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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
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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 of Signal
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 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
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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
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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 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
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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 tex t
t x t s t s te
x t
e
x a
( )a
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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
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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
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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
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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
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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
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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 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
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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 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
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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
• Assume N sets of doublets, i.e., 2N sensors
• Assume I sources, N > I
Δ
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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 is
the 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 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
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The ESPRIT Algorithm
• 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
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The ESPRIT Algorithm
• 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
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The ESPRIT Algorithm
• 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
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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 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
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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 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
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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 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.
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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 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