Principles of Communications Lecture 11: Stochastic ...

Microsoft PowerPoint - COMMI_lec11Chih-Wei Liu
Commun.-Lec11 [email protected] 2
Narrowband Noise
0, otherwise ( ) sinc(2 ).
White Noise: , ( ) ( ). 2
Therefore, the samples of ( ) are uncorrelated. ( , ) ( ) ( ) .n n n n n
N f B psd S f
R BN B NB R
n t R t t m t m t m m
τ τ
PAM with Random Delay ( ) ( ).
Assumptions: : WSS random process with [ ] 0 [ ] = a deterministic sequence of . : random delay, uniformly distributed over ( 2 2).
Mean of
k k
K k
a E a = E a a R m
- T / , T /
=−∞
If ( ) ( ) converges, it is independent
( )
( )
t t
R r mT
R r mT
function of the pulse function .
m m
p(t)
τ
T 1 2( )t k T− −
2
Case II: Correlated message (with memory) , where and are constants and is a sequence
, 0 of random variables that and [ ] ( ).
0, 0
a g A g A g g A
A m A A E A A R m
m tp t T
= Π unit pulse)
0 1 1 0 1 1 2 2 0 1 1 1 0 1 1 0 1 1
2 2 0 1 0 1 0 1
2 2 2 0 1
2 0 1
m k k m k k k m k m
k k m k k m k k m k k m
A A A A
m
R E a a E g A g A g A g A
E g A A g A A g g A A g g A A
g R m g R m g g R m g g R m
g g A m
R g A m
2 2 2 2 2 2 0 1 0 1
2 2 2 0 1
( ) ( ) {( ) ( ) [ ( ) ( )].
X m
X
T TR R A g g g g T T T T
S f A T fT g g g g e e
g g S f A T fT fT
π π
F.G. Stremler, Intr. Commu. Systems, p. 509, 3rd, ed, 1990.
Linear Systems and RP
System without memory: a r.v.→ a r.v. System with memory: correlated outputs
Now, we study only the statistics between inputs and outputs, e.g., my(t), Ry(τ), …
Assume X(t) stationary (or WSS at least) and H(⋅) is LTI.
H X(t) = X(t,ζ) Y(t) = Y(t,ζ)
y
X
XY
X
Y(t)
m t E Y t E h u X t u du
h u E X t u du m H
R E X t Y t E X t h u X t u du
h u E X t X t u du h u R u
ς ς
R R h R h R
S f S f H f S f H f S f
τ τ τ
∞
−∞
⇒ = ∗ ⇒ = • = − = − ∗ − = − ∗
= − = − =
∫
YX YX
Y YX
R E Y t Y t E Y t h t X t
E Y t h u X t u du h u E Y t X t u du
h u R u du h R
R h R h
) ( ) ( ).
( ) ( ) ( ) ( ) ( ) ( ). Remarks: (1) If ( ) is WSS and ( ) is LTI (no initial condition),
( ) is WSS too. (2) So far, we only consider 2nd order
X
h R
S f H f H f S f H f S f X t h t
Y t
τ τ∗ − ∗
= =
• ⇒
statistics (mean, correlation). In general, given the joint pdf of ( ), it is very difficult to find the joint pdf of ( ). But if ( ) is jointly Gaussian, then ( ) is also jointly Gaus
X t Y t X t Y t
sian and thus is completely characterized by mean and correlation functions.
Gaussian Random Process
Gaussian Random Process: X(t) has joint Gaussian pdf (of all orders).
Special Case 1: Stationary Gaussian random process
Mean = mx; auto-correlation = RX(τ).
White correlation in time Gaussian (or Lapacian,…) pdf on magnitude
0
( ) ( ) ( ) ( ) ( )
X t Y t X t
Y t h t X t X h t d
X k h t k τ
τ τ τ
τ τ τ
sian random variables. ( ) has a 1st-order Gaussian distribution. Similarly, the higher
order joint pdf of ( ) is jointly Gaussian. For example, 2, time = , ( , ) ( | ) ( ) Y Y Y
Y t Y t
∴
2 1
Case II: ( ) is Gaussian but not white (more realistic case). Claim: ( ) is produced by passing a white Gaussian random process through ( ). Then, define ( ) ( ) ( ), we have ( )
x t x t
•
= ∗ 2( ) ( ). Back to Case I. Therefore, ( ) is stationary Gaussian.
Note: We often use lower-case letters ( and instead of and ) for denoting random processes. You need to judge by the context
h t Z t y t
x y X Y
whether it is a deterministic signal or a random process.
Properties of Gaussian Processes
(1) X(t) Gaussian, H(⋅) stable, linear ⇒ Y(t) Gaussian. (2) X(t) Gaussian and WSS ⇒ X(t) is SSS. (3) Samples of a Gaussian process, X(t1), X(t2), …, are uncorrelated ⇒
They are independent. (4) Samples of a Gaussian process, X(t1), X(t2), …, have a joint
Gaussian pdf specified completely by the set of means and auto- covariance function. Remarks: Why do we use Gaussian model?
Easy to analyze. Central Limit Theorem: Many “independent” events combined together become a Gaussian random variable (random process).
)])()()([( ij XiXj mtXmtXE −−
0
0
0
3
3
1 1Filter: ( ) , 1 2 1
1 where =3 dB cutoff frequency = . 2
1Output: ( ) ( ) ( ) . 2 1 ( )
f RC
NR e RC
0 2
Mean: ( ) 0 (0) 0.
Another approach: ( ) lim ( ) 0.
N dx Nn t S f df RC x RC
n t H
n t R
f y t
2
0
2
0
1 .
2
Note: ( ) does not "completely" describes the behaviour of a random process.
RCy Ne
N RC
Noise Equivalent Bandwidth
Q: What is the bandwidth of an ideal fictitious filter that has the same mid-band gain as H(f) and that passes the same noise power?
( ) 2 2 0 0
NH f B H f df H
H
0
20
H f df N
H h t dt
H h t dt
Q
If the target system is a lowpass filter with maximum gain at f=0, then we have the following relationships.
Usage: BP noise (signal)
Interpretation: A baseband random process is shifted to a higher frequency.
2W
-f0 f0 f
0( ) ( )cos2 , ( ) : RP;cos() : carriern t R t f t R tπ=
0W f<<
0
0
However, a bandpass random signal can have a random phase: ( ) ( )cos(2 ( )).
In general, if we allow a time-invariant phase bias , then ( representation)
( ) ( )cos( ( ) ) o
n t R t t t
π φ θ
ω φ θ
c
n t n t t n t t n tR t n t n t t n t
ω θ ω θ
How to produce (t) and ( ): ( ) ( ) cos( ) ( ) sin( )
( in mean-square sense) Remarks: Here, we assume (initial phase) is a random variable, independent of ( ), uniformly distrib
c s
c s
n n t n t n t t n t t
n t
1 2
uted over (0, 2 ) or (- , ). If is not a random variable, ( ) and ( ) are not WSS. We cannot use LTI theory to predict the outputs of LPF's.
z t z t π π π
θ
25
n t n t n t
X t n t n t
R E X t X t n t n t n t n t
n t n t n t n t n t n t n t n t
R n t
(ii) { ( )} 2 ( ) cos( ) 0. { ( )} { ( )} (0) 0.
X X n
S n t d n t d
n t
E z t n t t E n t E z t H
τ τ τ τ
n n n n
S f S f Lowpass S f f S f f
S f f S f f W f W
S f j Lowpass S f f S f f
z t n t
cos( ). ( ) { ( ) ( )}
{4 ( ) ( ) cos( ) cos( ( ) )} 2 { ( ) ( )}cos 2 { ( ) ( ) cos(2 2 )} 2 ( ) cos 2 { ( ) ( )} {cos(2
Z
n
t R E z t z t
E n t n t t t E n t n t E n t n t t R E n t n t E t
ω θ τ τ
τ ω θ ω τ θ τ ω τ τ ω ω τ θ
τ ω τ τ ω ω τ
+ = +
= + + + + = + + + + + = + + +
Similar c
θ τ ω τ
+ =
= ∗ − + +
= − + +
∴ = − + +
0 0ly, ( ) { ( ) ( )}. sn n nS f Lowpass S f f S f f= − + +
Indept.
(iv) ( ) { ( ) ( )}
E n t n t t t R
S f j S f f S f f
R E n t n t
τ τ
τ τ
n n Z Z
E h u Z t u du h v Z t u dv
h u h v E Z t u Z t v dudv
h u h v R u v dudv
h h R
τ
τ
τ
n n
H f S f j H f S f f S f f
j Lowpass S f f S f f
= = − − +
= ⋅ − − +
( ) ( ) ( ) ( ) ( ) ( ).
(4) If ( ) is symmetric with respect to , then ( ) and ( ) are uncorrelated for all an
c s
c s
n c s
n t n t n t N
n t S f df S f df n t S f df n t
S f f n t n t t
∞ ∞ ∞
−∞ −∞ −∞
= = ≡
= = = = =∫ ∫ ∫
( ) and ( ) are uncorrelated. Remarks: If ( ) is NOT symmetric with res
c s
n t n t S f
τ τ
1 2 1 2
pect to , then ( ) and ( ) are uncorrelated at ( ) ( ( ) ( ) 2 ( )sin 0) ( ) those and such that ( ) 0
c s
c s
f n t n t
i t t h h R ii t t R t t
τ τ τ ω τ= − ∗ ∗ = − =
Q
is pdfjoint their t,independen are )( and )( If (6)
).( ofn combinatiolinear weightedare )( and )( :Proof Gaussian. are )( and )( then Gaussian, is )(If (5)
2
2/)(
2
22
0 0
0 0
(1) If 7 Hz, ( ) is symmetric with respect to . ( ) and ( ) are uncorrelated and
( ) 2 (2 6) 24.
(2) If 5 Hz, ( ) is NOT symmetric with respect to . ( ) and ( ) are co
n
N S f df
∞
−∞
= ⇒
= = ⋅ ⋅ =
= ⇒
∫
n n
S f S t Lowpass S f f S f f c
S f j Lowpass S f f S f f d
R τ τ πτ
0
Theorem: Given a WSS bandpass random process ( ) with BW = , then ( ) can be represented by ( )cos( ) ( )sin( ) in mean-square sense; that is, {[ ( ) ( ( )cos( ) ( )si
c s
c s
n t B n t n t t n t t
E n t n t t n t
ω θ ω θ
0 0 2
n( ))] } 0. Note that is a random variable uniformly distributed over (- , ) and is independent of ( ).
ˆProof: Let ( ) ( )cos( ) ( )sin( ). ˆWish to show {[ ( ) ( )] } 0. (
c s
n t n t n t t n t t
E n t n t
ω θ θ π π
ω θ ω θ
0
c s
c s
c s
E n t n t n t n t n t n t
n t E n t t n t t
n t t n t t
n t n t t
ω θ ω θ
ω θ ω θ
t
ω θ+

Principles of Communications Lecture 11: Stochastic ...

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