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# ECE-5200 Introduction to Digital Signal Processing Autumn 2012 Homework #9 Nov. 23, 2012 HOMEWORK ASSIGNMENT #9 Due Mon. 3, 2012 (in class) Consider...

This is my ece homework in Digital Signal Processing. I am willing to pay 20 dollars for the complete and correct answers. The due day could be monday 10 am.

ECE-5200 Introduction to Digital Signal Processing Autumn 2012 Homework #9 Nov. 23, 2012 HOMEWORK ASSIGNMENT #9 Due Mon. Dec. 3, 2012 (in class) Consider the problem of learning the MSE-optimal version of the Δ-delayed L -length inverse impulse response { w [ l ] } L - 1 l =0 of the LTI system with impulse response { h [ l ] } M - 1 l =0 from the processes { u[ n ] } and { d[ n ] } in the diagram below. The block z - Δ represents a system that delays the input by Δ samples. - + + z - Δ s[ n ] e[ n ] y[ n ] d[ n ] u[ n ] a[ n ] { w * [ l ] } { h [ l ] } Figure 1: Linear Channel Equalization Below, we will assume that { a[ n ] } is a zero-mean white random process with variance σ 2 a , that { s[ n ] } is a zero-mean white random process with variance σ 2 s , and that { a[ n ] } is uncorrelated with { s[ n ] } . 1. Write an expression for the ±lter input vector u [ n ] in terms of the noise vector a [ n ], the source vector s [ n ], and the (transposed) convolution matrix H , where u [ n ] = u[ n ] u[ n - 1] . . . u[ n - L +1] , a [ n ] = a[ n ] a[ n - 1] . . . a[ n - L +1] , s [ n ] = s[ n ] s[ n - 1] . . . s[ n - M - L +2] , and H = h [0] h [1] · · · h [ M - 1] h [0] h [1] · · · h [ M - 1] . . . . . . . . . h [0] h [1] · · · h [ M - 1] . Hint: First write u [ n ] using summations, then transform to vectors, then expand to u [ n ] using matrix H . 2. Write an expression for R u = E { u [ n ] u H [ n ] } in terms of H , σ 2 a , σ 2 s , I , and i Δ , where I is the identity matrix and i Δ is its Δ th column (where Δ = 0 gives the ±rst column). For example, I = 1 1 1 . . . 1 , i 0 = 1 0 0 . . . 0 , i 1 = 0 1 0 . . . 0 , etc. Hint: Because they are white random processes, E { a [ n ] a H [ n ] } = σ 2 a I and E { s [ n ] s H [ n ] } = σ 2 s I . 3. Write an expression for d[ n ] using i Δ and s [ n ]. P. Schniter, 2012 1
4. Write an expression for the cross-correlation vector r ud = E { u [ n ] d * [ n ] } using H , σ 2 s , and i Δ . 5. Write an expression for σ 2 d , the variance of { d[ n ] } , using σ 2 s . 6. Write an expression for the minimum MSE J mse ,⋆ using H , σ 2 a , σ 2 s , I , and i Δ . 7. Write an expression for the Wiener Flter w using H , σ 2 a , σ 2 s , I , and i Δ . 8. Using Matlab, compute both J mse ,⋆ and w assuming that h = b 1 - 0 . 5 B , Δ = 0 , L = 4 , σ 2 a = 0 . 01 , σ 2 s = 1 . Hint: Useful Matlab commands include convmtx , eye , pinv . 9. ±or the system parameters given in problem 8, what is the allowed range for the stepsize μ of gradient descent? Hint: Use the Matlab commands eig and max . 10. ±or the system parameters in problem 8, adapt the Flter w using gradient descent. Start from the initialization w = 0 , use the stepsize μ = 1 max (where λ max denotes the maximum eigenvalue of R u ), and use 50 iterations. (a) Plot the evolution of the Flter coe²cients w in Matlab using plot(W.’,’+-’) , where the n th column of matrix W contains the coe²cient vector w at iteration n . Do the Flter coe²cients converge to the Wiener solution w ? (b) Plot the evolution of the MSE cost J mse [ n ]. Does it converge to the minimum MSE J mse ,⋆ ? Your plots should look something like below. Note that the Wiener solution wstar is plotted using squares at the Fnal iteration N via plot(N,wstar,’s’) . 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 iteration filter coefficients gradient descent 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 iteration Jmse 11. ±or the parameters in problem 8, adapt the Flter w using LMS. Start from the initialization w = 0 , use the stepsize μ = 0 . 1 max , and use 500 iterations. Use the realizations of { u[ n ] } and { d[ n ] } provided in the Fle u.txt and d.txt on the course webpage. (a) Plot the evolution of the Flter coe²cients w in Matlab as above. Do the Flter coe²cients converge near to the Wiener solution w ? (b) Plot the evolution of the MSE cost J mse [ n ] as above. Does it converge near to the minimum MSE J mse ,⋆ ? Hint: Remember the correct (backwards) ordering of the samples within the u [ n ] vector. P. Schniter, 2012 2
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