basic deﬁnitions and spectral estimation

8/12/2019 Basic Definitions and Spectral Estimation

1/125

Basic Definitions

and

The Spectral Estimation Problem

Lecture 1

Lecture notes to accompanyIntroduction to Spectral Analysis Slide L11

by P. Stoica and R. Moses, Prentice Hall, 1997


2/125

Informal Definition of Spectral Estimation

Given:A finite record of a signal.

Determine:The distribution of signal power over

frequency.

t

signal

t=1, 2, ... +

spectral density

! = (angular) frequency in radians/(sampling interval)

f =! =

2 = frequency in cycles/(sampling interval)




3/125

Applications

Temporal Spectral Analysis

Vibration monitoring and fault detection

Hidden periodicityfinding

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




4/125

Deterministic Signals

f y ( t ) g

1

t =; 1

= discrete-time deterministic data

sequence

If:1

X

t =; 1

j y ( t ) j

2


5/125

Energy Spectral Density

Parseval's Equality:

1

X

t =; 1

j y ( t ) j

2 =1

2

Z

;

S ( ! )d !

where

S ( ! )

4

= j Y ( ! ) 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 )




6/125

Random Signals

Random Signal

t

random signal probabilistic statements about

future variations

current observation time

Here:1

X

t =; 1

j y ( t ) j

2

= 1

But:E

n

j y ( t ) j

2

o


7/125

First Definition of PSD

( ! )

=

1

X

k =; 1

r ( k ) e

;i ! k

where r ( k ) is theautocovariance sequence (ACS)

r ( k) =

E f y ( t ) y

( t ; k ) g

r ( k) =

r

( ; k ) r ( 0 ) j

r ( k ) j

Note that

r ( k) =

1

2

Z

;

( ! ) e

i ! k

d !(Inverse DTFT)

Interpretation:

r( 0 ) =

E

n

j y ( t ) j

2

o

=

1

2

Z

;

( ! )d !

so

( ! )d !

= infinitesimal signal power in the band

!

d !

2




8/125

Second Definition of PSD

( ! ) =l i m

N! 1

E

8

>

:

1

N

N

X

t= 1

y ( t ) e

;i ! t

2

9

>

=

>

Note that

( !) = l i m

N! 1

E

1

N

j Y

N

( ! ) j

2

where

Y

N

( !) =

N

X

t= 1

y ( t ) e

;i ! t

is thefinite DTFT off y ( t ) g .




9/125

Properties of the PSD

P1: ( !) =

( !+ 2

) for all ! .

Thus, we can restrict attention to

! 2 ;

]( )

f 2 ; 1 = 2 1 =2 ]

P2: ( ! ) 0

P3: Ify ( t ) is real,

Then: ( ! ) = ( ; ! )

Otherwise: ( ! ) 6= ( ; ! )




10/125

Transfer of PSD Through Linear Systems

System Function: H ( q) =

1

X

k= 0

h

k

q

; k

where q ; 1 = unit delay operator: q ; 1 y ( t) =

y ( t ;1 )

e ( t )

e

( ! )

y ( t )

y

( !) =

j H ( ! ) 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

( ! ) = j H ( ! ) j 2 e

( ! )




11/125

The Spectral Estimation Problem

The Problem:

From a sample f y( 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 the

PSD




12/125

Periodogram

and

Correlogram

Methods

Lecture 2




13/125


14/125

Correlogram

Recall 1st definition of ( ! ) :

( !) =

1

X

k =; 1

r ( k ) e

;i ! k

Truncate theP

and replace r ( k ) by r ( k ) :

c

( ! ) =N ; 1

X

k = ; ( N ;1 )

r ( k ) e

;i ! k




15/125

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


16/125

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

=

N ; 1

X

k = ; ( N ;1 )

r ( k ) e

;i ! k

=

c

( ! )

p

( !) =

c

( ! )

Consequence:

Both p

( ! ) and c

( ! ) can be analyzed simultaneously.




17/125

Statistical Performance of p

( ! ) and c

( ! )

Summary:

Both are asymptotically (for large N ) unbiased:

E

n

p

( ! )

o

! ( ! ) as N! 1

Both havelargevariance, even for large N .

Thus, p

( ! ) and c

( ! ) havepoor performance.

Intuitive explanation:

r ( k ) ; r ( k ) may be large for large j k j

Even if the errorsf r ( k ) ; r ( k ) g

N ; 1

j k j= 0 are small,

there areso manythat when summed in

p

( ! ) ; ( !) ]

, the PSD error is large.




18/125

Bias Analysis of the Periodogram

E

n

p

( ! )

o

= E

n

c

( ! )

o

=

N ; 1

X

k = ; ( N ;1 )

E f r ( k ) g e

;i !

=

N ; 1

X

k = ; ( N ;1 )

1 ;

j k j

N

!

r ( k ) e

;i ! k

=

1

X

k =; 1

w

B

( k ) r ( k ) e

;i ! k

w

B

( k) =

8


19/125

Bartlett Window WB

( ! )

W

B

( !) =

1

N

"

s i n ( ! N = 2 )

s i n ( ! = 2 )

#

2

W

B

( ! )= W

B

( 0 )

, forN

= 2 5

3 2 1 0 1 2 360

50

40

30

20

10

0

dB

ANGULAR FREQUENCY

Main lobe 3dB width 1= N

.

Forsmall N , WB

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




20/125

Smearing and Leakage

Main Lobe Width:smearingorsmoothing

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

resolvable.

()


21/125

Periodogram Bias Properties

Summary of Periodogram Bias Properties:

Forsmall N , severe bias

AsN

! 1

,W

B

( ! ) ! ( ! )

,so ( ! ) is asymptotically unbiased.




22/125

Periodogram Variance

As N ! 1

E

n h

p

( !

1

) ; ( !

1

)

i h

p

( !

2

) ; ( !

2

)

i o

=

(

2

( !

1

) !

1

= !

2

0 !

1

6= !

2

Inconsistent estimate

Erratic behavior

()^

1 st. dev = () too+-

asymptotic mean = ()

()^

Resolvability properties depend onbothbias and variance.




23/125

Discrete Fourier Transform (DFT)

Finite DTFT: YN

( !) =

N

X

t= 1

y ( t ) e

;i ! t

Let ! = 2 N

k and W = e ; i2

N .

Then YN

(

2

N

k ) is the Discrete Fourier Transform (DFT):

Y ( k) =

N

X

t= 1

y ( t ) W

t k

k = 0 : : : N ; 1

Direct computation off Y ( k ) g N ; 1k

= 0

from f y ( t ) g Nt

= 1

:

O ( N

2

) flops




24/125

Radix2 Fast Fourier Transform (FFT)

Assume: N= 2

m

Y ( k) =

N =2

X

t= 1

y ( t ) W

t k

+

N

X

t =N = 2 + 1

y ( t ) W

t k

=

N =2

X

t= 1

y ( t) +

y ( t +N = 2 )

W

N k

2

] W

t k

with WN k

2

=

+ 1 for even k

; 1 for odd k

Let ~N =N =

2 and ~W = W 2 = e ; i 2 =~

N .

For k = 0 2 4 : : : N ; 24

= 2p :

Y( 2

p) =

~

N

X

t= 1

y ( t) +

y ( t +

~

N) ]

~

W

t p

For k= 1

3 5 : : : N ;1 = 2

p+ 1

:

Y( 2

p+ 1 ) =

~

N

X

t= 1

f y ( t ) ; y ( t +

~

N) ]

W

t

g

~

W

t p

Each is a ~N =N =

2 -point DFT computation.




25/125

FFT Computation Count

Let ck

= number offlops for N = 2 k point FFT.

Then

c

k

=

2

k

2

+ 2c

k ; 1

) c

k

=

k 2

k

2

Thus,

c

k

=

1

2

Nl o g

2

N




26/125

Zero Padding

Append the given data by zeros prior to computing DFT

(or FFT):

f y( 1 ) : : : y

( N ) 0 : : :

0

| { z }

N

g

Goals:

Apply a radix-2 FFT (so N = power of 2)

Finer sampling of ( ! ) :

2

N

k

N ; 1

k= 0

!

2

N

k

N ; 1

k= 0

()^continuous curve

sampled, N=8




27/125

Improved Periodogram-Based

Methods

Lecture 3




28/125

Blackman-Tukey Method

Basic Idea:Weighted correlogram, with small weight

applied to covariances r ( k ) withlarge j k j .

B T

( !) =

M ; 1

X

k = ; ( M ;1 )

w ( k)

r ( k ) e

;i ! k

f w ( k ) g = Lag Window

1

-M M

w(k)

k




29/125

Blackman-Tukey Method, con't

B T

( !) =

1

2

Z

;

p

( ) W ( ! ; )d

W ( !) = DTFT f w ( k ) g

= Spectral Window

Conclusion: B T

( !) = locallysmoothed periodogram

Effect:

Variance decreases substantially

Bias increases slightly

By proper choice of M :

MSE = var + bias2 ! 0 as N! 1




30/125

Window Design Considerations

Nonnegativeness:

B T

( !) =

1

2

Z

;

p

( )

| { z }

0

W ( ! ; )d

IfW ( ! ) 0 (

, w ( k ) is a psd sequence)

Then: B T

( ! ) 0 (which is desirable)

Time-Bandwidth Product

N

e

=

M ; 1

X

k = ; ( M ;1 )

w ( k )

w( 0 )

= equiv time width

e

=

1

2

Z

;

W ( ! )d !

W( 0 )

=

equiv bandwidth

N

e

e

= 1




31/125

Window Design, con't

e

= 1 = N

e

= 0 ( 1 = M)

is the BT resolution threshold.

As M increases, bias decreases and variance

increases.

) Choose M as a tradeoff betweenvarianceand

bias.

Once M is given, Ne

(and hence e

) is essentially

fixed.

) Choose window shape to compromise between

smearing(main lobe width) andleakage(sidelobe

level).

The energy in the main lobe and in the sidelobescannot be reducedsimultaneously, once M is given.




32/125

Window Examples

Triangular Window,M

= 2 5

3 2 1 0 1 2 360

50

40

30

20

10

0

dB

ANGULAR FREQUENCY

Rectangular Window, M = 2 5

3 2 1 0 1 2 360

50

40

30

20

10

0

dB

ANGULAR FREQUENCY




33/125

Bartlett Method

Basic Idea:

1 2 . . .

average

. . . ()^

1 ()^

2 ()^

L

()^B

Mathematically:

y

j

( t) =

y

(

( j ;1 )

M + t

)

t= 1

: : : M

= the j th subsequence

( j= 1

: : : L

4

= N = M ] )

j

( !) =

1

M

M

X

t= 1

y

j

( t ) e

;i ! t

2

B

( !) =

1

L

L

X

j= 1

j

( ! )




34/125

Comparison of Bartlett

and Blackman-Tukey Estimates

B

( !) =

1

L

L

X

j= 1

8

>

:

M ; 1

X

k = ; ( M ;1 )

r

j

( k ) e

;i ! k

9

>

=

>

=

M ; 1

X

k = ; ( M ;1 )

8


35/125

Welch Method

Similar to Bartlett method, but

allow overlap of subsequences (gives more

subsequences, and thusbetteraveraging)

use data window for each periodogram; gives

mainlobe-sidelobe tradeoff capability

1 2 N

subseq#1

. . .

subseq#2

subseq#S

Let S = # of subsequences of length M .(Overlapping means

S >

N = M] ) better

averaging.)

Additional flexibility:

The data in each subsequence are weighted by atemporal

window

Welch is approximately equal to B T

( ! ) with a

non-rectangular lag window.




36/125

Daniell Method

By a previous result, for N 1 ,

f

p

( !

j

) g are (nearly) uncorrelated random variables for

!

j

=

2

N

j

N ; 1

j= 0

Idea: Local averagingof( 2 J + 1 ) samples in the

frequency domain should reduce the variance by about

( 2J

+ 1 ) .

D

( !

k

) =

1

2 J+ 1

k + J

X

j = k ; J

p

( !

j

)




37/125

Daniell Method, con't

As J increases:

Bias increases (more smoothing)

Variance decreases (more averaging)

Let = 2 J = N

. Then, for N 1 ,

D

( ! ) '

1

2

Z

;

p

( ! )d !

Hence: D

( ! ) '

B T

( ! ) with arectangular spectral

window.




38/125

Summary of Periodogram Methods

Unwindowed periodogram

reasonable bias unacceptable variance

Modified periodograms

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

BT periodogram

Local smoothing/averaging of p

( ! ) by a suitably selectedspectralwindow.

Implemented by truncating and weighting r ( k ) using alagwindow in

c

( ! )

Bartlett, Welch periodograms

Approximate interpretation: B T

( ! ) with a suitablelagwindow (rectangular for Bartlett; more general for Welch).

Implemented by averaging subsample periodograms.

Daniell Periodogram

Approximate interpretation: B T

( ! ) with a rectangularspectralwindow.

Implemented by local averaging of periodogram values.




39/125

Parametric Methods

for

Rational Spectra

Lecture 4




40/125

Basic Idea of Parametric Spectral Estimation

ObservedData

Assumedfunctional

form of (,)

Estimateparameters

in (,)

Estimate

PSD

() =(,)^ ^

possibly revise assumption on ()

Rational Spectra

( !) =

P

j kj

m

k

e

;i ! k

P

j kj

n

k

e

;i ! k

( ! ) is arational functionin e ; i ! .

ByWeierstrass theorem, ( ! ) can approximate arbitrarily

wellany continuous PSD, provided m and n are chosen

sufficiently large.

Note, however:

choice ofm and n is not simple

some PSDs arenotcontinuous




41/125

AR, MA, and ARMA Models

BySpectral Factorizationtheorem, a rational ( ! ) can befactored as

( !) =

B ( ! )

A ( ! )

2

2

A ( z) = 1 +

a

1

z

; 1

+

+ a

n

z

; n

B ( z) = 1 +

b

1

z

; 1

+

+ b

m

z

; m

and,e.g., A ( ! ) = A ( z ) jz = e

i !

Signal Modeling Interpretation:

e ( t )

e

( !) =

2

w h i t e n o i s e

y ( t )

y

( !) =

B ( ! )

A ( ! )

2

2

B ( q )

A ( q )

l t e r e d w h i t e n o i s e

--

ARMA: A ( q ) y ( t) =

B ( q ) e ( t )

AR:A ( q ) y ( t

) =e ( t )

MA: y ( t) =

B ( q ) e ( t )




42/125

ARMA Covariance Structure

ARMA signal model:

y ( t) +

n

X

i

= 1

a

i

y ( t ; i) =

m

X

j

= 0

b

j

e ( t ; j ) ( b

0

= 1 )

Multiply by y ( t ; k ) and take E f g to give:

r ( k) +

n

X

i= 1

a

i

r ( k ; i) =

m

X

j= 0

b

j

E f e ( t ; j ) y

( t ; k ) g

=

2

m

X

j= 0

b

j

h

j ; k

= 0for

k > m

where H ( q) =

B ( q )

A ( q )

=

1

X

k= 0

h

k

q

; k

( h

0

= 1 )




43/125

AR Signals: Yule-Walker Equations

AR: m= 0 .

Writing covariance equation in matrix form for

k= 1 : : : n

:2

6

6

6

4

r( 0 )

r ( ;1 ) : : : r

( ; n )

r( 1 )

r( 0 )

...... . . . r ( ;

1 )

r ( n ): : : r ( 0 )

3

7

7

7

5

2

6

6

6

4

1

a

1

...

a

n

3

7

7

7

5

=

2

6

6

6

4

2

0

...

0

3

7

7

7

5

R

"

1

#

=

"

2

0

#

These are theYuleWalker (YW) Equations.




44/125

AR Spectral Estimation: YW Method

Yule-Walker Method:

Replace r ( k ) by r ( k ) and solve for f ai

g and 2 :

2

6

6

6

4

r( 0 ) ^

r ( ;1 ) : : :

r ( ; n )

r( 1 ) ^

r( 0 )

...... . . . r ( ;

1 )

r ( n ): : :

r( 0 )

3

7

7

7

5

2

6

6

6

4

1

a

1

...

a

n

3

7

7

7

5

=

2

6

6

6

4

2

0

...

0

3

7

7

7

5

Then the PSD estimate is

( !) =

2

j

A ( ! ) j

2




45/125

AR Spectral Estimation: LS Method

Least Squares Method:

e ( t) =

y ( t) +

n

X

i= 1

a

i

y ( t ; i) =

y ( t) +

'

T

( t )

4

= y ( t) +

y ( t )

where ' ( t) =

y ( t ;1 )

: : : y ( t ; n) ]

T .

Find =

a

1

: : : a

n

]

T to minimize

f ( ) =

N

X

t = n+ 1

j e ( t ) j

2

This gives = ; ( Y Y ) ; 1 ( Y y ) where

y =

2

6

4

y ( n + 1 )

y ( n + 2 )

...

y ( N )

3

7

5

Y =

2

6

4

y ( n ) y ( n ; 1 ) y ( 1 )

y ( n + 1 ) y ( n ) y ( 2 )

... ...

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

3

7

5




46/125

LevinsonDurbin Algorithm

Fast, order-recursive solution to YW equations

2

6

6

6

4

0

; 1

; n

1

0

. . . ...... . . . . . .

; 1

n

1

0

3

7

7

7

5

| { z }

R

n + 1

2

6

6

6

4

1

n

3

7

7

7

5

=

2

6

6

6

4

2

n

0

...

0

3

7

7

7

5

k

= either r ( k ) or r ( k ) .

Direct Solution:

For one given value of n : O ( n3

) flops

For k= 1

: : : n : O ( n 4 ) flops

LevinsonDurbin Algorithm:

Exploits the Toeplitz form ofRn

+ 1

to obtain the solutions

for k = 1 : : : n in O ( n 2 ) flops!




47/125

Levinson-Durbin Alg, con't

Relevant Properties of R :

R x

= y $ R ~x= ~

y , where ~x=

x

n

: : : x

1

]

T

Nested structure

R

n+ 2

=

2

4

R

n + 1

n + 1

~r

n

n + 1

~r

n

0

3

5

~r

n

=

2

6

4

n

...

1

3

7

5

Thus,

R

n + 2

2

4

1

n

0

3

5

=

2

4

R

n + 1

n + 1

~r

n

n + 1

~r

n

0

3

5

2

4

1

n

0

3

5

=

2

4

2

n

0

n

3

5

where n = n

+ 1

+ ~r

n

n




48/125

Levinson-Durbin Alg, con't

R

n + 2

2

4

1

n

0

3

5

=

2

4

2

n

0

n

3

5

R

n + 2

2

4

0

~

n

1

3

5

=

2

4

n

0

2

n

3

5

Combining these gives:

R

n + 2

8


49/125

MA Signals

MA: n= 0

y ( t) =

B ( q ) e ( t )

= e ( t) +

b

1

e ( t ;1 ) +

+ b

m

e ( t ; m )

Thus,

r ( k) = 0

for j k j> m

and

( !) =

j B ( ! ) j

2

2

=

m

X

k = ; m

r ( k ) e

;i ! k




50/125

MA Spectrum Estimation

Two main ways to Estimate ( ! ) :

1. Estimate f bk

g and 2 and insert them in

( !) =

j B ( ! ) j

2

2

nonlinear estimation problem

( ! ) is guaranteed to be 0

2. Insert sample covariances f r ( k ) g in:

( !) =

m

X

k = ; m

r ( k ) e

;i ! k

This is B T

( ! ) with a rectangular lag window of

length 2 m+ 1 .

( ! ) isnotguaranteed to be 0

Both methods are special cases of ARMA methods

described below, with AR model order n= 0 .




51/125

ARMA Signals

ARMA models can represent spectra with both peaks

(AR part) and valleys (MA part).

A ( q ) y ( t) =

B ( q ) e ( t )

( !) =

2

B ( ! )

A ( ! )

2

=

P

m

k = ; m

k

e

;i ! k

j A ( ! ) j

2

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

=

n

X

j= 0

n

X

p= 0

a

j

a

p

r ( k + p ; j )




52/125

ARMA Spectrum Estimation

Two Methods:

1. Estimate f ai

b

j

2

g in ( !) =

2

B ( ! )

A ( ! )

2

nonlinear estimation problem; can use an approximatelineartwo-stage least squaresmethod

( ! ) is guaranteed to be 0

2. Estimate f ai

r( k ) g in ( !

) =

P

m

k = ; m

k

e

; i ! k

j A ( ! ) j

2

linear estimation problem (the Modified Yule-Walker

method).

( ! ) isnotguaranteed to be 0




53/125

Two-Stage Least-Squares Method

Assumption:The ARMA model is invertible:

e ( t) =

A ( q )

B ( q )

y ( t )

= y ( t) +

1

y ( t ;1 ) +

2

y ( t ;2 ) +

= AR( 1 ) with j k

j !0 as k

! 1

Step 1: Approximate, for some large K

e ( t ) ' y ( t) +

1

y ( t ;1 ) +

+

K

y ( t ; K )

1a) Estimate the coefficients f k

g

K

k= 1

by using AR

modelling techniques.

1b) Estimate the noise sequence

e ( t) =

y ( t) +

1

y ( t ;1 ) + +

K

y ( t ; K )

and its variance

2

=

1

N ; K

N

X

t = K+ 1

j e ( t ) j

2




54/125

Two-Stage Least-Squares Method, con't

Step 2: Replace f e ( t ) g by e ( t ) in the ARMA equation,

A ( q ) y ( t ) ' B ( q)

e ( t )

and obtain estimates of f ai

b

j

g by applying least squares

techniques.

Note that the ai

and bj

coefficients enter linearly in the

above equation:

y ( t ) ; e ( t ) ' ; y ( t ;1 ) : : :

; y ( t ; n )

e ( t ;1 ) : : :

e ( t ; m) ]

=

a

1

: : : a

n

b

1

: : : b

m

]

T




55/125

Modified Yule-Walker Method

ARMA Covariance Equation:

r ( k) +

n

X

i= 1

a

i

r ( k ; i) = 0 k > m

In matrix form for k = m+ 1

: : : m + M

2

6

4

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

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

... . . .

...

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

3

7

5

"

a

1

...

a

n

#

= ;

2

6

4

r ( m + 1 )

r ( m + 2 )

...

r ( m + M )

3

7

5

Replace f r ( k ) g by f r ( k ) g and solve for f ai

g .

IfM = n , fast Levinson-type algorithms exist for

obtaining f ai

g .

IfM > n overdetermined Y W system of equations; least

squares solution for f ai

g .

Note:For narrowband ARMA signals, the accuracy of

f a

i

g is often better forM > n




56/125

Sum

maryofParametricMethodsfor

RationalSpectra

Computational

Guarantee

Method

Burden

Accuracy

(

!

)

0

?

Usefor

AR:YWor

LS

low

medium

Yes

Spectrawith(narrow

)peaksbut

novalley

MA:BT

low

low-medium

No

Broadbandspectrapo

ssiblywith

valleysbutnopeaks

ARMA:MYW

low-medium

medium

No

Spectrawithbothpea

ksand(not

toodeep)valleys

ARMA:2-Sta

geLS

medium-high

medium-high

Yes

Asabove

LecturenotestoaccompanyIntroductiontoSpectralAnalysis

S

lideL418


57/125

Parametric Methods

for

Line SpectraPart 1

Lecture 5




58/125

Line Spectra

Many applications have signals with (near) sinusoidal

components. Examples:

communications

radar, sonar

geophysical seismology

ARMA model is apoor approximation

Better approximation byDiscrete/Line Spectrum Models

-

6

6

6

6

2

;

!

1

!

2

!

3

( 2

2

1

)

( 2

2

2

)

( 2

2

3

)

!

( ! )

AnIdealline spectrum




59/125

Line Spectral Signal Model

Signal Model:Sinusoidal components of frequencies

f !

k

g and powers f 2k

g , superimposed in white noise of

power 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, uniformly

distributed on ;

]

(realistic and mathematically convenient)

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

E f e ( t ) e

( s ) g =

2

t s

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

(can be achieved byslowsampling)




60/125

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 f y ( 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




61/125

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 least

squares method from

y ( t) =

n

X

k= 1

k

e

i !

k

t

+ residuals

with k

=

k

e

i '

k .




62/125

Nonlinear Least Squares (NLS) Method

m i n

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

i N !

1

e

i N !

n

3

7

5




63/125

Nonlinear Least Squares (NLS) Method, con't

Then:

F= (

Y ;B

)

( Y ;B ) =

k Y ;B

k

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

!= a r g m a x

!

Y

B ( B

B )

; 1

B

Y




64/125

NLS Properties

Excellent Accuracy:

var(

!

k

) =

6

2

N

3

2

k

( for N 1 )

Example: N= 3 0 0

SNRk

=

2

k

=

2

= 3 0 dB

Thenq

var(

!

k

) 1 0

; 5 .

Difficult Implementation:

The NLS cost function F is multimodal; it is difficult to

avoid convergence to local minima.




65/125

Unwindowed Periodogram as an

Approximate NLS Method

For a single (complex) sinusoid, the maximum of the

unwindowed periodogram is the NLS frequency estimate:

Assume: n= 1

Then: B B = N

B

Y =

N

X

t= 1

y ( t ) e

;i ! t

= Y ( ! ) (finite DTFT)

Y

B ( B

B )

; 1

B

Y =

1

N

j Y ( ! ) j

2

=

p

( ! )

= (Unwindowed Periodogram)

So, withno approximation,

!= a r g m a x

!

p

( ! )




66/125

Unwindowed Periodogram as an

Approximate NLS Method, con't

Assume:n >

1

Then:

f !

k

g

n

k= 1

' the locations of the n largest

peaks of p

( ! )

provided that

i n fj !

k

; !

p

j > 2 = N

which is the periodogram resolution limit.

If better resolution desired then use a High/Super

Resolutionmethod.




67/125

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 )

DegenerateARMA 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

( t ;1 ) +

'

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 )




68/125

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 )

givef !

k

g

n

k= 1

, along withL ; n

spuriousroots.




69/125

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 (ifL > n

) and overdetermined

(ifM > L

) system of YW equations.




70/125

High-Order and Overdetermined YW Equations,

con't

Fact: rank( ) =

n

SVD of : =

U V

U= (

M n ) with U U = In

V

= (n L ) with V V = I

n

= (

n n ) , diagonal and nonsingular

Thus,

( U V

) b = ;

The Minimum-Norm solution is

b = ;

y

= ; V

; 1

U

Important property:The additional ( L ; n ) spurious

zeros ofB ( q ) are located strictlyinsidethe unit circle, if

the Minimum-Norm solution b is used.




71/125

HOYW Equations, Practical Solution

Let =

but made from f r ( k ) g instead off r ( k ) g .

Let U , , V be defined similarly to U , , V from the

SVD of .

Compute b = ; V ; 1 U

Then f !k

g

n

k= 1

are found from the n zeroes of B ( q ) that

are closest to the unit circle.

When the SNR is low, this approach may give spurious

frequency estimates whenL > n

; this is the price paid for

increased accuracy whenL > n

.




72/125

Parametric Methods

for

Line SpectraPart 2

Lecture 6




73/125

The Covariance Matrix Equation

Let:

a ( !) = 1

e

;i !

: : : e

; i ( m ;1 )

!

]

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 ( t ; m+ 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 ( t ; m+ 1 ) ]

T

Then

R

4

= E f ~y ( t) ~

y

( t ) g =A P A

+

2

I

with

P = E f ~x ( t) ~

x

( t ) g =

2

6

4

2

1

0

. . .

0

2

n

3

7

5




74/125

Eigendecomposition of R and Its Properties

R =A P A

+

2

I (m > n

)

Let:

1

2

: : :

m

: eigenvalues of R

f s

1

: : : s

n

g : orthonormal eigenvectors associated

with f 1

: : :

n

g

f g

1

: : : g

m ; n

g : orthonormal eigenvectors associated

with f n

+ 1

: : :

m

g

S=

s

1

: : : s

n

] (m n )

G=

g

1

: : : g

m ; n

] (m ( m ; n

) )

Thus,

R= S G

]

2

6

4

1

. . .

m

3

7

5

"

S

G

#




75/125

Eigendecomposition of R and Its Properties,

con't

As rank(A P A

) =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:

R S=

A P A

S +

2

S = S

2

6

4

1

0

. . .

0

n

3

7

5

S = A (P A

S

; 1

)

4

=A C

with j Cj 6= 0

(since rank( S) = rank( A

) =n ).

Therefore, since S G= 0

,

A

G= 0




76/125

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 theunique solutionsof

a

( ! )G G

a ( !) = 0

.

Let:

R =

1

N

N

X

t = m

~y ( t) ~

y

( t )

S

G =S G made from the

eigenvectors of R




77/125

Spectral and Root MUSIC Methods

Spectral MUSIC Method:

f !

k

g

n

k= 1

= the locations of the n highest peaks of the

pseudo-spectrumfunction:

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

; ( m ;1 )

]

T

Note: Both variants of MUSIC may produce spurious

frequency estimates.




78/125

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




79/125

Min-Norm Method

Goals:Reduce computational burden, and reduce risk of

false frequency estimates.

Uses m n (as in MUSIC), but onlyonevector in

R ( G ) (as in Pisarenko).

Let"

1

g

#

=

the vector in R ( G ) , withfirst element equal

to one, that has minimum Euclidean norm.




80/125

Min-Norm Method, con't

Spectral Min-Norm

f ! g

n

k= 1

= the locations of the n highest peaks in the

pseudo-spectrum

1 =

a

( ! )

"

1

g

#

2

Root Min-Norm

f ! g

n

k= 1

= the angular positions of the n roots of the

polynomial

a

T

( z

; 1

)

"

1

g

#

that are closest to the unit circle.




81/125

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 ( S S ) ; 1

As: I = S S =

+

S

S , ( S S ) ; 1 exists iff

= k k

2

6= 1

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

Multiplying the above equation by gives:

( 1 ; k

k

2

) = (

S

S )

) (

S

S )

; 1

= = ( 1 ; k

k

2

)

) g = ;

S = ( 1 ; k k

2

)




82/125

ESPRIT Method

Let A1

= I

m ; 1

0 ]A

A

2

= 0I

m ; 1

] 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

m ; 1

0 ]S

S

2

= 0I

m ; 1

] S

Recall S = A C with j C j 6= 0 . Then

S

2

= A

2

C = A

1

D C= S

1

C

; 1

D C

| { z }

So has the same eigenvalues as D . is uniquely

determined as

= (

S

1

S

1

)

; 1

S

1

S

2




83/125

ESPRIT Implementation

From the eigendecomposition of

R

,find

S

, then

S

1 and

S

2

.

The frequency estimates are found by:

f !

k

g

n

k= 1

= ;a r g ( ^

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 MUSIC

or MinNorm)

accurate frequency estimates




84/125

SummaryofFre

quencyEstimationMethods

Computational

Accuracy/

RiskforFalse

Method

Burden

Resolution

FreqEstimates

Periodogr

am

small

medium-high

medium

Nonlinear

LS

veryhigh

veryhigh

veryhigh

Yule-Walker

medium

high

medium

Pisarenko

small

low

none

MUSIC

high

high

medium

Min-Norm

medium

high

small

ESPRIT

medium

veryhigh

none

Recommen

dation:

UsePeriodogramforme

dium-resolutionapplications

UseESPRITforhigh-res

olutionapplications


S

lideL613


85/125

Filter Bank Methods

Lecture 7




86/125

Basic Ideas

Two main PSD estimation approaches:

1. Parametric Approach:Parameterize ( ! ) by a

finite-dimensional model.

2. Nonparametric Approach:Implicitly smoothf ( ! ) g

! = ;

by assuming that ( ! ) is nearly

constant 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 compromise

associated with all nonparametric methods.




87/125

FilterBankApproachtoSpectra

lEstimation

-

-

-

-

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)consistentpowercalculation

(b)Idealpassbandfilterwithband

width

(c)

(

)

constanton

2

!

;

2

!

+

2

]

Notethatassumptions(a)and

(b),aswellas(b)a

nd(c),areconflicting.


SlideL73


88/125

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 ~! ( N ; t )

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

i N ( ~! ; ! )

; 1

e

i( ~

! ; ! )

; 1

center frequency ofH ( !

) = ~!

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




89/125

Filter Bank Interpretation of the Periodogram,

con't

j H ( ! ) j as a function of( ~

! ; ! ) , for N= 5 0 .

3 2 1 0 1 2 340

35

30

25

20

15

10

5

0

dB

ANGULAR FREQUENCY

Conclusion:The periodogram p

( ! ) is afilter bank PSD

estimator with bandpassfilter as given above, and:

narrowfilter passband,

power calculation from only1sample offilter output.




90/125

Possible Improvements to the Filter Bank

Approach

1. Split the available sample, and bandpassfilter each

subsample.

more data points for the power calculation stage.

This approach leads to Bartlett and Welch methods.

2. Useseveral bandpassfilters on the whole sample.

Eachfilter covers a small band centered on ~! .

provides several samples for power calculation.

Thismultiwindow approachis similar to the

Daniell method.

Both approachescompromise 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

filtering.




91/125

Capon Method

Idea:Data-dependent bandpassfilter 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 ( t ; m )

3

7

5

| { z }

~y ( t )

E

n

j y

F

( t ) j

2

o

= h

R h R = E f ~y ( t

) ~y

( t ) g

H ( !) =

m

X

k= 0

h

k

e

;i ! k

= h

a ( ! )

where a ( !) = 1 e

;i !

: : : e

;i m !

]

T




92/125


93/125

Capon Properties

m is the user parameter that controls the compromise

between bias and variance:

as m increases, bias decreases and variance

increases.

Capon uses one bandpassfilter only, but it splits theN -data point sample into ( N ; m ) subsequences of

length m with maximum overlap.




94/125

Relation between Capon and Blackman-Tukey

Methods

Consider B T

( ! ) withBartlett window:

B T

( !) =

m

X

k = ; m

m+ 1 ; j

k j

m+ 1

r ( k ) e

;i ! k

=

1

m+ 1

m

X

t= 0

m

X

s= 0

r ( t ; s ) e

;i !

( t ; s )

=

a

( ! )

R a( ! )

m+ 1

R=

r ( i ; j) ]

Then we have

B T

( !) =

a

( ! )

R a( ! )

m+ 1

C

( !) =

m+ 1

a

( ! )

R

; 1

a ( ! )




95/125

Relation between Capon and AR Methods

Let

ARk

( !) =

2

k

j

A

k

( ! ) j

2

be the k th order AR PSD estimate ofy ( t ) .

Then

C

( !) =

1

1

m+ 1

m

X

k= 0

1 =

ARk

( ! )

Consequences:

Due to the average over k , C

( ! ) generally hasless

statistical variabilitythan the AR PSD estimator.

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

C

( ! ) generally hasworse resolution and bias

propertiesthan the AR method.




96/125

Spatial MethodsPart 1

Lecture 8




97/125

The Spatial Spectral Estimation Problem

S o u r c e n

S o u r c e 2

S o u r c e 1

B

B

B

B

BN

@

@

@

@R

v

v

v

S e n s o r 1

@

@

;

;

@

@

;

;

@

@

;

;

S e n s o r 2

S e n s o r m

c

c

c

c

c

c

c

c

c

c

c

c

,

,

,

,

,

,

,

,

,

,

,

,

Problem:Detectandlocate 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 namespatial spectral analysis)




98/125

Simplifying Assumptions

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

sensors

Non-dispersive wave propagation

Hence:The waves are planar and the only location

parameter isdirection 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 withknown

transfer characteristicsandknown locations

(That is, the array iscalibrated.)




99/125

Array Model Single Emitter Case

x ( t) =

the signal waveform as measured at a reference

point (e.g., at thefirstsensor)

k

= the delay between the reference point and the

k th sensor

h

k

( t) =

the impulse response (weighting function) of

sensork

e

k

( t) = noiseat the k th sensor (e.g., thermal noise in

sensor electronics; background noise, etc.)

Note: t2 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 thetime delays f k

g with hk

( t )

known but x ( t ) unknown.

This is atime-delay estimation problemin the unknown

input case.




100/125

Narrowband Assumption

Assume: The emitted signals are narrowband with knowncarrier frequency !

c

.

Then: x ( t ) = ( t ) c o s !c

t + ' ( t) ]

where ( t ) '

( t ) varyslowly enoughso that

( t ;

k

) ' ( t ) '

( t ;

k

) ' ' ( t )

Time delay is now ' to aphase shift !c

k

:

x ( t ;

k

) ' ( t) c o s

!

c

t + ' ( t ) ; !

c

k

]

h

k

( t ) x ( t ;

k

)

' jH

k

( !

c

) j ( t) c o s

!

c

t + ' ( t ) ; !

c

k

+ a r g f H

k

( !

c

) g ]

where Hk

( !) = F f

h

k

( t ) g is the k th sensor's transfer

function

Hence, the k th sensor output is

y

k

( t) =

j H

k

( !

c

) j ( t )

c o s

!

c

t + ' ( t ) ; !

c

k

+ a r g H

k

( !

c

) ] + e

k

( t )




101/125

Complex Signal Representation

The noise-free output has the form:

z ( t) =

( t) c o s

!

c

t + ( t) ] =

=

( t )

2

n

e

i !

c

t + ( t) ]

+ e

; i !

c

t + ( t) ]

o

Demodulate z ( t ) (translate to baseband):

2 z ( t ) e

; !

c

t

= ( t ) f e

i ( t )

| { z }

lowpass

+ e

; i 2

!

c

t + ( t) ]

| { z }

highpass

g

Lowpassfilter 2 z ( t ) e ; i ! c t to obtain ( t ) e i ( t )

Hence, by low-passfiltering and sampling the signal

~y

k

( t ) =2 =

y

k

( t ) e

;i !

c

t

= y

k

( t) c o s (

!

c

t ) ; i y

k

( t) s i n (

!

c

t )

we get thecomplex representation: (for t2 Z )

y

k

( t) =

( t ) e

i ' ( t )

| { z }

s ( t )

j H

k

( !

c

) j e

i a r g 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 thecomplex envelopeofx ( t ) .




102/125

Vector Representation for a Narrowband Source

Let

= the emitter DOA

m = 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 f k

g and f Hk

( !

c

) g .

Foromnidirectionalsensors thef H

k

( !

c

) g

do not dependon .




103/125


104/125

Multiple Emitter Case

Given n emitters with

received signals: f sk

( 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, thearray equationis:

y ( t) = A s

( t) +

e ( t )

Use theplanar waveassumption tofind the dependence of

k

on .




105/125

Uniform Linear Arrays

S o u r c e

-

+

?

]

S e n s o r

m4321

d s i n

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 )

ds i n

c

where c = wave propagation speed




106/125

Spatial Frequency

Let:

!

s

4

= !

c

ds i n

c

= 2

ds i n

c = f

c

= 2

ds i n

=c = f

c

= signal wavelength

a ( ) = 1 e

;i !

s

: : : e

; i ( m ;1 )

!

s

]

T

By direct analogy with the vector a ( ! ) made from

uniform samples of asinusoidal time series,

!

s

= spatial frequency

The function ! s 7! a ( ) isone-to-onefor

j !

s

j $

d js i n

j

=2

1 d =

2

As

d = spatial sampling period

d =

2 is aspatialShannon sampling theorem.




107/125

Spatial MethodsPart 2

Lecture 9




108/125

Spatial Filtering

Spatialfiltering useful for

DOA discrimination (similar to frequency

discrimination of time-seriesfiltering)

Nonparametric DOA estimation

There is a strong analogy between temporalfiltering and

spatialfiltering.




109/125

Analogy between Temporal and Spatial Filtering

Temporal FIR Filter:

y

F

( t) =

m ; 1

X

k= 0

h

k

u ( t ; k) =

h

y ( t )

h=

h

o

: : : h

m ; 1

]

y ( t) =

u ( t ) : : : u ( t ; m+ 1 ) ]

T

Ifu ( t) =

e

i ! t then

y

F

( t) =

h

a ( !) ]

| { z }

filter transfer function

u ( t )

a ( !) = 1 e

;i !

: : : e

; i ( m ;1 )

!

]

T

We can select h to enhance or attenuate signals with

different frequencies ! .




110/125


Spatial Filter:

f y

k

( t ) g

m

k= 1

= thespatial samplesobtained with a

sensor 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 with

complex envelope s ( t ) is:

y ( t) =

a ( ) s ( t )

a ( ) = 1 e

;i !

c

2

: : : e

;i !

c

m

]

T

The correspondingfilter output is

y

F

( t) =

h

a ( ) ]

| { z }

filter transfer function

s ( t )

We can select h to enhance or attenuate signals coming

from different DOAs.




111/125


( T e m p o r a l s a m p l i n g )

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 ( m ; 1 ) !

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

m ; 1

h

a ( ! ) ] u ( t )

(a) Temporalfilter

Z

Z

Z

Z

Z~

n a r r o w b a n d s o u r c e w i t h D O A =

( S p a t i a l s a m p l i n g )

P

z

-

q

q

q

?

?

?

@

@

@

@

@R

-

-

-

-

-

!

!

a

a

-

!

!

a

a

-

!

!

a

a

q

q

q

r e f e r e n c e p o i n t

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

m ; 1

h

a ( ) ] s ( t )

(b) Spatialfilter




112/125

Spatial Filtering, con't

Example: The response magnitude j h a ( ) j of a spatial

filter (or beamformer) for a 10-element ULA. Here,

h = a (

0

)

, where

0

= 2 5

2 4 6 8 100

Magnitude

Theta(deg)

90

60

30

0

30

60

90




113/125

Spatial Filtering Uses

Spatial Filters can be used

To pass the signal of interest only, hence filtering out

interferences located outside thefilter's beam (but

possibly having the same temporal characteristics as

the signal).

To locate an emitter in thefield of view, by sweeping

thefilter through the DOA range of interest

(goniometer).




114/125

Nonparametric Spatial Methods

AFilter Bank Approachto DOA estimation.

Basic Ideas

Design afilter h ( ) such that for each

It passes undistorted the signal with DOA =

It attenuates all DOAs 6=

Sweep thefilter through the DOA range of interest,

and evaluate the powers of thefiltered signals:

E

n

j y

F

( t ) j

2

o

= E

n

j h

( ) y ( t ) j

2

o

= h

( )R h

( )

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

The (dominant) peaks ofh ( )R h

( ) give the

DOAs of the sources.




115/125

Beamforming Method

Assume the array output isspatially white:

R = E f y ( t ) y

( t ) g = I

Then: En

j y

F

( t ) j

2

o

= h

h

Hence:In direct analogy with the temporally whiteassumption forfilter bank methods, y ( t ) can be

considered as impinging on the array fromallDOAs.

Filter Design:

m i n

h

( h

h ) subject to h a ( ) = 1

Solution:

h = a ( )= a

( ) a ( ) =

a ( )= m

E

n

j y

F

( t ) j

2

o

= a

( )R a

( )= m

2




116/125

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 then

largest peaks ofa

( )

R a( ) .

This is the direct spatial analog of the Blackman-Tukey

periodogram.

Resolution Threshold:

i n fj

k

;

p

j >

wavelength

array length

= array beamwidth

Inconsistency problem:Beamforming DOA estimates are consistent if n

= 1 , but

inconsistent ifn >

1 .




117/125


118/125

Parametric Methods

Assumptions:

The array is described by the equation:

y ( t) = A s

( t) +

e ( t )

The noise is spatially white and has the same power in

all sensors:

E f e ( t ) e

( t ) g =

2

I

The signal covariance matrix

P = E f s ( t ) s

( t ) g

is nonsingular.

Then:

R = E f y ( t ) y

( t ) g =A P A

+

2

I

Thus: The NLS, YW, MUSIC, MIN-NORM and ESPRIT

methods of frequency estimation can be used, almost

without modification, for DOA estimation.




119/125

Nonlinear Least Squares Method

m i n

f

k

g f s ( t ) g

1

N

N

X

t= 1

k y ( t ) ;A s

( 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 ( A A ) ; 1 A ] R g

Thus, f k

g= a r g m a x

f

k

g

trf A ( A A ) ; 1 A ] R g

For N= 1 , this is precisely the form of the NLS method

of frequency estimation.




120/125

Nonlinear Least Squares Method

Properties of NLS:

Performance: high

Computational complexity: high

Main drawback: need for multidimensional search.




121/125

Yule-Walker Method

y ( t) =

"

y ( t )

~y ( t )

#

=

"

A

~

A

#

s ( t) +

"

e ( t )

~ e ( t )

#

Assume: E f e ( t) ~

e

( t ) g= 0

Then:;

4

= E f y ( t) ~

y

( t ) g =

A P

~

A

( M L )

Also

basic deﬁnitions and spectral estimation

Documents