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# Chapter 7 Generating and Processing Random Signals - PowerPoint PPT Presentation

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

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### Chapter 7Generating and Processing Random Signals

Outline

• Stationary and Ergodic Process

• Uniform Random Number Generator

• Mapping Uniform RVs to an Arbitrary pdf

• Generating Uncorrelated Gaussian RV

• Generating correlated Gaussian RV

• PN Sequence Generators

• Signal processing

• Noise, interference

• Random Number Generator- computational 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)

• strict-sense stationary (SSS)

• wide-sense stationary (WSS)

Gaussian

• SSS =>WSS ; WSS=>SSS

• Time average v.s ensemble average

• The ergodicity requirement is that the ensemble average coincide with the time average

• Sample function generated to represent signals, noise, interference should be ergodic

ensemble average

Time average v.s ensemble average

• 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)

• 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

• The mixed linear algorithm takes the form:

xi+1=[axi+c]mod(m)

- c≠0 and relative prime tom

- a-1 is a multiple of p, where p is the

prime factors of m

- a-1 is a multiple of 4 ifm is a

multiple of 4

• m=5000=(23)(54)

• c=(33)(72)=1323

• a-1=k1‧2 or k2‧5 or 4‧k3

so, a-1=4‧2‧5‧k =40k

• With k=6, we have a=241

xi+1=[241xi+ 1323]mod(5000)

• We can verify the period is 5000, so it’s full period

• The multiplicative generator defined as :

xi+1=[axi]mod(m)

- m isprime (usaually large)

- a is a primitive elementmod(m)

am-1/m = k =interger

ai-1/m ≠ k, i=1, 2, 3,…, m-2

• 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 x0is odd

a=3

c=0

m=16

x0=1

• Chi-square test, spectral test……

• Testing the randomness of a given sequence

• Scatterplots- a plot of xi+1 as a function of xi

• Durbin-Watson Test

-

Scatterplots ModulusExample 7.5

(i)rand(1,2048)

(ii)xi+1=[65xi+1]mod(2048)

(iii)xi+1=[1229xi+1]mod(2048)

Durbin-Watson Test (1) Modulus

Let X = X[n]

& Y = X[n-1]

Assume X[n] and X[n-1] are correlated and X[n] is an ergodic process

Let

Durbin-Watson Test (2) Modulus

X and Z are uncorrelated and zero mean

D>2 – negative correlation

D=2 –-uncorrelation (most desired)

D<2 – positive correlation

Example 7.6 Modulus

• 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.

Minimum Standards Modulus

• Full period

• Passes all applicable statistical tests for randomness.

• Easily transportable from one computer to another

• Lewis, Goodman, and Miller Minimum Standard (prior to MATLAB 5)

xi+1=[16807xi]mod(231-1)

• 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

Inverse Transform Method Modulus

• 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)

Example 7.8 (1) Modulus

• Rayleigh random variable with pdf –

Setting FR(R) = U

Example 7.8 (2) Modulus

∵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

The Histogram Method Modulus

• CDF andpdf are unknown

• Pi =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!

Rejection Methods (1) Modulus

• Having a targetpdf

• MgX(x)≧ fX(x), all x

Rejection Methods (2) Modulus

• 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

• 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

• 1.Central limit theorem

• 2.See next

.

• 3.

represent independent uniform R.V

is a constant that decides the var of

Y converges to a Gaussian R.V.

• Expectation and Variance

• We can set to any desired value

• Nonzero at

• Approximate Gaussian

• Maybe not a realistic situation.

• 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

• Transform

 let and

 and

• Examine the marginal pdf

R is Rayleigh RV and is uniform RV

The Polar Method Modulus

• From previous

• We may transform

The Polar Method Alothgrithm Modulus

• 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 step2

• 4.Form

• 5.Set and

• 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

• Mean，Variance，Correlation coefficient

• Covariance between X and Z

•  as desired

• PN generator produces periodic sequence that appears to be random

• Generated by algorithm using initial seed

• Although not random，but can pass many tests of randomness

• Unless algorithm and seed are known，the sequence is impractical to predict

• Nearly random with long period

• May have max period

• If output satisfy period ，is called max-length sequence or m-sequence

• We define generator polynomial as

• The coefficient to generate m-sequence can always be found

Property of m-sequence Modulus

• Has ones， zeros

• The periodic autocorrelation of a m-sequence is

• If PN has a large period，autocorrelation function approaches an impulse，and PSD is approximately white as desired

Signal Processing Modulus

• Relationship

1.mean of input and output

2.variance of input and output

3.input-output cross-correlation

4.autocorrelation and PSD

Input/Output Means Modulus

• Assume system is linearconvolution

• Assume stationarity assumption

• We can get

• and 

• 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

• Autocorrelation of the output

• Can’t be simplified without knowledge of

the Statistics of

• If input is delta-correlated(i.e. white noise)

substitute previous equation

Input/Output Variances Modulus

• By definition 

• Let m=0 substitute into

• But if is white noise sequence

The End Modulus

Thanks for listening