Chapter 7 Generating and Processing Random Signals

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Chapter 7Generating and Processing Random Signals

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• ??? B93902016 ???
• ??? B93902076 ???

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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
• PN Sequence Generators
• Signal processing

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Random Number Generator

• 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)

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Stationary and Ergodic Process

• strict-sense stationary (SSS)
• wide-sense stationary (WSS)
• 
  Gaussian
• SSS gtWSS WSSgtSSS
• 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

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Time average v.s ensemble average

• Time average

• ensemble average

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Example 7.1 (N100)
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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
• xi1axicmod(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
• xi1axicmod(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

• m5000(23)(54)
• c(33)(72)1323
• a-1k1?2 or k2?5 or 4?k3
• so, a-14?2?5?k 40k
• With k6, we have a241
• xi1241xi 1323mod(5000)
• We can verify the period is 5000, so its full
  period

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Technique B The Multiplication Algorithm With
Prime Modulus

• The multiplicative generator defined as
• xi1aximod(m)
• - m is prime (usaually large)
• - a is a primitive element mod(m)
• am-1/m k interger
• ai-1/m ? k, i1, 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
• xi1aximod(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
a3 c0 m16 x01
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Testing Random Number Generator

• Chi-square test, spectral test
• Testing the randomness of a given sequence
• Scatterplots- a plot of xi1 as a function of
  xi
• Durbin-Watson Test
• -

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ScatterplotsExample 7.5
(i) rand(1,2048) (ii)xi165xi1mod(2048) (iii
)xi11229xi1mod(2048)
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Durbin-Watson Test (1)
Let X Xn
Y Xn-1
Assume Xn and Xn-1 are correlated and Xn
is an ergodic process
Let
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Durbin-Watson Test (2)
X and Z are uncorrelated and zero mean
Dgt2 negative correlation D2 - uncorrelation
(most desired) Dlt2 positive correlation
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Example 7.6

• rand(1,2048) - The value of D is 2.0081 and ? is
  0.0041.
• xi165xi1mod(2048) - The value of D is 1.9925
  and ? is 0.0037273.
• xi11229xi1mod(2048) - The value of D is
  1.6037 and ? is 0.19814.

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Minimum Standards

• 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)
• xi116807ximod(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)

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Example 7.8 (1)

• Rayleigh random variable with pdf
• ?
• Setting FR(R) U

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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

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Example 7.8 (3)
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The Histogram Method

• CDF and pdf are unknown
• Pi Prxi-1 lt x lt 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!

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Rejection Methods (1)

• Having a target pdf
• MgX(x) ? fX(x), all 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)
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Example 7.9 (2)
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Generating Uncorrelated Gaussian RV

• Its CDF cant be written in closed form,so
  Inverse method cant 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 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.
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The Sum of Uniforms Method(2)

• Expectation and Variance
• We can set to any desired value
• Nonzero at

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The Sum of Uniforms Method(3)

• Approximate Gaussian
• Maybe 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
• ?

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Mapping a Rayleigh to Gaussian RV(2)

• Transform
• ? let and
• ? and
• ?
• ?

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Mapping a Rayleigh to Gaussian RV(3)

• Examine the marginal pdf
• ?R is Rayleigh RV and is uniform RV

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The Polar Method

• From previous
• We may transform

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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 step2
• 4.Form
• 5.Set and

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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

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Establishing a Given Correlation Coefficient(2)

• Mean,Variance,Correlation coefficient

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Establishing a Given Correlation Coefficient(3)

• Covariance between X and Z
• ? as desired

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Pseudonoise(PN) Sequence Genarators

• 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

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PN Generator implementation
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Property of Linear Feedback Shift Register(LFSR)

• 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

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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, 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

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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 linear?convolution
• Assume stationarity assumption
• ?
• We can get
• and ?

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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

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Output Autocorrelation Function(1)

• Autocorrelation of the output
• Cant be simplified without knowledge of
• the Statistics of

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Output Autocorrelation Function(2)

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

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Input/Output Variances

• By definition ?
• Let m0 substitute into
• But if is white noise sequence

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• The End
• Thanks for listening