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Chapter 7 Generating and Processing Random Signals. 第一組 電機四 B93902016 蔡馭理 資工四 B93902076 林宜鴻. Outline. Outline. Stationary and Ergodic Process Uniform Random Number Generator Mapping Uniform RVs to an Arbitrary pdf Generating Uncorrelated Gaussian RV Generating correlated Gaussian RV - PowerPoint PPT Presentation
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1
Chapter 7Generating and Processing
Random Signals
第一組電機四 B93902016 蔡馭理資工四 B93902076 林宜鴻
2
Outline
Stationary and Ergodic ProcessUniform Random Number GeneratorMapping Uniform RVs to an Arbitrary pdfGenerating Uncorrelated Gaussian RVGenerating correlated Gaussian RVPN Sequence GeneratorsSignal processing
Outline
3
Random Number Generator
Noise, interferenceRandom Number Generator- computation
al or physical device designed to generate a sequence of numbers or symbols that lack any pattern, i.e. appear random, pseudo-random sequence
MATLAB - rand(m,n) , randn(m,n)
4
Stationary and Ergodic Process
strict-sense stationary (SSS)wide-sense stationary (WSS) Gaussian
SSS =>WSS ; WSS=>SSSTime average v.s ensemble average The ergodicity requirement is that the ensemble
average coincide with the time averageSample function generated to represent signals,
noise, interference should be ergodic
5
Time average v.s ensemble average
Time average ensemble average
6
Example 7.1 (N=100)
0 0.5 1 1.5 2-1
0
1
x(t)
0 0.5 1 1.5 2-0.5
0
0.5
x ensem
ble-
avar
age(
t)0 0.5 1 1.5 2
-1
0
1
y(t)
0 0.5 1 1.5 2-1
0
1
y ensem
ble-
avar
ag(t
)
0 0.5 1 1.5 2-2
0
2
z(t)
0 0.5 1 1.5 2-2
0
2z ens
embl
e-av
arag
(t)
)2cos()(),( iii φπftμ1Aξtx
)2cos(),( ii φπftAξtx
7
Uniform Random Number Genrator
Generate a random variable that is uniformly distributed on the interval (0,1)
Generate a sequence of numbers (integer) between 0 and M and the divide each element of the sequence by M
The most common technique is linear congruence genrator (LCG)
8
Linear Congruence
LCG is defined by the operation:
xi+1=[axi+c]mod(m)
x0 is seed number of the generator
a, c, m, x0 are integer
Desirable property- full period
9
Technique A: The Mixed Congruence Algorithm
The mixed linear algorithm takes the form:
xi+1=[axi+c]mod(m)
- c≠0 and relative prime to m
- a-1 is a multiple of p, where p is the
prime factors of m
- a-1 is a multiple of 4 if m is a
multiple of 4
10
Example 7.4
m=5000=(23)(54)c=(33)(72)=1323a-1=k1‧2 or k2‧5 or 4‧k3 so, a-1=4‧2‧5‧k =40kWith k=6, we have a=241
xi+1=[241xi+ 1323]mod(5000)We can verify the period is 5000, so it’s full
period
11
Technique B: The Multiplication Algorithm With Prime Modulus
The multiplicative generator defined as :
xi+1=[axi]mod(m)
- m is prime (usaually large)
- a is a primitive element mod(m)
am-1/m = k =interger
ai-1/m ≠ k, i=1, 2, 3,…, m-2
12
Technique C: The Multiplication Algorithm With Nonprime Modulus
The most important case of this generator having m equal to a power of two :
xi+1=[axi]mod(2n)
The maximum period is 2n/4= 2n-2
the period is achieved if
- The multiplier a is 3 or 5
- The seed x0 is odd
13
Example of Multiplication Algorithm With Nonprime Modulus
a=3
c=0
m=16
x0=1
0 5 10 15 20 25 30 351
2
3
4
5
6
7
8
9
10
11
14
Testing Random Number Generator
Chi-square test, spectral test……Testing the randomness of a given sequen
ceScatterplots
- a plot of xi+1 as a function of xi
Durbin-Watson Test
-
N
n
N
n
nXN
nXnXN
2
2
2
2
][)/1(
])1[][()/1(D
15
ScatterplotsExample 7.5
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(i) rand(1,2048)
(ii)xi+1=[65xi+1]mod(2048)
(iii)xi+1=[1229xi+1]mod(2048)
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Durbin-Watson Test (1)
N
n
N
n
nXN
nXnXND
2
2
2
2
][)/1(
])1[][()/1(
}({1
}{
}({ 22x
2
2
Y)-XEXE
Y)-XED
Let X = X[n] & Y = X[n-1]
ZρρXY 21 11 ρ
Let
Assume X[n] and X[n-1] are correlated and X[n] is an ergodic process
17
Durbin-Watson Test (2)
222222
)1()1()1(2)1(1
ZρXZρρXρEσ
D
)1(2)1()1(
2
2222
ρσ
σρσρD
X and Z are uncorrelated and zero mean
D>2 – negative correlation
D=2 –- uncorrelation (most desired)
D<2 – positive correlation
18
Example 7.6
rand(1,2048) - The value of D is 2.0081 and ρ is 0.0041.
xi+1=[65xi+1]mod(2048) - The value of D is 1.9925 and ρ is 0.0037273.
xi+1=[1229xi+1]mod(2048) - The value of D is 1.6037 and ρ is 0.19814.
19
Minimum Standards
Full period Passes all applicable statistical tests for
randomness.Easily transportable from one computer to
anotherLewis, Goodman, and Miller Minimum
Standard (prior to MATLAB 5)xi+1=[16807xi]mod(231-1)
20
Mapping Uniform RVs to an Arbitrary pdf
The cumulative distribution for the target random variable is known in closed form – Inverse Transform Method
The pdf of target random variable is known in closed form but the CDF is not known in closed form – Rejection Method
Neither the pdf nor CDF are known in closed form – Histogram Method
21
Inverse Transform Method
CDF FX(X) are known in closed form
U = FX (X) = Pr { X ≦ x }
X = FX-1
(U)
FX (X) = Pr { FX-1
(U) ≦ x } = Pr {U ≦ FX (x) }= FX (x)
FX(x)
1
U
FX-1(U) x
22
Example 7.8 (1)
Rayleigh random variable with pdf –
∴
Setting FR(R) = U
)(2
exp)(2
2
2ru
σ
r
σ
rrf R
2
2
2
2
0 2 2exp1)(
σ
rdy
2σ
yexp
σ
yrF
r
R
Uσ
r
2
2
2exp1
23
Example 7.8 (2)
∵ RV 1-U is equivalent to U (have same pdf) ∴
Solving for R gives
[n,xout] = hist(Y,nbins) - bar(xout,n) - plot the histogram
Uσ
r
2
2
2exp
)ln(2R 2 Uσ
24
Example 7.8 (3)
0 1 2 3 4 5 6 7 8 90
500
1000
1500
Num
ber
of S
ampl
es
Independent Variable - x
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
Pro
babi
lity
Den
sity
Independent Variable - x
true pdf
samples from histogram
25
The Histogram Method
CDF and pdf are unknownPi = Pr{xi-1 < x < xi} = ci(xi-xi-1)
FX(x) = Fi-1 + ci(xi-xi-1)
FX(X) = U = Fi-1 + ci(X-xi) more samples
more accuracy!
1
1111 }Pr{
i
jiii PXXF
)(1
11 ii
i FUc
xX
26
Rejection Methods (1)
Having a target pdf MgX(x) ≧ fX(x), all x
otherwise
0
,0
a/)(
axMb xMg X
}max{ (x)fa
Mb X
axx+dx
M/a=b
1/a
0
0
MgX(x)
fX(x)
gX(x)
27
Rejection Methods (2)
Generate U1 and U2 uniform in (0,1)
Generate V1 uniform in (0,a), where a is the maximum value of X
Generate V2 uniform in (0,b), where b is at least the maximum value of fX(x)
If V2 ≦ fX(V1), set X= V1. If the inequality is not satisfied, V1 and V2 are discarded and the process is repeated from step 1
28
Example 7.9 (1)
R0
0
MgX(x)
fX(x)
gX(x)
πRR
M 4
R
1
otherwise0,
Rx0xRπRxf X
222
4)(
29
Example 7.9 (2)
0 1 2 3 4 5 6 70
50
100
150
Num
ber
of S
ampl
es
Independent Variable - x
0 1 2 3 4 5 6 70
0.05
0.1
0.15
0.2
Pro
babi
lity
Den
sity
Independent Variable - x
true pdf
samples from histogram
30
Generating Uncorrelated Gaussian RV
Its CDF can’t be written in closed form , so Inverse method can’t be used and rejection method are not efficient
Other techniques
1.The sum of uniform method
2.Mapping a Rayleigh to Gaussian RV
3.The polar method
31
The Sum of Uniforms Method(1)
1.Central limit theorem2.See next
.
3.
0
1( )
2
N
ii
Y B U
iU 1,2..,i N represent independent uniform R.V
B is a constant that decides the var of Y
N Y converges to a Gaussian R.V.
32
The Sum of Uniforms Method(2)
Expectation and Variance
We can set to any desired valueNonzero at
1{ }
2iE U 0
1{ } ( { } ) 0
2
N
ii
E Y B E U
1/ 2 2
1/ 2
1 1var{ }
2 12iU x dx
2
2 2
1
1var{ }
2 12
N
y ii
NBB U
12yB
N
123
2y y
NN
N
33
The Sum of Uniforms Method(3)
Approximate GaussianMaybe not a realistic situation.
34
Mapping a Rayleigh to Gaussian RV(1)
Rayleigh can be generated by
U is the uniform RV in [0,1] Assume X and Y are indep. Gaussian RV
and their joint pdf
22 lnR U
2 2
2 2
1 1( , ) exp( ) exp( )
2 22 2XY
x xf x y
2 2
2 2
1exp( )
2 2
x y
35
Mapping a Rayleigh to Gaussian RV(2)
Transform
let and
and
cosx r siny r 2 2 2x y r 1tan ( )
y
x
( , ) ( , )R R XY XYf r dA f x y dA
/ /( , )
/ /( , )XY
R
dx dr dx ddA x yr
dy dr dy ddA r
2
2 2( , ) exp( )
2 2R
r rf r
36
Mapping a Rayleigh to Gaussian RV(3)
Examine the marginal pdf
R is Rayleigh RV and is uniform RV
2 22
2 2 2 20( ) exp( ) exp( )
2 2 2R
r r r rf r d
0 r
2
2 20
1( ) exp( )
2 2 2
r rf dr
0 2
cosX R 2
1 22 ln( ) cos 2X U U
sinY R 21 22 ln( ) sin 2Y U U
37
The Polar Method
From previous
We may transform
21 22 ln( ) cos 2X U U 2
1 22 ln( ) sin 2Y U U
2 2 2 ( )s R u v R s
1cos 2 cosu u
UR s
2sin 2 sinv v
UR s
22 2
1 2
2 ln( )2 ln( ) cos 2 2 ln( )( )
u sX U U s u
ss
22 2
1 2
2 ln( )2 ln( ) sin 2 2 ln( )( )
v sY U U s v
ss
38
The Polar Method Alothgrithm
1.Generate two uniform RV , and and they are all on the interval (0,1) 2.Let and , so they are independent and uniform on (-1,1)3.Let if continue , else back to step24.Form 5.Set and
1U 2U
1 12 1V U 2 22 1V U
2 21 2S V V 1S
2( ) ( 2 ln ) /A S S S
1( )X A S V 2( )Y A S V
39
Establishing a Given Correlation Coefficient(1)
Assume two Gaussian RV X and Y , they are zero mean and uncorrelated
Define a new RV We also can see Z is Gaussian RV Show is correlation coefficient relating
X and Z
21Z X Y | | 1
40
Establishing a Given Correlation Coefficient(2)
Mean , Variance , Correlation coefficient { } { } { } 0E Z E X E Y
2 2 2 2 2{ } 2 1 { } (1 ) { }E X E XY E Y
{ } { } { } 0E XY E X E Y 2 2 2 2 2 2 2{ } ( { }) { } { }X Y E X E X E X E Y
2 2 2 2 2(1 )
2 2 2{[ 1 ] }Z E X Y
41
Establishing a Given Correlation Coefficient(3)
Covariance between X and Z
as desired
{ } { [ (1 ) ]}E XZ E X X Y
2{ } (1 ) { }E X E XY
2 2{ }E X
2
2
{ }XZ
X Z
E XZ
42
Pseudonoise(PN) Sequence Genarators
PN generator produces periodic sequence that appears to be random
Generated by algorithm using initial seedAlthough not random , but can pass man
y tests of randomnessUnless algorithm and seed are known , t
he sequence is impractical to predict
43
PN Generator implementation
44
Property of Linear Feedback Shift Register(LFSR)
Nearly random with long periodMay have max period If output satisfy period , is called
max-length sequence or m-sequenceWe define generator polynomial as
The coefficient to generate m-sequence can always be found
45
Example of PN generator
46
Different seed for the PN generator
47
Family of M-sequences
48
Property of m-sequence
Has ones , zerosThe periodic autocorrelation of a m-se
quence is
If PN has a large period , autocorrelation function approaches an impulse , and PSD is approximately white as desired
1
49
PN Autocorrelation Function
50
Signal Processing
Relationship
1.mean of input and output
2.variance of input and output
3.input-output cross-correlation
4.autocorrelation and PSD
51
Input/Output Means
Assume system is linearconvolution
Assume stationarity assumption
We can getand
[ ] [ ] [ ]k
k
y n h k x n k
{ [ ]} { [ ] [ ]} [ ] { [ ]}
k k
E y n E h k x n k h k E x n k
{ [ ]} { [ ]}E x n k E x n
{ } { } [ ]k
E y E x h k
[ ] (0)
k
h k H
{ } (0) { }E y H E x
52
Input/Output Cross-Correlation
The Cross-Correlation is defined by
This use is used in the development of a number of performance estimators , which will be developed in chapter 8
{ [ ] [ ]} [ ] { [ ] [ ] [ ]}xyj
E x n y n m R m E x n h j x n j m
[ ] [ ] { [ ]}xy
j
R m h j E x n j m
[ ] [ ]xxj
h j R m j
53
Output Autocorrelation Function(1)
Autocorrelation of the output
Can’t be simplified without knowledge of the Statistics of
{ [ ] [ ]} [ ]yyE y n y n m R m
{ [ ] [ ] [ ] [ ]}j k
E h j x n j h k x n k m
[ ] [ ] [ ] { [ ] [ ]}yy
j k
R m h j h k E x n j x n m k
[ ] [ ] ( )xxj k
h j h k R m k j
[ ]x n
54
Output Autocorrelation Function(2)
If input is delta-correlated(i.e. white noise)
substitute previous equation
2
[ ] { [ ] [ ]}0
xxxR m E x n x n m
20
[ ]0 x
mm
m
[ ]yyR m
2[ ] [ ] [ ] ( )yy xj k
R m h j h k m k j
2 [ ] [ ]x
j
h j h j m
55
Input/Output Variances
By definition Let m=0 substitute into
But if is white noise sequence
2[0] { [ ]}yyR E y n
[ ]yyR m
2 [0] [ ] [ ] [ ]y yy xxj k
R h j h k R j k
[ ]x n
2 2 2[0] [ ]y yy xj
R h j
56
The EndThanks for listening