelation function
is

R XX    exp   2

Write an expressions for the autocorrelation function of Z(t).
RZZ t1 , t 2   EZ t1   Z t 2 
RZZ t1 , t 2   EX t1   X t1   1  X t 2   X t 2   1 
RZZ t1 , t 2   EX t1   X t 2   X t1   X t 2   1   X t1   1   X t 2   X t1   1   X t 2   1 
RZZ t1 , t 2   E  X t1   X t 2   E X t1   X t 2   1 
 E X t1   1   X t 2   E X t1   1   X t 2   1 
Note: based on the autocorrelation definition,   t 2  t1
RZZ t1 , t 2   R XX t 2  t1   R XX t 2   1  t1   R XX t 2  t1   1   R XX t 2  t1 
RZZ t1 , t 2   2  R XX    R XX    1   R XX    1 





R ZZ t1 , t 2   2  exp   2  exp     1   exp     1 
2
2

Note that for this example, the autocorrelation is strictly a function of tau and not t1 or t2. As a
result, we expect that the random variable is stationary.
Helpful in defining a stationary random process:
A stationary random variable will have an autocorrelation that is dependent on the time
difference, not the absolute times.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Autocorrelation of a Binary Process
Binary communications signals are discrete, non-deterministic signals with plenty of random
variables involved in their generations.
Nominally, the binary bit last for a defined period of time, ta, but initially occurs at a random
delay (uniformly distributed across one bit interval 0<=t0<ta). The amplitude may be a fixed
constant or a random variable.
ta
A
t0
-A
bit(0)
bit(1)
bit(2)
bit(3)
bit(4)
The bit values are independent, identically distributed with probability p that amplitude is A and
q=1-p that amplitude is –A.
1
pdf t 0   , for 0  t 0  t a
ta
Determine the autocorrelation of the bipolar binary sequence, assuming p=0.5.
 
t 
 t   t0  a 2   k  ta


xt    a k  rect 

ta
k  









R XX t1 , t 2   EX t1   X t 2 
For samples more than one period apart, t1  t 2  t a , we must consider


E a k  a j  p  A  p  A  p  A  1  p    A  1  p    A  p  A  1  p    A  1  p    A



E a k  a j  A 2  p 2 2  p  1  p   1  p 



2


E a k  a j  A 2  4  p 2 4  p  1
For p=0.5




E a k  a j  A 2  4  p 2 4  p  1  0
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
13 of 27
ECE 3800
For samples within one period, t1  t 2  t a ,
Ea k  a k   p  A 2  1  p    A  A 2
2


Ea k  a k 1   A 2  4  p 2 4  p  1  0
there are two regions to consider, the sample bit overlapping and the area of the next bit. But the
overlapping area … should be triangular. Therefore
R XX   
1

ta

ta
2
 Ea
k
 ak 1   dt 
ta
 2
1
R XX    
ta
ta
1

ta
2
1
t Eak  ak  dt  t a 
a
 ta 2
 Ea

ta
k
 ak   dt ,
for  t a    0
k
 ak 1   dt ,
for 0    t a
2
 t a 2
 2
 Ea
ta
2
or
R XX    A 2 
1

ta
 t a 2
1  dt,

1
R XX    A  
ta
2
ta
for  t a    0
2
ta
2
1  dt,
for 0    t a
ta
 2
Therefore
 2 ta  
A  t ,

a
R XX    

t
 A2  a  ,

ta
for  t a    0
for 0    t a
or recognizing the structure
 
R XX    A 2  1 
 ta

, for  t a    t a


This is simply a triangular function with maximum of A2, extending for a full bit period in both
time directions.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
14 of 27
ECE 3800
For unequal bit probability
 2  ta  

 4  p 2 4  p  1 
 A  
ta
R XX    
 ta
 2
2
 A  4  p 4  p  1 ,





,


for  t a    t a
for t a  
As there are more of one bit or the other, there is always a positive correlation between bits (the
curve is a minimum for p=0.5), that peaks to A2 at  = 0.
Note that if the amplitude is a random variable, the expected value of the bits must be further
evaluated. Such as,
Eak  ak    2   2
Ea k  a k 1    2
In general, the autocorrelation of communications signal waveforms is important, particularly
when we discuss the power spectral density later in the textbook.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Examples of discrete waveforms used for communications, signal processing, controls, etc.
(a) Unipolar RZ & NRZ, (b) Polar RZ & NRZ , (c) Bipolar NRZ , (d) Split-phase Manchester,
(e) Polar quaternary NRZ.
From Cahp. 11: A. Bruce Carlson, P.B. Crilly, Communication Systems,
5th ed., McGraw-Hill, 2010. ISBN: 978-0-07-338040-7
In general, a periodic bipolar “pulse” that is shorter in duration than the pulse period will have
the autocorrelation function
R XX    A 2 
tw
tp
  
 1  ,
 tw 
for  t w    t w
for a tw width pulse existing in a tp time period, assuming that positive and negative levels are
equally likely.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Exercise 6-3.1
a) An ergodic random process has an autocorrelation function of the form
R XX    9  exp  4     16  cos 10     16
Find the mean-square value, the mean value, and the variance of the process.
The mean-square (2nd moment) is
 
E X 2  R XX 0   9  16  16  41   2   2
The constant portion of the autocorrelation represents the square of the mean. Therefore
E X    2  16 and   4
2
Finally, the variance can be computed as,
 2  E X 2   E X 2  R XX 0    2  41  16  25
b) An ergodic random process has an autocorrelation function of the form
R XX   
4  2  6
 2 1
Find the mean-square value, the mean value, and the variance of the process.
The mean-square (2nd moment) is
 
E X 2  R XX 0  
6
 6  2  2
1
The constant portion of the autocorrelation represents the square of the mean. Therefore
E X 
2
4  2  6 4
  4 and   2
   lim 2
t    1
1
2
Finally, the variance can be computed as,
 2  E X 2   E  X 2  R XX 0    2  6  4  2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
The Crosscorrelation Function
For a two sample function defined by samples in time of two random processes, how alike are
the different samples?
Define:
X 1  X t1  and Y2  Y t 2 
The cross-correlation is defined as:
R XY t1 , t 2   E  X 1Y2  
RYX t1 , t 2   E Y1 X 2  








 dx1  dy2  x1 y2 f x1, y2 
 dy1  dx2  y1x2 f  y1, x2 
The above function is valid for all processes, jointly stationary and non-stationary.
For jointly WSS processes:
R XY t1 , t 2   E X t Y t     R XY  
RYX t1 , t 2   EY t X t     RYX  
Note: the order of the subscripts is important for cross-correlation!
If the processes are jointly ergodic, the time average is equivalent to the probabilistic
expectation, or
1
 XY    lim
T   2T
1
YX    lim
T   2T
T
 xt   yt     dt 
xt   y t   
T
T
 yt   xt     dt 
y t   xt   
T
and
 XY    R XY  
YX    RYX  
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Properties of Crosscorrelation Functions
1)
The properties of the zoreth lag have no particular significance and do not represent
mean-square values. It is true that the “ordered” crosscorrelations must be equal at 0. .
R XY 0  RYX 0 or  XY 0  YX 0
2)
Crosscorrelation functions are not generally even functions. However, there is an
antisymmetry to the ordered crosscorrelations:
R XY    RYX   
For
T
1
 XY    lim
T   2T
 xt   yt     dt 
xt   y t   
T
t  
Substitute
1
 XY    lim
T   2T
1
T   2T
 XY    lim
T 
 x     y   d  x     y  
T 
T 
 y   x     d  y   x   
  YX   
T 
3)
The crosscorrelation does not necessarily have its maximum at the zeroth lag. This makes
sense if you are correlating a signal with a timed delayed version of itself. The crosscorrelation
should be a maximum when the lag equals the time delay!
It can be shown however that
R XY    R XX 0   R XX 0 
As a note, the crosscorrelation may not achieve the maximum anywhere …
4)
If X and Y are statistically independent, then the ordering is not important
R XY    E  X t   Y t     E  X t   E Y t     X  Y
and
R XY    X  Y  RYX  
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
5)
If X is a stationary random process and is differentiable with respect to time, the
crosscorrelation of the signal and it’s derivative is given by
R XX   
dR XX  
d
Defining derivation as a limit:
X t  e   X t 
X    lim
e
e 0
and the crosscorrelation

X t    e   X t    

R XX    E X t   X t     E  X t    lim

e
 e 0




E  X t   X t    e   X t   X t   
e
e 0
R XX    lim
E X t   X t    e   E X t   X t   
e
e 0
R XX    lim
R XX   e   R XX  
e
e 0
R XX    lim
R XX   
dR XX  
d
Similarly,
R XX    
d 2 R XX  
d 2
That is all the properties for the crosscorrelation function.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
Example: A delayed signal in noise … cross correlated with the original signal …
X t 
Y t   a  X t  t d   N t 
Find
R XY    EX t Y t   
Then
R XY    E X t   a  X t  t d     N t   
R XY    a  R XX   t d   R XN  
For noise independent of the signal,
R XN    X  N  R NX  
and therefore
R XY    a  R XX   t d   X  N
For either X or N a zero mean process, we would have
R XY    a  R XX   t d 
Does this suggest a way to determine the time delay based on what you know of the
autocorrelation?
The maximum of the autocorrelation is at zero; therefore, the maximum of the crosscorrelation
can reveal the time delay!

any electronic system based on the time delay of a known, transmitted signal pulse …
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
21 of 27
ECE 3800
Back to Autocorrelation examples, using time a time average
Example: Rect. Function
From Dr. Severance’s notes;
X t  
 Ak  Re ct
k
t  t 0  kT 

T

where A(k) are zero mean, identically distributed, independent R.V., t0 is a R.V. uniformly
distributed over possible bit period T, and T is a constant. k describes the bit positions.


 t  t 0  kT 
 t    t 0  jT 

R XX t , t     E
Ak  Re ct 
A j  Re ct 



T
T




j
 k


 t  t 0  kT 
 t    t 0  jT 
R XX t , t    
E  Ak  A j  Re ct 

  Re ct 

T
T







 
k
R XX t , t    
j
   EAk  A j  E Re ct

k
j
but
 
 E A 2
E Ak  A j  
0


t  t 0  kT 
 t    t 0  jT 
  Re ct 

T
T



for k  j
for k  j
Therefore (one summation goes away)
 
R XX t , t     E A 2 
k

 t  t 0  kT 
 t    t 0  kT 
 E Re ct 
  Re ct 

T
T





Now using the time average concept
T
1
 XX    lim 
T  T

2
T xt   xt     dt 
xt   xt   
2
Establishing the time average as the average of the kth bit, (notice that the delay is incorporated
into the integral, and assuming that the signal is Wide-Sense Stationary)
T t0
2
  T1  
 XX    E A 2 
T  t 0
2

 t 
1  Re ct  T



  dt

Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
22 of 27
ECE 3800
for T>>=0
T t0
2
  T1   1 1 dt  EA  T1  t 
 XX    E A 2 
2
 T  t 0 
2
T t0
2
 T  t 0 
2
  T1  T 2  t   T 2  t     E A  T1  T   
 
    E A  1  ,
for   0
 T
 XX    E A 2 
XX
0
0
2
2
Due to symmetry,  XX     XX    , we have
 
  
 XX    E A 2  1  
 T
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
23 of 27
ECE 3800
Example: Rect. Function that does not extend for the full period
X t  
 Ak  Re ct
k
t  t 0  kT 

 T 
where A(k) are zero mean, identically distributed, independent R.V., t0 is a R.V. uniformly
distributed over possible bit period T, and T and alpha are constants. k describes the bit
positions.
From the previous example, the average value during the time period was defined as:
T t0
2
  T1  
 XX    E A 2 

 t 
1  Re ct  T


T  t 0
2

  dt

For this example, the period remains the same, but the integration period due to the rect functions
change to:
T  t 0
2
  T1  

 XX    E A 2 
 T t0
2
for T>>=0
  T1 
 XX    E A 2 
T  t 0
2

  dt

1

1  1  dt  E A 2   t  T2


T  t 0
T
 T  t 0  
2
2
 t 0 
  T1  T 2  t   T 2  t     E A  T1  T   
   E A  T  1     E A    1   ,
for   0
T  T 
 T 
 XX    E A 2 
 XX

 t 
1  Re ct  T


Due to symmetry, we have
0
0
2
2
2
 

 
 XX    E A 2    1 

 T 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
24 of 27
ECE 3800
Measurement of the Autocorrelation Function
We love to use time average for everything. For wide-sense stationary, ergodic random
processes, time average are equivalent to statistical or probability based values.
T
1
 XX    lim
T   2T
 xt   xt     dt 
xt   xt   
T
Using this fact, how can we use short-term time averages to generate auto- or cross-correlation
functions?
An estimate of the autocorrelation is defined as:
Rˆ XX   
1
T 
T 
 xt   xt     dt
0
Note that the time average is performed across as much of the signal that is available after the
time shift by tau.
In most practical cases, function is performed in terms of digital samples taken at specific time
intervals, t . For tau based on the available time step, k, with N equating to the available time
interval, we have:
Rˆ XX kt  
1
N  1t   kt 
Rˆ XX kt   Rˆ XX k  
N k
 xit   xit  kt   t
i 0
1
N 1 k
N k
 xi   xi  k 
i 0
In computing this autocorrelation, the initial weighting term approaches 1 when k=N. At this
point the entire summation consists of one point and is therefore a poor estimate of the
autocorrelation. For useful results, k<<N!
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
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ECE 3800
It can be shown that the estimated autocorrelation is equivalent to the actual autocorrelation;
therefore, this is an unbiased estimate.
N k


1
ˆ
ˆ

E R XX kt   E R XX k   E
xi   xi  k 
 N 1 k

i 0



 



E Rˆ XX kt  




1

N 1 k
1

N 1 k
N k
 Exi   xi  k 
i 0
N k
 R XX kt 
i 0
1
  N  k  1  R XX kt 
N 1 k

E Rˆ XX kt   R XX kt 
As noted, the validity of each of the summed autocorrelation lags can and should be brought into
question as k approaches N. As a result, a biased estimate of the autocorrelation is commonly
used. The biased estimate is defined as:
~
R XX k  
1
N 1
N k
 xi   xi  k 
i 0
Here, a constant weight instead of one based on the number of elements summed is used. This
estimate has the property that the estimated autocorrelation should decrease as k approaches N.
The expected value of this estimate can be shown to be


n 
~

E R XX k   1 
  R XX kt 
N  1

The variance of this estimate can be shown to be (math not done at this level)


2
~
Var R XX k   
N
M
 R XX kt 2
k  M
This equation can be used to estimate the number of time samples needed for a useful estimate.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
26 of 27
ECE 3800
Matlab Example
%%
% Figure 6_2
%
clear;
close all;
nsamples = 1000;
n=(0:1:nsamples-1)';
%%
% rect
rect_length = 100;
x = zeros(nsamples,1);
x(100+(1:rect_length))=1;
% Scale to make the maximum unity
Rxx1 = xcorr(x)/rect_length;
Rxx2 = xcorr(x,'unbiased');
nn = -(nsamples-1):1:(nsamples-1);
figure
plot(n,x)
figure
plot(nn,Rxx1,nn,Rxx2)
legend('Raw','Unbiased')
1.2
Raw
Unbiased
1
0.8
0.6
0.4
0.2
0
-0.2
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
27 of 27
ECE 3800