# 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:** lanterma@ece.gatech.edu (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**: users.ece.gatech.edu/~lanterma/ece6279

**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!

<!–

# Syllabus

–>

# News

- 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.

# Quizzes

- 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

# Homeworks

- Homework 1,

due 1/26/07 (campus) or 2/2/07 (video)

- Homework 2,

due 2/7/07 (campus) or 2/14/07 (video) - Homework 3,

due 2/19/07 (campus) or 2/26/07 (video)- MATLAB files:

look_vector.m,

steering_vector.m,

makecross.m,

steered_response_hw3_07.m,

crandn.m

- MATLAB files:
- Homework 4,

due 3/26/07 (campus) or 4/2/07 (video) - Homework 5,

due 4/11 (campus) or 4/18 (video) (I forgot to mention this in the writeup,

but of course, provide your code!)- MATLAB file: hw5_data.mat

- Homework 6
- Homework 7<!–
- Homework 2,

originally due 2/7/05 (campus) or 2/21/05 (video);**extended to**

2/9/05 (campus) or 2/23/05 (video) - Homework 4,

due 3/18/05 (campus) or 4/1/05 (video) - Homework 6, due

4/29/05 (campus) or 5/13/05 (video) - Homework 7, due

5/6/05 (campus) or 5/20/05 (video)

–># Lectures

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

(pdf) - 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

(pdf)

(MATLAB) - 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)

(MATLAB) - 1/26: Lecture 7: Delay-and-Sum Beamforming for Spherical Waves

(B&W pdf,

color pdf, revised 1/28)

(MATLAB,

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)
- B.D. Rao and K.V.S. Hari,

“Performance

Analysis of Root-Music,”

*IEEE Trans. on Acoustics, Speech, and Signal Proc.*, Vol. 37, No. 12,

Dec. 1989.

- B.D. Rao and K.V.S. Hari,
- 3/12: Lecture 21: ESPRIT, Part I (The Setup)
- A.L. Swindlehurst, B. Ottersten, R. Roy, and T. Kailath,

“Multiple

Invariance ESPRIT“,

*IEEE Trans. on Signal Proc.*, Vol. 40, No. 4, 1992. (See Section II). - A. Paulraj, R. Roy, and T. Kailath,

“A Subspace Rotation Approach to Signal Parameter Estimation“,

Proc. IEEE, Vol. 74, No. 7, July 1986, pp. 1044-1045.

- A.L. Swindlehurst, B. Ottersten, R. Roy, and T. Kailath,
- 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,

“Maximum-likelihood

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)- M. Wax and T. Kailath,

“Determining

the Number of Signals by Information Theoretic

Criteria“, IEEE Trans. on Acoustics, Speech, and Signal Proc.,

Vol. 33, No. 2, April 1985, pp. 387-392.

- M. Wax and T. Kailath,
- 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
- P. Stoica and A. Nehorai,

“MUSIC, Maximum Likelihood, and

Cramer Rao Bound“, IEEE Trans. on Acoustics, Speech, and Signal

Proc., Vol. 37, No. 5, May 1989, pp. 720-741. - P. Stoica and A. Nehorai,

“MUSIC, Maximum Likelihood, and

Cramer-Rao Bound: Further Results and Comparisons“, IEEE Trans. on

Acoustics, Speech, and Signal Proc., Vol. 38, No. 12, Dec. 1990, pp.

2140-2150.

- P. Stoica and A. Nehorai,
- 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!

# References

**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

should

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

zero?” - “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 studentTo 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 - 1/10: Lecture 1: Introduction and Sales Pitch