Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
1
Chapter 7. Random Process – Spectral Characteristics
0. Introduction
1. Power density spectrum and its properties
2. Relationship between power spectrum and autocorrelation function
3. Cross-power density spectrum and its properties
4. Relationship between cross-power spectrum and cross-correlation function
5. Power spectrums for discrete-time processes and sequences
6. Some noise definitions and other topics
7. Power spectrums of complex processes
1
7.1 Power density spectrum and its properties
1( ) ( )
2
j tx t X e d
ω
ω ω
π
∞
−∞
= ∫Inverse FT
Fourier transform ( ) ( )j t
X x t e dtω
ω
∞−
−∞
= ∫
Goal: Find the frequency component of RP X(t).
For deterministic signal x(t) it is simple
You can also get the time domain representation of the signal using
This is not the case with RP X(t). Why?
How to find the frequency spectrum of RP X(t)?
2
2
7.1 Power density spectrum and its properties
22 2 1( ) ( ) ( ) ( )
2
T
T TT
E T x t dt x t dt X dω ω
π
∞ ∞
−∞ − −∞
= = =∫ ∫ ∫
( ) ( ) ( )T
j t j t
T TT
X x t e dt x t e dtω ω
ω
∞− −
−∞ −
= =∫ ∫
( ),( )
0, /T
x t T t Tx t
o w
− < <=
Energy contained in ( ) in the interval ( , )x t T T−
Assume ( ) , for all finite .T
TT
x t dt T−
< ∞∫
Energy saving property
average power in ( ) over the interval ( , )x t T T−
2
2( )1 1
( ) ( )2 2 2
TT
T
XP T x t dt d
T T
ω
ω
π
∞
− −∞
= =∫ ∫Power=Energy/time
3
For a RP X(t), we find the average power using the same equation as
but with two modifications:
1. We take the expectation of X(t) to include all possible sample values.
2. We let T� infinity to cover the whole time domain
Thus we can write the average power of RP X(t) as
7.1 Power density spectrum and its properties
2{ [ ( ) ]}
XXP A E X t=
power density spectrum
2
2[ ( ) ]1 1
[ ( ) ]2 2 2
lim limT
T
XXT
T T
E XP E X t dt d
T T
ω
ω
π
∞
− −∞→∞ →∞
= =∫ ∫1
( )2
XX XXP S dω ω
π
∞
−∞
= ∫2
[ ( ) ]
2lim
T
XX
T
E XS
T
ω
→∞
=
Power Density Spectrum
Power Spectral Density
PSD4
3
7.1 Power density spectrum and its properties
2 2
2 2 2 0 0
0 0 0
2 2 2 2
0 0 0 02 20 0 00
2 2
0 0
0
[ ( ) ] [ cos ( )] [ cos(2 2 )]2 2
2cos(2 2 ) sin(2 2 )
2 2 2 2
sin(2 )2
A AE X t E A t E t
A A A At d t
A At
π π
θ
ω ω
ω θ θ ω θπ π
ωπ
=
= +Θ = + + Θ
= + + = + +
= −
∫
2 2 2
2 0 0 0
0
1{ [ ( ) ]} [ sin(2 )]
2 2 2lim
T
XXT
T
A A AP A E X t t dt
Tω
π−→∞
= = − =∫
2{ [ ( ) ]}
XXP A E X t=
w.s.s. (0)XX XX
P R⇒ =
Example 7.1-1:0 0
( ) cos( )X t A tω= +Θ -- uniformly distributed on (0, )2
π
Θ
5
7.1 Power density spectrum and its properties
power density spectrum
Example 7.1-2: 0 0( ) cos( )X t A tω= +Θ
0 0
0 0
0 0 0
( ) ( )0 0
0 00 0
0 0
1( ) cos( ) [ ]
2
2 2
sin[( ) ] sin[( ) ]
( ) ( )
T Tj t j tj t j j j t
TT T
T Tj t j tj j
T T
j j
X A t e dt A e e e e e dt
A Ae e dt e e dt
T TA Te A Te
T T
ω ωω ω
ω ω ω ω
ω ω
ω ω ω ω
ω ω ω ω
−− Θ − Θ −
− −
− − +Θ − Θ
− −
Θ − Θ
= +Θ = +
= +
− += +
− +
∫ ∫
∫ ∫
1 sin( )2
j T j TT T
j t j t
t TT
e e Te dt e T
j j T
β ββ β β
β β β
−
=−−
−
= = =∫2
[ ( ) ]
2lim
T
XX
T
E XS
T
ω
→∞
=
6
4
7.1 Power density spectrum and its properties
2 22 * 2 2 20 0
0 2 2 2 2
0 0
2 2 2 2 0 0
0
0 0
sin [( ) ] sin [( ) ]( ) ( ) ( ) [ ]
( ) ( )
sin[( ) ] sin[( ) ]( )
( ) ( )
T T T
j j
T TX X X A T T
T T
T TA T e e
T T
ω ω ω ω
ω ω ω
ω ω ω ω
ω ω ω ω
ω ω ω ω
Θ − Θ
− += = +
− +
− ++ +
− +
0 0
0 0
0 0
sin[( ) ] sin[( ) ]( )
( ) ( )
j j
T
T TX A Te A Te
T T
ω ω ω ω
ω
ω ω ω ω
Θ − Θ− +
= +
− +
2 2 / 22
00
2 2[ ] [2cos2 ] 2cos2 sin 2 0j j
E e e E d
π
π
θ θ θπ π
Θ − Θ+ = Θ = = =∫
22 2 2
0 0 0
2 2 2 2
0 0
[ ( ) ] sin [( ) ] sin [( ) ][ ]
2 2 ( ) ( )
TE X A T TT T
T T T
ω π ω ω ω ω
π ω ω π ω ω
− +
= +
− +
* 0 0
0 0
0 0
sin[( ) ] sin[( ) ]( )
( ) ( )
j j
T
T TX A Te A Te
T T
ω ω ω ω
ω
ω ω ω ω
− Θ Θ− +
= +
− +
7
7.1 Power density spectrum and its properties
22
0
0 0
[ ( ) ]( ) lim [ ( ) ( )]
2 2
T
XXT
E X AS
T
ω πω δ ω ω δ ω ω
→∞
= = − + +
2
2
sin (C-54)
xdx
xπ
∞
−∞
=∫
2
2
, if 0sin ( )lim (b)
0, if 0( )T
T T
T
αα
απ α→∞
∞ ==
≠2
2
sin ( )(a) & (b) lim ( )
( )T
T T
T
αδ α
π α→∞
⇒ =
2 2
2 2
sin ( ) sin 11 (a)
( )
T T T xd dx
T x T
α
α
π α π
∞ ∞
−∞ −∞
= =∫ ∫
2 2
0 0
0 0
1 1( ) [ ( ) ( )]
2 2 2 2XX XX
A AP S d d
πω ω δ ω ω δ ω ω ω
π π
∞ ∞
−∞ −∞
= = − + + =∫ ∫
8
5
7.1 Power density spectrum and its properties
Properties of the power density spectrum:
(1) ( ) 0XX
S ω ≥
(2) ( ) real ( ) ( )XX XX
X t S Sω ω⇒ − =
(3) ( ) is realXX
S ω
21(4) ( ) { [ ( ) ]}
2XX
S d A E X tω ω
π
∞
−∞
=∫
( ) ( )T
j t
TT
X X t e dtω
ω−
−
= ∫
* *[ ( ) ( ) ] [ ( ) ( )]( ) lim lim ( )
2 2
T T T T
XX XXT T
E X X E X XS S
T T
ω ω ω ω
ω ω→∞ →∞
− −
− = = =
PF of (2):
* *( ) ( ) ( ) ( )
T Tj t j t
T TT T
X X t e dt X t e dt Xω ω
ω ω− −
= = = −∫ ∫
2
[ ( ) ]( ) lim
2
T
XXT
E XS
T
ω
ω→∞
=
9
7.1 Power density spectrum and its properties
Properties of the power density spectrum
2(5) ( ) ( )
XXXXS Sω ω ω=ɺ ɺ
22 2
2 2[ ( ) ] [ ( ) ] [ ( ) ]
( ) lim lim lim ( )2 2 2
T T T
XXXXT T T
E X E j X E XS S
T T T
ω ω ω ω
ω ω ω ω→∞ →∞ →∞
= = = =ɺ ɺ
ɺ
PF of (5):
0
( ) ( )( ) lim
d X t X tX t
dt ε
ε
ε→
+ −=
0
( ) ( )lim ,
( )
0, o/wT
X t X tT t T
X t ε
ε
ε→
+ −− < <
=
ɺ
FT
0
( ) ( )( ) lim = ( )
j
T TT T
X e XX t j X
ωε
ε
ω ω
ω ω
ε→
−←→
ɺ
( ) ( )FT j af t a F e ω
ω−
− ←→
10
6
7.1 Power density spectrum and its properties
Bandwidth of the power density spectrum
( ) real ( ) evenXX
X t S ω⇒
( ) lowpass form XX
S ω ⇒
2
2
rms
( )
( )
XX
XX
S d
W
S d
ω ω ω
ω ω
∞
−∞
∞
−∞
=
∫
∫
2
02 0
rms
0
4 ( ) ( )
( )
XX
XX
S d
W
S d
ω ω ω ω
ω ω
∞
∞
−
=
∫
∫
root mean square Bandwidth
mean frequency
rms BW
( ) bandpass form XX
S ω ⇒0
0
0
( )
( )
XX
XX
S d
S d
ω ω ω
ω
ω ω
∞
∞=
∫
∫
11
7.1 Power density spectrum and its properties
Example 7.1-3: ( ) lowpass formXX
S ω2 2
10( )
[1 ( /10) ]XX
S ω
ω
=
+
/ 22
2 2 2 2/ 2
/ 2 / 2 / 22
2/ 2 / 2 / 2
10 10( ) 10sec
[1 ( /10) ] [1 tan ]
100 1 cos2100cos 100 50
sec 2
XXS d d d
d d d
π
π
π π π
π π π
ω ω ω θ θω θ
θθ θ θ θ π
θ
∞ ∞
−∞ −∞ −
− − −
= =
+ +
+= = = =
∫ ∫ ∫
∫ ∫ ∫
2
2
rms
( )100
( )
XX
XX
S d
W
S d
ω ω ω
ω ω
∞
−∞
∞
−∞
= =
∫
∫rms
10 rad/secW =rms BW
12
7
7.2 Relationship between power spectrum and autocorrelation function
(6)
1( ) [ ( , )]
2
( ) [ ( , )]
j
XX XX
j
XX XX
S e d A R t t
S A R t t e d
ωτ
ωτ
ω ω τ
π
ω τ τ
∞
−∞
∞−
−∞
= +
= +
∫
∫1 2
1 2
1 2
*
1 1 2 2
( )
1 2 2 1
( )
1 2 2 1
[ ( ) ( )] 1( ) lim lim [ ( ) ( ) ]
2 2
1lim [ ( ) ( )]
2
1lim ( , )
2
T Tj t j tT T
XXT TT T
T Tj t t
T TT
T Tj t t
XXT TT
E X XS E X t e dt X t e dt
T T
E X t X t e dt dtT
R t t e dt dtT
ω ω
ω
ω
ω ω
ω−
− −→∞ →∞
−
− −→∞
−
− −→∞
= =
=
=
∫ ∫
∫ ∫
∫ ∫
1 2
1 2
( )
1 2 2 1
( )
1 2 2 1
1 1 1( ) lim ( , )
2 2 2
1 1lim ( , )
2 2
T Tj t tj j
XX XXT TT
T Tj t t
XXT TT
S e d R t t e dt dt e dT
R t t e d dt dtT
ωωτ ωτ
ω τ
ω ω ω
π π
ω
π
∞ ∞−
−∞ −∞ − −→∞
∞+ −
− − −∞→∞
=
=
∫ ∫ ∫ ∫
∫ ∫ ∫13
7.2 Relationship between power spectrum and autocorrelation function
( ) [ ( , )]j
XX XXS A R t t e d
ωτ
ω τ τ
∞−
−∞
= +∫
1 2 1 2 2 1
1 1 1
1 1( ) lim ( , ) ( )
2 2
1 1lim ( , ) lim ( , )
2 2
[ ( , )]
T Tj
XX XXT TT
T T
XX XXT TT T
XX
S e d R t t t t dt dtT
R t t dt R t t dtT T
A R t t
ωτ
ω ω δ τπ
τ τ
τ
∞
−∞ − −→∞
− −→∞ →∞
= + −
= + = +
= +
∫ ∫ ∫
∫ ∫
( ) 1FT
tδ ←→
( ) 1
1( )
2
j t
j t
t e dt
t e d
ω
ω
δ
δ ωπ
∞−
−∞
∞
−∞
=
=
∫
∫
[ ( , )] ( )FT
XX XXA R t t Sτ ω+ ←→
14
8
7.2 Relationship between power spectrum and autocorrelation function
( ) ( )j
XX XXS R e dωτ
ω τ τ
∞−
−∞
= ∫
1( ) ( )
2
j
XX XXR S e dωτ
τ ω ω
π
∞
−∞
= ∫
( ) ( )FT
XX XXR Sτ ω←→
( ) w.s.s. [ ( , )] ( )XX XX
X t A R t t Rτ τ⇒ + =
15
7.2 Relationship between power spectrum and autocorrelation function
2
0
0 0( ) [ ( ) ( )]
2XX
AS
πω δ ω ω δ ω ω= − + +
1 2 ( )FT
πδ ω←→
Example 7.2-1:0
( ) cos( )X t A tω= +Θ
2
0
0by Ex 6.2-1, ( ) cos( )
2XX
AR τ ω τ=
( ) ( )FTj t
x t e Xα
ω α←→ −
0 0
2
0( ) ( )4
j j
XX
AR e e
ω τ ω τ
τ−
= +
16
9
7.2 Relationship between power spectrum and autocorrelation function
0 00 0
( ) (1 )( ) 2 (1 )cos( )T T
j j
XXS A e e d A d
T T
ωτ ωττ τ
ω τ ωτ τ−
= − + = −∫ ∫
0
0 00
( ) ( ) (1 ) (1 )T
j j j
XX XXT
S R e d A e d A e dT T
ωτ ωτ ωττ τ
ω τ τ τ τ
∞− − −
−∞ −
= = + + −∫ ∫ ∫
Example 7.2-2: ( ) -- w.s.s.X t
0[1 ] ,
( )
0 , elsewhere
XX
A T TR T
τ
τ
τ
− − < <
=
1cos( ) sin( )u uωτ ωτ
ω
′ = ⇒ =1
1 v vT T
τ −′= − ⇒ =
0 0
0
(1 ) (1 ) ( ) (1 )T
j j j
T Te d e d e d
T T T
ωτ ωα ωττ α τ
τ α τ−
−
+ = − − = −∫ ∫ ∫
17
7.2 Relationship between power spectrum and autocorrelation function
0
0
0
0
0 0
2 2
0
2 2
0 02 2
2
0
sin( )( ) 2 (1 )
2 sin( )
2 2cos( )[1 cos( )]
sin ( / 2) sin ( / 2)4
( / 2)
Sa ( / 2)
T
XX
T
T
S AT
Ad
T
A AT
T T
T TA A T
T T
A T T
τ ωτ
ω
ω
ωτ
τ
ω
ωτ
ω
ω ω
ω ω
ω ω
ω
= −
+
−= = −
= =
=
∫
18
10
7.6 Some noise definitions and other topics
white noise & colored noise ( )N t
colored noise
1( ) unrealizable
2NN
S dω ω
π
∞
−∞
= ∞ ⇒∫
white noise
19
7.6 Some noise definitions and other topics
sin( )( )
NN
WR P
W
τ
τ
τ
=
lowpass type
bandpass type
0
sin( / 2)( ) cos( )
( / 2)NN
WR P
W
τ
τ ω τ
τ
=
band-limited white noise
20
11
7.6 Some noise definitions and other topics
3
( )NN
R Peτ
τ−
=
3
0(3 ) (3 )
0
0
(3 ) (3 )
0
2
( )
1 1
3 3
1 1 6[ ]3 3 9
j
NN
j j
j j
S Pe e d
Pe d Pe d
P e P ej j
PP
j j
τ ωτ
ω τ ω τ
ω τ ω τ
ω τ
τ τ
ω ω
ω ω ω
∞− −
−∞
∞− − +
−∞
∞
− − +
−∞
=
= +
−= +
− +
= + =
− + +
∫
∫ ∫
Example 7.6-1: w.s.s. ( )N t
21
7.6 Some noise definitions and other topics
0 0( ) ( ) cos( )Y t X t A tω=
2
0 0 0 0
2
0
0 0 0
( , ) [ ( ) ( )] [ ( ) ( )cos( )cos( )]
( , )[cos( ) cos(2 )]2
YY
XX
R t t E Y t Y t E A X t X t t t
AR t t t
τ τ τ ω ω ω τ
τ ω τ ω ω τ
+ = + = + +
= + + +
( ) -- w.s.s. ( ) -- NOT w.s.s. X t Y t⇒
2
0
0 0( ) [ ( ) ( )]
4YY XX XX
AS S Sω ω ω ω ω= − + +
2
0
0[ ( , )] ( )cos( )
2YY XX
AA R t t Rτ τ ω τ+ =
22
12
7.6 Some noise definitions and other topics
2
0 0 0
2
0 0
2
0 0 0
/ 8, 2 / 2
/ 4, / 2( )
/8, 2 / 2
0, o/w
RF
RF
YY
RF
N A W
N A WS
N A W
ω ω
ω
ω
ω ω
+ <
<=
− <
2
0 0/ 4, / 2
( )0, o/w
RF
YY
N A WS
ω
ω
<=
0 0( ) ( ) cos( )Y t X t A tω=
Example 7.6-2: 0 0
0 0
/ 2, / 2
( ) / 2, / 2
0, o/w
RF
XX RF
N W
S N W
ω ω
ω ω ω
+ <
= − <
After lowpass filtering,
2 2/ 2
0 0 0 0
/ 2
1output noise power =
2 4 8
RF
RF
WRF
W
N A N A Wdω
π π−
=∫23