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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

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)

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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

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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