chapter 8. linear systems with random inputs 1 0. introduction 1. linear system fundamentals 2....
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Chapter 8. Linear Systems with Random Inputs1
Chapter 8. Linear Systems with Random Inputs
0. Introduction
1. Linear system fundamentals
2. Random signal response of linear systems
3. ..
4. Spectral characteristics of system response
Chapter 8. Linear Systems with Random Inputs2
8.1 Linear system fundamentals
( ) ( ) ( ) ( ) ( )y t x h t d h x t d
( ) ( ) ( , )y t x h t d
Linear System
( ) ( , ) impulse responset h t
Linear Time-Invariant System (LTI system)
convolution integral( ) ( ) ( ) ( ) ( )y t x t h t h t x t
( ) ( ) ( )Y X H ( )( )( )
( ) ( ) ( )( )
j t
j t jj t
h e dy tx t e h e d H
x t e
Chapter 8. Linear Systems with Random Inputs3
8.1 Linear system fundamentals
Example 8.1-1: ( )R
H ssL R
( )R
Hj L R
LTI causal ( ) 0 for 0h t t
LTI stable ( )h t dt
Chapter 8. Linear Systems with Random Inputs4
8.1 Linear system fundamentals
Ideal lowpass filter
0 ,( )
0, o/w
jte WH
0 0
0
0 0
( )
( )
0
( ) ( )
0
0
0
1 1( )
2 2
1 1
2 ( )
1
2 ( )
sin[( ) ]
( )
W Wjt j t tjt
W W
W
j t t
W
j t t W j t t W
h t e e d e d
ej t t
e e
j t t
t t WW
t t W
Not causal Not physically realizable
Chapter 8. Linear Systems with Random Inputs5
8.2 Random signal response of linear systems
( ) ( ) ( )Y t h X t d
( ) -- w.s.s. random inputX t
( ) w.s.s. ( ) w.s.s.X t Y t
[ ( )] [ ( ) ( ) ] ( ) [ ( )]
( )
E Y t E h X t d h E X t d
X h d Y
1 1 1 2 2 2
1 2 1 2 1 2
1 2 1 2 1 2
( , ) [ ( ) ( )]
[ ( ) ( ) ( ) ( ) ]
[ ( ) ( )] ( ) ( )
( ) ( ) ( )
YY
XX
R t t E Y t Y t
E h X t d h X t d
E X t X t h h d d
R h h d d
Chapter 8. Linear Systems with Random Inputs6
8.2 Random signal response of linear systems
Example 8.2-1: white noise ( )X t 0( ) ( / 2) ( )XXR N
20 1 2 1 2 1 2
20 2 2
[ ( ) ] ( / 2) ( ) ( ) ( )
( / 2) ( )
E Y t N h h d d
N h d
1 2
1 2 1 1 2 2
2 1 1 1 2 2
1 1 2 2
( ) [ ( ) ( ) ] ( )
[ ( ) ( ) ] ( )
( ) ( ) ( )
( ) ( ) ( )
YY XX
XX
XX
XX
R R h d h d
R h d h d
R h h d
R h h
21 2 1 2 1 2[ ( ) ] ( ) ( ) ( )XXE Y t R h h d d
Chapter 8. Linear Systems with Random Inputs7
8.2 Random signal response of linear systems
( , ) [ ( ) ( )] [ ( ) ( ) ( ) ]
[ ( ) ( )] ( )
( ) ( )
( ) ( ) ( )
XY
XX
XX XY
R t t E X t Y t E X t h X t d
E X t X t h d
R h d
R h R
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
YX XY XX XX
XX
R R R h R h
R h d
( ) w.s.s. ( ) & ( ) jointly w.s.s.X t X t Y t
( ) ( ) ( ) ( ) ( )YY XY YXR R h R h
Chapter 8. Linear Systems with Random Inputs8
8.2 Random signal response of linear systems
Example 8.2-2: white noise ( )X t 0( ) ( / 2) ( )XXR N
0 0
( ) ( ) ( ) ( ) ( )
( / 2) ( ) ( ) ( / 2) ( )
XY XX XXR R h R h d
N h d N h
0( ) ( ) ( / 2) ( )YX XYR R N h
Chapter 8. Linear Systems with Random Inputs9
8.4 Spectral characteristics of system response
( ) ( ) ( )XY XXR R h ( ) ( ) ( )XY XXS S H
*( ) ( ) ( ) ( ) ( )YX XX XXS S H S H
2* *( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )YY XY XX XXS S H S H H S H
( ) ( )FTh H
*( ) ( ) ( )FTh H H ( ) real h
( ) ( ) ( )YX XXR R h
( ) ( ) ( ) ( ) ( ) ( )YY XY XXR R h R h h
Chapter 8. Linear Systems with Random Inputs10
8.4 Spectral characteristics of system response
2 02
/ 2( ) ( ) ( )
1 ( / )YY XX
NS S H
L R
21 1( ) ( ) ( )
2 2YY YY XXP S d S H d
average power
Example 8.4-1:0( )
2XX
NS
1( )
1 ( / )H
j L R
02
/ 2 / 220 0 02/ 2 / 2
1 1( )
2 4 1 ( / )
1sec
4 1 tan 4 4
YY YY
NP S d d
L R
N N R N RRd d
L L L
Chapter 8. Linear Systems with Random Inputs11
8.4 Spectral characteristics of system response
1( )
1 ( / )H
j L R
2 2 2 / 2 /0 0 0 0
00( ) ( / )
2 2 4 4Rt L Rt L
YY
N N N R N RP h t dt R L e dt e
L L
/( ) ( / ) ( ) FTRt Lh t R L u t e
By Example 8.2-1,