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Random Processes
ECE460
Spring, 2012
Power Spectral Density
Generalities :
Example:
X  t ,  i   A cos  2  f 0 t     i  
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Example
Given a process Yt that takes the values ±1 with equal
probabilities:
P  Yt  1   P  Yt   1   1 / 2



P Yt  1 | Yt 2   1  P Yt   1 | Yt 2  1
1
1
 
,

  2T
1 / 2,


 T
 T
w h ere   t 2  t1
Find R Y  t1 , t 2 
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Ergodic
1. A wide-sense stationary (wss) random process is ergodic in
the mean if the time-average of X(t) converges to the
ensemble average:
lim
t 
1
T

T /2
T /2
g ( x ( t ;  i )) d t  E [ g ( X ( t ))]
2. A wide-sense stationary (wss) random process is ergodic in
the autocorrelation if the time-average of RX(τ) converges to
the ensemble average’s autocorrelation
R X 

x t  x t  

 E  X  t  X  t      R X  

3. Difficult to test. For most communication signals,
reasonable to assume that random waveforms are ergodic in
the mean and in the autocorrelation.
4. For electrical engineering parameters:
1.
x  x t 
2.
x
3.
R XX  0   x
4.
X  x
5.
 X is eq u al to th e rm s valu e o f th e ac co m p o n en t
2
2
is eq u al to th e d c level o f th e sig n a l
 x t 
2
t 
2
eq u als th e n o rm alized d c p o w er
2
t 
is eq u al to th e to tal avg . n o rm alize d p o w er
 x t 
2
is th e avg . n o rm alized ac p o w er
4
Multiple Random Processes
h (t )
Filter
X (t )
Y (t )
Multiple Random Processes
• Defined on the same sample space (e.g., see X(t) and Y(t)
above)
• For communications, limit to two random processes
Independent Random Processes X(t) and Y(t)
– If random variables X(t1) and Y(t2) are independent
for all t1 and t2
Uncorrelated Random Processes X(t) and Y(t)
– If random variables X(t1) and Y(t2) are uncorrelated
for all t1 and t2
Jointly wide-sense stationary
– If X(t) and Y(t) are both individually wss
– The cross-correlation function RXY(t1, t2) depends
only on τ = t2 - t1
R XY  t , t  

E  X  t  Y  t      R X Y  

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Transfer Through a Linear System
X t 
h t 
Y t 
Let X(t) be a wss random process and h(t) be the impulse
response of a stable filter.
Find E{Y(t)}
Find the cross-correlation function RXY(t1,t2).
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Transfer Through a Linear System
X t 
h t 
Y t 
For a wss X(t) with autocorrelation RX(τ) and a stable (bibo)
filter h(t), X(t) and Y(t) are jointly wide sense stationary:
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Example
X t 
h t 
X  t   A co s  2  f 0 t   
Y t 
h  t   differentiator     t 
w h ere  is a ran d o m variab le
u n ifo rm ly d istrib u ted o n [0 ,2  ]
mY :
R X  t1 , t 2  :
SY
 f :
SXY
 f :
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Energy Processes
Recall that the energy of signal x(t) was calculated by
Ex 


x

2
 t  dt
if Ex < ∞ then this is an energy signal
Define for a random process
EX 



X
2
 t  dt
Then the energy content of this signal can be given by
EX  E EX


2

 E  X  t  dt 
  









E  X
2
 t   d t
R X  t, t  dt
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Power Processes
Recall that the power of signal x(t) was calculated by
Px  lim
T 
T
1

T
2

x
T
2
 t  dt
2
if Px > 0 then this is a power signal
Define for a random process
P X  lim
1
T 
T
T

2

T
2
X
 t  dt
2
Then the power of signal x(t) can be given by
PX  E  P X

1

 E  lim
T   T
 lim
T 
 lim
T 
1
T
1
T
T

X
2
E  X
2
2

T

2
T

2

T
 t   d t
2
T


 t  dt 
2

T
R X  t, t  dt
2
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Example
Show that SX(f0) ≈ power of X(t) in [f0, f0+Δf]
X t 
H
h t 
 
f
 1,

 0,
f   f0 , f0  f
Y t 

o th erw ise
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Thermal Noise
R
N t 
Sn
f
2K T R
W /H z
R - valu e o f resister in O h m s
T - tem perature of resister in K elvin
23
K - B o ltzm an n 's co n stan t 1 .3 8  1 0 J/K 
Because of the wide-band of thermal noise, it is usually
modeled as white noise:
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Gaussian Processes
A random process X(t) is Gaussian if for every t1, t2, …, tn, and
every n, the random variables
X(t1), X(t2)…, X(tn)
Are jointly Gaussian.
The Gaussian random process is completely determined by its
mean and autocorrelation functions, i. e., by
m X  t   E  X  t 
R X  t1 , t 2   E  X  t 1  X  t 2  
If a Gaussian process X(t) is passed through a linear filter, the
output process is Gaussian
If X(t) is a wss Gaussian process with mean mX(t),
autocorrelation RX(τ), and an LTI filter with input response h(t),
then Y(t) = X(t)* h(t) is a wss Gaussian process with
mY  t   m X  t  H  0 
RY 

R X     h    * h  

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Zero-Mean White Gaussian Noise
A zero mean white Gaussian noise, W(t), is a random process
with
1.
E W  t   0  t
2.
RW  
3.
SW

 E W  t  W  t  
f
No


No
2
 

W att/H z
2
4. For any n and any sequence t1, t2, …, tn the random
variables W(t1), W(t2), …, W(tn), are jointly Gaussian with
zero mean
E W  t i   0
fo r i  1, 2, ..., n
and covariances




co v W  t i  W  t j   E W  t i  W  t j 
 RW  t j  t i 

No
2
  t j  ti 
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Bandpass Processes
Similar to Chapter 2.5
X(t) is a bandpass process
 R X 
an d S X

is a d eterm in istic b an d p ass sig n al
f
F  R X     is n o n -zero ab o u t f 0
Filter X(t) using a Hilbert Transform:
h t  
H
f
1
t
 j sg n  f

and define
X c  t   X  t  co s  2  f 0 t   X  t  sin  2  f 0 t 
X s  t   X  t  co s  2  f 0 t   X  t  sin  2  f 0 t 
If X(t) is a zero-mean stationary bandpass process, then Xc(t)
and Xs(t) will be zero-mean jointly stationary processes:
E  X c  t    E  X s  t    0
R X c  t   , t   R X c 

R X s  t   , t   R X s 

R X c X s  t   , t   R X c X s 

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Bandpass Processes
This results in two key formulas for future use:
R X c 

R X s 

R X    co s  2  f 0   R X    sin  2  f 0
R X c X s 

R X    sin  2  f 0   R X    co s  2  f 0

Note: Xc(t) and Xs(t) are lowpass processes; i.e., their power
spectrum vanishes for |f| ≥ W.
Find the power spectrum of the in-phase and quadrature
components:
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
Bandpass Example 4.6.1
The white Gaussian noise process N(t) with power spectrum
N0/2 passes through an ideal bandpass filter with frequency
response
H
 
f
 1,

 0,
f  fc  W
O th erw ise
where W << fc. The output process is denoted by X(t). Find the
power spectrum and the cross-spectral density of the in-phase
and quadrature components in the following two cases:
1. f0 is chosen to be equal to fc.
2. f0 is chosen to be equal to fc-W.
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