Spectral Estimation 2

Basic Denitions and The Spectral Estimation Problem

Lecture 1

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

Slide L11

Informal Denition of Spectral Estimation

Given: A nite record of a signal. Determine: The distribution of signal power over frequency.signal t t=1, 2, ... + spectral density

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


Slide L12

Applications Temporal Spectral Analysis Vibration monitoring and fault detection Hidden periodicity nding Speech processing and audio devices Medical diagnosis Seismology and ground movement study Control systems design Radar, Sonar

Spatial Spectral Analysis Source location using sensor arraysLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L13

Deterministic Signals

fy(t)g1 ;1 = t=If:

discrete-time deterministic data sequence

X

1

t=;1

jy(t)j2 < 1X

Then:

Y (! ) =

1

t=;1

y(t)e;i!t

exists and is called the Discrete-Time Fourier Transform (DTFT)


Slide L14

Energy Spectral Density

Parseval's Equality:

jy(t)j2 = 21 ; S (!)d! t=;1X Z

1

where

4 S (!) = jY (!)j2 = Energy Spectral DensityWe can write

S (!) =where

X

1

k=;1X

(k)e;i!k

(k) =

1

t=;1

y(t)y (t ; k)Slide L15


Random Signals

Random Signalrandom signal probabilistic statements about future variations t current observation time

Here:

X

1

t=;1n

jy(t)j2 = 1o

But:

E jy(t)j2 < 1

E f g = Expectation over the ensemble of realizations E jy(t)j2n o

= Average power in y

(t)

PSD = (Average) power spectral density


Slide L16

First Denition of PSD

(! ) =where r

X

1

k=;1

r(k)e;i!k

(k) is the autocovariance sequence (ACS) r(k) = E fy(t)y (t ; k)g r(k) = r (;k) r(0) jr(k)jZ

Note that

r(k) = 21 ; (!)ei!k d!n o

(Inverse DTFT)

Interpretation:

so

r(0) = E jy(t)j2 = 21 ; (!)d!Z

(!)d! =

innitesimal signal power in the band

! d! 2


Slide L17

Second Denition of PSD

1 N y(t)e;i!t (!) = Nlim E N !1 t=1X > :

8 > = >

Note that

1 jY (!)j2 (!) = Nlim E N N !1YN (!) =X

where

N

is the nite DTFT of

fy(t)g.

t=1

y(t)e;i!t


Slide L18

Properties of the PSD

P1:

(!) = (! + 2 ) for all !.Thus, we can restrict attention to

!2 ;P2:

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

(!) 0 (t) is real, Then: (! ) = (;! ) Otherwise: (! ) 6= (;! )

P3: If y


Slide L19

Transfer of PSD Through Linear Systems

System Function: where q ;1

H (q) =

X

1

k=0

hk q;k

= unit delay operator: q;1y(t) = y(t ; 1)H (q ) y (t) 2 y (! ) = jH (! )je(

e(t) e (! )Then

!)

y(t) =

X

1

k=0

hk e(t ; k) hk e;i!k

H (! ) =

X

1

y (!) = jH (!)j2 e(!)

k=0


Slide L110

The Spectral Estimation Problem

The Problem: From a sample

fy(1) : : : y(N )g(!): f ^(!) ! 2 ; ]g

Find an estimate of

Two Main Approaches :

Nonparametric: Derived from the PSD denitions.

Parametric: Assumes a parameterized functional form of the PSD


Slide L111

Periodogram and Correlogram Methods

Lecture 2


Slide L21

Periodogram Recall 2nd denition of (! ):

1 N y(t)e;i!t (!) = Nlim E N !1 t=1X > :

8 > = >

Given : Drop

fy(t)gN t=1lim and E

N !1

f g to getX

^p(!) = 1 y(t)e;i!t N t=1Natural estimator

N

2

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


Slide L22

Correlogram

Recall 1st denition of (! ):

(! ) =P

X

1

Truncate the and replace r

k=;1 N ;1 X

r(k)e;i!k

(k) by ^(k): rr ^(k)e;i!k

^c(!) =

k=;(N ;1)


Slide L23

Covariance Estimators (or Sample Covariances)

Standard unbiased estimate:

1 N y(t)y (t ; k) k 0 ^ r(k) = N ; k t=k+1X

Standard biased estimate:

1 N y(t)y (t ; k) k 0 r(k) = N ^ t=k+1X

For both estimators:

r ^(k) = ^ (;k) k < 0 r


Slide L24

Relationship Between ^p (! ) and ^c(! )

If: the biased ACS estimator r Then:

^(k) is used in ^c(!),N

^p(!) = 1 y(t)e;i!t N t=1X

2

=

N ;1 X

= ^c(!)

k=;(N ;1)

r ^(k)e;i!k

^p(!) = ^c(!)Consequence: Both p ! and

^( )

^c(!) can be analyzed simultaneously.Slide L25


Statistical Performance of ^p(! ) and ^c(! ) Summary: Both are asymptotically (for large N ) unbiased:

E ^p(!) ! (!) as N ! 1n o

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 Êven if the errors r k r k Nj;1 are small, jk =0 there are so many that when summed in ! , the PSD error is large. p!

f^( ) ; ( )g

^ ( ) ; ( )]


Slide L26

Bias Analysis of the Periodogram

E ^p(!) = E ^c(!) = E f^(k)g e;i!k r k=;(N ;1) N ;1 jkj r(k)e;i!k = 1; N k=;(N ;1) 1 = wB (k)r(k)e;i!k k=;1n o n o X ! X

N ;1 X

wB (k) =

8 < :

=Thus,n o

0

k 1 ; jNj

jkj N ; 1 jkj N

Bartlett, or triangular, window

E ^p(!) = 21 ; ( )WB (! ; ) dZ

Ideally:

WB (!) = Dirac impulse (!).Slide L27


Bartlett Window WB (! )

1 sin(!N=2) 2 WB (!) = N sin(!=2)" #0

WB (!)=WB (0), for N = 25

10

20

dB

30

40

50

60

3

2

1 0 1 ANGULAR FREQUENCY

2

3

Main lobe 3dB width For small N , WB

1=N .

(!) may differ quite a bit from (!).Slide L28


Smearing and Leakage

Main Lobe Width: smearing or smoothing Details in resolvable.() smearing

(!) separated in f by less than 1=N are not^ ()

< > :

M ;1 X

=

'Thus:

k=;(M ;1) M ;1 X k=;(M ;1)

k=;(M ;1) 8 M ;1 < 1 X L X:

r ^j (k)e;i!k9 =

9 > = >

L j =1

r ^j (k) e;i!k

r ^(k)e;i!k

^B (!) ' ^BT (!)Since B ! implicitly uses method has

with a rectangular lag window wR k

()

^( )

fwR(k)g, the Bartlett

High resolution (little smearing) Large leakage and relatively large varianceLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L38

Welch Method Similar to Bartlett method, but allow overlap of subsequences (gives more subsequences, and thus better averaging) use data window for each periodogram; gives mainlobe-sidelobe tradeoff capability

1

2 . . . subseq subseq #1 #2 subseq #S

N

Let S # of subsequences of length M . (Overlapping means S > N=M better averaging.)

=

])

Additional exibility: The data in each subsequence are weighted by a temporal window Welch is approximately equal to non-rectangular lag window.

^BT (!) with aSlide L39


Daniell Method

By a previous result, for N

1,

f ^p(!j )g are (nearly) uncorrelated random variables for 2 j N ;1 ! =N j =0 Idea: Local averaging of (2J + 1) samples in thej

(2 + 1)

frequency domain should reduce the variance by about J .

k +J 1 ^D(!k ) = ^p (!j ) 2J + 1 j=k;JX


Slide L310

Daniell Method, con't

As J increases: Bias increases (more smoothing) Variance decreases (more averaging)

= 2J=N . Then, for N 1, ^D(!) ' 1 ^p(!)d! 2 ; Hence: ^D (! ) ' ^BT (! ) with a rectangular spectralLetZ

window.


Slide L311

Summary of Periodogram Methods Unwindowed periodogram reasonable bias unacceptable variance

Modied periodograms Attempt to reduce the variance at the expense of (slightly) increasing the bias.

BT periodogram Local smoothing/averaging of spectral window.

^p(!) by a suitably selected ^(k) using a lag

Implemented by truncating and weighting r window in c !

^( )

Bartlett, Welch periodograms

Approximate interpretation: BT ! with a suitable lag window (rectangular for Bartlett; more general for Welch).

^ ()

Implemented by averaging subsample periodograms.

Daniell Periodogram Approximate interpretation: spectral window.

^BT (!) with a rectangularSlide L312

Implemented by local averaging of periodogram values.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997

Parametric Methods for Rational Spectra

Lecture 4


Slide L41

Basic Idea of Parametric Spectral EstimationObserved Data Assumed functional form of (,) Estimate parameters in (,)

^

Estimate PSD

^ ^ =(,) ()

possibly revise assumption on ()

Rational Spectra

jkj m k e;i!k (!) = P jkj n k e;i!k (!) is a rational function in e;i! .P

By Weierstrass theorem, ! can approximate arbitrarily well any continuous PSD, provided m and n are chosen sufciently large. Note, however: choice of m and n is not simple some PSDs are not continuousLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L42

( )

AR, MA, and ARMA Models

By Spectral Factorization theorem, a rational factored as

(!) can be

B (!) 2 2 (!) = A(!) A(z) = 1 + a1z;1 + + anz;n B (z) = 1 + b1z;1 + + bmz;m and, e.g., A(! ) = A(z )jz =ei!Signal Modeling Interpretation:

e(t) e (! ) =

2

-

B (q) A(q)

white noise

B (!) 2 2 A(!) ltered white noise

y(t)y (! ) =

ARMA: AR: MA:

A(q)y(t) = B (q)e(t) A(q)y(t) = e(t) y(t) = B (q)e(t)Slide L43


ARMA Covariance Structure

ARMA signal model:

y(t)+

X

n

i=1 n

aiy(t ; i) =

X

m

j =0

bj e(t ; j )

(b0 = 1)

Multiply by y

(t ; k) and take E f g to give:X

r (k ) +

X

2 m bj hj ;k = j =0 = 0 for k > m B (q) = 1 h q;k (h = 1) where H (q ) = A(q ) 0 k k=0X X

i=1

air(k ; i) =

m

j =0

bj E fe(t ; j )y (t ; k)g


Slide L44

AR Signals: Yule-Walker Equations

AR: m

= 0.2 1 a1 = 0 . . . . an 03 7 7 7 5 2 6 6 6 4

Writing covariance equation in matrix form for k : : : n:

=1 r(0) r(;1) : : : r(;n) . r(1) r(0) . . . . . r (;1) . r(n) : : : r(0) 1 = 2 R 02 6 6 6 4 7 7 7 5 " # "

32 6 6 6 4

3 7 7 7 5

#

These are the YuleWalker (YW) Equations.


Slide L45

AR Spectral Estimation: YW Method

Yule-Walker Method: Replace r2 6 6 6 4

(k) by r(k) and solve for fîg and ^2: ^ a r ^(0) ^(;1) : : : ^(;n) 1 r r ^2 . . r ^(1) r(0) ^ ^.1 = 0 a . . . . . ^(;1) . . r . r ^(n) : : : r(0) ^ ^n a 032 7 7 7 5 6 6 6 4 3 7 7 7 5 2 6 6 6 4

3 7 7 7 5

Then the PSD estimate is

^2 ^(!) = ^ 2 jA(!)j


Slide L46

AR Spectral Estimation: LS Method

Least Squares Method:

e(t) = y(t) +

X

n

4 = y(t) + y(t) ^ where '(t) = y (t ; 1) : : : y (t ; n)]T .Find

i=1

aiy(t ; i) = y(t) + 'T (t)

= a1 : : : an]T to minimizef( ) =X

N

This gives2

y=6 4

y(n + 1) y(n + 2) y(N ). . .

^ = ;(Y Y );1(Y y) where3 7 5 2 6

t=n+1

je(t)j2y(1) y(2) y(N ; n). . .3 7 5

Y

=4

y(n) y(n ; 1) y(n + 1) y(n). . .

y(N ; 1) y(N ; 2)


Slide L47

LevinsonDurbin Algorithm

Fast, order-recursive solution to YW equations2 6 6 6 4 |

0 ;1 1 .0 ..

n

..

... ...{z

;n . ;1. .

32 7 7 7 5 } 6 6 6 4

1n

3 7 7 7 5

2 6 6 6 4

^ k = either r(k) or r(k).Direct Solution: For one given value of n: For k

Rn+1

1

0

= 0 . . 0

2 n

3 7 7 7 5

O(n3) ops

= 1 : : : n: O(n4) ops

LevinsonDurbin Algorithm: Exploits the Toeplitz form of Rn+1 to obtain the solutions for k : : : n in O n2 ops!

=1

( )


Slide L48

Levinson-Durbin Alg, con't

Relevant Properties of R:

Rx = y $ Rx = y, where x = xn : : : x1]T ~ ~ ~Nested structure2

Rn+2 =Thus,2

4

Rn+1n+1

r ~n

n+1 r ~n0

3 5

2

r ~n =

6 4

. .

n

3 7 5

1

Rn+2 4

1 n = 03 5

2 4

Rn+1n+1

r ~n

n+1 r ~n0

32 54

1 n n = 0 0 n3 5 2 4

2

3 5

where

r n = n+1 + ~n n


Slide L49

Levinson-Durbin Alg, con't2

Rn+2 482 < 4 +2 :

1 n n = 0 0 n3 5 2 4

2

3 5

2

Rn+2 439 = 5 2

0 ~n = 0n 2 1 n3 5 2 4 3 5 2

3 5

Combining these gives:

Rn

0 1 n + kn ~n 0 13 5 2 4

=

4

n + kn n2

0 n + kn

2 n

=

4

n

2 +1

3 5

0 0

Thus,

2 kn = ; n = n )n+1 =

0 + kn 2+1 = n + kn n = n(1 ; jknj2) 2 2 n

"

n

#

"

~n 1

#

Computation count: k ops for the step k

2 )

n2 ops

!k+1 2 to determine f k

k gn=1 kSlide L410

This is O

(n2) times faster than the direct solution.


MA Signals

MA: n

=0 y(t) = B (q)e(t) = e(t) + b1e(t ; 1) +

+ bme(t ; m)

Thus,

r(k) = 0 for jkj > mand

2 2 = m r(k)e;i!k (!) = jB(!)j k=;mX


Slide L411

MA Spectrum Estimation Two main ways to Estimate (! ): 1. Estimate

fbkg and 2 and insert them in (!) = jB(!)j2 2

nonlinear estimation problem

^(!) is guaranteed to be 02. Insert sample covariances

(! ) =This is length

X

m

f^(k)g in: rr(k)e;i!k

k=;m

^BT (!) with a rectangular lag window of 2m + 1. ^(!) is not guaranteed to be 0 =0

Both methods are special cases of ARMA methods described below, with AR model order n .Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L412

ARMA Signals

ARMA models can represent spectra with both peaks (AR part) and valleys (MA part).

A(q)y(t) = B (q)e(t) B (!) 2 = m=;m k e;i!k k (!) = 2 A(!) jA(!)j2P

where

k

= E f B(q)e(t)] B(q)e(t ; k)] g = E f A(q)y(t)] A(q)y(t ; k)] g =X

n

X

n

j =0 p=0

aj ap r(k + p ; j )


Slide L413

ARMA Spectrum Estimation

Two Methods:

2g in (!) = 2 B(!) 2 1. Estimate fai bj A(!)nonlinear estimation problem; can use an approximate linear two-stage least squares method

^(!) is guaranteed to be 02. Estimate

fai r(k)g in (!) =

P

m ;i!k k=;m k e

jA(!)j2

linear estimation problem (the Modied Yule-Walker method).

^(!) is not guaranteed to be 0Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L414

Two-Stage Least-Squares Method Assumption: The ARMA model is invertible:

A e(t) = B(q) y(t) (q ) = y(t) + 1y(t ; 1) + 2y(t ; 2) + = AR(1) with j kj ! 0 as k ! 1 Step 1: Approximate, for some large K e(t) ' y(t) + 1y(t ; 1) + + K y(t ; K )1a) Estimate the coefcients modelling techniques.

f kgK=1 by using AR k

1b) Estimate the noise sequence

^(t) = y(t) + ^1y(t ; 1) + eand its variance

+ ^K y(t ; K )

^2 =

N 1 j^(t)j2 e N ; K t=K +1X


Slide L415

Two-Stage Least-Squares Method, con't

fe(t)g by ^(t) in the ARMA equation, e A(q)y(t) ' B (q)^(t) e and obtain estimates of fai bj g by applying least squaresStep 2: Replace techniques. Note that the ai and bj coefcients enter linearly in the above equation:

y(t) ; ^(t) ' ;y(t ; 1) : : : ; y(t ; n) e ^(t ; 1) : : : ^(t ; m)] e e = a1 : : : an b1 : : : bm]T


Slide L416

Modied Yule-Walker Method

ARMA Covariance Equation:

r(k) +2 6 4

X

n

In matrix form for k

i=1

air(k ; i) = 0 k > m3 7 5 "

r(m) r(m + 1). . .

: : : r(m ; n + 1) r(m ; n + 2)... . . .

= m + 1 ::: m + Ma1 an. . .#

2

=

r(m + M ; 1) : : : r(m ; n + M )

;6 4

Replace

fr(k)g by f^(k)g and solve for faig. r f^ g

. 5 . . r(m + M )

r(m + 1) r(m + 2)

3 7

If M n, fast Levinson-type algorithms exist for obtaining ai . If M > n overdetermined squares solution for ai .

=

f^ g

Y W system of equations; least

f^ g

Note: For narrowband ARMA signals, the accuracy of ai is often better for M > n


Slide L417

Summary of Parametric Methods for Rational Spectra

Method AR: YW or LS low low-medium medium-high medium-high medium low-medium No No Yes

Computational Burden low Accuracy medium

Guarantee ^(! ) 0 ? Yes

MA: BT

ARMA: MYW

ARMA: 2-Stage LS

Use for Spectra with (narrow) peaks but no valley Broadband spectra possibly with valleys but no peaks Spectra with both peaks and (not too deep) valleys As above


Slide L418

Parametric Methods for Line Spectra Part 1

Lecture 5


Slide L51

Line Spectra Many applications have signals with (near) sinusoidal components. Examples: communications radar, sonar geophysical seismology ARMA model is a poor approximation Better approximation by Discrete/Line Spectrum Models

2 6(2 1) 2

6

(!)6

(2 2) 2

6

(2 2) 3!Slide L52

; !1

!2

!3

-

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

Line Spectral Signal Model Signal Model: Sinusoidal components of frequencies !k and powers 2 , superimposed in white noise of k 2. power

f g

f g

y(t) = x(t) + e(t) t = 1 2 : : : n x(t) = k ei(!k t+ k ) k=1 xk (t)X | {z }

Assumptions: A1: A2:

k>0

!k 2 ;

]

(prevents model ambiguities)

f'k g = independent rv's, uniformly distributed 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)Slide L53


Covariance Function and PSD

Note that:

E E

n

i'p e;i'j o = 1, for p = j e i'p e;i'j o = E nei'p o E ne;i'j o eZ

n

1 i' d' 2 = 0, for p 6= j =2 ; eE xp(t)xj (t ; k) = 2 ei!pk p j pn o

Hence,

r(k) = E fy(t)y (t ; k)g = n=1 2ei!pk + 2 k 0 p pP

and

(! ) = 2

X

n

p=1

2 (! ; !p) + 2 pSlide L54


Parameter Estimation Estimate either:

f!k

k 'k gn=1 k

2

(Signal Model)

f!k 2gn=1 2 k kOnce

(PSD Model)

^ Major Estimation Problem: f!k g

f!kg are determined: ^X

f ^2g can be obtained by a least squares method from k n 2ei!pk + residuals ^ ^ r(k) = pp=1OR: Both k and 'k can be derived by a least squares method from

f^ g

f^ gX

y(t) =with

n

k=

k=1 k ei'k .

^ ei!k t + residuals k


Slide L55

Nonlinear Least Squares (NLS) Method

min' g f!Let:

X

N

k k k t=1 |

y(t) ;

X

n

k=1 {z F (! ')

k ei(!k t+'k )

2}

= kei'k = 1 : : : n]T Y = y(1) : : : y(N )]T ei!1 ei!n . . . . B = eiN!1 eiN!nk2 6 4

3 7 5


Slide L56

Nonlinear Least Squares (NLS) Method, con't

Then:

F = (Y ; B ) (Y ; B ) = kY ; B k2 = ; (B B);1B Y ] B B] ; (B B);1B Y ] +Y Y ; Y B(B B);1B YThis gives:

^ = (B B);1B Y !=^ !and

! = arg max Y B (B B );1B Y ^ !


Slide L57

NLS Properties

Excellent Accuracy:

6 2 (for N var (^k ) = 3 2 ! N k Example: N = 300 SNRk = 2= 2 = 30 dB kq

1)

Then

var

(^k ) 10;5. !

Difcult Implementation: The NLS cost function F is multimodal; it is difcult to avoid convergence to local minima.


Slide L58

Unwindowed Periodogram as an Approximate NLS Method

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

n=1X

B B=NN t=1

BY=

y(t)e;i!t = Y (!)

(nite DTFT)

1 Y B (B B );1B Y = N jY (!)j2 = ^p(!) = (Unwindowed Periodogram)So, with no approximation,

! = arg max ^p(!) ^ !


Slide L59

Unwindowed Periodogram as an Approximate NLS Method, con't

Assume: Then:

n>1

f!kgn=1 ' the ^ kprovided that

locations of the peaks of p !

^( )

n

largest

inf j!k ; !pj > 2 =Nwhich is the periodogram resolution limit. If better resolution desired then use a High/Super Resolution method.


Slide L510

High-Order Yule-Walker Method Recall:

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

k ei(!k t+'k ) +e(t) | {z } k=1 xk (t)X

n

Degenerate ARMA equation for y (t):n

(1 ; ei!k q;1)xk (t) = k ei(!kt+'k) ; ei!k ei !k(t;1)+'k] = 0o

Let

B (q) = 1 +

4 bk q;k = A(q)A(q) k=1 A(q) = (1 ; ei!1 q;1) (1 ; ei!n q;1) A(q) = arbitraryX

L

Then B

(q)x(t) 0 ) B (q)y(t) = B (q)e(t)Slide L511


High-Order Yule-Walker Method, con't

Estimation Procedure: Estimate

fîgL=1 using an ARMA MYW technique b i^( )

Roots of B q give spurious roots.

f!kgn=1, along with L ; n ^ k


Slide L512

High-Order and Overdetermined YW Equations

ARMA covariance:

r (k ) +

X

L

i=1

bir(k ; i) = 0 k > L

In matrix form for k2 6 6 6 4 |

= L + 1 ::: L + M3 7 7 7 5 2 6 6 6 4 3 7 7 7 5 }

: : : r(1) r(L + 1) : : : r(2) b = ; r(L + 2) . . . . . . r(L + M ; 1) : : : r(M ) r(L + M ) 4 4 = ={z } | {z

r(L) r(L + 1)

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


Slide L513

High-Order and Overdetermined YW Equations, con't Fact: SVD of rank :

( )=n =U V

U = (M n) with U U = In V = (n L) with V V = In

= (n n), diagonal and nonsingularThus,

(U V )b = ;The Minimum-Norm solution is

b = ; y = ;V ;1U Important property: The additional (L ; n) spurious zeros of B (q ) are located strictly inside the unit circle, if the Minimum-Norm solution b is used.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L514

HOYW Equations, Practical Solution

Let

^= ^ ^ ^ ^

but made from

f^(k)g instead of fr(k)g. r, V from the

Let U , , V be dened similarly to U , SVD of . Compute

^ = ;V ^ ;1U ^ ^ ^ b ^(q) that

Then !k n=1 are found from the n zeroes of B k are closest to the unit circle.

f^ g

When the SNR is low, this approach may give spurious frequency estimates when L > n; this is the price paid for increased accuracy when L > n.


Slide L515

Parametric Methods for Line Spectra Part 2

Lecture 6


Slide L61

The Covariance Matrix Equation Let:

a(!) = 1 e;i! : : : e;i(m;1)! ]T A = a(!1) : : : a(!n)] (m n)rank

Note: Dene

(A) = ny(t) y(t ; 1)

(for m3 7 7 7 5

n)

2

46 y(t) = 6 ~6 4

y(t ; m + 1)

. .

= Ax(t) + ~(t) ~ e

where

x(t) = x1(t) : : : xn(t)]T ~ ~(t) = e(t) : : : e(t ; m + 1)]T e4 R = E fy(t)~ (t)g = APA + 2I ~ y

Then

with2

P = E fx(t)~ (t)g = ~ x

6 4

2 1

0

...

02 n

3 7 5


Slide L62

Eigendecomposition of R and Its Properties

R = APA + 2I (m > n)Let:

1

2 :::

m: eigenvalues of R

fs1 : : : sng: orthonormal eigenvectors associated with f 1 : : : ng fg1 : : : gm;ng: orthonormal eigenvectors associated with f n+1 : : : m g S = s1 : : : sn] (m n) G = g1 : : : gm;n] (m (m ; n))Thus,2

R = S G]

6 4

1

3

"

...

m

7 5

S G

#


Slide L63

Eigendecomposition of R and Its Properties, con't As rank

(APA ) = n: k > 2 k = 1 ::: n k = 2 k = n + 1 ::: m 0 1; 2 . .. = = nonsingular 0 n; 22 6 4 3 7 5 2 6 4

Note:

RS = APA S + 2S = S

1

0

...

0n

3 7 5

with C (since rank S , Therefore, since S G

j j 6= 0

4 ;1) = AC S = A(PA S

( ) = rank(A) = n). =0 A G=0Slide L64


MUSIC Method

a (!1) . . A G= G=0 a (!n) ) fa(!k )gn=1 ? R(G) k2 6 4 3 7 5

Thus,

f!kgn=1 are the unique solutions of k a (!)GG a(!) = 0.Let:

1 N y(t)~ (t) ^ ~ y R = N t=m ^^ S G = S G made from the ^ eigenvectors of RX


Slide L65

Spectral and Root MUSIC Methods Spectral MUSIC Method:

f!kgn=1 = the locations of the n highest peaks of the ^ kpseudo-spectrum function:

1 ^G a(!) ! 2 ; a (!)G ^Root MUSIC Method:

]

f!kgn=1 = the angular positions of the n roots of: ^ k^^ aT (z;1)GG a(z) = 0that are closest to the unit circle. Here,

a(z) = 1 z;1 : : : z;(m;1)]TNote: Both variants of MUSIC may produce spurious frequency estimates.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L66

Pisarenko Method

Pisarenko is a special case of MUSIC with m (the minimum possible value). If:

=n+1

m=n+1

Then:

^ g G = ^1, ) f!kgn=1 can be found from the roots of ^ k aT (z;1)^1 = 0 gno problem with spurious frequency estimates computationally simple (much) less accurate than MUSIC with m

n+1


Slide L67

Min-Norm Method

Goals: Reduce computational burden, and reduce risk of false frequency estimates. Uses m n (as in MUSIC), but only one vector in G (as in Pisarenko).

R( )Let"

1 = the vector in R(G), with rst element equal ^ ^ to one, that has minimum Euclidean norm. g#


Slide L68

Min-Norm Method, con't

Spectral Min-Norm

f!gn=1 = ^k

the locations of the n highest peaks in the pseudo-spectrum

2 1 = a (!) 1 ^ g" #

Root Min-Norm

f!gn=1 = ^k

the angular positions of the n roots of the polynomial

aT (z;1) 1 ^ g"

#

that are closest to the unit circle.


Slide L69

Let S

^= S g1 ;1 gm" # " #

Min-Norm Method: Determining g ^

Then:

1 2 R(G) ) S 1 = 0 ^ ^ ^ ^ g g ) S ^= ; g" #

Min-Norm solution: As:

^ = ;S(S S );1 g

+ S S , (S S );1 exists iff = k k2 6= 1 (This holds, at least, for N 1.) ^^ I=S S=Multiplying the above equation by gives:

(1 ; k k2) = (S S ) ) (S S );1 = =(1 ; k k2) ) ^ = ;S =(1 ; k k2) gLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L610

ESPRIT Method

Let

A1 = Im;1 0]A A2 = 0 Im;1]A

Then A2

Also, let

= A1D, where e;i!1 0 ... D= 0 e;i!n S1 = Im;1 0]S S2 = 0 Im;1]S2 6 4 |

3 7 5

Recall S

= AC with jC j 6= 0. Then S2 = A2C = A1DC = S1 C ;1DC{z

}

So has the same eigenvalues as D . determined as

is uniquely

= (S1S1);1S1S2Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L611

ESPRIT Implementation From the eigendecomposition of R, nd S , then S1 and S2 .

^

^

^

^

The frequency estimates are found by:

where

f^kgn=1 are the eigenvalues of k ^ = (S1S1);1S1S2 ^^ ^^

f!kgn=1 = ; arg(^k ) ^ k

ESPRIT Advantages: computationally simple no extraneous frequency estimates (unlike in MUSIC or MinNorm) accurate frequency estimates


Slide L612

Summary of Frequency Estimation Methods

Method Periodogram Nonlinear LS Yule-Walker Pisarenko MUSIC Min-Norm ESPRIT

Computational Accuracy / Risk for False Burden Resolution Freq Estimates small medium-high medium very high very high very high medium high medium small low none high high medium medium high small medium very high none

Recommendation:

Use Periodogram for medium-resolution applications

Use ESPRIT for high-resolution applicationsSlide L613


Filter Bank Methods

Lecture 7


Slide L71

Basic Ideas

Two main PSD estimation approaches: 1. Parametric Approach: Parameterize nite-dimensional model.

(!) by a

2. Nonparametric Approach: Implicitly smooth ! !=; by assuming that ! is nearly constant over the bands

f ( )g

( )

!;

!+ ] N >1

1

2 is more general than 1, but 2 requires

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

=2 2 =1

N > 1 leads to the variability / resolution compromiseassociated with all nonparametric methods.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L72

Filter Bank Approach to Spectral EstimationH 6 (!)1

!c

-!Filtered Signal

y (t)

- with varying !Power Calculation

Bandpass Filter c and xed bandwidth

-

Power in the band

-

Division by lter bandwidth

^(!c )

-

6y

y

(! )

;!c

@Q Q -!;Z

6 ~ (! ) @ -!(b)Z

!c

^FB (!) ' 1 2(a)

;

jH ( )j2 ( )d = ' 21

!+ !;

c ( )d = (') (!)

(a) consistent power calculation

(b) Ideal passband lter with bandwidth

(c)

( ) constant on 2 ! ; 2 ! + 2 ]

Note that assumptions (a) and (b), as well as (b) and (c), are conicting.Slide L73


Filter Bank Interpretation of the Periodogram

1 N y(t)e;i!t 2 ~ ^p(~) = ! 4 N t=1 2 N 1 ~ = N y(t)ei!(N ;t) t=1X X

= Nwhere(

X

1

k=0

hk y(N ; k)

2

! ;i!k = 1 eiN (~;!) ; 1 H (!) = hk e ! N ei(~;!) ; 1 k=0X

1 ei!k k = 0 : : : N ; 1 ~ hk = N otherwise 01

center frequency of H 3dB bandwidth of H

(!) = ! ~

(!) ' 1=NSlide L74


Filter Bank Interpretation of the Periodogram, con't

jH (!)j as a function of (~ ; !), for N = 50. !0 5 10

15 dB

20

25

30

35

40

3

2

1 0 1 ANGULAR FREQUENCY

2

3

Conclusion: The periodogram p ! is a lter bank PSD estimator with bandpass lter as given above, and: narrow lter passband, power calculation from only 1 sample of lter output.

^( )


Slide L75

Possible Improvements to the Filter Bank Approach 1. Split the available sample, and bandpass lter each subsample. more data points for the power calculation stage. This approach leads to Bartlett and Welch methods. 2. Use several bandpass lters on the whole sample. Each lter covers a small band centered on ! .

~

provides several samples for power calculation. This multiwindow approach is similar to the Daniell method.

Both approaches compromise bias for variance, and in fact are quite related to each other: splitting the data sample can be interpreted as a special form of windowing or ltering.


Slide L76

Capon Method

Idea: Data-dependent bandpass lter design.

yF (t) =

X

m

k=0|

hk y(t ; k)2 6 4 |

. = h0 h1 : : : hm] . y(t ; m) h y(t) ~ E jyF (t)j2 = h Rh R = E fy(t)~ (t)g ~ y{z } 7 5 {z } n o

y(t)

3

H (! ) =where a

X

m

(! ) = 1

k=0 e;i! : : : e;im! ]T

hk e;i!k = h a(!)


Slide L77

Capon Method, con't Capon Filter Design Problem:

min(h Rh) hSolution:

subject to h

a(!) = 1

h0 = R;1a=a R;1ao

The power at the lter output is:

E jyF (t)j2 = h0Rh0 = 1=a (!)R;1a(!) which should be the power of y (t) in a passband centered on ! .n

The Bandwidth

1 ' m+1 = (lter 1 length)

Conclusion Estimate PSD as:

+1 ^(!) = m^;1 a (!)R a(!)with

N 1 ^ y(t)~ (t) ~ y R = N ;m t=m+1X


Slide L78

Capon Properties

m is the user parameter that controls the compromisebetween bias and variance: as m increases, bias decreases and variance increases.

Capon uses one bandpass lter only, but it splits the N -data point sample into (N m) subsequences of length m with maximum overlap.

;


Slide L79

Relation between Capon and Blackman-Tukey Methods

Consider

^BT (!) with Bartlett window: m m + 1 ; jkj ^BT (!) = r(k)e;i!k ^ k=;m m + 1 1 m m ^(t ; s)e;i!(t;s) = m+1 r t=0 s=0 ^ a (!)Ra(!) R = ^(i ; j )] ^ r = m+1X X X

Then we have

^ a (!)Ra(!) ^BT (!) = m+1 m+1 ^C (!) = ^ a (!)R;1a(!)


Slide L710

Relation between Capon and AR Methods Let

2 ^k ^k (!) = ^ 2 jAk(!)j be the kth order AR PSD estimate of y (t).AR Then

^C (!) =

1 1 m 1= ÂR(!) m + 1 k=0 kX

Consequences: Due to the average over k, C ! generally has less statistical variability than the AR PSD estimator. Due to the low-order AR terms in the average, C ! generally has worse resolution and bias properties than the AR method.

^( )

^( )


Slide L711

Spatial Methods Part 1

Lecture 8


Slide L81

The Spatial Spectral Estimation ProblemSource 1

vSource 2B B B B N B cccc ccc c c c c c

v

@ ,,, @, ,,@ , ,R , , ,@ ,

v Source n

@; @;Sensor 1

@; @; @; @;Sensor 2 Sensor

m

Problem: Detect and locate n radiating sources by using an 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 over space (thus the name spatial spectral analysis)Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L82

Simplifying Assumptions

Far-eld sources in the same plane as the array of sensors Non-dispersive wave propagation

Hence: The waves are planar and the only location parameter is direction of arrival (DOA) (or angle of arrival, AOA). The number of sources n is known. (We do not treat the detection problem) The sensors are linear dynamic elements with known transfer characteristics and known locations (That is, the array is calibrated.)


Slide L83

Array Model Single Emitter Case

x(t) =k=

the signal waveform as measured at a reference point (e.g., at the rst sensor) the delay between the reference point and the kth sensor the impulse response (weighting function) of sensor k noise at the kth sensor (e.g., thermal noise in sensor electronics; background noise, etc.)

hk (t) = ek (t) =Note:

t 2 R (continuous-time signals). Then the output of sensor k is yk (t) = hk (t) x(t ; k) + ek (t) ( = convolution operator).Basic Problem: Estimate the time delays known but x t unknown.

()

f kg with hk(t)

This is a time-delay estimation problem in the unknown input case.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L84

Narrowband Assumption

Assume: The emitted signals are narrowband with known carrier frequency !c . Then:

x(t) = (t) cos !ct + '(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 !ct + '(t) ; !c k ] hk (t) x(t ; k ) ' jHk (!c)j (t)cos !ct + '(t) ; !c k + argfHk (!c)g]

where Hk function

(!) = Ffhk(t)g is the kth sensor's transfer

Hence, the kth sensor output is

yk (t) = jHk (!c)j (t) cos !ct + '(t) ; !c k + arg Hk (!c)] + ek(t)Slide L85


Complex Signal Representation The noise-free output has the form:

z(t) = (t) cos !ct + (t)] = = (t) ei !ct+ (t)] + e;i !ct+ (t)] 2 Demodulate z (t) (translate to baseband): 2z(t)e;!ct = (t)f ei (t) + e;i 2!ct+ (t)] gn o | {z } | {z }

Lowpass lter

2z(t)e;i!ct to obtain (t)ei (t)

lowpass

highpass

Hence, by low-pass ltering and sampling the signal

yk (t)=2 = yk(t)e;i!ct ~ = yk(t) cos(!ct) ; iyk(t) sin(!ct) we get the complex representation: (for t 2 Z ) yk (t) = (t) ei'(t) jHk (!c)j ei arg H (! )] e;i! + ek (t)|

s(t)

{z

} |

Hk (!c)

{z

k

c

}

c k

yk (t) = s(t)Hk (!c) e;i!c k + ek (t) where s(t) is the complex envelope of x(t).Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L86

or

Vector Representation for a Narrowband Source

Let

= m =a( ) =2 6 4

the emitter DOA the number of sensors2 6 4

H1(!c) e;i!c. .

1

3 7 5

Hm(!c) e;i!c m y1(t) e1(t) . . y(t) = e(t) = . . ym(t) em(t)3 7 5 2 6 4

3 7 5

Then

y(t) = a( )s(t) + e(t)NOTE: enters a via both For omnidirectional sensors the on .

()

f kg and fHk(!c)g. fHk(!c)g do not depend


Slide L87

Analog Processing Block Diagram

Analog processing for each receiving array elementSENSOR DEMODULATION

A/D CONVERSION

-

-

Internal Noise External Noise XXXXX ? XX ll z l - Cabling and , Filtering : ,, Incoming Transducer Signal x(t ; k )

Re yk (t)] Lowpass Re yk (t)] - !! ~ Filter A/D cos(!ct) 6 y k (t) Oscillator -sin(!ct) ? ~ - Im yk (t)] Lowpass Im yk (t)] - !! Filter A/D

-


Slide L88

Multiple Emitter Case Given n emitters with received signals: DOAs:

fsk(t)gn=1 k

k

Linear sensors

) y(t) = a( 1)s1(t) +

+ a( n)sn(t) + e(t)

Let

A = a( 1) : : : a( n)] (m n) s(t) = s1(t) : : : sn(t)]T (n 1)Then, the array equation is:

y(t) = As(t) + e(t)Use the planar wave assumption to nd the dependence of k on .Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L89

Uniform Linear Arrays

Source h

JJ J J

J J

JJ J ^ J

J

JJ

JJ+ JJ

JJ2

] J d sin J ^ JJ J JJ 3 J Jv4

1

v ]? vd -

v

ppp

m

vSensor

ULA Geometry Sensor #1 = time delay reference Time Delay for sensor k:

where c = wave propagation speed

d sin k = (k ; 1) c


Slide L810

Spatial Frequency

Let:

sin 4 !s = !c d sin = 2 dc=f = 2 d sin c c = c=fc = signal wavelength a( ) = 1 e;i!s : : : e;i(m;1)!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!sj $ dj sin j 1 d =2As

=2

d

=2 is a spatial Shannon sampling theorem.Slide L811

d = spatial sampling period


Spatial Methods Part 2

Lecture 9


Slide L91

Spatial Filtering

Spatial ltering useful for DOA discrimination (similar to frequency discrimination of time-series ltering) Nonparametric DOA estimation

There is a strong analogy between temporal ltering and spatial ltering.


Slide L92

Analogy between Temporal and Spatial Filtering

Temporal FIR Filter:

yF (t) = hk u(t ; k) = h y(t) k=0 h = ho : : : hm;1] y(t) = u(t) : : : u(t ; m + 1)]TIf u

m;1 X

(t) = ei!t then yF (t) = h a(!)] u(t)| {z }

lter transfer function

a(!) = 1 e;i! : : : e;i(m;1)! ]T We can select h to enhance or attenuate signals with different frequencies ! .


Slide L93

Analogy between Temporal and Spatial Filtering Spatial Filter:

fyk(t)gm=1 = k

the spatial samples obtained with a sensor array.

Spatial FIR Filter output:

yF (t) =

X

m

k=1

hk yk (t) = h y(t)

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

() y(t) = a( )s(t) a( ) = 1 e;i!c 2 : : : e;i!c m ]TyF (t) = h a( )] s(t)| {z }

The corresponding lter output is

lter transfer function

We can select h to enhance or attenuate signals coming from different DOAs.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L94

Analogy between Temporal and Spatial Filteringu(t) = ei!t

z1

s q; s qq

h-

?1

?

-

m;1

q ;1

? -

1 - @ 0 B 1 C h @@ C B B e;i! C @ h a(!)]u(t) R @ C - ? B -P C B C B B B .. Cu(t) C B . C C B C B C B C B C B A @ ;i m; ! hm; | e {z } ? a(!)1

?

0

qq

(

1)

1

(Temporal sampling)

(a) Temporal lternarrowband source with DOA=Z

z

Z

Z

Z ~ aa 1reference !!

h

0

0 1 B aa 2- B ;i! Be 2 B !! B B B B .. B . B B @ ;i! m | e {z a( ) aa m!

point

( )

qq

( )

1 C C C C C Cs(t) C C C C C A }

-

? @ ? @ @

h-

1

@ R @ -

P

h a( )]s(t) -

hm;?

qq

1

!

(Spatial sampling)

(b) Spatial lterLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L95

Spatial Filtering, con't

Example: The response magnitude h a of a spatial lter (or beamformer) for a 10-element ULA. Here, h a 0 , where 0

j ( )j

= ( )

= 25

0

30

30

Th et a (d eg )

60

60

90 0 2 4 6 Magnitude 8 10

90


Slide L96

Spatial Filtering Uses

Spatial Filters can be used To pass the signal of interest only, hence ltering out interferences located outside the lter's beam (but possibly having the same temporal characteristics as the signal). To locate an emitter in the eld of view, by sweeping the lter through the DOA range of interest (goniometer).


Slide L97

Nonparametric Spatial Methods

A Filter Bank Approach to DOA estimation. Basic Ideas Design a lter h

( ) such that for each

It passes undistorted the signal with DOA = It attenuates all DOAs

6=

Sweep the lter through the DOA range of interest, and evaluate the powers of the ltered signals:

E jyF (t)j2 = E jh ( )y(t)j2 = h ( )Rh( ) with R = E fy (t)y (t)g.n o n

o

The (dominant) peaks of h DOAs of the sources.

( )Rh( ) give theSlide L98


Beamforming Method

Assume the array output is spatially white:

Then:

R = E fy(t)y (t)g = I E jyF (t)j2 = h hn o

Hence: In direct analogy with the temporally white assumption for lter bank methods, y t can be considered as impinging on the array from all DOAs.

()

Filter Design:

min (h h) hSolution:

subject to

h a( ) = 1

h = a( )=a ( )a( ) = a( )=m E jyF (t)j2 = a ( )Ra( )=m2n o


Slide L99

Implementation of Beamforming

1 N y(t)y (t) ^ R=N t=1X

The beamforming DOA estimates are:

f^kg =

the locations of the n largest peaks of a Ra .

( )^ ( )

This is the direct spatial analog of the Blackman-Tukey periodogram. Resolution Threshold:

inf j k ; pj > =

wavelength array length array beamwidth

Inconsistency problem: Beamforming DOA estimates are consistent if n inconsistent if n > .

1

= 1, butSlide L910


Capon Method Filter design:

min(h Rh) subject to h a( ) = 1 hSolution:

h = R;1a( )=a ( )R;1a( ) E jyF (t)j2 = 1=a ( )R;1a( )n o

Implementation:

f^kg =

the locations of the n largest peaks of

^ 1=a ( )R;1a( ):

Performance: Slightly superior to Beamforming. Both Beamforming and Capon are nonparametric approaches. They do not make assumptions on the covariance properties of the data (and hence do not depend on them).Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L911

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 in all sensors:

E fe(t)e (t)g = 2IThe signal covariance matrix

P = E fs(t)s (t)gis nonsingular.

Then:

R = E fy(t)y (t)g = APA + 2IThus: The NLS, YW, MUSIC, MIN-NORM and ESPRIT methods of frequency estimation can be used, almost without modication, for DOA estimation.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L912

Nonlinear Least Squares Method

1 N ky(t) ; As(t)k2 min(t)g N f k g fs t=1X |

Minimizing f over s gives

f ( s)

{z

}

^(t) = (A A);1A y(t) t = 1 : : : N sX

Then

1 N k I ; A(A A);1A ]y(t)k2 f ( ^) = N s t=1 1 N y (t) I ; A(A A);1A ]y(t) = N t=1 ^ = trf I ; A(A A);1A ]Rg ^ Thus, f ^k g = arg max trf A(A A);1A ]RgX

f kg

For N , this is precisely the form of the NLS method of frequency estimation.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L913

=1

Nonlinear Least Squares Method

Properties of NLS: Performance: high Computational complexity: high Main drawback: need for multidimensional search.


Slide L914

Yule-Walker Method

e y(t) = y(t) = A s(t) + ~(t) ~ y(t) ~ e(t) A" # " # "

#

Assume: Then:

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

4 E fy(t)~ (t)g = AP A ~ ;= y

(M L)

Also assume:

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

~ (A) = rank(A) = n" # " #

(;) = n, and the SVD of ; is ; = U1 U2 ] 0 n n 0 V1 g n ;n 0 V2 g L n M ;n ~ ~ Properties: A V2 = 0 V1 2 R(A)Then: rank|{z} |{z}


Slide L915

YW-MUSIC DOA Estimator

f^kg =where

the n largest peaks of

1=~ ( )V2V2 ~( ) a ^â

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

^ V2 is dened similarly to V2, using 1 N y(t)~ (t) ^ ;=N y t=1X

Properties: Computational complexity: medium Performance: satisfactory if m

2n

Main advantages: weak assumption on e t the subarray A need not be calibrated

f ( )g


Slide L916

MUSIC and Min-Norm Methods

Both MUSIC and Min-Norm methods for frequency estimation apply with only minor modications to the DOA estimation problem. Spectral forms of MUSIC and Min-Norm can be used for arbitrary arrays Root forms can be used only with ULAs MUSIC and Min-Norm break down if the source signals are coherent; that is, if rank

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

Modications that apply in the coherent case exist.


Slide L917

ESPRIT Method Assumption: The array is made from two identical subarrays separated by a known displacement vector. Let

m = # sensors in each subarray A1 = Im 0]A (transfer matrix of subarray 1) A2 = 0 Im]A (transfer matrix of subarray 2)Then

A2 = A1D, where e;i!c ( 1) 0 ... D= 0 e;i!c ( n) ( ) = the time delay from subarray 1 to2 6 4

3 7 5

subarray 2 for a signal with DOA = : where d is the subarray separation and is measured from the perpendicular to the subarray displacement vector.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L918

( ) = d sin( )=c

ESPRIT Method, con't ESPRIT Scenariosource

known

subarray 1

displac ement v

ector

subarray 2

Properties: Requires special array geometry Computationally efcient

No risk of spurious DOA estimatesDoes not require array calibration Note: For a ULA, the two subarrays are often the rst m and last m array elements, so m m and

;1

;1 A1 = Im;1 0]A

= ;1 A2 = 0 Im;1]A


Slide L919

Spectral Estimation 2

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