1 Chapter 7 Generating and Processing Random Signals 第 第 一 第第第 B93902016 第第第 第第第 B93902076 第第第
Jan 05, 2016
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Chapter 7Generating and Processing
Random Signals
第一組電機四 B93902016 蔡馭理資工四 B93902076 林宜鴻
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Outline
Stationary and Ergodic ProcessUniform Random Number GeneratorMapping Uniform RVs to an Arbitrary pdfGenerating Uncorrelated Gaussian RVGenerating correlated Gaussian RVPN Sequence GeneratorsSignal processing
Outline
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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)
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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
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Time average v.s ensemble average
Time average ensemble average
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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)
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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
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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
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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
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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
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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
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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
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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)
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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
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Example 7.9 (1)
R0
0
MgX(x)
fX(x)
gX(x)
πRR
M 4
R
1
otherwise0,
Rx0xRπRxf X
222
4)(
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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
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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
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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.
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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
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The Sum of Uniforms Method(3)
Approximate GaussianMaybe not a realistic situation.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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PN Generator implementation
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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
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Example of PN generator
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Different seed for the PN generator
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Family of M-sequences
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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
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PN Autocorrelation Function
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Signal Processing
Relationship
1.mean of input and output
2.variance of input and output
3.input-output cross-correlation
4.autocorrelation and PSD
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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
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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
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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
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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
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The EndThanks for listening