8/10/2019 SSP Assignment Problems_Final

http://slidepdf.com/reader/full/ssp-assignment-problemsfinal 1/2

8/10/2019 SSP Assignment Problems_Final

http://slidepdf.com/reader/full/ssp-assignment-problemsfinal 2/2

13. What is meant by a stationary random process? Differentiate between SSS and WSS random process and mention it’s properties. (5)

14. Using the Yule Walker equation, determine the coefficient values for an AR (1) process. (5)

15. 

Determine the mean and autocorrelation of the signal x(n) = Ae j(n0 + Φ)

 where A and 0 arefixed  constants and Φ is a random variable that is uniformly distributed over the interval - to,i.e., the probability density function for Φ is

f Φ () = (2)-1 ; -  ≤  ≤  0 ; otherwise (5)

16. With necessary derivation, explain briefly the periodogram smoothing using Blackman- Tukeymethod.

17. The input to a LSI filter with unit sample response h(n) = δ(n) + 0.5δ(n-1) + 0.25δ(n-2) is a zeromean wide sense stationary process with autocorrelation r x(k) = (0.5) |k| 

(i) 

What is the variance of the output process?(ii)  Find the autocorrelation of the output process, r y(k) for all k. (5)

18. Consider a first order AR process that is generated by the difference equationy(n) = ay(n-1) + w(n) where |a|<1 and w(n) is a zero mean white noise random process withvariance σw

2. Determine the(i)  Unit sample response of the filter that generates y(n) from w(n)(ii)  Autocorrelation of y(n)(iii)  Power spectrum of y(n). (5)

19. Given r x(0) = 1 and the first three reflection coefficients are Γ1 = Γ2 = 0.5 and Γ3 = 0.25. Find

the corresponding autocorrelation sequence. (6)

20. Suppose that the first five values in the autocorrelation sequence for the process x(n) are r x = [3,9/4, 9/8, 9/16, 9/32,….]T. Use modified Yule Walker equation method to find ARMA(1,1)model for x(n). (5)

21. Derive the normal equation and the minimum error for all pole model using Prony’s method (5)

22. Suppose that a signal d(n) is corrupted by noise, x(n) = d(n) + w(n) where r w(k)=0.5δ(k) and

r dw(k) = 0. The signal d(n) is an AR(1) process that satisfies the difference equation d(n) =0.5d(n-1) + v(n) , where v(n) is white noise with variance σv

2= 1. Design a 1st order FIR linear

 predictor W(z) = w(0) + w(1)z-1

  for d(n) and also find the minimum mean square predictionerror ε = E{[d(n+2) – d(n+2)]2}. Assume that w(n) and v(n) are uncorrelated. (6)

23. Consider the system shown in the figure below for estimating a process d(n) from x(n)(Refer Exercise Problem in Optimum Filter chapter)If σd2= 4 and rx = [1, 0.5, 0.25]T ; rdx =[-1,1]T. Find the value of a(1) that minimizes the meansquare error ε = E{|e(n)|2} and also find the minimum mean square error. (6)