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Basic Deﬁnitions and The Spectral Estimation Problem Lecture 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L1–1 by P. Stoica and R. Moses, Prentice Hall, 1997
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Spectral Analysis by Petre Stoica

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• Basic Definitionsand

The Spectral Estimation Problem

Lecture 1

Lecture notes to accompany Introduction to Spectral Analysis Slide L11by P. Stoica and R. Moses, Prentice Hall, 1997

• Informal Definition of Spectral Estimation

Given: A finite record of a signal.

Determine: The distribution of signal power overfrequency.

t

signal

t=1, 2, . . . +

spectral density

! = (angular) frequency in radians/(sampling interval)f = !=2 = frequency in cycles/(sampling interval)

Lecture notes to accompany Introduction to Spectral Analysis Slide L12by P. Stoica and R. Moses, Prentice Hall, 1997

• Applications

Temporal Spectral Analysis

Vibration monitoring and fault detection

Hidden periodicity finding

Speech processing and audio devices

Medical diagnosis

Seismology and ground movement study

Control systems design

Spatial Spectral Analysis

Source location using sensor arrays

Lecture notes to accompany Introduction to Spectral Analysis Slide L13by P. Stoica and R. Moses, Prentice Hall, 1997

• Deterministic Signals

fy(t)g

1

t=1

= discrete-time deterministic datasequence

If:1

X

t=1

jy(t)j

2

• Energy Spectral Density

Parseval's Equality:

1

X

t=1

jy(t)j

2

=

1

2

Z

S(!)d!

where

S(!)

4

= jY (!)j

2

= Energy Spectral Density

We can write

S(!) =

1

X

k=1

(k)e

i!k

where

(k) =

1

X

t=1

y(t)y

(t k)

Lecture notes to accompany Introduction to Spectral Analysis Slide L15by P. Stoica and R. Moses, Prentice Hall, 1997

• Random Signals

Random Signal

t

random signal probabilistic statements about future variations

current observation time

Here:1

X

t=1

jy(t)j

2

=1

But: En

jy(t)j

2

o

• First Definition of PSD

(!) =

1

X

k=1

r(k)e

i!k

where r(k) is the autocovariance sequence (ACS)r(k) = E fy(t)y

(t k)g

r(k) = r

(k); r(0) jr(k)j

Note that

r(k) =

1

2

Z

(!)e

i!k

d! (Inverse DTFT)

Interpretation:

r(0) = E

n

jy(t)j

2

o

=

1

2

Z

(!)d!

so

(!)d! = infinitesimal signal power in the band!

d!

2

Lecture notes to accompany Introduction to Spectral Analysis Slide L17by P. Stoica and R. Moses, Prentice Hall, 1997

• Second Definition of PSD

(!) = lim

N!1

E

8

>

:

1

N

N

X

t=1

y(t)e

i!t

2

9

>

=

>

;

Note that

(!) = lim

N!1

E

1

N

jY

N

(!)j

2

where

Y

N

(!) =

N

X

t=1

y(t)e

i!t

is the finite DTFT of fy(t)g.

Lecture notes to accompany Introduction to Spectral Analysis Slide L18by P. Stoica and R. Moses, Prentice Hall, 1997

• Properties of the PSD

P1: (!) = (!+2) for all !.

Thus, we can restrict attention to

! 2 [; ] () f 2 [1=2;1=2]

P2: (!) 0

P3: If y(t) is real,

Then: (!) = (!)

Otherwise: (!) 6= (!)

Lecture notes to accompany Introduction to Spectral Analysis Slide L19by P. Stoica and R. Moses, Prentice Hall, 1997

• Transfer of PSD Through Linear Systems

System Function: H(q) =1

X

k=0

h

k

q

k

where q1 = unit delay operator: q1y(t) = y(t 1)

e(t)

e

(!)

y(t)

y

(!) = jH(!)j

2

e

(!)

H(q)

--

Then

y(t) =

1

X

k=0

h

k

e(t k)

H(!) =

1

X

k=0

h

k

e

i!k

y

(!) = jH(!)j2

e

(!)

Lecture notes to accompany Introduction to Spectral Analysis Slide L110by P. Stoica and R. Moses, Prentice Hall, 1997

• The Spectral Estimation Problem

The Problem:

From a sample fy(1); : : : ; y(N)g

Find an estimate of (!): f^(!); ! 2 [; ]g

Two Main Approaches :

Nonparametric:

Derived from the PSD definitions.

Parametric:

Assumes a parameterized functional form of thePSD

Lecture notes to accompany Introduction to Spectral Analysis Slide L111by P. Stoica and R. Moses, Prentice Hall, 1997

• Periodogramand

CorrelogramMethods

Lecture 2

Lecture notes to accompany Introduction to Spectral Analysis Slide L21by P. Stoica and R. Moses, Prentice Hall, 1997

• Periodogram

Recall 2nd definition of (!):

(!) = lim

N!1

E

8

>

:

1

N

N

X

t=1

y(t)e

i!t

2

9

>

=

>

;

Given : fy(t)gNt=1

Drop limN!1

and E fg to get

^

p

(!) =

1

N

N

X

t=1

y(t)e

i!t

2

Natural estimator

Used by Schuster (1900) to determine hiddenperiodicities (hence the name).

Lecture notes to accompany Introduction to Spectral Analysis Slide L22by P. Stoica and R. Moses, Prentice Hall, 1997

• Correlogram

Recall 1st definition of (!):

(!) =

1

X

k=1

r(k)e

i!k

Truncate the P and replace r(k) by r^(k):

^

c

(!) =

N1

X

k=(N1)

r^(k)e

i!k

Lecture notes to accompany Introduction to Spectral Analysis Slide L23by P. Stoica and R. Moses, Prentice Hall, 1997

• Covariance Estimators(or Sample Covariances)

Standard unbiased estimate:

r^(k) =

1

N k

N

X

t=k+1

y(t)y

(t k); k 0

Standard biased estimate:

r^(k) =

1

N

N

X

t=k+1

y(t)y

(t k); k 0

For both estimators:

r^(k) = r^

(k); k < 0

Lecture notes to accompany Introduction to Spectral Analysis Slide L24by P. Stoica and R. Moses, Prentice Hall, 1997

• Relationship Between ^p

(!) and ^c

(!)

If: the biased ACS estimator r^(k) is used in ^c

(!),

Then:

^

p

(!) =

1

N

N

X

t=1

y(t)e

i!t

2

=

N1

X

k=(N1)

r^(k)e

i!k

=

^

c

(!)

^

p

(!) =

^

c

(!)

Consequence:Both ^

p

(!) and ^c

(!) can be analyzed simultaneously.

Lecture notes to accompany Introduction to Spectral Analysis Slide L25by P. Stoica and R. Moses, Prentice Hall, 1997

• Statistical Performance of ^p

(!) and ^c

(!)

Summary:

Both are asymptotically (for large N ) unbiased:

E

n

^

p

(!)

o

! (!) as N !1

Both have large variance, even for large N .

Thus, ^p

(!) and ^c

(!) have poor performance.

Intuitive explanation:

r^(k) r(k) may be large for large jkj

Even if the errors fr^(k) r(k)gN1jkj=0

are small,there are so many that when summed in[

^

p

(!) (!)], the PSD error is large.

Lecture notes to accompany Introduction to Spectral Analysis Slide L26by P. Stoica and R. Moses, Prentice Hall, 1997

• Bias Analysis of the Periodogram

E

n

^

p

(!)

o

= E

n

^

c

(!)

o

=

N1

X

k=(N1)

E fr^(k)g e

i!k

=

N1

X

k=(N1)

1

jkj

N

!

r(k)e

i!k

=

1

X

k=1

w

B

(k)r(k)e

i!k

w

B

(k) =

8

n overdetermined YW system of equations; leastsquares solution for fa^

i

g.

Note: For narrowband ARMA signals, the accuracy offa^

i

g is often better for M > n

Lecture notes to accompany Introduction to Spectral Analysis Slide L417by P. Stoica and R. Moses, Prentice Hall, 1997

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• Parametric Methodsfor

Line Spectra Part 1

Lecture 5

Lecture notes to accompany Introduction to Spectral Analysis Slide L51by P. Stoica and R. Moses, Prentice Hall, 1997

• Line Spectra

Many applications have signals with (near) sinusoidalcomponents. Examples:

communications

geophysical seismology

ARMA model is a poor approximation

Better approximation by Discrete/Line Spectrum Models

-

6

6

6

6

2

!

1

!

2

!

3

(2

2

1

)

(2

2

2

)

(2

2

3

)

!

(!)

An Ideal line spectrumLecture notes to accompany Introduction to Spectral Analysis Slide L52by P. Stoica and R. Moses, Prentice Hall, 1997

• Line Spectral Signal Model

Signal Model: Sinusoidal components of frequenciesf!

k

g and powers f2k

g, superimposed in white noise ofpower 2.

y(t) = x(t) + e(t) t = 1;2; : : :

x(t) =

n

X

k=1

k

e

i(!

k

t+

k

)

| {z }

x

k

(t)

Assumptions:

A1: k

> 0 !

k

2 [; ]

(prevents model ambiguities)A2: f'

k

g= independent rv's, uniformlydistributed on [; ]

(realistic and mathematically convenient)

A3: e(t) = circular white noise with variance 2

E fe(t)e

(s)g =

2

t;s

E fe(t)e(s)g = 0

(can be achieved by slow sampling)

Lecture notes to accompany Introduction to Spectral Analysis Slide L53by P. Stoica and R. Moses, Prentice Hall, 1997

• Covariance Function and PSD

Note that:

E

n

e

i'

p

e

i'

j

o

= 1, for p = j

E

n

e

i'

p

e

i'

j

o

= E

n

e

i'

p

o

E

n

e

i'

j

o

=

1

2

Z

e

i'

d'

2

= 0, for p 6= j

Hence,

E

n

x

p

(t)x

j

(t k)

o

=

2

p

e

i!

p

k

p;j

r(k) = E fy(t)y

(t k)g

=

P

n

p=1

2

p

e

i!

p

k

+

2

k;0

and

(!) = 2

n

X

p=1

2

p

(! !

p

) +

2

Lecture notes to accompany Introduction to Spectral Analysis Slide L54by P. Stoica and R. Moses, Prentice Hall, 1997

• Parameter Estimation

Estimate either:

f!

k

;

k

; '

k

g

n

k=1

;

2 (Signal Model)

f!

k

;

2

k

g

n

k=1

;

2 (PSD Model)

Major Estimation Problem: f!^k

g

Once f!^k

g are determined:

f^

2

k

g can be obtained by a least squares method from

r^(k) =

n

X

p=1

2

p

e

i!^

p

k + residuals

OR:

Both f^k

g and f'^k

g can be derived by a leastsquares method from

y(t) =

n

X

k=1

k

e

i!^

k

t + residuals

with k

=

k

e

i'

k

.

Lecture notes to accompany Introduction to Spectral Analysis Slide L55by P. Stoica and R. Moses, Prentice Hall, 1997

• Nonlinear Least Squares (NLS) Method

min

f!

k

;

k

;'

k

g

N

X

t=1

y(t)

n

X

k=1

k

e

i(!

k

t+'

k

)

2

| {z }

F(!;;')

Let:

k

=

k

e

i'

k

= [

1

: : :

n

]

T

Y = [y(1) : : : y(N)]

T

B =

2

6

4

e

i!

1

e

i!

n

.

.

.

.

.

.

e

iN!

1

e

iN!

n

3

7

5

Lecture notes to accompany Introduction to Spectral Analysis Slide L56by P. Stoica and R. Moses, Prentice Hall, 1997

• Nonlinear Least Squares (NLS) Method, con't

Then:

F = (Y B)

(Y B) = kY Bk

2

= [ (B

B)

1

B

Y ]

[B

B]

[ (B

B)

1

B

Y ]

+Y

Y Y

B(B

B)

1

B

Y

This gives:

^

= (B

B)

1

B

Y

!=!^

and

!^ = arg max

!

Y

B(B

B)

1

B

Y

Lecture notes to accompany Introduction to Spectral Analysis Slide L57by P. Stoica and R. Moses, Prentice Hall, 1997

• NLS Properties

Excellent Accuracy:

var (!^k

) =

6

2

N

3

2

k

(for N 1)

Example: N = 300SNR

k

=

2

k

=

2

= 30 dB

Thenq

var(!^k

) 10

5

.

Difficult Implementation:

The NLS cost function F is multimodal; it is difficult toavoid convergence to local minima.

Lecture notes to accompany Introduction to Spectral Analysis Slide L58by P. Stoica and R. Moses, Prentice Hall, 1997

• Unwindowed Periodogram as anApproximate NLS Method

For a single (complex) sinusoid, the maximum of theunwindowed periodogram is the NLS frequency estimate:

Assume: n = 1

Then: BB = N

B

Y =

N

X

t=1

y(t)e

i!t

= Y (!) (finite DTFT)

Y

B(B

B)

1

B

Y =

1

N

jY (!)j

2

=

^

p

(!)

= (Unwindowed Periodogram)So, with no approximation,

!^ = argmax

!

^

p

(!)

Lecture notes to accompany Introduction to Spectral Analysis Slide L59by P. Stoica and R. Moses, Prentice Hall, 1997

• Unwindowed Periodogram as anApproximate NLS Method, con't

Assume: n > 1

Then:

f!^

k

g

n

k=1

' the locations of the n largestpeaks of ^

p

(!)

provided that

inf j!

k

!

p

j > 2=N

which is the periodogram resolution limit.

If better resolution desired then use a High/SuperResolution method.

Lecture notes to accompany Introduction to Spectral Analysis Slide L510by P. Stoica and R. Moses, Prentice Hall, 1997

• High-Order Yule-Walker Method

Recall:

y(t) = x(t) + e(t) =

n

X

k=1

k

e

i(!

k

t+'

k

)

| {z }

x

k

(t)

+e(t)

Degenerate ARMA equation for y(t):

(1 e

i!

k

q

1

)x

k

(t)

=

k

n

e

i(!

k

t+'

k

)

e

i!

k

e

i[!

k

(t1)+'

k

]

o

= 0

Let

B(q) = 1+

L

X

k=1

b

k

q

k

4

= A(q)

A(q)

A(q) = (1 e

i!

1

q

1

) (1 e

i!

n

q

1

)

A(q) = arbitrary

Then B(q)x(t) 0 )

B(q)y(t) = B(q)e(t)

Lecture notes to accompany Introduction to Spectral Analysis Slide L511by P. Stoica and R. Moses, Prentice Hall, 1997

• High-Order Yule-Walker Method, con't

Estimation Procedure:

Estimate f^bi

g

L

i=1

using an ARMA MYW technique

Roots of ^B(q) give f!^k

g

n

k=1

, along with L nspurious roots.

Lecture notes to accompany Introduction to Spectral Analysis Slide L512by P. Stoica and R. Moses, Prentice Hall, 1997

• High-Order and Overdetermined YW Equations

ARMA covariance:

r(k) +

L

X

i=1

b

i

r(k i) = 0; k > L

In matrix form for k = L+1; : : : ; L+M

2

6

6

6

4

r(L) : : : r(1)

r(L+1) : : : r(2)

.

.

.

.

.

.

r(L+M 1) : : : r(M)

3

7

7

7

5

| {z }

4

=

b =

2

6

6

6

4

r(L+1)

r(L+2)

.

.

.

r(L+M)

3

7

7

7

5

| {z }

4

=

This is a high-order (if L > n) and overdetermined(if M > L) system of YW equations.

Lecture notes to accompany Introduction to Spectral Analysis Slide L513by P. Stoica and R. Moses, Prentice Hall, 1997

• High-Order and Overdetermined YW Equations,con't

Fact: rank() = n

SVD of : = UV

U = (M n) with UU = In

V

= (n L) with V V = In

= (n n), diagonal and nonsingular

Thus,

(UV

)b =

The Minimum-Norm solution is

b =

y

= V

1

U

Important property: The additional (L n) spuriouszeros of B(q) are located strictly inside the unit circle, ifthe Minimum-Norm solution b is used.

Lecture notes to accompany Introduction to Spectral Analysis Slide L514by P. Stoica and R. Moses, Prentice Hall, 1997

• HOYW Equations, Practical Solution

Let ^U , ^, ^V be defined similarly to U , , V from theSVD of ^.

Compute ^b = ^V ^1^U^

Then f!^k

g

n

k=1

are found from the n zeroes of ^B(q) thatare closest to the unit circle.

When the SNR is low, this approach may give spuriousfrequency estimates when L > n; this is the price paid forincreased accuracy when L > n.

Lecture notes to accompany Introduction to Spectral Analysis Slide L515by P. Stoica and R. Moses, Prentice Hall, 1997

• Parametric Methodsfor

Line Spectra Part 2

Lecture 6

Lecture notes to accompany Introduction to Spectral Analysis Slide L61by P. Stoica and R. Moses, Prentice Hall, 1997

• The Covariance Matrix Equation

Let:

a(!) = [1 e

i!

: : : e

i(m1)!

]

T

A = [a(!

1

) : : : a(!

n

)] (m n)

Note: rank(A) = n (for m n )

Define

~y(t)

4

=

2

6

6

6

4

y(t)

y(t 1)

.

.

.

y(tm+1)

3

7

7

7

5

= A~x(t) + ~e(t)

where

~x(t) = [x

1

(t) : : : x

n

(t)]

T

~e(t) = [e(t) : : : e(tm+1)]

T

Then

R

4

= E f~y(t)~y

(t)g = APA

+

2

I

with

P = E f~x(t)~x

(t)g =

2

6

4

2

1

0

...

0

2

n

3

7

5

Lecture notes to accompany Introduction to Spectral Analysis Slide L62by P. Stoica and R. Moses, Prentice Hall, 1997

• Eigendecomposition of R and Its Properties

R = APA

+

2

I (m > n)

Let:

1

2

: : :

m

: eigenvalues of R

fs

1

; : : : s

n

g: orthonormal eigenvectors associatedwith f

1

; : : : ;

n

g

fg

1

; : : : ; g

mn

g: orthonormal eigenvectors associatedwith f

n+1

; : : : ;

m

g

S = [s

1

: : : s

n

] (m n)

G = [g

1

: : : g

mn

] (m (m n))

Thus,

R = [S G]

2

6

4

1

...

m

3

7

5

"

S

G

#

Lecture notes to accompany Introduction to Spectral Analysis Slide L63by P. Stoica and R. Moses, Prentice Hall, 1997

• Eigendecomposition of R and Its Properties,con't

As rank(APA) = n:

k

>

2

k = 1; : : : ; n

k

=

2

k = n+1; : : : ;m

=

2

6

4

1

2

0

...

0

n

2

3

7

5

= nonsingular

Note:

RS = APA

S+

2

S = S

2

6

4

1

0

...

0

n

3

7

5

S = A(PA

S

1

)

4

= AC

with jCj 6= 0 (since rank(S) = rank(A) = n).Therefore, since SG= 0,

A

G = 0

Lecture notes to accompany Introduction to Spectral Analysis Slide L64by P. Stoica and R. Moses, Prentice Hall, 1997

• MUSIC Method

A

G=

2

6

4

a

(!

1

)

.

.

.

a

(!

n

)

3

7

5

G = 0

) fa(!

k

)g

n

k=1

? R(G)

Thus,

f!

k

g

n

k=1

are the unique solutions ofa

(!)GG

a(!) = 0.

Let:

^

R =

1

N

N

X

t=m

~y(t)~y

(t)

^

S;

^

G = S;G made from theeigenvectors of ^R

Lecture notes to accompany Introduction to Spectral Analysis Slide L65by P. Stoica and R. Moses, Prentice Hall, 1997

• Spectral and Root MUSIC Methods

Spectral MUSIC Method:

f!^

k

g

n

k=1

= the locations of the n highest peaks of thepseudo-spectrum function:

1

a

(!)

^

G

^

G

a(!)

; ! 2 [; ]

Root MUSIC Method:

f!^

k

g

n

k=1

= the angular positions of the n roots of:

a

T

(z

1

)

^

G

^

G

a(z) = 0

that are closest to the unit circle. Here,

a(z) = [1; z

1

; : : : ; z

(m1)

]

T

Note: Both variants of MUSIC may produce spuriousfrequency estimates.Lecture notes to accompany Introduction to Spectral Analysis Slide L66by P. Stoica and R. Moses, Prentice Hall, 1997

• Pisarenko Method

Pisarenko is a special case of MUSIC with m = n+1(the minimum possible value).

If: m = n+1

Then: ^G = g^1

,

) f!^

k

g

n

k=1

can be found from the roots of

a

T

(z

1

)g^

1

= 0

no problem with spurious frequency estimates

computationally simple

(much) less accurate than MUSIC with m n+1

Lecture notes to accompany Introduction to Spectral Analysis Slide L67by P. Stoica and R. Moses, Prentice Hall, 1997

• Min-Norm Method

Goals: Reduce computational burden, and reduce risk offalse frequency estimates.

Uses m n (as in MUSIC), but only one vector inR(G) (as in Pisarenko).

Let"

1

g^

#

=

the vector in R( ^G), with first element equalto one, that has minimum Euclidean norm.

Lecture notes to accompany Introduction to Spectral Analysis Slide L68by P. Stoica and R. Moses, Prentice Hall, 1997

• Min-Norm Method, con't

Spectral Min-Norm

f!^g

n

k=1

= the locations of the n highest peaks in thepseudo-spectrum

1 =

a

(!)

"

1

g^

#

2

Root Min-Norm

f!^g

n

k=1

= the angular positions of the n roots of thepolynomial

a

T

(z

1

)

"

1

g^

#

that are closest to the unit circle.

Lecture notes to accompany Introduction to Spectral Analysis Slide L69by P. Stoica and R. Moses, Prentice Hall, 1997

• Min-Norm Method: Determining g^

Let ^S ="

S

#

g 1

g m 1

Then:"

1

g^

#

2 R(

^

G) )

^

S

"

1

g^

#

= 0

)

S

g^ =

Min-Norm solution: g^ = S(SS)1

As: I = ^S^S = + SS, (SS)1 exists iff

= kk

2

6= 1

(This holds, at least, for N 1.)

Multiplying the above equation by gives:

(1 kk

2

) = (

S

S)

) (

S

S)

1

= =(1 kk

2

)

) g^ =

S=(1 kk

2

)

Lecture notes to accompany Introduction to Spectral Analysis Slide L610by P. Stoica and R. Moses, Prentice Hall, 1997

• ESPRIT Method

Let A1

= [I

m1

0]A

A

2

= [0 I

m1

]A

Then A2

= A

1

D, where

D =

2

6

4

e

i!

1

0

...

0 e

i!

n

3

7

5

Also, let S1

= [I

m1

0]S

S

2

= [0 I

m1

]S

Recall S = AC with jCj 6= 0. Then

S

2

= A

2

C = A

1

DC = S

1

C

1

DC

| {z }

So has the same eigenvalues as D. is uniquelydetermined as

= (S

1

S

1

)

1

S

1

S

2

Lecture notes to accompany Introduction to Spectral Analysis Slide L611by P. Stoica and R. Moses, Prentice Hall, 1997

• ESPRIT Implementation

From the eigendecomposition of ^R, find ^S, then ^S1

and^

S

2

.

The frequency estimates are found by:

f!^

k

g

n

k=1

= arg(^

k

)

where f^k

g

n

k=1

are the eigenvalues of

^

= (

^

S

1

^

S

1

)

1

^

S

1

^

S

2

computationally simple

no extraneous frequency estimates (unlike in MUSICor MinNorm)

accurate frequency estimates

Lecture notes to accompany Introduction to Spectral Analysis Slide L612by P. Stoica and R. Moses, Prentice Hall, 1997

• Sum

mar

yo

fFre

quen

cyEs

timat

ion

Met

hods

Com

puta

tiona

lA

ccur

acy

/R

iskfo

rFa

lseM

etho

dBu

rden

Res

olut

ion

Freq

Estim

ates

Perio

dogr

amsm

all

med

ium

-hig

hm

ediu

mN

onlin

earL

Sver

yhi

ghver

yhi

ghver

yhi

ghYu

le-W

alke

rm

ediu

mhi

ghm

ediu

mPi

sare

nko

smal

llo

wn

on

e

MU

SIC

high

high

med

ium

Min

-Nor

mm

ediu

mhi

ghsm

all

ESPR

ITm

ediu

mver

yhi

ghn

on

e

Rec

omm

enda

tion:

Use

Peri

odog

ram

form

ediu

m-re

solu

tion

appl

icat

ions

Use

ESPR

ITfo

rhig

h-re

solu

tion

appl

icat

ions

Lect

ure

note

sto

acco

mpa

ny

Intro

duct

ion

toSp

ectra

lAnal

ysis

Slid

eL6

13by

P.St

oic

aan

dR.

Mose

s,Pr

entic

eH

all,

1997

• Filter Bank Methods

Lecture 7

Lecture notes to accompany Introduction to Spectral Analysis Slide L71by P. Stoica and R. Moses, Prentice Hall, 1997

• Basic Ideas

Two main PSD estimation approaches:

1. Parametric Approach: Parameterize (!) by afinite-dimensional model.

2. Nonparametric Approach: Implicitly smoothf(!)g

!=

by assuming that (!) is nearlyconstant over the bands

[! ; !+ ]; 1

2 is more general than 1, but 2 requires

N > 1

to ensure that the number of estimated values(= 2=2 = 1=) is < N .

N > 1 leads to the variability / resolution compromiseassociated with all nonparametric methods.

Lecture notes to accompany Introduction to Spectral Analysis Slide L72by P. Stoica and R. Moses, Prentice Hall, 1997

• Filte

rBan

kA

ppro

ach

toSp

ectra

lEst

imat

ion

-

-

-

-

D

i

v

i

s

i

o

n

b

y

l

t

e

r

b

a

n

d

w

i

d

t

h

P

o

w

e

r

C

a

l

c

u

l

a

t

i

o

n

B

a

n

d

p

a

s

s

F

i

l

t

e

r

w

i

t

h

v

a

r

y

i

n

g

!

c

a

n

d

x

e

d

b

a

n

d

w

i

d

t

h

^

(

!

c

)

P

o

w

e

r

i

n

t

h

e

b

a

n

d

F

i

l

t

e

r

e

d

S

i

g

n

a

l

y

(

t

)

-

6

!

c

!

y

(

!

)

@

@

Q

Q

-

6

!

c

!

~

y

(

!

)

@

@

-

6

1

!

c

!

H

(

!

)

^

F

B

(

!

)

(a) '1

2

Z

j

H

(

)

j

2

(

)

d

=

(b) '1

2

Z

!

+

!

(

)

d

=

(c) '

(

!

)

(a)co

nsis

tent

pow

erca

lcul

atio

n

(b)Id

ealp

assb

and

filte

rwith

band

wid

th

(c)

(

)

const

anto

n

2

[

!

2

;

!

+

2

]

Not

eth

atas

sum

ptio

ns(a)

and

(b),a

sw

ella

s(b)

and

(c),a

reco

nfli

ctin

g.

Lect

ure

note

sto

acco

mpa

ny

Intro

duct

ion

toSp

ectra

lAnal

ysis

Slid

eL7

3by

P.St

oic

aan

dR.

Mose

s,Pr

entic

eH

all,

1997

• Filter Bank Interpretation of the Periodogram

^

p

(~!)

4

=

1

N

N

X

t=1

y(t)e

i~!t

2

=

1

N

N

X

t=1

y(t)e

i~!(Nt)

2

= N

1

X

k=0

h

k

y(N k)

2

where

h

k

=

(

1

N

e

i~!k

; k = 0; : : : ; N 1

0; otherwise

H(!) =

1

X

k=0

h

k

e

i!k

=

1

N

e

iN(~!!)

1

e

i(~!!)

1

center frequency of H(!) = ~!

3dB bandwidth of H(!) ' 1=N

Lecture notes to accompany Introduction to Spectral Analysis Slide L74by P. Stoica and R. Moses, Prentice Hall, 1997

• Filter Bank Interpretation of the Periodogram,con't

jH(!)j as a function of (~! !), for N = 50.

3 2 1 0 1 2 340

35

30

25

20

15

10

5

0

dB

ANGULAR FREQUENCY

Conclusion: The periodogram ^p

(!) is a filter bank PSDestimator with bandpass filter as given above, and:

narrow filter passband,

power calculation from only 1 sample of filter output.

Lecture notes to accompany Introduction to Spectral Analysis Slide L75by P. Stoica and R. Moses, Prentice Hall, 1997

• Possible Improvements to the Filter BankApproach

1. Split the available sample, and bandpass filter eachsubsample.

more data points for the power calculation stage.

This approach leads to Bartlett and Welch methods.

2. Use several bandpass filters on the whole sample.Each filter covers a small band centered on ~!.

provides several samples for power calculation.

This multiwindow approach is similar to theDaniell method.

Both approaches compromise bias for variance, and in factare quite related to each other: splitting the data samplecan be interpreted as a special form of windowing orfiltering.

Lecture notes to accompany Introduction to Spectral Analysis Slide L76by P. Stoica and R. Moses, Prentice Hall, 1997

• Capon Method

Idea: Data-dependent bandpass filter design.

y

F

(t) =

m

X

k=0

h

k

y(t k)

= [h

0

h

1

: : : h

m

]

| {z }

h

2

6

4

y(t)

.

.

.

y(tm)

3

7

5

| {z }

~y(t)

E

n

jy

F

(t)j

2

o

= h

Rh; R = E f~y(t)~y

(t)g

H(!) =

m

X

k=0

h

k

e

i!k

= h

a(!)

where a(!) = [1; ei! : : : eim!]T

Lecture notes to accompany Introduction to Spectral Analysis Slide L77by P. Stoica and R. Moses, Prentice Hall, 1997

• Capon Method, con't

Capon Filter Design Problem:

min

h

(h

Rh) subject to ha(!) = 1

Solution: h0

= R

1

a=a

R

1

a

The power at the filter output is:

E

n

jy

F

(t)j

2

o

= h

0

Rh

0

= 1=a

(!)R

1

a(!)

which should be the power of y(t) in a passband centeredon !.

The Bandwidth' 1m+1

=

1

(filter length)

Conclusion Estimate PSD as:

^

(!) =

m+1

a

(!)

^

R

1

a(!)

with

^

R =

1

N m

N

X

t=m+1

~y(t)~y

(t)

Lecture notes to accompany Introduction to Spectral Analysis Slide L78by P. Stoica and R. Moses, Prentice Hall, 1997

• Capon Properties

m is the user parameter that controls the compromisebetween bias and variance:

as m increases, bias decreases and varianceincreases.

Capon uses one bandpass filter only, but it splits theN -data point sample into (N m) subsequences oflength m with maximum overlap.

Lecture notes to accompany Introduction to Spectral Analysis Slide L79by P. Stoica and R. Moses, Prentice Hall, 1997

• Relation between Capon and Blackman-TukeyMethods

Consider ^BT

(!) with Bartlett window:

^

BT

(!) =

m

X

k=m

m+1 jkj

m+1

r^(k)e

i!k

=

1

m+1

m

X

t=0

m

X

s=0

r^(t s)e

i!(ts)

=

a

(!)

^

Ra(!)

m+1

;

^

R = [r^(i j)]

Then we have

^

BT

(!) =

a

(!)

^

Ra(!)

m+1

^

C

(!) =

m+1

a

(!)

^

R

1

a(!)

Lecture notes to accompany Introduction to Spectral Analysis Slide L710by P. Stoica and R. Moses, Prentice Hall, 1997

• Relation between Capon and AR Methods

Let

^

ARk

(!) =

^

2

k

j

^

A

k

(!)j

2

be the kth order AR PSD estimate of y(t).

Then

^

C

(!) =

1

1

m+1

m

X

k=0

1=

^

ARk

(!)

Consequences:

Due to the average over k, ^C

(!) generally has lessstatistical variability than the AR PSD estimator.

Due to the low-order AR terms in the average,^

C

(!) generally has worse resolution and biasproperties than the AR method.

Lecture notes to accompany Introduction to Spectral Analysis Slide L711by P. Stoica and R. Moses, Prentice Hall, 1997

• Spatial Methods Part 1

Lecture 8

Lecture notes to accompany Introduction to Spectral Analysis Slide L81by P. Stoica and R. Moses, Prentice Hall, 1997

• The Spatial Spectral Estimation Problem

Source n

Source 2

Source 1

B

B

B

B

BN

@

@

@

@R

v

v

v

Sensor 1

@

@

@

@

@

@

Sensor 2

Sensor m

c

c

c

c

c

c

c

c

c

c

c

c

,

,

,

,

,

,

,

,

,

,

,

,

Problem: Detect and locate n radiating sources by usingan array of m passive sensors.

Emitted energy: Acoustic, electromagnetic, mechanical

Receiving sensors: Hydrophones, antennas, seismometers

Applications: Radar, sonar, communications, seismology,underwater surveillance

Basic Approach: Determine energy distribution overspace (thus the name spatial spectral analysis)

Lecture notes to accompany Introduction to Spectral Analysis Slide L82by P. Stoica and R. Moses, Prentice Hall, 1997

• Simplifying Assumptions

Far-field sources in the same plane as the array ofsensors

Non-dispersive wave propagation

Hence: The waves are planar and the only locationparameter is direction of arrival (DOA)(or angle of arrival, AOA).

The number of sources n is known. (We do not treatthe detection problem)

The sensors are linear dynamic elements with knowntransfer characteristics and known locations

(That is, the array is calibrated.)

Lecture notes to accompany Introduction to Spectral Analysis Slide L83by P. Stoica and R. Moses, Prentice Hall, 1997

• Array Model Single Emitter Case

x(t) = the signal waveform as measured at a referencepoint (e.g., at the first sensor)

k

= the delay between the reference point and thekth sensor

h

k

(t) = the impulse response (weighting function) ofsensor k

e

k

(t) = noise at the kth sensor (e.g., thermal noise insensor electronics; background noise, etc.)

Note: t 2 R (continuous-time signals).

Then the output of sensor k is

y

k

(t) = h

k

(t) x(t

k

) + e

k

(t)

( = convolution operator).

Basic Problem: Estimate the time delays fk

g with hk

(t)

known but x(t) unknown.

This is a time-delay estimation problem in the unknowninput case.Lecture notes to accompany Introduction to Spectral Analysis Slide L84by P. Stoica and R. Moses, Prentice Hall, 1997

• Narrowband Assumption

Assume: The emitted signals are narrowband with knowncarrier frequency !

c

.

Then: x(t) = (t) cos[!c

t+ '(t)]

where (t); '(t) vary slowly enough so that

(t

k

) ' (t); '(t

k

) ' '(t)

Time delay is now' to a phase shift !c

k

:

x(t

k

) ' (t) cos[!

c

t+ '(t) !

c

k

]

h

k

(t) x(t

k

)

' jH

k

(!

c

)j(t) cos[!

c

t+ '(t) !

c

k

+argfH

k

(!

c

)g]

where Hk

(!) = Ffh

k

(t)g is the kth sensor's transferfunction

Hence, the kth sensor output is

y

k

(t) = jH

k

(!

c

)j(t)

cos[!

c

t+ '(t) !

c

k

+ argH

k

(!

c

)] + e

k

(t)

Lecture notes to accompany Introduction to Spectral Analysis Slide L85by P. Stoica and R. Moses, Prentice Hall, 1997

• Complex Signal Representation

The noise-free output has the form:

z(t) = (t) cos [!

c

t+ (t)] =

=

(t)

2

n

e

i[!

c

t+ (t)]

+ e

i[!

c

t+ (t)]

o

Demodulate z(t) (translate to baseband):

2z(t)e

!

c

t

= (t)f e

i (t)

| {z }

lowpass+ e

i[2!

c

t+ (t)]

| {z }

highpassg

Lowpass filter 2z(t)ei!ct to obtain (t)ei (t)

Hence, by low-pass filtering and sampling the signal

~y

k

(t)=2 = y

k

(t)e

i!

c

t

= y

k

(t) cos(!

c

t) iy

k

(t) sin(!

c

t)

we get the complex representation: (for t 2 Z)y

k

(t) = (t) e

i'(t)

| {z }

s(t)

jH

k

(!

c

)j e

iarg[H

k

(!

c

)]

| {z }

H

k

(!

c

)

e

i!

c

k

+ e

k

(t)

or

y

k

(t) = s(t)H

k

(!

c

) e

i!

c

k

+ e

k

(t)

where s(t) is the complex envelope of x(t).

Lecture notes to accompany Introduction to Spectral Analysis Slide L86by P. Stoica and R. Moses, Prentice Hall, 1997

• Vector Representation for a Narrowband Source

Let

= the emitter DOAm = the number of sensors

a() =

2

6

4

H

1

(!

c

) e

i!

c

1

.

.

.

H

m

(!

c

) e

i!

c

m

3

7

5

y(t) =

2

6

4

y

1

(t)

.

.

.

y

m

(t)

3

7

5

e(t) =

2

6

4

e

1

(t)

.

.

.

e

m

(t)

3

7

5

Then

y(t) = a()s(t) + e(t)

NOTE: enters a() via both fk

g and fHk

(!

c

)g.

For omnidirectional sensors the fHk

(!

c

)g do not dependon .

Lecture notes to accompany Introduction to Spectral Analysis Slide L87by P. Stoica and R. Moses, Prentice Hall, 1997

• Ana

log

Pro

cess

ing

Blo

ckDi

agra

m

Ana

log

proc

essin

gfo

reac

hre

ceiv

ing

arra

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t

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i

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S

i

g

n

a

l

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(

t

k

)

S

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,

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a

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u

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)

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i

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-

s

i

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(

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c

t

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c

o

s

(

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c

t

)

- -

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m

[

y

k

(

t

)

]

R

e

[

y

k

(

t

)

]

- -

I

m

[

~

y

k

(

t

)

]

R

e

[

~

y

k

(

t

)

]

A

/

D

C

O

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V

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R

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!

!

!

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A

/

D

A

/

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Lect

ure

note

sto

acco

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ny

Intro

duct

ion

toSp

ectra

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ysis

Slid

eL8

8by

P.St

oic

aan

dR.

Mose

s,Pr

entic

eH

all,

1997

• Multiple Emitter Case

Given n emitters with

(t)g

n

k=1

DOAs: k

Linear sensors )

y(t) = a(

1

)s

1

(t) + + a(

n

)s

n

(t) + e(t)

Let

A = [a(

1

) : : : a(

n

)]; (m n)

s(t) = [s

1

(t) : : : s

n

(t)]

T

; (n 1)

Then, the array equation is:

y(t) = As(t) + e(t)

Use the planar wave assumption to find the dependence of

k

on .

Lecture notes to accompany Introduction to Spectral Analysis Slide L89by P. Stoica and R. Moses, Prentice Hall, 1997

• Uniform Linear Arrays

Source

-

+

?

]

p p p

Sensor

m4321

d sin

J

J^

J

J]

d

-

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J^

h

v v v v v

ULA Geometry

Sensor #1 = time delay reference

Time Delay for sensor k:

k

= (k 1)

d sin

c

where c = wave propagation speed

Lecture notes to accompany Introduction to Spectral Analysis Slide L810by P. Stoica and R. Moses, Prentice Hall, 1997

• Spatial Frequency

Let:

!

s

4

= !

c

d sin

c

= 2

d sin

c=f

c

= 2

d sin

= c=f

c

= signal wavelength

a() = [1; e

i!

s

: : : e

i(m1)!

s

]

T

By direct analogy with the vector a(!) made fromuniform samples of a sinusoidal time series,

!

s

= spatial frequency

The function !s

7! a() is one-to-one for

j!

s

j \$

dj sin j

=2

1 d =2

As

d = spatial sampling period

d =2 is a spatial Shannon sampling theorem.

Lecture notes to accompany Introduction to Spectral Analysis Slide L811by P. Stoica and R. Moses, Prentice Hall, 1997

• Spatial Methods Part 2

Lecture 9

Lecture notes to accompany Introduction to Spectral Analysis Slide L91by P. Stoica and R. Moses, Prentice Hall, 1997

• Spatial Filtering

Spatial filtering useful for

DOA discrimination (similar to frequencydiscrimination of time-series filtering)

Nonparametric DOA estimation

There is a strong analogy between temporal filtering andspatial filtering.

Lecture notes to accompany Introduction to Spectral Analysis Slide L92by P. Stoica and R. Moses, Prentice Hall, 1997

• Analogy between Temporal and Spatial Filtering

Temporal FIR Filter:

y

F

(t) =

m1

X

k=0

h

k

u(t k) = h

y(t)

h = [h

o

: : : h

m1

]

y(t) = [u(t) : : : u(tm+1)]

T

If u(t) = ei!t then

y

F

(t) = [h

a(!)]

| {z }

filter transfer functionu(t)

a(!) = [1; e

i!

: : : e

i(m1)!

]

T

We can select h to enhance or attenuate signals withdifferent frequencies !.

Lecture notes to accompany Introduction to Spectral Analysis Slide L93by P. Stoica and R. Moses, Prentice Hall, 1997

• Analogy between Temporal and Spatial Filtering

Spatial Filter:

fy

k

(t)g

m

k=1

= the spatial samples obtained with asensor array.

Spatial FIR Filter output:

y

F

(t) =

m

X

k=1

h

k

y

k

(t) = h

y(t)

Narrowband Wavefront: The array's (noise-free)response to a narrowband ( sinusoidal) wavefront withcomplex envelope s(t) is:

y(t) = a()s(t)

a() = [1; e

i!

c

2

: : : e

i!

c

m

]

T

The corresponding filter output is

y

F

(t) = [h

a()]

| {z }

filter transfer functions(t)

We can select h to enhance or attenuate signals comingfrom different DOAs.Lecture notes to accompany Introduction to Spectral Analysis Slide L94by P. Stoica and R. Moses, Prentice Hall, 1997

• Analogy between Temporal and Spatial Filtering

(Temporal sampling)

z

-

q

q

q

?

?

?

-

-

-

-

@

@

@

@

@R

-

?

?

q

q

q

-s

-

?

s

P

u(t) = e

i!t

q

1

q

1

1

m 1

0

B

B

B

B

B

B

B

B

B

B

B

B

B

B

@

1

e

i!

.

.

.

e

i(m1)!

1

C

C

C

C

C

C

C

C

C

C

C

C

C

C

A

| {z }

a(!)

u(t)

h

0

h

1

h

m1

[h

a(!)]u(t)

(a) Temporal filter

Z

Z

Z

Z

Z~

narrowband source with DOA=

(Spatial sampling)

P

z

-

q

q

q

?

?

?

@

@

@

@

@R

-

-

-

-

-

!

!

a

a

-

!

!

a

a

-

!

!

a

a

q

q

q

reference point

1

2

m

0

B

B

B

B

B

B

B

B

B

B

B

@

1

e

i!

2

()

.

.

.

e

i!

m

()

1

C

C

C

C

C

C

C

C

C

C

C

A

| {z }

a()

s(t)

h

0

h

1

h

m1

[h

a()]s(t)

(b) Spatial filterLecture notes to accompany Introduction to Spectral Analysis Slide L95by P. Stoica and R. Moses, Prentice Hall, 1997

• Spatial Filtering, con't

Example: The response magnitude jha()j of a spatialfilter (or beamformer) for a 10-element ULA. Here,h = a(

0

), where 0

= 25

2 4 6 8 100Magnitude

Theta (deg)

90

60

30

0

30

60

90

Lecture notes to accompany Introduction to Spectral Analysis Slide L96by P. Stoica and R. Moses, Prentice Hall, 1997

• Spatial Filtering Uses

Spatial Filters can be used

To pass the signal of interest only, hence filtering outinterferences located outside the filter's beam (butpossibly having the same temporal characteristics asthe signal).

To locate an emitter in the field of view, by sweepingthe filter through the DOA range of interest(goniometer).

Lecture notes to accompany Introduction to Spectral Analysis Slide L97by P. Stoica and R. Moses, Prentice Hall, 1997

• Nonparametric Spatial Methods

A Filter Bank Approach to DOA estimation.

Basic Ideas

Design a filter h() such that for each

It passes undistorted the signal with DOA =

It attenuates all DOAs 6=

Sweep the filter through the DOA range of interest,and evaluate the powers of the filtered signals:

E

n

jy

F

(t)j

2

o

= E

n

jh

()y(t)j

2

o

= h

()Rh()

with R = E fy(t)y(t)g.

The (dominant) peaks of h()Rh() give theDOAs of the sources.

Lecture notes to accompany Introduction to Spectral Analysis Slide L98by P. Stoica and R. Moses, Prentice Hall, 1997

• Beamforming Method

Assume the array output is spatially white:

R = E fy(t)y

(t)g = I

Then: En

jy

F

(t)j

2

o

= h

h

Hence: In direct analogy with the temporally whiteassumption for filter bank methods, y(t) can beconsidered as impinging on the array from all DOAs.

Filter Design:

min

h

(h

h) subject to ha() = 1

Solution:

h = a()=a

()a() = a()=m

E

n

jy

F

(t)j

2

o

= a

()Ra()=m

2

Lecture notes to accompany Introduction to Spectral Analysis Slide L99by P. Stoica and R. Moses, Prentice Hall, 1997

• Implementation of Beamforming

^

R =

1

N

N

X

t=1

y(t)y

(t)

The beamforming DOA estimates are:

f

^

k

g= the locations of the n largest peaks ofa

()

^

Ra().

This is the direct spatial analog of the Blackman-Tukeyperiodogram.

Resolution Threshold:

inf j

k

p

j >

wavelengtharray length

= array beamwidth

Inconsistency problem:Beamforming DOA estimates are consistent if n= 1, butinconsistent if n > 1.

Lecture notes to accompany Introduction to Spectral Analysis Slide L910by P. Stoica and R. Moses, Prentice Hall, 1997

• Capon Method

Filter design:

min

h

(h

Rh) subject to ha() = 1

Solution:

h = R

1

a()=a

()R

1

a()

E

n

jy

F

(t)j

2

o

= 1=a

()R

1

a()

Implementation:

f

^

k

g = the locations of the n largest peaks of1=a

()

^

R

1

a():

Performance: Slightly superior to Beamforming.

Both Beamforming and Capon are nonparametricapproaches. They do not make assumptions on thecovariance properties of the data (and hence do notdepend on them).Lecture notes to accompany Introduction to Spectral Analysis Slide L911by P. Stoica and R. Moses, Prentice Hall, 1997

• Parametric Methods

Assumptions:

The array is described by the equation:

y(t) = As(t) + e(t)

The noise is spatially white and has the same power inall sensors:

E fe(t)e

(t)g =

2

I

The signal covariance matrix

P = E fs(t)s

(t)g

is nonsingular.

Then:

R = E fy(t)y

(t)g = APA

+

2

I

Thus: The NLS, YW, MUSIC, MIN-NORM and ESPRITmethods of frequency estimation can be used, almostwithout modification, for DOA estimation.Lecture notes to accompany Introduction to Spectral Analysis Slide L912by P. Stoica and R. Moses, Prentice Hall, 1997

• Nonlinear Least Squares Method

min

f

k

g; fs(t)g

1

N

N

X

t=1

ky(t)As(t)k

2

| {z }

f(;s)

Minimizing f over s gives

s^(t) = (A

A)

1

A

y(t); t = 1; : : : ; N

Then

f(; s^) =

1

N

N

X

t=1

k [I A(A

A)

1

A

]y(t)k

2

=

1

N

N

X

t=1

y

(t)[I A(A

A)

1

A

]y(t)

= trf[I A(AA)1A] ^Rg

Thus, f^k

g= argmax

f

k

g

trf[A(AA)1A] ^Rg

For N = 1, this is precisely the form of the NLS methodof frequency estimation.

Lecture notes to accompany Introduction to Spectral Analysis Slide L913by P. Stoica and R. Moses, Prentice Hall, 1997

• Nonlinear Least Squares Method

Properties of NLS:

Performance: high

Computational complexity: high

Main drawback: need for multidimensional search.

Lecture notes to accompany Introduction to Spectral Analysis Slide L914by P. Stoica and R. Moses, Prentice Hall, 1997

• Yule-Walker Method

y(t) =

"

y(t)

~y(t)

#

=

"

A

~

A

#

s(t) +

"

e(t)

~e(t)

#

Assume: E fe(t)~e(t)g = 0

Then:

4

= E fy(t)~y

(t)g =

AP

~

A

(M L)

Also assume:

M > n; L > n () m =M + L > 2n)

rank( A) = rank( ~A) = n

Then: rank() = n, and the SVD of is

= [U

1

|{z}

n

U

2

|{z}

Mn

]

"

nn

0

0 0

# "

V

1

V

2

#

g n

g Ln

Properties: ~AV2

= 0 V

1

2 R(

~

A)

Lecture notes to accompany Introduction to Spectral Analysis Slide L915by P. Stoica and R. Moses, Prentice Hall, 1997

• YW-MUSIC DOA Estimator

f

^

k

g= the n largest peaks of1=~a

()

^

V

2

^

V

2

~a()

where

~a(), (L 1), is the array transfer vector for ~y(t)at DOA

^

V

2

is defined similarly to V2

, using

^

=

1

N

N

X

t=1

y(t)~y

(t)

Properties:

Computational complexity: medium

Performance: satisfactory if m 2n

Main advantages: weak assumption on fe(t)g the subarray A need not be calibrated

Lecture notes to accompany Introduction to Spectral Analysis Slide L916by P. Stoica and R. Moses, Prentice Hall, 1997

• MUSIC and Min-Norm Methods

Both MUSIC and Min-Norm methods for frequencyestimation apply with only minor modifications to theDOA estimation problem.

Spectral forms of MUSIC and Min-Norm can be usedfor arbitrary arrays

Root forms can be used only with ULAs

MUSIC and Min-Norm break down if the sourcesignals are coherent; that is, if

rank(P) = rank(E fs(t)s(t)g) < n

Modifications that apply in the coherent case exist.

Lecture notes to accompany Introduction to Spectral Analysis Slide L917by P. Stoica and R. Moses, Prentice Hall, 1997

• ESPRIT Method

Assumption: The array is made from two identicalsubarrays separated by a known displacement vector.

Let

m = # sensors in each subarrayA

1

= [I

m

0]A (transfer matrix of subarray 1)A

2

= [0 I

m

]A (transfer matrix of subarray 2)

Then A2

= A

1

D, where

D =

2

6

4

e

i!

c

(

1

)

0

...

0 e

i!

c

(

n

)

3

7

5

() = the time delay from subarray 1 tosubarray 2 for a signal with DOA = :

() = d sin()=c

where d is the subarray separation and is measured fromthe perpendicular to the subarray displacement vector.

Lecture notes to accompany Introduction to Spectral Analysis Slide L918by P. Stoica and R. Moses, Prentice Hall, 1997

• ESPRIT Method, con't

ESPRIT Scenario

subarray 1

subarray 2

source

known displacement vector

Properties:

Requires special array geometry

Computationally efficient

No risk of spurious DOA estimates

Does not require array calibration

Note: For a ULA, the two subarrays are often the firstm1 and last m1 array elements, so m = m1 and

A

1

= [I

m1

0]A; A

2

= [0 I

m1

]A

Lecture notes to accompany Introduction to Spectral Analysis Slide L919by P. Stoica and R. Moses, Prentice Hall, 1997