PHYS 628, Spring 2010 — Course Outline

Week #0: 22 Jan 2010

∅ Introductions, syllabus and text(s), motivations for the course and DSP in general. ⊕ No HW this week ∇ Throughout the semester, most solutions to the HW assignments will be posted on-line, directly accessible from this page. Other solutions will be posted in the course display case (somewhere in the Physics Dept. hallway).

Week #1: 25 Jan–29 Jan 2010

∅ We'll begin by undertaking a review of some statistical concepts, beginning with continuous theory and moving to discrete forms. ∅ A correction to Wednesday's lecture notes is available here. Also, note that for a Gaussian, P(x) = erfc( -1/sqrt(2) )/2 = ( 1 + erf( 1/sqrt(2) ) )/2. ⊕ Homework #1 due 1 Feb, and the solution is now posted ⊗ Here's an example of an excellent HW solution: Student HW #1 and code ∇ No lecture on Friday – we'll arrange a make-up on Tuesday, 2 Feb.

Week #2: 1–5 Feb 2010

∅ We'll discuss some more measures of likeness and begin our investigation of least squares ⊕ Homework #2 due 8 Feb, and the solution is now posted ∇ Make-up lecture, 17:15 Tuesday 2 Feb in the 7th floor GI "bullpen" – just outside my office in 706D, SW corner of the building © Here are some MATLAB functions I've written to help with common tasks on the past few HW assignments, plus some others for the work ahead... cols.m — this Little gem makes multiple copies of a vector into each column rows.m — same as above, but as rows hist_pdf.m — this normalizes the MATLAB hist function to give p(x) powerpointFig.m — changes the bounding box of a plot suitable for the landscape orientation of a presentation powerpoint.m — ever notice how some graphics are illegible when converted to a projected presentation?, this allows you to make text, axes and plot symbols bold, in effect

Week #3: 8–12 Feb 2010

∅ This week we'll explore some connections between least squares fitting and more general concepts regarding the modeling of data ⊕ Homework #3 due 15 Feb, and the solution is now posted

Week #4: 15–19 Feb 2010

∅ This week we'll finish LLSF with a look at the orthogonality principle and lead our way to the discrete Fourier transform (DFT) ⊕ Homework #4 due 24 Feb (submit this one via email as I'll be on travel next week, and here's the solution for the HW set along with the code used to create & solve the problems

Week #5: 22–26 Feb 2010

∅ This week Prof. Newman will ably guide you through some more aspects of the DFT, its inverse and how to apply it to rudimentary spectral estimation ⊕ Homework #5 due 03 Mar

Week #6: 1–5 Mar 2010

∅ This week we'll continue to look at the DFT, as well as some theorems relating it to other techniques. We'll introduce convolution. ⊕ No HW this week

Spring Break: 8–12 Mar 2010

Week #7: 15–19 Mar 2010

⊕ No HW this week © Here are a pair of cross correlation codes in case you don't have access to MATLAB's Signal Processing Toolbox and xcorr.m (each has the same normalizations available as MATLAB's version): xcor.m — this code computes the cross correlation function estimate via sums, but slightly faster than nested for loops due to a trick using circular shifting of the data in each vector xcorf.m — this code uses MATLAB's built-in conv.m to compute the cross correlation, and since it's FFT-based, it is quite a bit faster than xcor.m &empty We'll relate convolution to an important concept; cross-correlation, and then continue with PSD estimation. ¤ Mid-Term Exam (take-home) will be posted here beginning 19 Mar. It is due at the beginning of lecture on 26 Mar. A solution is now posted.

Week #8: 23–26 Mar 2010

∅ This week we'll cover cross correlation and its relation to PSD estimation. We'll also look at how leakage occurs and begin to see a way to control it in spectrum estimation. Finally, we'll also look at the uncertainty in our spectral estimates. ⊕ No HW this week, but your Mid-Term Exam is due on Friday, 26 March.

Week #9: 29 Mar–2 Apr 2010

∅ This week we'll finish up spectral estimation via the DFT ¤ NO CLASS ON FRIDAY - comp time for Take-Home mid term. ⊕ Homework #6 due 07 Apr; and a solution is now posted © After too much delay, here are some codes that illustrate concepts we covered in class and also add some useful tools to your kit: chi2_old.m — this code was written in 1994, when I was taking PHYS 628, and it calculates the quantiles to a desired confidence (upper and lower bounds) of a χ²-distributed variable; this code doesn't require any special MATLAB toolboxes chi2.m — this is an improvement on the primitive code above, but only in terms of some speed and compact code; it only calculates a single quantile and uses MATLAB's built-in fminsearch optimization code; effectively, you should see no performance difference between the codes leaks.m — I ran this demo for you in class; it depicts some common windows and their DFT, as well as their off-index FT values, showing the effects of leakage for each type of window; if you don't have the Signal Processing Toolbox, you can hack out the various windows from the code welch.m — this is an example of how to write a window function if you don't have them built-in from MATLAB; this window is commonly used, but not a MATLAB feature

Week #10: 5–09 Apr 2010

∅ After that brief Maui visit (work is tough!) we'll consider the improvements in spectral estimation via the Welch method, as well as a rigorous statistical description of the uncertainty bounds on the PSD as a function of the degrees of freedom (ν) in our estimates. ⊗ Here is a scan of Fig. 3.10 from Jenkins & Watts, showing a graphical look at the uncertainties in spectral estimators ⇒ Rather than do a graphical look-up in Fig. 3.10, you can use either of the χ² quantile codes to estimate the uncertainties in a PSD (given ν > 2 d.o.f. & a desired uncertainty &alpha &isin (0.5, 1) ), you execute either of the following in MATLAB: lims = nu./chi2_old(nu,1-(1-alpha)/2); lims = nu./[chi2(nu,(1-alpha)/2) chi2(nu,1-(1-alpha)/2)]; ⇐ Then your PSD & bounds can be expressed as [PSD*lims(1) PSD PSD*lims(2)]

Week #11: 12–16 Apr 2010

∅ I'll be in Huntsville AL this week, but Prof. Olson will take you through the spectral matrix and its applications ⊕ Homework #7 due 21 Apr, and a solution is now posted

Week #12: 19–23 Apr 2010

∅ I'm actually here this week! And for a few days we'll look at an alternative functional expansion, similar to the DFT, called the Discrete Karhunen-Loéve Transform (or DKLT, for short) © Here are some notes (excerpted from my thesis) on DSP in general and the DKLT ⊕ Homework #8 due 26 Apr, and a solution is now posted ⊄ 23 Apr: Spring Fest, no classes

Week #13: 26–30 Apr 2010

∅ This week we'll finish up with the DKLT and take a swing through the territory of parameter estimation via array processing ⊄ An elegant proof of the positive, semi-definite nature of the eigenvalues of the matrices P & R (used in the DLKT), graciously provided by my esteemed colleague, Dr. Roger Waxler of the National Center for Physical Acoustics at The University of Mississippi. © Here is a code for computing DKLT, including the proper normalizations, for use on HW #9 ® And an improved version of the code I ran for you in class on Friday, egDKLT.m – I corrected the problem with letting M > N; play with this before you do the HW assignment ⊕ Homework #9 due 7 May, and a solution is now posted

Week #14: 3–7 May 2010

∅ Last week of class – we'll discuss a bit of array processing and the ability to estimate the back-azimuth of a source (acoustic, seismic, etc.) from an array of omni-directional sensors whose output is sampled in time © Here's a link to the lecture on acoustic signal processing I gave for PASS 2008 (the Physical Acoustics Summer School) – the movies and sounds aren't included, but I can provide them if you really want 'em

Week #F: 12 May 2010

¤ Final Exam (take-home) will be posted here beginning 7 May. It is due NLT 10:00 on 12 May, per the UAF Final Exam Schedule. ∅ And here's the solution for the final exam – have a great summer!