0.
Time-Continous
Time
Continous Stochastic Processes
process X(t)
single realisation of
X(t) : sample function x(t)
A process is stationary, if the ensemble averages (moments,
autocorrelation function, cross-correlation function,
probability density functions) are independent of time.
If you can calculate the expected values of a process
by averaging one sample function in time domain
instead of averaging an ensemble of sample functions
functions,
then the process is ergodic.
An ergodic process is always stationary too ,
but a stationary process need not to be ergodic.
We will only
y consider ergodic
g
p
processes on the next slides.
0-1
Correlation Functions
Time-Continous
Time
Continous Stochastic Processes
process X (t ) , function g ( X (t )) :
moments
E{g ( X (t ))} = g (x ) p X ( x ) dx = lim
1st moment:
E{ X } =
T /2
1
T
g (x(t )) dt
T / 2
x p (x ) dx =
X
2nd moment:
{ } x
E X2 =
p X ( x ) dx
variance:
E X X
}= E{ X } (E{ X })
2
= =
2
X
0-2
Power Density Spectrum
2
X
p X ( x ) dx
Correlation Functions of Continous Processes
autocorrelation function (ACF) of a complex-valued process X(t):
rXX ( 1 , 2 ) = E X ( 1 ) X ( 2 ) = E{( X R ( 1 ) jX I ( 1 )) ( X R ( 2 ) + jX I ( 2 ))}
stationary processes: 1 t , 2 t +
autocovariance function:
{[
(t + ) X ]} = rXX ( ) X
c XX ( ) = rXX ( )
c XX ( ) = E X (t )
zero mean processes
rXX ( ) = E X (t ) X (t + )
] [X
cross correlation function (CCF) of two processes X(t) and Y(t):
cross-correlation
rXY ( 1 , 2 ) = E X ( 1 ) Y ( 2 )
0-3
stationary rXY
( ) = E{X (t ) Y (t + )}
Correlation Functions
Correlation Functions of Continous Processes
characteristics of ACF
(
)
r
=
r
XX
XX ( )
real processes
rXX ( ) = rXX ( )
max{rXX ( )} = rXX (0 )
{
(
)
(
)
(
)
(
)
rXX 0 = E{X t X t } = E X t }
zero mean
ACF is even
rXX (0) = X2
characteristics of CCF
rXY ( ) = rYX ( )
real processes
c XY ( ) = rXY ( ) X Y
rXY ( ) = rYX ( )
cross-covariance
uncorrelated processes: c XY ( ) = 0 rXY ( ) = X Y
orthogonal processes:
0-4
rXY ( ) = 0
uncorr. processes with
Correlation Functions
or
Y = 0
Power Density Spectrum
(Power Density Spectrum, PDS)
definition (Wiener-Khintshine theorem)
S XX ( j ) = F{ rXX ( )} =
j
r
(
)
e
d
XX
explanation of plausibility:
see slide 9
ACF is conjugate even power density spectrum is real
process average power (zero mean):
Var{X (t )} =
2
X
1
2
XX
( j ) d
= rXX (0)
white noise: PDS is a constant ( infinite power only a model)
S XX ( j ) = N 0 / 2 with - < <
rXX ( ) = F1{N 0 / 2} = N 0 / 2 0 ( )
0-5
Power Density Spectrum
Definition of Wiener-Khintschine : explanation of
plausibility
x(t ) if T t T
xT (t ) =
finite sample function:
X T ( j )
otherwise
0
T
finite energy:
xT (t ) dt =
2
average power:
1
2
X T ( j ) d
2
Theorem of Parseval
T
1
2T
xT (t ) dt =
2
1
2
1
2T
X T ( j ) d PDS of sample func.
2
definition: PDS of the ergodic process X(t)
mean off a sample
l ffunction
ti with
ith iinfinite
fi it llength
th :
S XX ( j ) = lim
0-6
1
2T
X T ( j )
Power Density Spectrum
Definition of Wiener-Khintschine : explanation of
plausibility
S XX ( j ) = F{rXX ( )} = rXX ( ) e j d
Wiener-Khintschine:
ergodic processes: rXX ( ) = lim
S XX ( j )
T
= lim
0-7
1
2T
lim
1
2T
(
)
(
)
(t ) X (t + )}
{
x
t
x
t
+
dt
E
X
T
T
e
d =
(
)
(
)
x
t
x
t
+
dt
T T
x ( t ) x ( t + )e
T T
1
2T
1
dtd = Tlim
2T
j
(
t
)
(
t
)
d dt
x
x
+
e
T T
Power Density Spectrum
Definition of Wiener-Khintschine : explanation of
plausibility
67
8
j
+ j t
(
)
(
)
x
t
x
t
+
e
e
T
T
= lim 21T
T
4
6
47
8
( t + )
d dt
(substitution: = t + ; d = d )
T
= lim 21T
T
0-8
j t
j
1
(
)
(
)
x
t
e
dt
x
e
=
( )
lim
T
T
T
T
2T X T j
T
14
4244
3 1442443
X T j
X T ( j )
Power Density Spectrum
(see slide 8)
Response of a linear system to a random input signal
random input signal : X (t )
h(t )
random output signal : Y (t )
impulse response:
(Energy-) ACF :
rhhE ( ) = h (t ) h(t + ) dt = h( ) h ( )
ACF of the output:
rYY ( ) = rXX ( ) rhhE ( ) = rXX ( ) h( ) h ( )
CCF off ini / output:
t t rXY ( ) = rXX ( )
PDS of output process:
h( )
SYY ( j ) = S XX ( j ) H ( j )
no phase information!
Cross-power density spectrum: S XY ( j ) = S XX ( j ) H ( j )
((between input
p and output)
p )
0-9
Response of a Linear System
Response of a linear system to a random input signal
white noise as input signal of a linear system:
rYY ( ) = N 0 / 2 0 ( ) rhhE ( ) = N 0 / 2 rhhE ( )
SYY ( j ) = N 0 / 2 H ( j )
rXY ( ) = N 0 / 2 0 ( ) h( ) = N 0 / 2 h( )
S XY ( j ) = N 0 / 2 H ( j )
0-10
Response of a Linear System
