Note: homework assignments that are past due will not be accepted unless you have made prior arrangements with the cruise director (dbp@apl.washington.edu).

Homework Assignments

• assignment 1 (due Monday 1/14):
  1. Exercise [1.3] (page 26 of textbook)
  2. Exercise [2.9] (page 55)
  3. See one of the following for the statement of this exercise: html; PostScript; PDF
  4. See one of the following for the statement of this exercise: html; PostScript; PDF
• assignment 2 (due Wednesday 1/23):
  1. Exercise [3.3] (page 120)
  2. Exercise [3.8] (page 121)
  3. See one of the following for the statement of this exercise: html; PostScript; PDF
  4. See one of the following for the statement of this exercise: html; PostScript; PDF
• assignment 3 (due Monday 1/28):
  1. Exercise [4.1] (page 149)
  2. Exercise [4.5] (page 150)
  3. Exercise [5.7] (page 185)
  4. See one of the following for the statement of this exercise: html; PostScript; PDF
• assignment 4 (due Monday 2/4): do exercises 1, 2 and 3 below, and then also do either 4, 5 or 6 (or do five or six exercises in all for extra credit (3 points for each extra exercise successfully done))
  1. Exercise [6.1] (page 322)
  2. parts a and b of Exercise [6.2] (pages 322-3); you can do part c for 1 point of extra credit if you so desire
  3. parts a and b of Exercise [6.6] (pages 324-5)
  4. See one of the following for the statement of this exercise on aliasing: html; PostScript; PDF
  5. See one of the following for the statement of this exercise on filtering: html; PostScript; PDF
  6. See one of the following for the statement of this exercise on the statistical properties of the periodogram: html; PostScript; PDF
• assignment 5 (due Monday 2/11): do exercises 1, 2 and 3 below, and then also do either 4, 5 or 6 (or do five or six exercises in all for extra credit (3 points for each extra exercise successfully done))
  1. Exercise [6.3] (page 323)
  2. Exercise [6.5] (pages 323-4)
  3. Exercise [6.10] (pages 325-6)
  4. See one of the following for the statement of this exercise about the periodogram: html; PostScript; PDF
  5. See one of the following for the statement of this exercise about prewhitening: html; PostScript; PDF
  6. Repeat Exercise 6 of Assignment 4, with the following modifications:
     • Use a direct spectral estimate with a Hanning data taper (see parts a and b of Exercise [6.10]) instead of the periodogram.
     • In comparing your sample values to the corresponding large sample values, replace Equation (222b) by Equation (223c), and replace Equation (222c) by Equation (224) with c = 2.
• assignment 6 (due Wednesday 2/20): Do either (A) the first four exercises or (B) the fifth exercise. Option (B) is computational, so any software that you develop might be useful if you plan to do a spectral analysis of a time series for your class project. If you do choose (B), you might want to test out the software that you develop by replicating the example given in Figure 288 (note that all of the numbers used to create this figure can be accessed from the middle of the DATA AND DATA TAPERS page for the course Web site). If you want to do both (A) and (B), you can earn up to 6 points extra credit.
  1. Exercise [6.17] (page 327)
  2. Exercise [6.19] (page 327)
  3. Exercise [6.20] (pages 327-8)
  4. See this two page PDF file for this exercise about measuring and making use of the spectral bandwidth.
  5. This exercise is designed to give you some practice in applying nonparametric spectral analysis to a `mystery' time series of length N = 2048 that has a sampling interval of 1/1250 second.
     • (a) Plot the time series to get an idea of what you are dealing with. Comment on anything special you note about the series that might affect the way you compute the sdf estimates called for in the next three parts.
     • (b) Compute and plot an appropriate periodogram for the time series. Compute and plot a corresponding direct spectral estimate that uses the Hanning data taper. Is tapering useful here (explain briefly why or why not)?
     • (c) Suppose someone with prior experience with this `mystery' series tells you that it has a spectral bandwidth B_S of 25 Hertz (cycles/second). Based upon what you deem to be the most appropriate of the two estimates computed in part b, compute and plot a Parzen lag window estimate that has a smoothing window bandwidth B_W dictated by the rule of thumb B_W = B_S / 2. How many equivalent degrees of freedom nu does this estimate have?
     • (d) Repeat part (c), but now use a Daniell lag window estimate with the same smoothing window bandwidth B_W. Explain any differences that you observe between the Parzen and Daniell sdf estimates.
• assignment 7 (due Monday 2/25): Do either (A) the first three exercises or (B) the last two exercises (both of these are computational). If you want to do both (A) and (B), you can earn up to 6 points extra credit.
  1. Exercise [7.2] (pages 375-6)
  2. Exercise [7.3] (page 376)
  3. Exercise [7.5] (page 376)
  4. Using the sine tapers of orders k=0, ..., K-1 defined by
     h_{t,k} = (2/[N+1])^{1/2} sin([k+1] pi t/[N+1]), t = 1, ..., N
     (here N = 2048), compute multitaper spectral estimates using K = 6 and K = 10 data tapers for the `mystery' time series used in assignment 6. Plot both estimates so that you can compare them with the sdf estimates computed in that assignment. Compute and plot (on the same plots showing the spectral estimates) a crisscross indicating the approximate bandwidth of each multitaper spectral estimate and the width of a 95% confidence interval for the true sdf based upon each estimate (note: the bandwidth for the sine multitaper estimator can be taken to be (K+1)/[(N+1)*(Delta t)], where Delta t = 1/1250 second is the sampling interval). How well do the sine multitaper estimates agree with the various estimates computed in assignment 6?
  5. Repeat the previous exercise, but now compute WOSA spectral estimates using a Hanning data taper with 50% overlap and the following two choices for block sizes: N_S = 256 and 512. In computing the crisscross, take the bandwidth of the WOSA estimator to be twice the spacing dictated by the Fourier frequencies for the block size, i.e., 2/[N_S*(Delta t)].
• assignment 8 (due Monday 3/3): Do either (A) the first three exercises or (B) the fourth (computational) exercise. If you want to do both (A) and (B), you can earn up to 6 points extra credit.
  1. Exercise [9.3] (page 454)
  2. See this PDF file for an exercise about the Yule-Walker method.
  3. See this PDF file for an exercise about Burg's algorithm and the forward/backward least squares method.
  4. Use the Yule-Walker method and Burg's algorithm to compute AR(10) and AR(30) spectral estimates for the `mystery' time series used in assignments 6 and 7. Plot the Yule-Walker and Burg spectral estimates for the same order p together so that you can compare them with each other. In addition, turn in a list of the AR coefficients estimates and innovations variance estimates that you get in all four cases. Plot the Burg estimates together with the Daniell estimate computed in assignment 6 using the direct spectral estimate with the Hanning data taper. How well do the two Burg AR estimates agree with the Daniell estimate? Note: here are numerical examples that you can use to test your implementation of the Yule-Walker method and Burg's algorithm for estimating the parameters of an autoregressive process.