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EE567 一 Advanced Digital Signal Processing - FinalDate : 25lhJune Thursday, 2015 Instructor : Hiiseyin OZKARAMANLI Due:30,h June Tuesday, 10:00amAnswer All Questions(Group work will be penalized)(Please attach all your matlab codes)QI. Consider the noise cancellation system shown below. The useful signal is a sinusoidS() = COS(Vt； +。),S() = COS(Vt； +。),W=7T/Swhere the phase is a random variable uniformlydistributed from 0 to 2兀.The noise signals are given byv, (n) = 0.8v, ( 一 1) + w(n) andv2(n) = -0.7v2(n-l) + w()where the sequence w(n) is additive white Gausian noise with zero mean and variance 1.a) Design an LMS filter with an appropriate length and choose the step size p. to achieve a 10% misadjustment.b) Plot the signals s(n), s(n)+vi(n), V2(n), the clean signal e()(n) using the optimum filter, and the clean signal eimS(n) using the LMS filter and comment upon the obtained results.c) Repeat part (b) and (c) for the normalized LMS.d) Re-solve the above noise cancellation problem using RLS adaptive system.e) Plot the error curve vs iterations for LMS, normalized LMS and RLS and comment on the differences between RLS, normalized LMS and LMS in terms of speed of convergence and excess error.inpute(n)Q2. Consider the process x(n) generated using AR(2) model53xn) x(n 1)xn 2) + w(/i)88where w(n) is WGN(O,1). We want to design a linear predictor of x(n) using the Steepest Descent Algorithm (SDA). Let= x() = /x(n 1)4- h.xfn 2) 4- l%x(n 3) + h4x(n 4)a) Determine the 4x4 autocorrelation matrix R ofx(n) and compute its eigenvalues h .b) Determine the 4x 1 cross correlation vector d .c) Choose the step size g so that the resulting response is overdamped. Simulate the mean square error vs iteration number performance for the chosen step size.九=hk_x + 2x/(d-R/V1)d) What value of p results in an underdamped response. Simulate for the chosen step size.e) Use appropriate estimators to estimate the autocorrelation matrix and cross-correlation vector and implemenl the SDA. Comment on the differences you observe when compared to part (c ) where the exact information is used.f) Use the following recursive estimators for R and d and implement the SDA.Ra =4Ri+xX，1 =汨+Xa (为人where X is a forgetting factor and ds is the desired signal. Comment on the results.g) Repeat the above prediction problem using RLS and compare with recursive SDA in terms of speed of convergence and excess error.Q3. In this problem an orthogonal 2 band filterbank will be designed.Figure 3 shows a 2 band filterbank with analysis filters H(z), and /)()and synthesis filters %(z), and 已(z).It is well known that filterbanks cannot be orthogonal and symmetric at the same time. Thus the filters you will design will not be symmetric. Orthogonal filterbanks have the perfect reconstruction property.v/7 = Dwhen the high pass analysis filter h)(n) is formed by the alternate flip construction and the synthesis filters are transposes of the analysis filters. Here A and D are constants (D an integer).In this problem we would like to design a 2 band orthogonal filter bank where the filters satisfy the orthogonality property妃妃 + 2幻=5伙.output of the low pass channeloutput of the high pass channelFigure 3: Orthogonal FilterbankFurthermore we would like to design the low pass analysis filter ho(n) to be of length 10 (i.e., 10 tap filter). We also want the filter ho(n) to be able to interpolate polynomials upto degree 4. This condition implies that the z-transform of h()(n), H()(z) should be of the following formH(z) = B(z)Q(z)where B(z) is a polynomial which has 4 zeros at z=-l (i.e., b(z)=(i + z ) ) and Q(z) is an arbitrary polynomial chosen in such a way that orthogonality condition is satisfied.a) Derive a choice for the filter ho(n).b) Complete the filter-bank by finding all the other filters.c) Verify that the system has the perfect reconstruction property by letting x(n) be a linearsequence of length 2048 (i.c., x(n)= 0123 2047) and tracing it through the twochannels and finally reconstructing Xhat. That is to say show that x(n) = A Xhat (n - D). (verification of perfect reconstruction should be done in MATLAB)for the linear signal in part (c) observe and comment on the output in the low and high pass channels.d) Now take the 5 level wavelet transform of the above linear signal. Plot on separate figures the wavelet subband coefficients and the final low resolution coefficients. What do you observe for the magnitude of the wavelet coefficients.e) Plot the mother wavelet and the scaling function for the orthogonal system you designed. (To do this take the forward wavelet transform of an arbitrary signal, set all coefficients to zero ( except one coefficient in the last wavelet subband (lor the mother wavelet) and one coefficient in the last resolution level (for the scaling function) ) and take the inverse wavelet transform.
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