ECE 6279: Spatial Array Processing

ECE 6279: Spatial Array Processing
Georgia Institute of Technology
Spring 2013

Instructor: Prof. Aaron Lanterman
Official Office: Van Leer W431
Phone: 404-385-2548 (I only occasionally check messages – e-mail
is by far the best way to get in touch with me)
E-mail: (If you’re e-mailing me about a 6279
related topic, please put “6279” in your
subject line so I can sort 6279 related e-mails easily. Similarly, when
I send class related e-mail, which I tend to do a lot, I’ll try to put
6279 in the subject, although sometimes I will forget.)
Course website:
(actually you should know that already if you’re reading this)
When and where: TuTh, 12:05-1:25, Van Leer C457
Syllabus: You’re looking at it right now. It’s a living
syllabus. I’m not going to pass out a dead tree version, nor do I suggest you
print it out when you can always get the latest version by looking at the
website. Go electrons!


  • 11/30: Homework 10 posted. Last one!


  • Quiz 1: October 10
  • Quiz 2: November 19


  • Homework 2,
    due Tuesday, 9/17/13
  • Homework 3,
    due Tuesday, 10/1/13
  • Homework 4,
    due Tuesday, 10/8/13

  • Homework 5,
    due Tuesday, 10/22/13
  • Homework 6,
    due Tuesday, 10/29/13
  • Homework 7,
    due Tuesday, 11/5/13

  • Homework 8,
    due Friday, 11/15/13 (only worth half as much as a regular homework)
  • Homework 9,
    due Friday, 11/22/13
  • Homework 10,
    due Friday, 12/6/13


    Credit to where it’s due: These slides draw heavily from Johnson and
    Dudgeon, and are also heavily influenced by notes provided by Doug Williams
    for earlier versions of this course, and the version of this course I
    took from Dan Fuhrmann at Washington University many years ago.

    In addition to the PDF, I’ll include the MATLAB code used to make the
    various figures on the slides and for various in-class demos. Note that
    I whipped up each code fragment as quickly as possible to get the result
    I needed, with no thought to re-use, good coding style, or even any
    sense of common decency. The comments may be leftovers from previous
    scripts, victims of cut and paste, and hence may not accurate.
    Use at your own risk!


  • 10/29 ESPRIT
  • 10/31 Robust Constrained Est
  • 11/5 Intro Est Th Part 1
  • 11/7 Stochastic Signal Guassian Model
  • 11/12 Det Sig Gaussian Model
  • 11/14 Special Cases of ML Est
  • 11/19 Intro to CR Bounds
  • 11/21 CRB for Arrays
  • 11/26 Transform of CR Bounds
  • 12/3 Model Order Est
  • 12/6 Unconstrained Cov Est
  • 10/1: Lecture 11: Conventional Wideband Beamforming
    slides 4-up)
  • 10/15: Fall Break
  • 10/29: Lecture 18: Pisarenko Harmonic Decomposition
    slides 4-up)
  • 11/12: Lecture 22: ESPRIT, Part 2 (Total Least Squares)
    slides 4-up)
  • 11/21: Lecture 25: Introduction to Estimation Theory, Part II
    slides 4-up)

  • 4/25: Lecture 32: Model Order Estimation
    slides 4-up)
  • 4/13: Lecture 33: Unconstrained Covariance Estimation
    slides 4-up)
  • 4/15: Lecture 34: Where to Go from Here?
    slides 4-up)


    • Only required text: D.H. Johnson and J.E. Dudgeon,
      Array Signal Processing: Concepts and Techniques, Prentice Hall, 1993.
      Available in the bookstore. (I also saw a few used copies at a reasonable
      price on Amazon.) If the bookstore has run out and you want a copy from them,
      let me know so I can tell them to order more.
    • Good to get if you’re rolling in extra $$$ and plan to do
      serious research in array signal processing:

      H.L. Van Trees, Optimum Array Processing (Part IV of Detection,
      Estimation, and Modulation Theory), Wiley 2002 (not required!)
    • The Matrix Cookbook (thanks to
      Aaron Albin for the tip)

    Administrative Details

    Laptops down during lecture: This isn’t because I’m offended that
    you’re paying more attention to your e-mail than to me; it has to do with
    the degree to which I am easily distracted. When I see laptops up, I find
    myself wondering what people are looking at and I too easily wander off course.

    Prerequisities: Officially, ECE4270: Introduction to Digital
    Signal Processing. In reality, I won’t use much more theory than you
    would find in ECE2025 or ECE2026+ECE3084.
    You will need to know basic probability theory
    (Gaussian distributions, covariances,
    Bayes rule, conditional expectations) at the
    level of ECE3075, and not be afraid of a little linear algebra (i.e.
    eigenvectors and eigenvalues should be good friends), although you need
    not have had an entire class on linear algebra. I will try to make this
    course as accessible to a wide variety of backgrounds as possible; hence,
    if I start throwing around some theory you’ve never seen before, let me
    know. I wouldn’t mind either doing a little review in class, or pointing
    you to where you should look to get caught up quickly.

    Note that at
    other universities, a detection and estimation class is usually a prerequisite
    for an spatial array processing. That is not the case at Georgia Tech.
    Hence, I will give Reader’s Digest versions of some topics from ECE7251,
    just focusing on the results you’ll need, without the proofs.

    We will be doing a lot of hacking in MATLAB. You can use another language
    if you really really really want, but you will find MATLAB will make your
    life much easier. The rather inexpensive student version is available in
    the bookstore. Also,
    you can look into Octave or
    which are pretty impressive open-source programming languages designed to
    be compatible with MATLAB.

    Office hours: Shortly before
    homeworks and exams, I will send out an e-mail describing when and
    where I will be sure to be available for questions. This will tend to
    change slightly from week to week, so look for that e-mail. (Also,
    if you walk by my office and
    happen to see my office door open, you are welcome to pop in with
    questions about the class and/or life in general.)

    Of course you are always welcome to e-mail me and we can set up a specific
    time to meet. Again, put “6279” in your subject.

    Homeworks: Homeworks will be assigned every one to two weeks, and
    be turned in at the beginning of the class they are due. If you need to
    miss class, make other arrangements to get the homework to me.
    Late homeworks will be penalized, usually 30% per day (or something
    like that; details will be provided on each homework),
    and not be accepted at all after
    solutions are handed out to the class. The class is rather large and was
    given no T.A. support.
    Hence, I will probably not be able to do heavily
    detailed grading and to a certain extent will be grading by zen. Please have
    mercy on me and do not quibble over minor partial credit issues – these things
    tend to average out in the end.

    Honor code:
    This course will be conducted under the rules and guidelines of
    the Georgia Tech Honor Code; infractions will be reported to the
    Dean of Students.

    Backfile policy: Use of homework solutions and quizzes from previous
    versions of ECE6279 is forbidden. The material is highly complex, so it is
    extremely difficult to come up with 100% new problems on each offering.
    Please be fair to students who may not have access to the same old materials.
    Detection of the use of backfiles will result in significant wrath. I have
    substantial experience with this matter on recycled lab reports in ECE2025.
    I have never had to report an honor code violation in any of my graduate
    classes; please let me continue that trend.

    Collaboration policy:
    You are encouraged to discuss the homework problems with one other
    at the “whiteboard” level – i.e., you are free to exchange ideas about
    how to approach a problem, and doodle equations to each other on napkins
    at lunch. Working homework in groups is OK (you are encouraged to learn
    from each other as much as you learn from me!), but you must keep the
    discussion at a conceptual level, and the work you turn in should be your
    own. You shouldn’t look at someone else’s completed solution, as the
    temptation to just copy it would be too high.

    You can help one another debug at the level of:

    • “Why is MATLAB giving me
      this error on line 52?”
    • “Hey, what’s the MATLAB command to find
      the eigenvalues of an unlaiden European swallow?”
    • “Glancing at this horrendous recursion,
      can you think of why I’m running out of memory?”
    • “Why does my linear beamforming code complain about me dividing by
    • “Why don’t my matrix dimensions match in this multiplication?”

    …but you should not spend a whole lot of time looking
    at the screen of a fellow student when they are
    working on their code. In particular:
    Under no circumstances
    should you give your computer code
    to another student

    To my knowledge, this pretty much matches the approach most grad
    classes in ECE
    take towards what collaboration is and isn’t allowed. Ask me if you
    are uncertain about any of this.

    Quizzes: We will have two midterm quizzes given during the usual
    lecture period.

    The quizzes will be targeted to be about an hour in length, but
    to avoid punishing people who understand the material but have difficulties
    working under the stress of time pressure, I will give you the full hour and a
    twenty minutes to work them. The quizzes will be open book and open notes.

    Tentative grade breakdown: Quiz 1: 20%, Quiz 2: 20%, Final: 25%,
    Homeworks 35%.

    Note the homework is worth more than you’ll find in most classes –
    that’s because the homework will include some valuable computer exercises
    that I cannot replicate in an exam setting,
    and I want to make sure you don’t blow them off.

    The class divides nicely into three chunks, and the final will primarily
    on the last chunk
    (hence it’s not worth much more than the two earlier quizzes).

    Major emergencies: If you have some sort of major life emergency
    – serious illness or injury, death in the family, house
    burns down or is flooded, etc. – that seriously
    your progress in the class,
    please let me know as soon as possible so we can work something
    out. You will find professors can be quite reasonable if you keep us in the
    loop. Please don’t disappear with no warning
    half way through, making me think that you
    dropped the class, and then reappear out of nowhere
    during dead week asking what you can do to make
    things up. (Yes, this has happened quite a bit, in both undergrad and
    grad classes.)

    Tentative Topics

    Here are the topics I covered in previous offerings of ECE6279, in
    the order I covered them, and the number of lectures I spent on each topic.
    I’ll probably rearrange things as we go along, adding and deleting a topic
    here or there, sometimes swapping things aroung if I think of a better order,
    but this will give you an idea of where we’re going:

    Introduction and Sales Pitch – Propagating Waves – Wavenumber-Frequency Space
    – Apertures (two lectures) – Delay-and-Sum Beamforming for Plane Waves –
    Delay-and-Sum Beamforming for Spherical Waves –
    Filter-and-Sum Beamforming –
    Bandlimited Signals –
    Conventional Narrowband Beamforming –
    Conventional Wideband Beamforming –
    Stochastic Narrowband Models –
    Signal to Noise –
    Temporal Averaging –
    Spatial Averaging and Co-Arrays –
    Constrained Optimization –
    MVDR Beamforming (two lectures) –
    Pisarenko Harmonic Decomposition –
    Subspace Decompositions –
    Eigenvalue Method and MUSIC –
    Root MUSIC –
    ESPRIT (three lectures) –
    Robust Constrained Optimization –
    Nonrandom Parameter Estimation –
    Properties of Estimators –
    “Stochastic Signal” Gaussian Model –
    “Deterministic Signal” Gaussian Model –
    Special Cases of Maximum-Likelihood Estimation –
    Introduction to Cramer-Rao Bounds –
    Cramer-Rao Bounds for Direction Finding –
    Model Order Estimation – Unconstrained Covariance Estimation –
    Where to Go from Here