ECE 6279: Spatial Array Processing
Georgia Institute of Technology
Spring 2007

Instructor: Prof. Aaron Lanterman
Main Office: Centergy 5212
Phone: 404-385-2548
AOL Instant Messager ID: DrAaronL (Even if you don’t like AOL, there
are many chat clients that will use that protocol. I use iChat on the Mac.
I’m on quite frequently, pretty much whenever I’m at a computer,
and welcome questions over AIM.)
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:
When and where: MWF, 12:05-12:55,
Weber SST III 1 (That’s what it says on Oscar – I haven’t actually figured
out where the building is yet!)
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!





  • Homework 7 (due 4/25 for in class students, 5/2 for video). Please get it
    in on time so I can post solutions immediately – I’ve been too lenient on this
    in the past.


  • Quiz I: Tuesday, 2/20, 7:00 PM; room Van Leer 241
  • Quiz II: Tuesday, 3/27, 7:00 PM; room Van Leer 241
  • Quiz III: Thursday, 4/26, 7:00 PM; room TBA



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’ve included 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!

  • 1/10: Lecture 1: Introduction and Sales Pitch

  • 1/12: Lecture 2: Propagating Waves
    (pdf, revised 1/17)

  • 1/17: Lecture 3: Wavenumber-Frequency Space
    (pdf, revised 1/17)

  • 1/19: Lecture 4: Apertures, Part I

  • 1/21: Lecture 5: Apertures, Part II
    (B&W pdf,
    color pdf)
    (MATLAB, revised 1/28)

  • 1/24: Lecture 6: Delay-and-Sum Beamforming for Plane Waves
    (B&W pdf,
    color pdf, revised 1/28)

  • 1/26: Lecture 7: Delay-and-Sum Beamforming for Spherical Waves
    (B&W pdf,
    color pdf, revised 1/28)
    21.5 MB AVI movie)

  • 1/29: Lecture 8: Filter-and-Sum Beamforming
    (B&W pdf,
    color pdf, revised 2/1)

  • 1/30: Lecture 9: Quadrature Demodulation (whiteboard)
  • 2/2: Lecture 10: Conventional Narrowband Beamforming
    (B&W pdf,
    color pdf, second revision posted 2/16)

  • 2/5: Lecture 11: Conventional Wideband Beamforming
    (B&W pdf,
    color pdf)

  • 2/7: Lecture 12: Stochastic Narrowband Models
    (B&W pdf,
    color pdf, revised 2/16)

  • 2/9: Lecture 13: Signal to Noise
    (B&W pdf,
    color pdf)

  • 2/12: Lecture 14: Time Averaging
    (B&W pdf,
    color pdf, link to revision fixed 3/22)

  • 2/14: Lecture 15: Spatial Averaging and Co-arrays
    (B&W pdf,
    color pdf, revised 3/22)

  • 2/19-2/21: Lecture 16: Constrained Optimization
    (B&W pdf,
    color pdf, revised 3/22)

  • 2/23: Lecture 17: MVDR Beamforming
    (B&W pdf,
    color pdf)

  • 2/26: Lecture 18: Pisarenko Harmonic Decomposition
    (B&W pdf,
    color pdf)

  • 2/28: Lecture 19: Subspace Methods: Eigenvalue Method and MUSIC
    (B&W pdf,
    color pdf)

  • 3/2, 3/5, 3/7: No class – Aaron out of town
  • 3/9: Lecture 20: Root MUSIC (whiteboard)
  • 3/12: Lecture 21: ESPRIT, Part I (The Setup)
  • 3/14: Lecture 22: ESPRIT, Part II (Total Least Squares)
  • 3/16: Lecture 23: Robust Constrained Estimation
  • 3/19-3/23: Spring break
  • 3/26: Review for Quiz 2
  • 3/28: Lecture 24: Introduction to Maximum-Likelihood Estimation
  • 3/30: No class – Aaron out of town
  • 4/2: Lecture 25: Properties of Estimators (Bias and Variance)
  • Remaining dates are tentative (I’m just trying to plot out a roadmap
    to make sure the endgame works out nicely)

  • 4/4: Lecture 26: Bias/Variance Tradeoffs
  • Useful paper covering some material in Lectures 27, 28, and 29:
    M.I. Miller and D.R. Fuhrmann,
    narrow-band direction finding and the EM algorithm
    “, IEEE Trans. on
    Acoustics, Speech, and Signal Proc., Vol. 38, No. 9, Sept. 1990,
    pp. 1560-1577.

  • 4/6: Lecture 27: “Stochastic Signal” Gaussian Model
    (B&W pdf,
    color pdf)

  • 4/9: Lecture 28: “Deterministic Signal” Gaussian Model
    (B&W pdf,
    color pdf)

  • 4/11: Lecture 29: Special Cases of Maximum-Likelihood Estimation
    (B&W pdf,
    color pdf)

  • 4/13: Lecture 30: Model Order Estimation, Part 1 (Introduction
    and Nonsensical Derivation)

  • 4/16: Lecture 31: Model Order Estimation, Part 2 (Sensical Derivation);
    Introduction to Cramer-Rao Bounds

  • 4/18: Lecture 32: Transformation of Cramer-Rao Bounds
  • 4/20: Lecture 33: Cramer-Rao Bounds for Sensor Arrays
  • 4/23: Lecture 34: Where to Go from Here (and Maybe Random Quiz Hints)
    (B&W pdf,
    color pdf)

  • 4/25: TBD
  • 4/27: Last class! Woo hoo!


  • 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!)

Administrative Details

Prerequisities: Officially, ECE4270: Introduction to Digital
Signal Processing. In reality, I won’t use much more theory than you
would find in ECE2025. You will need to know basic probability theory
(Gaussian distributions, 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 result’s 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
which is a pretty impressive open-source programming language designed to
be compatible with MATLAB.

Office hours:
I generally don’t have official locked in
“office hours.”
I tend to bounce back and forth between Centergy, Van Leer,
and Energy Coffee on the corner 14th and State. (I highly recommend that you
check out Energy Coffee, it’s a great place.) If you happen to already be in
Centergy and find that my door is
open, come on in and say hi, I always welcome questions. However,
I’m hard to “catch” in my office, so I
don’t recommend making a special trip to Centery without checking with me
first to make sure I’ll be there.

I’ll usually head to lunch in the student center after class.
People are welcome to join me and discuss things.

I will make sure I spend some specific
time both in Centergy and Bunger Henry before
homeworks are due.
I’ll announce such things in class.

You’re of course 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 20%, and not accepted at all after
solutions are handed out to the class. This is the largest graduate
class I’ve ever taught; there are over 50 students signed up. The last time
I ran 6279, there were 15. 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.

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.

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.

Tentative grade breakdown: Quiz 1: 25%, Quiz 2: 25%, Quiz 3: 25%,
Homeworks 25%. (That is a lot of quizzes for a grad class, but the material
divides nicely into three sections; also, this helps reduce the “bad day”
effect. If you have a bad day on one quiz, it’s not as devastating as
in a class where there’s just a midterm and a final.)

The “no slacking” rule: The
homeworks in this class are important; in them, you will explore concepts
more deeply than I can possibly put on a quiz. They have a cosmic importance
beyond the 25% described above. Hence, you must make a decent good-faith
effort on every homework, i.e. don’t ditch any of the homeworks. Turn in
something. If you seem to be slacking on the homework,
I reserve the right to lower your letter grade beyond that indicated by the
percentages in a way you won’t like. I will give you fair warning if I start
to get the impression that you are in the slacking zone. It will be easiest
on everyone if you just don’t go into the slacking zone to begin with.

Why aren’t we doing a final?:
The scheduled final is Friday at 8 AM. I can’t get excited about that. I
imagine you’re not terribly excited about that time slot either.
Doing all
the quizzes during the semester ensures that you can focus on on studying
for other finals during finals week itself. Also, for any given quiz, you
only have to study an amount of material equivalent to 50 minutes worth
of quizzing, instead of having to
jam down nearly three hours worth of material. (I’m not 100% set on this
idea. If the class really wants a final exam at 8 AM,
I’ll entertain the possibility
of giving one. I also reserve the right to give some sort of take-home final,
in which case I’ll adjust the grade percentages given above accordingly).

Tentative Topics

Here are the topics I covered in the Spring 2005 offering 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 –
Where to Go from Here