Chapter 7 Generating and Processing Random Signals

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 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

3. 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)

4. Stationary and Ergodic Process • 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

5. Time average ensemble average Time average v.s ensemble average

6. Example 7.1 (N=100)

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 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

10. Example 7.4 • 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

11. Technique B: The Multiplication Algorithm With Prime Modulus • 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

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 x0is odd

13. Example of Multiplication Algorithm With Nonprime Modulus a=3 c=0 m=16 x0=1

14. Testing Random Number Generator • 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 -

15. ScatterplotsExample 7.5 (i)rand(1,2048) (ii)xi+1=[65xi+1]mod(2048) (iii)xi+1=[1229xi+1]mod(2048)

16. Durbin-Watson Test (1) 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

17. Durbin-Watson Test (2) 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 another • Lewis, 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)

22. Example 7.8 (1) • Rayleigh random variable with pdf – ∴ Setting FR(R) = U

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

24. Example 7.8 (3)

25. The Histogram Method • 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!

26. Rejection Methods (1) • Having a targetpdf • MgX(x)≧ fX(x), all 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)

29. Example 7.9 (2)

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 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.

32. The Sum of Uniforms Method(2) • Expectation and Variance • We can set to any desired value • Nonzero at

33. The Sum of Uniforms Method(3) • Approximate Gaussian • Maybe 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 

35. Mapping a Rayleigh to Gaussian RV(2) • Transform  let and  and  

36. Mapping a Rayleigh to Gaussian RV(3) • Examine the marginal pdf R is Rayleigh RV and is uniform RV

37. The Polar Method • From previous • We may transform

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 step2 • 4.Form • 5.Set and

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

40. Establishing a Given Correlation Coefficient(2) • Mean，Variance，Correlation coefficient

41. Establishing a Given Correlation Coefficient(3) • Covariance between X and Z •  as desired

42. 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

43. PN Generator implementation

44. 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

45. Example of PN generator

46. Different seed for the PN generator

47. Family of M-sequences

48. 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

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