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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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The ESPRIT Algorithm
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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The ESPRIT Algorithm
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 ESPRIT Algorithm
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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The ESPRIT Algorithm
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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The ESPRIT Algorithm
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 ESPRIT Algorithm
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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The ESPRIT Algorithm
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
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