Response of Linear Systems to Random Signals
Properties of Random Signals
 Introduction: Since random signals are defined in terms of probability, one way to
characterize a random signal is to analyze the long-term observation of the process.
The following analysis assumes the (weakly) stationary/ergodic property.
 The four basic properties of random signals:
o Autocorrelation function – A measure of the degree of randomness of a signal.
o Cross-correlation function – A measure of correlation between two signals.
o Autospectral density function – A measure of the frequency distribution of a
signal.
o Cross-spectral density function – A measure of the frequency distribution of
one signal (output) due to another signal (input).
 Correlation functions:
o Autocorrelation function: The autocorrelation function Rxx(t) of a random
signal X(t) is a measure of how well the future values of X(t) can be predicted
based on past measurements. It contains no phase information of the signal.

Rxx ( )  E[ X (t ) X (t   )]   x1 x2 P( x1 , x2 ) dx1dx2
0 0
where x1  X (t ), x2  X (t   ), and P( x1 , x2 )  joint PDF of x1 and x2
o Cross-correlation function: The cross-correlation function Rxy(t) is a measure
of how well the future values of one signal can be predicted based on past
measurements of another signal.
Rxy ( )  E[ X (t )Y (t   )]
o Properties of correlations functions:
 Rxx(-t) = Rxx(t) (even function)

Rxy(-t) = Ryx(t)

Rxx(0) = x2 +  x2 = mean square value of x(t)


Rxx() =  x2
The cross-correlation inequality: Rxy (t )  Rxx (t ) R yy (t )
o Example:
 Rxx(t) of a sine wave process:
1
0    2

X (t )  A sin( 2 f t   ) where P( )   2
0
elsewhere
Let x1 ( )  A sin[ 2 f t   ] and x2 ( )  A sin[ 2 f (t   )   ]
A2
 Rxx ( )  E[ x1 ( ) x2 ( )] 
2
 Rxx ( ) 
2
A
cos( 2 f  )
2
2
 sin[ 2 f t   ] sin[ 2 f (t   )   ] d
0
2
R_xx( t )
0
2
0

0.005
t
Since the envelop of the autocorrelation function is constant as ,
we can conclude that any future values of a sine wave process can be
accurately predicted based on past measurements.
A time-delay system:
n(t)

x(t)
z-1
y(t)
Rxy ( )  E[ x(t ) y (t   )]
y (t )  x(t  t 0 )  n(t )



 Rxy ( )  E  x(t ) x(t    t 0 )  n(t   )   E x(t ) x(t    t 0 )
 


mean0  

 Rxy ( )  Rxx (  t 0 ) R yy ( )  E[ y (t ) y (t   )]  Rxx ( )  Rnn ( )

Spectral density functions:
o Autospectral density function (autospectrum): The autospectral density
function Sxx(f) of a random signal X(t) is a measure of its frequency
distribution. It is the Fourier transform of the autocorrelation function.
S xx ( f ) 
R xx (t ) 




0
 j 2 f t
dt  2  R xx (t ) e  j 2 f t dt
 Rxx (t ) e

j 2 f t
df  2  S xx ( f ) cos( 2 f t ) df
 S xx ( f ) e

0
o Cross-spectral density function (cross-spectrum): The cross-spectral density
function Sxy(f) is a measure of the portion of the frequency distribution of one
signal that is a result of another signal. It is the Fourier transform of the crosscorrelation function.

S xy ( f ) 
R
xy

(t ) e
2 f t

dt
Rxy (t )  2 S xy ( f ) cos(2 f t ) df
0
o Symmetry: Sxx(-f) = Sxx(f) (even function) Sxy(-f) = Sxy*(f) = Syx(f)
o Example:
 White noise: The autocorrelation function of a white noise process is
an impulse function. The implication here is that white noise is totally
unpredictable since its autocorrelation function decays to zero at t > 0.
Of course, this is what we expect to see in a white noise process, a
random process with infinite energy.
 Low-pass white noise: Assuming the low-pass white noise has a
magnitude of 1 and the bandwidth will vary. The autocorrelation
function is computed using IFFT.
S_nn f 
f  200
1 if
R_nn  ifft ( S_nn )
0 otherwise
2
S_nnf
10
1
0
R_nnt
0
200
0
10
400
0
50
f
S_nn f
1 if
f
100
R_nn
0 otherwise
2
S_nn
f
100
t
ifft ( S_nn )
5
R_nn
1
t
0
0
5
0
S_nn f
200
f
1 if
400
f
0
50
t
100
50
R_nn
0 otherwise
ifft ( S_nn )
2
2
S_nn
f
R_nn
1
t
0
2
0
0
200
f
400
0
50
t
100


The autocorrelation function of a low-pass white noise process is a
sinc [sin(x)/x] function. The implication here is that the predictability
of a low-pass white noise process is inversely proportional to the
bandwidth of the noise. Since the noise energy is proportional to the
bandwidth, this conclusion is consistent with common sense.
Band-pass white noise: Assuming the band-pass white noise has a
magnitude of 1 and the bandwidth will vary. The autocorrelation
function is computed using IFFT.
S_nn f
1 if ( f
100 )  ( f
200 )
R_nn
0 otherwise
2
S_nn
f
5
R_nn
1
t
0
0
5
0
S_nn f
200
f
1 if ( f
400
100 )  ( f
0
R_nn
2
f
50
t
100
110 )
0 otherwise
S_nn
ifft ( S_nn )
ifft ( S_nn )
0.5
R_nn
1
t
0
0
0.5
0
S_nn f
200
f
1 if ( f
400
100 )  ( f
0
50
t
100
101 )
R_nn
0 otherwise
ifft ( S_nn )
2
S_nn
f
R_nn
1
t
0
0
0
200
f
400
0
50
t
100

The autocorrelation function of a band-pass white noise process has
the similar property – the predictability is inversely proportional to the
bandwidth of the noise. In addition, the autocorrelation function
approaches the sine wave process as the bandwidth decreases, a
conclusion that can easily be drawn using LTI system theory for
deterministic systems.
LTI Systems
 Introduction: LTI systems are analyzed using the correlation/spectral technique. The
inputs are assumed to be stationary/ergodic random with mean = 0.
 Ideal system:
H(f)
x(t)
y(t)
h(t)
t
y(t )   x( ) h(t   ) d
0
Y( f )  X ( f ) H( f )

Ideal model:
o Correlation and spectral relationships:


R yy (t )   h( ) h(  ) Rxx (t    ) d d 
Rxy (t )   h( ) Rxx (t  ) d
00
0
S yy ( f )  H ( f ) S xx ( f )
S xy ( f )  H ( f ) S xx ( f )
2

Total output noise energy:  y2   S yy ( f ) df

o Example: LPF to white noise
1
H( f ) 
 H ( f ) e  j ( f )
1  j 2 f K
K  RC
(LPF)
t
1 
1
 h(t )  e K  u (t )  H ( f ) 
, and  ( f )  tan 1 (2 f K )
K
1  (2 f K ) 2
For white noise : S xx ( f )  A
A
2
 S yy ( f )  H ( f ) S xx ( f ) 
t

and
R yy (t ) 
1  (2 f K ) 2
 S yy ( f ) e
j 2 f t

 2y 


A K
df 
e
2K
A
 S yy ( f ) df  2   1  (2 f K ) 2 df

 2y  A
0

h
0

2
(t ) dt  A 
1
0K
2
2t
A
e K dt 

2K

A
, or
2K
One-sided spectrum: G(f) = 2 S(f) for f  0
o Example: LPF to a sine process
1
H( f ) 
 H ( f ) e  j ( f ) K  RC
1  j 2 f K
 1  Kt
 e
 h(t )   K
0

 H( f ) 
(LPF)
t0
t0
1
1  (2 f K )
2
 ( f )  tan 1 (2 f K )
1 2
A  ( f  f0 )
2
A2  ( f  f 0 )
2
 G yy ( f )  H ( f ) G xx ( f ) 
2[1  (2 f K ) 2 ]
For sine wave : G xx ( f ) 

R yy (t )   G yy ( f ) cos (2 f t ) df
and
0
 R yy (t ) 
A2
cos (2 f 0t )
2[1  (2 f 0 K ) 2 ]


A2  ( f  f 0 )
   G yy ( f ) df  
df
2[1  (2 f K ) 2 ]
0
0
2
y
  y2 

A2
2[1  (2 f 0 K ) 2 ]
Models uncorrelated input and output noise: Gmn(f) = Gum(f) = Gvn(f) = 0
n(t)
H(f)
h(t)
u(t)
m(t)

x(t )  u (t )  m(t )
v(t)

y(t)
x(t)
y (t )  v(t )  n(t )
 G xx ( f )  Guu ( f )  Gmm ( f ) G yy ( f )  Gvv ( f )  Gnn ( f )
G xy ( f )  Guv ( f )  H ( f ) Guu ( f ) Gvv ( f )  H ( f ) Guu ( f )
2
 H( f ) 
G xy ( f )
Guu ( f )

G xy ( f )
G xx ( f )  Gmm ( f )
H( f ) 
2
Gvv ( f ) G yy ( f )  Gnn ( f )

Guu ( f ) G xx ( f )  Gmm ( f )