Spectral Analysis

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)


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

Radar, Sonar

Spatial Spectral Analysis

Source location using sensor arrays


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)


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


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.


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= (!)


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

(!)


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


Periodogramand

CorrelogramMethods

Lecture 2


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).


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


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


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.


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.


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


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Parametric Methodsfor

Line Spectra Part 1

Lecture 5


Line Spectra

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

communications

radar, sonar

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)


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


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

.


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


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


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.


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

(!)


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.


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)


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.


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.


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.


HOYW Equations, Practical Solution

Let ^ = but made from fr^(k)g instead of fr(k)g.

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.


Parametric Methodsfor

Line Spectra Part 2

Lecture 6


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


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

#


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


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


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


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.


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.


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

)


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


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

ESPRIT Advantages:

computationally simple

no extraneous frequency estimates (unlike in MUSICor MinNorm)

accurate frequency estimates


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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


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.


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

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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


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.


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.


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


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)


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.


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(!)


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.


Spatial Methods Part 1

Lecture 8


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)


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.)


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)


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).


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 .


Ana

log

Pro

cess

ing

Blo

ckDi

agra

m

Ana

log

proc

essin

gfo

reac

hre

ceiv

ing

arra

yel

emen

t

X

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:

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r

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i

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c

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i

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i

g

n

a

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x

(

t

k

)

S

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,

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,

l

l

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-

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a

b

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i

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i

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u

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(

t

)

D

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D

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s

c

i

l

l

a

t

o

r

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w

p

a

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s

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i

l

t

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r

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o

w

p

a

s

s

F

i

l

t

e

r

- -

6 ?

-

s

i

n

(

!

c

t

)

c

o

s

(

!

c

t

)

- -

I

m

[

y

k

(

t

)

]

R

e

[

y

k

(

t

)

]

- -

I

m

[

~

y

k

(

t

)

]

R

e

[

~

y

k

(

t

)

]

A

/

D

C

O

N

V

E

R

S

I

O

N

!

!

!

!

- -

A

/

D

A

/

D

Lect

ure

note

sto

acco

mpa

ny

Intro

duct

ion

toSp

ectra

lAnal

ysis

Slid

eL8

8by

P.St

oic

aan

dR.

Mose

s,Pr

entic

eH

all,

1997

Multiple Emitter Case

Given n emitters with

received signals: fsk

(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 .


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


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.


Spatial Methods Part 2

Lecture 9


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.


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 !.



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


(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


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).


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.


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


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.


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.


Nonlinear Least Squares Method

Properties of NLS:

Performance: high

Computational complexity: high

Main drawback: need for multidimensional search.


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)


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


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.


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.


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


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