ADSP ADSP ADSP ADSP. Advanced Digital Signal Processing (18-792) Spring Fall Semester, Department of Electrical and Computer Engineering

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1 Advanced Digital Signal rocessing (18-792) Spring Fall Semester, Department of Electrical and Computer Engineering ROBLEM SET 8 Issued: 10/26/18 Due: 11/2/18 Note: This problem set is due Friday, in part because of the holiday weekend, the lack of recitations for the past two weeks, and the fact that we are a little behind the syllabus (thanks in part to my multiple absences). I encourage you to work on roblem 8.1 and C8.1 over the weekend. You may want to wait until the Monday lecture for the remaining problems, during which time I will introduce the topic of linear prediction, including solutions using Levinson-Durbin recursion. You can get a heads up on these topics by reviewing the class notes online on linear prediction. Reading: This problem set is centered on parametric spectral estimation techniques, as we begin to look at maximum entropy spectral estimation and linear prediction. As you know, our discussion of power spectral analysis has been drawn from several sources. Our discussion of classical nonparametric spectral estimation techniques based on the periodogram, periodogram smoothing, periodogram windowing, and periodogram averaging is based on the material in Secs and 11.4 in OS1 and Secs and 10.6 in OSY. Although this material was not discussed in class, I also recommend looking over Secs and 11.6 in OS1. We also highly recommend Kay and Marple s very comprehensive tutorial papers that compares many spectral estimation techniques: Kay, S. M., and Marple, L. M. (1981), Spectrum analysis a modern perspective, roceedings of the IEEE, vol. 69, no. 11, pp This review also spawned two good books published by rentice-hall of very different styles: the more accessible and tutorial book Digital Spectral Analysts with Applications by Marple in 1987, and the more thorough and scholarly Modern Spectral Estimation: Theory and Application by Kay in Much of this material is also covered in Kay s chapter in Lim and Oppenheim (LO), and LO Sec. 2.4 is especially relevant to our in-class discussions. (Reprints of the LO chapter are also available online.) We have also introduced the philosophy of maximum entropy spectral estimation following the discussion in the review/tutorial article Robinson, E.A. (1982), A historical perspective of spectrum estimation, roceedings of the IEEE, vol.70, no.9, pp Class notes from the MEM lecture are also available online. The material on MEM spectral estimation leads into our discussion of linear prediction that will be largely covered by class notes, along with Chapter 8 of Rabiner and Schafer (1978), both of which are available on the Web.

2 roblem Set 8-2- Fall, 2018 roblem 8.1: A bandlimited continuous-time signal has a bandlimited power spectrum that is zero for rad/s. The signal is sampled at a rate of 16,000 samples/s over a time interval of 30 s. The power spectrum of the signal is estimated by the Bartlett method of averaging periodograms. (a) What is the length Q (number of samples) of the data record? (b) If a radix-2 FFT program is used to compute the periodograms, what is the minimum length N if we wish to obtain estimates of the power spectrum at equally spaced frequencies no more than 10 Hz apart? (c) If the segment length L is equal to the FFT length N in part (b), how many segments K are available if the segments do not overlap? (d) Suppose we wish to reduce the variance of the spectral estimates by a factor of 10 while maintaining the frequency spacing of part (b). Give two methods of doing this, one using the Bartlett method and the other using windowing of the original random process. Do these two methods give the same results? If not, explain how they differ. roblem 8.2: An all-zero filter has a transfer function of Xz = = z 1 Wz A white noise WSS random process wn with variance equal to 1 is input to the filter. (a) Using random process properties and your knowledge of the filter s response, obtain the values of xx m, the autocorrelation function of xn, the system s output for lags m 3. (b) Using the Levinson-Durbin recursion technique, and the autocorrelation function for xn that you calculated in part (a), obtain the and the k coefficients for the first-, second-, and third-order LC model of the filter. lease obtain this result by hand (using a calculator or its computer equivalent), rather than using routines in MATLAB. For each order of the model, sketch the locations of the poles of the z-transform of the system. Using MATLAB, sketch the magnitude of the DTFT in decibels (db) for each of the three models. roblem 8.3: (a former quiz problem) In class we developed the basic LC equations that minimized the forward mean square error Ee pn E xn k xn k = =

3 roblem Set 8-3- Fall, 2018 and we will introduce the concept of backward error in our discussion of lattice filters. It is also possible to consider a non-causal operation that predicts the current sample of past and future samples in a two-sided fashion: xˆ n = k xn k + xn + k The corresponding mean-square error for such a system would be xn from both 2 2 Exn xˆ n 2 2 E xn k xn k + xn + k = = (a) Obtain a set of matrix-vector equations that will specify the set of coefficients k when solved. Your equations should be similar to (but not identical to) the standard LC equations that are solved, for example, by Levinson-Durbin recursion. The matrices and vectors in the equations should be expressed in terms of the coefficients k and m xx m = Exnxn + m, the auto-correlation function of xn. You may assume that xn is a real wide-sense stationary random process. (b) Can your answer to part (a) be solved using Levinson-Durbin recursion? Why or why not? (c) Can your answer to part (a) be solved using Cholesky decomposition? Why or why not? roblem 8.4: The impulse response of a causal all-pole model of the form G = k z k with system parameters G and k hn = k hn k + Gn satisfies the difference equation (a) The autocorrelation function of the impulse response of the system is hh m = hn hn + m n =

4 roblem Set 8-4- Fall, 2018 By substituting the equation above for hn into the equation for hh m, and using the fact that hh m = hh m show that k hh m k = hh m, m = 123 (b) Using the same approach as in (a), now show that hh 0 k hh k = G 2 MATLAB roblems Note: In working the problems, turn in a printout of your results, a copy of the MATLAB code you developed to work the problem, as well as any additional comments you'd like to add. WARNING!! The texts by Oppenheim and Schafer, and the book by Rabiner and Schafer have all consistently used the filter notation convention = M b k z k k = N 1 a k z k We have used this notation in class with some variants (such as using the coefficients k instead of a k ). Unfortunately, MATLAB uses the convention used in a classic review article on linear prediction and lattice filtering by John Makhoul some years ago: = M b k z k k = N 1 + a k z k This means that the coefficients of the denominator polynomial a k and the reflection coefficients k i produced by MATLAB are opposite in sign to what we have seen in Rabiner and Schafer and used in class discussions. In turning in your homework beginning with roblem C8.1 you must use the OSB/RS/ conventions! lease save yourself and us a lot of aggravation by being careful about this annoying inconsistency! (Thanks!)

5 roblem Set 8-5- Fall, 2018 Note: You may wish to create helper functions so you don t have to repeat writing the same code. roblem C8.1: In roblem C7.1 last week you examined traditional estimates of the power spectral density of the random process described above. In this problem we will explore the effects of various means to reduce the variance, as described in OS , which is available on the Web. You will be provided with a main file called main_8_1.m that you must complete. Consider again estimates of SD drawn from 500, 1000, and 4000 samples of noise from noisgen. You can either use the same samples as in roblem C7.1 or you can create new ones. (a) In this part of the problem we look into the effects of periodogram averaging using the Bartlett method. For each of the three lengths of data, subdivide the data into abutting subsegments of length N 4, and N 50, where N is the number of data points. As described in OS1 Sec , form periodograms for each of the segments using rectangular windows, and obtain power spectral density estimates by averaging the periodograms obtained from the subsegments. Repeat the whole process four times, as in roblem C7.1, so we can obtain estimates of the global mean and variance of the estimated SDs. In each case, 1. lot the first of the four resulting SD functions from the averaged periodograms. 2. Calculate the expected value of the SD estimate for = 0.25 and 0.75 by averaging the values of the four samples at those frequencies. Also calculate the corresponding bias of the estimates. 3. Calculate the variance of the SD estimate for = 0.25 and (b) Now repeat part (a), but instead of obtaining abutting subsegments of the original data, obtain a series of subsegments with 50-percent overlap as in certain aspects of the Welch method.. (For example, if the lengths of the subsegments were 10 samples, the first one would be samples 0-9, the second would be 5-14, the third would be 10-19, etc.) This will approximately double the total number of periodograms to be averaged for each SD estimate. What are the advantages and disadvantages that you observe with abutting the segments in the fashion described? (c) Now repeat part (a) for SDs drawn from 500 and 4000 samples of noise only, but using a Hamming window instead of the rectangular window that is implied by doing no windowing in particular. What are the advantages and disadvantages that you observe with the windowing? This is an example of the, well, the windowing method. (d) As in roblem C7.1, comment on the general success or failure that this approach has in describing the periodic components of the input data. roblem C8.2: You will be provided with a main file called main_8_2.m that you must complete for this problem. Consider again the all-zero filter you examined in roblem 8.2, = 1 0.9z 1 Using the MATLAB routines lpc or levinson, along with freqz, sketch the magnitude of the DTFT

6 roblem Set 8-6- Fall, 2018 of the best-fit all-pole approximation to with 5 poles, 10 poles, 15 poles, and 20 poles. How many poles do you think are needed for an adequate all-pole approximation to the original transfer function with a single zero? roblem C8.3: *log10(SD) Figure C8.3. True power spectral density function of noise process to be used in roblem C8.3. / You will be provided with a main file called main_8_3.m that you must complete. In this and the following problem we will be modeling the colored noise samples examined last week using maximum entropy techniques. The true power spectral density function for this noise is given in the figure above. Note that it contains aperiodic noise components (which are represented by the smooth curve in the SD function) and periodic components (which are represented by the three spikes in the SD function). (a) We will begin by considering various ways of estimating the power spectral density function of a random process that is very similar but that does not include the periodic components of the process shown in Figure C8.3 above. Use the MATLAB function noisegen2.m to generate a sample function of this process that is 1000 samples long. The function noisegen2.m, as well as the original function noisegen.m, is available on the course Website. 1. lot the periodogram of this process using any method you choose. 2. lot the estimate of the power spectral density function obtained for this process using the Bartlett method, averaging 4 abutting subsections of the data. (b) Now use the full 1000 samples of the process to compute the first several coefficients of the auto-correlation function of the process. (Because the number of lags is small compared to the length of the data it should not matter much whether you use the biased or unbiased estimate of the auto-correlation function.) Using MATLAB, obtain the coefficients of the best-fit all-pole model for the random process. (You can use

7 roblem Set 8-7- Fall, 2018 a routine you write yourself to implement the Levinson-Durbin recursion or the built-in MATLAB function levinson. If you use levinson, be sure to read the help file and be mindful of the sign convention noted above.) 1. lot the estimate of the power spectral density function using three values of the model order parameter of your choosing that illustrate how the value of affects the quality of the estimate. 2. What value of the model order parameter would you select as reasonable to model this process? Explain how you made this choice. 3. Compare the nature of these estimates with the estimates of the power spectrum that you obtained in part (a). (c), (d) Repeat parts (a) and (b) but using the samples of noise generated by the MATLAB function noisegen.m, which includes the periodic components whose effects are indicated by the spikes in Fig. C8.3 above. (e) Compare the effectiveness of the all-pole model for characterizing the power spectral density function of the random process produced by noisegen.m (with the periodic components) with the characterization of the power spectral density function of the random process produced by noisegen2.m (which is identical but without the periodic components.) What to turn in on Gradescope: Completed versions of the routines main_8_1.m, main_8_2.m, and main_8_3.m. Answers to the questions and all the plots.