Random Processes
ECE460
Spring, 2012
Random (Stocastic) Processes
2
Random Process Definitions
Example: X  t , i   A cos  2 f 0t   i  
Notation:
Mean:
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Random Process Definitions
Example: X  t , i   A cos  2 f 0t   i  
Autocorrelation
Auto-covariance
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Stationary Processes
Strict-Sense Stationary (SSS)
A process in which for all n, all (t1, t2, …,tn,), and all Δ
f X t1 , X t1 ,..., X tn   x1 , x2 ,..., xn   f X t1  , X t1  ,..., X tn    x1 , x2 ,..., xn 
Wide-Sense Stationary (WSS)
A process X(t) with the following conditions
1.
mX(t) = E[X(t)] is independent of t.
2.
RX(t1,t2) depends only on the difference τ = t1 - t2
and not on t1 and t2 individually.
Cyclostationary
A random process X(t) is cyclostationary if both the mean,
mx(t), and the autocorrelation function, RX(t1+τ, t2), are
periodic in t with some period T0: i.e., if
mX  t  T0   mX T0 
and
RX  t    T0   RX  t   , t 
for all t and τ.
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Wide-Sense Stationary
Example: X  t , i   A cos  2 f 0t   i  
Mean:
Autocorrelation:
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Power Spectral Density
Generalities :
Example:
X  t , i   A cos  2 f 0t   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| Yt2  1  P Yt  1| Yt2  1
1
1

 ,
  2T
1/ 2,

 T
 T
where   t2  t1
Find RY  t1 , t2 
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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:
1
t  T
lim

T /2
T /2
g ( x(t ; i ))dt  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      RX  
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  is equal to the dc level of the signal
2.
x
2
 x t 
2
equals the normalized dc power
3. RXX  0   x 2  t  is equal to the total avg. normalized power
4.  X2  x 2  t   x  t 
2
is the avg. normalized ac power
5.  X is equal to the rms value of the ac component
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Multiple Random Processes
h(t )
X (t )
Filter
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
RXY  t , t     E  X  t  Y  t      RXY  
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