Autocorrelation - F.T - power spectrum
¾ Discrete and continuous
Power spectrum density
¾ realization by time averagey g
Review
¾ Even function property of autocorrelation
¾ Even function property of autocorrelation
The spectrum of the time function
: The weighting function of the Fourier series or transform
A sample function of a random process
: selected from an ensemble of allowable time functions
The weighting function or spectrum for a random process
¾ The average rate of change of the ensemble of allowable time functions
functions
¾ The autocorrelation function RX (τ ) is an appropriate measure for the avg. rate of change of a random process
Einstein-Wiener-Khinchin theorem
: the power spectral density of a wide-sense stationary random process is given by the Fourier transform of the autocorrelation function
is given by the Fourier transform of the autocorrelation function
X(t)
¾ a continuous-time WSS random processp
¾ mean = mX
¾ autocorrelation function = RX (τ )
¾ power spectral density of X(t)
= R τ f
SX ( ) F{ X ( )}
∫
−∞∞= RX (τ)e−j2πfτdτ
RX (
τ
) = RX (−τ
) : an even function ofτ
with assumption of a real valued random process∴
τ τ π τ
τ τ π τ
π τ
d f R
d f j
f R
f
SX X
∫
∫
∞
∞
−
=
−
=
2 cos ) (
) 2
sin 2
)(cos (
) (
SX ( f )
τ τ π
τ f d
RX
∫
−∞= ( )cos2
¾ real-valued
¾ an even function of f
¾ S ( f ) 0 f ll f
¾ SX ( f ) ≥ 0 for all f
The inverse Fourier transform of the power spectral
The inverse Fourier transform of the power spectral density
df f
S
f S R
f j X
X
∫
∞= −
τ π
τ
2 1
) (
)}
( {
)
( F
df e
f
SX j f
∫
−∞= ( ) 2πτ
The average power of X(t)
df f
S R
t X
E[X 2(t)] = R (0) =
∫
∞ S ( f )dfE[ ( )] = X (0) =
∫
−∞ X ( ) RX (
τ
) = CX (τ
) + mX2} )
( {
)
( f C 2
S
) ( )}
( {
} )
( {
) (
2 2
f m
C
m C
f S
X X
X X
X
δ τ
τ
+
=
+
= F F
⇒ mX : the “dc” component of X(t)
X X
Cross power spectral density SX,Y ( f )
: two jointly wide-sense stationary processesj y y p )}
( {
)
( ,
,Y X Y
τ
X f R
S =F
)]
( ) (
[ )
( E X t Y t
R +
where
SX Y ( f ) : a complex function of f even if X(t) and Y(t)
)]
( ) (
[ )
, ( E X t Y t
RX Y τ = +τ
where
X,Y ( f ) p f ( ) ( )
are both real-valued.
Ex. 7.1
Find the power spectral density of the random telegraph Find the power spectral density of the random telegraph signal
l) R ( ) −2α τ sol) RX ( ) e 2 ,
α
α
= α τsignal the
of rate transition
average the
:
2
0 2 2 2
)
( f e e d e e d
S
∫
ατ −j πfττ
+∫
0∞ − ατ −j πfττ
4)
( f e e d e e d
SX j f j f
α
τ τ
=
+
=
∫
−∞∫
2 2
2 4
4
α
+π
f=
Ex. 7.2
is where
,
Let X (t) = a cos(2
π
f0t + Θ) Θ .
Find distributed in the interval .
uniformly ,
)
( (0,2 )
) (
)
( 0
f S
f
X
π
sol)
τ
2 cos 2π
0τ
) 2
( a f
RX =
∴ {cos2 } ) 2
( 0
2
a f f
SX = F
π τ
) 4 (
) 4 (
2
0 2
0 2
f a f
f
a f − + +
=
δ δ
4 4
Note
¾ The average power of the signalg p g ) 2
0 (
a2
RX =
¾ All of this power is concentrated at the frequencies ±f0
Xn : a discrete-time WSS random process with mean mX and autocorrelation function RXX (k)( )
Power spectral density of X
Power spectral density of Xn
= X
X f R k
S ( ) F{ ( )}
∑
∞−∞
=
= − k
fk j
X k e
R ( ) 2π
∞ k=
Note
¾ Only consider frequencies in the range . 2 1 2
1 ≤ ≤
− f
y q g
∵ SX ( f ) is periodic in f with period 1.
¾ a real valued, nonnegative, even function of f .
2
2 f
The inverse Fourier transform of SX ( f ) df
e f S
k
RX =
∫
−12 X j fk 21
) 2
( )
( π
Note
RX (k) : the coefficients of the Fourier series of the periodic functions S ( f )
2
functions SX ( f )
The cross-power spectral density SX,Y ( f ) of two jointly WSS discrete-time processes Xp nn and Ynn
)}
( {
)
( ,
, f R k
SX Y =F X Y
h RX,Y (k) = E[Xn+kYn] where
Ex. 7.7
Let the process Y be defined by Y = X +
α
X 1Let the process Yn be defined by Yn Xn +
α
Xn−1, where Xn is the white noise processFind S ( f ) Find SY ( f ).
sol)
2 2
[ ] 0n E Y =
⎧ 2 2
2
(1 ) 0
( ) [ ] 1
X
n n k X
k
R k E Y Y k
α σ
+ ασ
⎧ + =
= = ⎪⎨ = ±
⎪0
⎪⎩ otherwise
∴ SY ( f ) = (1+
α
2)σ
X2 +ασ
X2 {e j2πf + e− j2πf } }2 cos 2
) 1
{( 2
2 f
X
α α π
σ
+ +=
¾ For α = 1
Let X0,…,Xk−1 be k (time) observations from the discrete-time WSS process p
Let ~xk ( f ) : the discrete Fourier transform of this
Let : the discrete Fourier transform of this sequence
∑
− −= 1 2
)
~x ( f k X e j πfm )
( f xk
Note
∑
==
0
) (
m
m
k f X e
x
Note
: a complex-valued random variable measure of the energy at f
)
~xk( f
measure of the energy at f
The magnitude squared of
: a measure of the energy at the frequency f )
~xk ( f
: a measure of the energy at the frequency f
The “power” at the frequency f
The power at the frequency f
9 (time average)
th i d ti t f th t l d it
) 2
~ ( ) 1
~ ( x f
f k
pk = k
: the periodogram estimate for the power spectral density
¾ Note
k
¾ Note
Divide the energy by the total “time” k.
The expected value of the periodogram estimate
~∗
1 ~
~
−
−
∗
⎤
⎡
=
1
1 1
)]
~ ( )
~ ( 1 [
)]
~ ( [
k k
k k
k E x f x f
f k p
E
∑
∑
=−
=
− ⎥⎦
⎢ ⎤
⎣
= ⎡ 1
0 1 2
0
1 2 k
i
fi j i k
m
fm j
me X e
X k E
π π
∑∑
−=
−
=
−
= 1 − 0
1
0
) (
] 2
1 k [
m k
i
i m f j i
mX e
X k E
π
∑∑
− − − − −= 1 1 2 ( )
0 0
)
1 k k ( j f m i
X
m i
e i m k R
π
=0 =0
m i
k
cf)
¾ R ( i) is constant along the diagonal ’ i
¾ RX (m−i) is constant along the diagonal m’ = m−i.
¾ m’ ranges from −(k−1) to k−1
¾ k |m’| terms along the diagonal m’ = m i
¾ k−|m | terms along the diagonal m = m−i
∴
∑
−
−
′=
′ −
− ′
=
) 1 (
) 2
( }
{ )]
~ ( [
k m
m f j X
k k m R m e
f k p
E π
∑
−′
− ′
⎭ ′
⎬⎫
⎩⎨
⎧ ′
−
= 1
) 1 (
) 2
(
k 1
k
m f j
X m e
k R
m π
Note
−
−
′= (k 1)⎩ ⎭
m k
∞ 2
[ ( )] ( ) ( )
2
j fk
k X X
k
E p f S f R k e
T k
π
−
=−∞
≠ =
∑
Differences (replace with in Chap. 6.7) 1 m
k
⎧ ′ ⎫
⎪ − ⎪
⎨ ⎬
⎪ ⎪
⎩ ⎭
∑
1. term
2 li it f ∑ 2. limits of
Note
¾ ~pk( f ) is a "biased" estimator for SX ( f )
¾ k
k goes infinite, m approaches
As 1 1
⎭⎬
⎫
⎩⎨
⎧ ′
−
¾ E pk f SX f as k
.
approaches summation
of limits
and
) ( )]
~ (
[ → → ∞
∞
±
¾
f f
x f
p
f f
SX
all for e
nonnegativ is
all for e
nonnegativ is
) 2
~ ( ) 1
~ ( ) (
∵
¾
f f
k x f
pk k
should estimate
m periodogra the
of variance The
all for e
nonnegativ ) is
( )
(
∵ =
zero.
approaches also
Chapter 7
4,8,12,14,16
4,8,12,14,16
