Gird Systems Intro to Music Esprit

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GIRD Systems Inc. http://www.girdsystems.comEngineering Research and Development

An Introduction to MUSIC and ESPRIT

GIRD Systems, Inc.310 Terrace Ave.

Cincinnati, Ohio 45220

Based on

R. O. Schmidt, Multiple emitter location and signal parameter estimation,

IEEE Trans. Antennas & Propagation, vol. 34, no. 3, March 1986, and

R. Roy and T. Kailath, ESPRIT Estimation of signal parameters via rotation invariancetechniques,IEEE Trans. Acoust., Speech, Signal Proc., vol. 17, no. 7, July 1989


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Introduction

Well known high-resolution DOA algorithms

Able to find DOAs of multiple sources

High spatial resolution compared with other alg.(I.e., a few antennas can result in high accuracy)

MUSIC stands for Multiple Signal Classifier

ESPRIT stands for Estimation ofSignal

Parameters via Rotational Invariance Technique

Apply to only narrowband signal sources


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Narrowband Signal Sources

A complex sinusoid

( ) j j t j t s t e e e

A real sinusoid is a sum of two sinusoids

1 2cos( )

2 2

j j t j j t j t j t t e e e e e e

A delay of a sinusoid is a phase shift

0 0

0( ) ( ) j t j t j ts t t e e e s t

Apply approximately to narrowband signals


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Narrowband Signal Sources

ConsiderInarrowband signal sources

1 2

1 1 2 2( ) , ( ) , , ( )I j t j t j t

I Is t e s t e s t e

Assume that all frequencies are different

Assume that all amplitudes are uncorrelated

2;

{ }0;

i

i j

i jE

i j


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A Uniform Linear Array

1( ) ( )x t s t If the received signal at sensor 1 is

Then the received signal at sensor i is( 1) sin

1( ) ( ) ( ) ( )i i

i dj

j j c

ix t e s t e s t e s t

Then it is delayed at sensor i by ( 1) sini i dc

A signal source

impinges on the array

with an angle

c: propagation speed

( ) j ts t e


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Signal Model

Put received signals at all N sensors together:

is called a steering vector

sin1

22 sin

3

( 1) sin

1

( )

( )

( ) ( ) ( ) ( ) ( )

( )

dj

c

dj

c

N dN jc

x t

ex t

t x t s t s t e

x t

e

x a

( )a


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Signal Model

If there areIsources signals received by the array, we geta signal model:

x(t) --- received signal vector (Nby 1 )

s(t) --- source signal vector (Iby 1 )

n(t) --- noise vector (Nby 1 )

(NbyI)1( ) [ ( ), , ( )]

T

It s t s t s

Sources are independent, noises are uncorrelated

Column ofA can also be normalized


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The MUSIC Algorithm

Compute theNxNcorrelation matrix

sR

2 2

1{ ( ) ( )} .{ , , }H

I E t t diag sR s s

2

0{ ( ) ( )}H H

E t t x sR x x AR A I

where

If the sources are somewhat correlated so is not

diagonal, it will still work if has full rank.

If the sources are correlated such that is rank deficient,

then it is a problem. A common solution is spatialsmoothing.

Q: Why is the rank of (beingI) so important?

A: It defines the dimension of the signal subspace.

sR

sR

sR


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The MUSIC Algorithm

ForN>I, the matrix is singular, i.e.,HsAR A

2 2

0[ ] ; 1,2, ,H

i i i ii N x sR u AR A I u u

But this implies that is an eigenvalue of

Since the dimension of the null space of isN-I, there areN-Isuch eigenvalues of

Since both and are non-negative definite,

there areIother eigenvalues such that Let be the ith eigenvector of corresponding to

2 2

00

i

2

0

2

0

2

i

2

0det[ ] det[ ] 0H s xAR A R I

xR

HsAR A

H

sAR AxR

iu xR

2

i

xR

2 2 2 2

0 00, 1, , ; , 1, ,i ii I i I N


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The MUSIC Algorithm

This implies

Partition theN-dimensional vector space into thesignal subspace and the noise subspacesU nU

2 2

0[ ] ; 1,2, ,H

i i i ii N x sR u AR A I u u

2 2

0( ) ; 1,2, ,H

i i ii N sAR A u u

2 2

0( ) ; 1,2, ,

0; 1, ,

H i i

i

i I

i I N

s

uAR A u

2 2

0

1 1

:0 eigenvalues: ( ) 0 eigenvalues

[ ]

i

I I N

ns

s n

UU

U U u u u u


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The MUSIC Algorithm

The steering vector is in the signal subspace

Signal subspace is orthogonal to noise subspace

sU

nU

( )i

a

(1) meansIlinear combinations of columns ofAequal the signal subspace spanned by columns of

(2) means the linear combinations of columns ofA,i.e., the signal subspace, is orthogonal to

2 2

0

( ) ; 1,2, , (1)

0; 1, , (2)

Hi ii

i I

i I N

s

uAR A u


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The MUSIC Algorithm

The steering vector is in the signal subspace

Signal subspace is orthogonal to noise subspace

This implies that

So the MUSIC algorithm searches through allangles , and plots the spatial spectrum

( )Hi

na U 0

( )i

a

1( )

( )H

P

n

a U

Wherever exhibits a peak

Peak detection will give spatial angles of allincident sources

, ( )i P


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The MUSIC Algorithm

MUSIC spatial spectrum compared with other methods


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Pros/Cons of The MUSIC Algorithm

Works for other array shapes, need to know sensorpositions

Very sensitive to sensor position, gain, and phaseerrors, need careful calibration to make it work well

Searching through all could be computationallyexpensive

The ESPRIT algorithm overcomes such

shortcomings to some degree ESPRIT relaxes the calibration task somewhat

ESPRIT takes much less computation

But ESPRIT takes twice as many sensors


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The ESPRIT Algorithm

Based on doublets of sensors, i.e., in each pair of

sensors the two should be identical, and all doubles

should line up completely in the same direction with a

displacement vector having magnitude Otherwise there are no restrictions, the sensor patterns

could be very different from one pair to another

The positions of the doublets are also arbitrary

This makes calibration a little easier

AssumeNsets of doublets, i.e., 2Nsensors

AssumeIsources,N>I


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A sensor array of doublets

The amount of delay

between two sensors in

each doublet for a

given incident signal isthe same for all

doublets, which is

sin /c


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This array is consisted of two identical subarrays,and , displaced from each other by

xZ

yZ

1

( ) ( ) ( ) ( ) ( ) ( )I

i i

i

t s t t t t

x xx a n As n

1

( ) ( ) ( ) ( ) ( ) ( )iI

j

i i

i

t e s t t t t

x xy a n As n

1 2.{ , , , }I j j jdiag e e e 0 sin / i i c

The steering vector depends on the array

geometry, and should be known just like in MUSIC

( )a


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The objective is to estimate , thereby obtaining

Define

i

( )( )( ) ( ) ( ) ( )

( )( )

ttt t t t

tt

x

z

y

nx Az s As n

ny A Compute the 2Nx 2Ncorrelation matrix

2

0{ ( ) ( )}H H

E t t z sR z z AR A I

Since there areIsources, theIeigenvectors of

corresponding to theIlargest eigenvalues form the

signal subspace ; The remaining 2N-I

eigenvectors form the noise subspace

zR

sU

nU


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is 2NxI, and its span is the same as the span of

Therefore, there exists a unique nonsingularIxI

matrix such that (A needs to be known here)

A

Partition into twoNxIsubmatrices

The columns of both and are linear

combinations of , so each of them has a column

rankI

sU

T

sU AT

sU

x

sy

U AT

U U AT

Ax

U yU

G S


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Define anNx 2Imatrix, which has rankI

Therefore, has a null space with dimensionI,

i.e., there exists a 2IxImatrix F such that

Then the above gives, since T has full column rank

xy x yU U U

x

xy x y x x y y

y

x y y x

FU F 0 U U U F U F 0

F

ATF ATF 0 ATF ATF

xyU

1 1

1 1 1

x y x y x yAT ATF F A ATF F T TF F T

GIRD S I


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The final algorithm is

In practice, the measurement could be noisy, and

there could be array calibration errors, so a total least

squares ESPRIT is used

The estimation is one shot, even with matrix

inversions the computation is much less than theMUSIC search

We must haveN>Ifor ESPRIT to work with 2N

sensors, so need twice as many sensors as MUSIC

1 1x y TF F T

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Simulation results of ESPRIT comparing with MUSIC

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Conclusions

Both methods are high-resolution, much better spatial

resolution than beamforming and other methods

Both methods are able to detect multiple sources

If sensors are expensive and few, and if computationis not of concern, MUSIC is suitable

If there are plenty of sensors compared with the

number of sources to detect, and if computationalpower is limited, ESPRIT is suitable

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About GIRD Systems, Inc.

Founded in 2000 - based in Cincinnati

Specializes in communications and

signal processing, especially

developing novel algorithms to solvechallenging problems

As of Nov. 2009, won more than 13

Phase I awards and 6 Phase II awards

from Navy, Air Force, Army Partnerships with many large

contractors including Northrop

Grumman, L-3 Communications, etc.

GIRD S t I


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About GIRD Systems, Inc.

Key Technology Areas

Interference Mitigation (no reference, in-band)

Direction Finding (wideband, high-resolution)

Location/Navigation (Assisted GPS, GPS denied,

signals of opportunity)

Wireless Network Security (physical layer)

Power Amplifier LinearizationNovel Communications systems/modeling

Contact Information: GIRD Systems, Inc. Phone: (513) 281-2900310 Terrace Ave. Fax: (513) 281-3494

Cincinnati, OH 45220 E-mail: [email protected] Web: www.girdsystems.com
mailto:[email protected]:[email protected]

Gird Systems Intro to Music Esprit

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