An Introduction to MUSIC and ESPRIT - GIRD Systems

http://www.girdsystems.com

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 invariance techniques,” IEEE Trans. Acoust., Speech, Signal Proc., vol. 17, no. 7, July 1989


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 of Signal

Parameters via Rotational Invariance Technique

• Apply to only narrowband signal sources


Narrowband Signal Sources

• A complex sinusoid

( ) j j t j ts 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 tt e e e e e e

• A delay of a sinusoid is a phase shift

0 0

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

• Apply approximately to narrowband signals


Narrowband Signal Sources

• Consider I narrowband signal sources

1 2

1 1 2 2( ) , ( ) , , ( ) Ij 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


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 cix t e s t e s t e s t

• Then it is delayed at sensor i by ( 1) sin

i

i d

c

A signal source

“impinges” on the array

with an angle

c: propagation speed

( ) j ts t e


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 tex t

t x t s t s te

x t

e

x a

( )a


Signal Model

• If there are I sources signals received by the array, we get

a “signal model”:

x(t) --- received signal vector ( N by 1 )

s(t) --- source signal vector ( I by 1 )

n(t) --- noise vector ( N by 1 )

( N by I )

1( ) [ ( ), , ( )]T

It s t s ts

• Sources are independent, noises are uncorrelated

• Column of A can also be normalized


The MUSIC Algorithm

• Compute the N x N correlation matrix

sR

2 2

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

IE t t diag sR s s

2

0{ ( ) ( )}H HE 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 “spatial

smoothing”.

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

A: It defines the dimension of the signal subspace.

sR

sR

sR


The MUSIC Algorithm

• For N > I, the matrix is singular, i.e., H

sAR A

2 2

0[ ] ; 1,2, ,H

i i i i i 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 is N-

I, there are N-I such eigenvalues of

• Since both and are non-negative definite,

there are I other eigenvalues such that

• Let be the ith eigenvector of corresponding to

2 2

0 0i

2

0

2

0

2

i

2

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

xR

H

sAR A

H

sAR AxR

iu xR2

i

xR

2 2 2 2

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


The MUSIC Algorithm

• This implies

• Partition the N-dimensional vector space into the

signal subspace and the noise subspace sU nU

2 2

0[ ] ; 1,2, ,H

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

2 2

0( ) ; 1,2, ,H

i i i i 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


The MUSIC Algorithm

• The steering vector is in the signal subspace

• Signal subspace is orthogonal to noise subspace

sU

nU

( )ia

(1) means I linear combinations of columns of A equal the signal subspace spanned by columns of

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

2 2

0( ) ; 1,2, , (1)

0; 1, , (2)

H i i

i

i I

i I N

s

uAR A u


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 all angles , and plots the “spatial spectrum”

( )H

i na U 0

( )ia

1( )

( )HP

na U

• Wherever exhibits a peak

• Peak detection will give spatial angles of all incident sources

, ( )i P


The MUSIC Algorithm

MUSIC spatial spectrum compared with other methods


Pros/Cons of The MUSIC Algorithm

• Works for other array shapes, need to know sensor positions

• Very sensitive to sensor position, gain, and phase errors, need careful calibration to make it work well

• Searching through all could be computationally expensive

• 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


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

• Assume N sets of doublets, i.e., 2N sensors

• Assume I sources, N > I

Δ



A sensor array of doublets

The amount of delay

between two sensors in

each doublet for a

given incident signal is

the same for all

doublets, which is

sin / c



• This array is consisted of two identical subarrays,

and , displaced from each other by

xZ

yZ Δ

1

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

i i

i

t s t t t t

x xx a n As n

1

( ) ( ) ( ) ( ) ( ) ( )i

Ij

i i

i

t e s t t t t

x xy a n AΦs n

1 2.{ , , , }Ij j jdiag e e e Φ0 sin /i i c

• The steering vector depends on the array

geometry, and should be known just like in MUSIC

( )a



• The objective is to estimate , thereby obtaining

• Define

iΦ

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

( )( )

ttt t t t

tt

x

z

y

nx Az s As n

ny AΦ

• Compute the 2N x 2N correlation matrix

2

0{ ( ) ( )}H HE t t z sR z z AR A I

• Since there are I sources, the I eigenvectors of

corresponding to the I largest eigenvalues form the

signal subspace ; The remaining 2N-I

eigenvectors form the noise subspace

zR

sU

nU



• is 2N x I, and its span is the same as the span of

• Therefore, there exists a unique nonsingular I x I

matrix such that (A needs to be known here)

A

• Partition into two N x I submatrices

• The columns of both and are linear

combinations of , so each of them has a column

rank I

sU

T

sU AT

sU

x

s

y

U ATU

U AΦT

AxU yU



• Define an N x 2I matrix, which has rank I

• Therefore, has a null space with dimension I,

i.e., there exists a 2I x I matrix 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 AΦTF 0 AΦTF ATF

xyU

1 1 1 1 1

x y x y x yAΦT ATF F AΦ ATF F T Φ TF F T



• 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 the

MUSIC search

• We must have N > I for ESPRIT to work with 2N

sensors, so need twice as many sensors as MUSIC

1 1

x yΦ TF F T



Simulation results of ESPRIT comparing with MUSIC


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 computation

is not of concern, MUSIC is suitable

• If there are plenty of sensors compared with the

number of sources to detect, and if computational

power is limited, ESPRIT is suitable


About GIRD Systems, Inc.

• Founded in 2000 - based in Cincinnati

• Specializes in communications and

signal processing, especially

developing novel algorithms to solve

challenging 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.


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 Linearization

– Novel Communications systems/modeling

• Contact Information: GIRD Systems, Inc. Phone: (513) 281-2900

310 Terrace Ave. Fax: (513) 281-3494

Cincinnati, OH 45220 E-mail: [email protected]

USA Web: www.girdsystems.com

mailto:[email protected]

An Introduction to MUSIC and ESPRIT - GIRD Systems

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