1992 04 application od uniform linear array bearing estimation techniques to uniform circular...
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I008 I E E E T RANS ACT I ONS ON S I G N A L PROCESSING, V O L . 40. NO. 4. A P R IL 1992
171 C . S . Burms nd T. W . Parks,
D F T I F F T a n d C o n t d u t i o n
AIXoriihrn.7.
New
York: Wiley, 1985.
[8]
D.
M .
W .
Evans,
A
second im proved digit-reversal permutation
al-
gorithm for fast transforms. €€€
Trans . A c o u s r . . Speech ,
Signcl/
Processing, vol . 37, pp. 1288-1291, 1989.
On
the Application of Uniform Linear Array Bearing
Estimation Techniques to Uniform Circular Arrays
A . H. Tewfik and W. Hong
Abstract-The proble m of estimatin g the directions of arrivals of
narrow-band plane waves impinging on a uniform circular array with
M
identical sensors uniformly distributed around a circle is consid-
ered. It
is
shown that if the numb er of sensors
M
is large enough then
a reordering of the inverse discrete Fourier transform of the sensor
outputs yields a sequence of measurements
z q)
=
ZA A , Jq (2n R
in $ / I )
exp
( - O n ) ,
where 0, and
q5
are the azimu th and elevation angles of
arrival of the nth plane wave, respectively. For ea ch candidate eleva-
tion angle
6
this sequence
is
processed using
ROOT
MUSIC or any
other modern l ine spectral estimation technique as if i t came from
a
uniform linear array. Any root estimated via
ROOT
MUSIC which is
on or close to the unit circle then indicates the presence of a source at
the elevation under consideration and an azimuth equal to the phase
of the root. Experimen tal results are provided to demonstrate the ad-
vantages of processing the transformed data.
I . I NTRODUCTI ON
Circular ar rays are known to be equivalent to nonuniform l inear
ar rays
[ l ] .
Unfor tunately , some of the modem ar ray processing
a l g o ri t h m s ( e .g . , R OOT MUS I C
[ 2 ] )
an only be applied to uni-
form linear arrays. Furthermore, other algorithms have been ob-
served to perform be tter when applied to uniform linea r arrays than
when applied to circular arrays (e.g., in terms of signal-to-noise
ratio resolution thresholds)
[3].
In this correspondence we show that i t is possible to use ROOT
MUSIC and other procedures which were designed to work wi th
l inear ar rays to determine the azimuths and elevat ions of
N
sources
using a circular array which consists of
M ( M
> N ) dentical sen-
s rs
uniformly distributed around a c ircle of radius
R
( F i g .
I ) .
Ou r
proof is based on an expansion of a plane w ave in an infinite series
of Bessel functions of the first kind. The series may be considered
to be the discrete time Fourier transform of a discrete time signal.
The sensors provide samples of that Fourier transform at the fre-
quencies 27rm/M, m = 0 , M 1. By computing the inverse dis-
crete Fourier transform of the sensor outputs we obtain a tempo-
rally aliased version of the discrete time signal. We show that if
the number of sensors
M
is larger than
4 7 r R / h
where
h
the wave-
length of the plane wave, then a reordering
of
the inverse discrete
Fourier transform of the sensor outputs yields a sequence of mea-
surements z ( q )
=
E
A,,Jq(2*R
sin
I ~ / A )
ex p
( - j q O , ) ,
where
O, ,
and
O,, are the azimuth and elevation angles of arrival of the nth plane
wave and A , is a complex constant proportional to the nth plane
wave magni tude and phase and to the sensor response. To f ind 0,,
and
d,,
a search is conducted o ver possib le e levat ions I n . In partic-
z
axis
y ax1\
Fig. 1 . Circular array
ular, for each candidate e levat ion angle I ,l the sequence z ( q ) is
processed using RO OT M USIC o r any other modem l ine spect ral
estimation technique as if i t came from a uniform linea r array. Any
root est imated via R OOT MUS IC which is on o r c lose to the uni t
circle then indic ates the presence o f a source at the elevation un der
consideration and an azimuth equal to the phase of the root.
11.
T R A N S F O R M A T I O N
F
P L A N EW A V E S
Consider f irst the case where a single narrow-band plane wave
of wavelength h is impinging in the absence of noise on the array
at an azimuth 8 and an elevat ion I as shown in F i g . 1. Using the
comple x notation for narrow-band signals
[ 4 ]
we may write the
output of sensor m as
x ( m , r =
A ( r )
exp
j ~
sin
I
cos
i2: e ) ) 1)
i
2:R
where the complex parameter A ( r ) denotes the complex ampl i tude
of the plane wave at t ime
t
measured at the center of the array.
Using the addition theore m for Bessel function [ 5 ]we can rewrite
1 )
as
x ( m ,
t )
= A ( ? ) = - r n Jk Fin
0
exp j k ,
-
F 2)
where J k ( . denotes a Bessel function of the first type and order k
[5] and 8 = 7r /2 . Observe that according to ( 2 ) ,x ( m , ) may
be interpreted as providing the value of a sample of the Fourier
transform of the sequence y ( k ) =
A( r) Jk ( 27rR
in ,,/A) ex p ( - j k O )
at the frequency 2 7 r k / M . Using this interpretation w e find that the
inverse discrete Fourier transform of x ( m , r , x q ,
t )
0
5
q 5 M
1 ,
is given by
[ 6 ]
Manuscript received May 9, 1990; revised February 15. 1991.
The authors are with the Department of Electrical Engineering, Univer-
IEEE
Log
Number 9
1060
2 ,
sity
of Minnesota, Minneapolis,
M N
55455.
The above equation may also be found in
[ 7 ] .
If the number of sensors M is much larger than
4 7 r R / h ,
then
J u + k M ( 2 7 r R
in
I /h)= 0
for
1 5 k
and
k 5 - 2 [ 5 ] .
Under those
1053-587X/92$03.00 992 IEEE
m
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IEEE T R A N S A C T I O N S O N S I G N A L P R O C E S S I N G . VOL. 40 N O .
4.
A P R I L 1992
I009
condi t ions (3) may be rewrit ten as
( 4 )
where we have used the fact that
J -
.
)
=
-
.
[ 5 ] .
Note
that if M
q
i s l a rge enough then J M - , ( 2 a R s in 4 / X )
=
0 and
x q ,
t ) = A( t ) J , ( 2nR s in d/X) e xp - j q 0 ) . On the other hand, if
q
is large enough then
J q ( 2 n R
sin
4/A) = 0
and x q , t )
=
A ( t ) ( - 1 ) M - 4 J M - y ( 2 ~ Rin / A ) e x p - j q M ) @ . Hence , by
properly reordering the sequence x q , r we can obta in another se -
quenc e that is proportional to A( t ) J , ( 2aR s in 4/A) e xp
- j q 0 ) .
In
part icular, using the fact that M i s much la rger than 4 a R / X we can
define a new sequence z ( q , t ) as fol lows:
where L M / 2 J denotes the largest integer that is smaller than o r
equa l to M / 2 .
If the array response is l inear then
5)
implies that the trans-
formed array outputs
z q ,
t ) corresponding to N narrow-band plane
waves of wavelength X impinging on the array with angles
8,, 1
n
5 N
is of the form
Equation (6) with 4 = a / 2 ( i . e . , for the case where al l the
sources and the array are coplanar) appears already in [ 8 ] .H o w -
ever , the deriva t ion of [ 8 ] s incorrect as i t claim s that (6) (wi th 4,,
= a / 2 ) may be obta ined by comput ing a di sc re te Fourie r t rans-
form of the sensor outputs . As ment ioned above , a discre te Fourie r
transform of the sensor outputs wil l in fact yield (3) and not (6) .
Note that since
J ,(x ) =
0 whe n q
>>
2 x , ( 6 ) also implies that
the effective number
of
sources that may be resolved by a circular
array at a fixed elevation is a function of the radius R of the array
and the elevation angle. In part icular, that number may be smaller
than M , the number of sensors. The effect of this implication on
the procedure presented here and other high resolution bearing es-
t imat ion techniques when app l ied to uni form c i rcula r a rrays i s cur-
rently under investigation.
Now observe tha t z q, t ) s the sum of exponent ia l s that a re mod-
ulated by the sequence o f A , ( t ) J q ( 2 a R sin &/A ) . To determine the
elevation and azimuth angles of the sources we use a
ROOT
M U -
SIC
based approach as fol lows. W e begin by di sc re t iz ing the range
of possible elevation angles
q5
and est imating a matrix E , whose
columns form an orthogonal bas i s for the noise-only subspace
[ 2 ] . As in other e igendecomposi t ion techniques , E, may be es t i -
mated either by using an eigendecomposit ion of the transformed
data correlat ion matrix when the background noise is white or a
generalized eigendecomposit ion of that matrix and the correlat ion
matrix
of
the transformed noise process when the noise is colored
wi th a known corre la t ion mat r ix. F or each candida te e leva t ion 6
we compute the roots of the equa t ion:
Z H D H ( 4 ) E , E D ( 4 ) Z=
0
(7)
where z is an M X 1 vector given by
=
[1
z ?
.
.
.
Z M - l ~ 7
and D ( 4 ) s the diagonal M x M matrix
A
root close to the unit circle would then indicate the presence of
a source at the elevation currently under consideration and an azi-
muth equa l to the phase of the root .
111.
TR A N S FO R M A TIO N
F A BA C K G R O U N DOISE
Consider now the case where the a rray opera tes in the presence
of a background noise process only and observ e that the array sam-
ples the noise process at the points
7,
=
( R , 27rm/M, 7 r /2 ) ,
0
5
m 5 M 1. Denote by K,( m, m )
= E { x ( m , t ) x * ( m ' ,
t ) } the
autocorrelat ion of the sensor outputs
x ( m ,
) . n genera l , K , ( m , m ' )
will not be a function of
m m ' ,
i .e . , the process
x ( m , t )
will not
be stat ionary in the m variable, even if the background noise field
is spatial ly homogeneous. However, if the underlying noise pro-
cess is isotropic, isotropic and homog eneou s, or cylindrically sym -
metric [ 9 ] hen
x ( m ,
t ) will be stat ionary in the m variable. Specif-
ically, if the underlying field is isotropic and stat ionary then
K,(m,
m ) =
K(I
?
? A I
=
K(2R
sin
( a ( m m ' ) / M )
( 8 )
where K ( r ) is the rotat ionally invariant correlat ion of the underly-
ing isotropic and homogeneous background noise field. In part ic-
ular, K,( m, m ) = 0 2 6 ( m m ' ) when the background noise field
is spatial ly white and homogeneous with a variance of
u 2 .
The covariance of the inverse di sc re te Fourie r t ransform x q , t )
of
x ( m , t )
can be di rec t ly computed from tha t of x ( m ,
t ) .
If we
denote by
R k
and
R ,
the correlat ion matrices of the sensor outputs
and the inverse di sc re te Fourie r t ransform of the sensor outputs in
the presence of the noise field only respectively, we can relate
Rk
and R , through the equation
R,
= UR,
U H
( 9 )
where denotes U he complex conjuga te t ranspose of
U
and U =
[U,,] is the M x M discrete Fourier transform matrix with
U,, =
1
exp j
T .
Now observe tha t s ince K,( m, m ) = K ( 2 R s in ( a( m m ' ) / M )
for a homogen eous and i sot ropic backgroun d noise f i e ld, the mat r ix
Rk is circulant [ l o ] when the background noise f i e ld i s homoge-
neous and isotropic. Since a circulant matrix is diagonalized by the
discrete Fourier transform matrix, we conclude that R , will be a
diagonal m at r ix. Thus , an i sot ropic and homogeneous background
noise is mapped into a possibly no nstationary w hite noise process.
In part icular, a spatial ly white and homogeneous noise process is
mapped into a stat ionary white noise with variance 0 2 / M .
If we assume that the real and imaginary parts of
x ( m ,
t ) are
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1010
IEEE TRANSACTIONS ON SI G N A L PR O C ESSI N G .
VOL. 40,
NO.
4.
A PR I L 1992
- rans-MUSIC
IS--. Conv-MUSIC
- onv-MIN NORM
-10
0 10 20
30
Fig. 2 . Probability of resolution of MUSIC, proposed approach and Min-
Norm applied to the array data and transformed array data.
S N R dB )
uncorrelated a t any given time
t
and have identical covarianc e func-
tions then E { x ( n , t ) x ( m ,
t } = 0.
Combining this fact with the
above discussion and ( 5 ) we find by direct substitution that for a
homoge neous and isotropic backgrou nd noise field the transformed
process
z ( q , t )
is a white noise process which is possibly nonsta-
tionary
.
Combining the results of this section with those of Section 11,
we find that the transformed output z ( q ,
1 )
corresponding to a cir-
cular array that is operating in the presenc e of M narrow-band plane
waves and a background noise field is
where the O,,'s are the azimuths o f the angles of arrival of the plane
waves,
q5,,
the elevations, and
w q , )
is generally a nonstationary
process whose c ovariance is related to that of the background noise
field as explained above and which is white if the background noise
field is homogeneous and stationary.
IV .
S I M U L A T I O NEsuLrs
Simulation experiments were conducted to compare the results
of applying ROOT MUSIC and Minimum Norm
[
111 to the trans-
formed outputs of a
23
element circular array of radius 5 h / 2 a with
those that are obtained by applying MUSIC
[ I 2 1
and Minimum
Norm to the array outputs directly. The sensors were equally spaced
along the c i rcumference of the ar ray . T wo sources of equal power
were assumed to be in the far f ield of the array at azimuths and
elevations (e,
5)
of (60 , 40 ) and (80 , 45 ). The probability of
resolution of MUSIC and Minimum Norm applied to the array out-
puts and transformed array outputs as well as the average mean-
square errors in the elevations and azimuths estimated via those
procedures were computed from
100
independent tr ials at each of
the signal-to-noise ratios that we considered. The corresponding
results for RO OT M USIC app lied to the transformed array outputs
were computed f rom
50
independent tr ials at e ach signal-to-noise
ratio. In each trial the covariance matrices of the array outputs and
the transformed array outputs were estimated using
100
snapshots.
The signal-to-noise ratio was defined in terms of the ratio of a sin-
gle source power to the variance of the additive white noise pro-
cess. The sources were considered to be resolved if two estimates
of the direction of arrivals were obtained and each was located
within
k2'
of the true elevation and azimuth angles. Fig. 2 shows
the probability of resolution at various signal-to-noise ratios for the
0.30
E
.
*
>
-
=
0.20
;;
5
0.10
W
2 -
W O
0.00
-
rans-Root MUSIC
-X- Trans-MUSIC
Conv-MUSIC
--) --.
e onv-MIN NORM
0 10
20 30 40 50
S N R
dB
)
Fig. 3 . Average mean-square error in elevation estimates
b Trans-Root MUSIC
- rans-MUSIC
---I ---
Conv-MUSIC
Conv-MIN NORM
.6
0
10
20 30
40 50
S N R
( d B
)
Fig.
4 .
Average mean-square error in azimuth estimates
ROOT MUSIC based technique described in Section
111,
Minimum
Norm applied to the transformed array output data and
for
M U S I C
and Minimum Norm applied to the sensors output data. Figs. 3 and
4 show the average mean-square errors in the estimated elevations
and azimuths. Note that both figures confirm the improvement in
performance gained by using ROOT MUSIC with the transformed
data.
V .
C O N C L U S I O N
In this correspondence we have studied a transformed data se-
quence that is equal to a reordering of the inverse discrete Fourier
transform sequenc e corresponding to the outputs of a circu lar array
with M elements uniformly distributed around the array circumfe r-
ence. We have shown that the transformed sequence may be pro-
cessed using a ROO T MU SIC based approach to est imate the e le-
vations and azimuths of the observed sources. A computationally
less expensive search over the elevation angles
q5
may also be done
using the multichannel Schur algorithm
[13]
and will be reported
in the near future together with a statistical performance analysis
of the proposed approach.
REF ERENCES
[
11 R . A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays.
N e w
York: Wiley. 1980.
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101
IEEE TRANSACTIONS
ON SIGNAL
PROCESSING,
VOL. 40 NO.
4,
APRIL
1992
[2] A. J. Barabell, “Improving the resolution performance of eigenstruc-
ture-based direction-finding algorithm s,” in
Proc . 1983IEEE
Con5
Acoust. , Speech, Signal Processing
(Boston, MA), 1983, pp. 336-
339.
[3] K. M. Buckley and X. L. Xu, “A comparison of element and beam
space spatial-spectrum estimation for multiple source clust ers,” in
Proc . 1990 IEEE Conf: Acoust. , Speech, Signal Processing
(Albu-
querque, NM), 1990, pp. 2643-2646.
[4] S . Haykin,
Communications Systems,
2nd ed . New York: Wiley,
1983.
[ 5 ]
W. Magnus,
Formulas and Theorems fo r the Special Functions of
Mathematical Physics.
[6] A.
V .
Oppenheim and R. W. Schafer,
Discrete-Time Signal Proce ss-
ing.
[7] J. N . Maks ym, “Directional accuracy of small ring arrays, ” J .
Acousr. Soc. Amer., vol. 61, no. 1, pp. 105-109, Jan . 1977.
[8]M.
P .
Moody, “Resolution of coherent sources incident on a circular
array,”
Proc. IEEE,
vol. 68, no. 2, pp. 276-277, F eb. 1980.
[9] M.
I.
Yadrenko, Spectral Theory
of
Random Fields. New York:
Optimization Software, 1986.
[IO] J. H . McClellan and T. W. Parks, “Eigenvalue and eigenvector de-
composition of the discrete Fourier transform,”
IEEE Trans. Audio
Elec troacoust . ,vol. AU-20, pp. 66-74, Mar. 1972.
[ l l ] R . Kumaresan and D. W . Tufts, “Estimating the angles of arrival of
multiple plane waves,” IEEE Trans. Aerosp. Electron. S y s t . , vol.
AES-19, no. 1, pp. 134-139, Jan. 1983.
[12] R.
0
Schmidt, “Multiple emitter location and signal parameter es-
timation,” IEEE Trans. Antennas Pro pag ., vol.
AP-34,
pp. 276-280,
March 1986.
[
131 J . Rissanen, “Algorithms for triangular decomposition of block Han-
kel and Toeplitz matrices with applications to factoring positive ma-
trix polynomials,”
Mat. Comput . ,
vol. 27, no. 121, pp. 147-154,
1973.
New York: Springer, 1966.
Englewood Cliffs, NJ: Prentice-Hall, 1989.
Counterexamples to “On Estimating Noncausal
Nonminimum Phase ARMA M odels of
Non-Gaussian Processes”
Jitendra
K .
Tugnai t
Abstract-In
the above-pap er,’ an order selection procedure has been
proposed
for
parameter estimation for noncausal, nonminimum phase
ARMA models
of
non-Gaussian processes. We will show that it has
been derived under a n erroneous as sumption , and we also give a coun-
terexamp le to show that it does not yield a consistent order es timate in
general. Two linear approaches
for
parameter estimation have also
been presented in the paper.’ We p oint out that an existing counter-
example to an earlier version
of
one
of
the algorithms also applies to
both the approaches of the above paper.
H ( z )
is the transfer function of the underlying noncausal ARMA
model ( see ( IC ) and ( I d) in the paper ) , and the sym bol * denotes
convolution. This c la im is false . The cor rect maximum ord erp , i s
given by
The
p +
causal poles
of
H(z) yield the
p’(p’
+ 1) /2 causal poles
of H 2 ( z ,
n),
and similarly for the anticausal poles. The “interac-
tion” between the causal and the anticausal poles of H(z) does not
contribute any poles to H2(z, n). We refer the reader to [ l ] for
details. Here we will i l lustrate this with a simple example of a
noncausal AR(2) model that has one causal pole and one anticausal
pole , i .e . , p + = 1 , p - = 1 , a n d p = 2. (Note that since this is an
AR model, we have B(z) = 1 so that there is no cancellation be-
tween B ( z ) and A2(z- ’ , n) for any n . )
We consider an AR(2) model with the transfer function
It has a causal pole at 0.5 and an ant icausal pole a t 2 .5 . We have
where
H u ) =
[u’][(u
0 . 5 ) ~
2.5) ] - ’
(4)
H(zt4-I)
= [ZZ][1.25(U 2z)(u 0.4z)I-’
(5)
The closed contour in (3) is the unit circle running counterclock-
wise since H ( z ) is analytic in an annular region enclosing the unit
circle. The poles of the integrand in (3) encircled by the unit circle
are located at
0.5,
0.42, and 0 if
n
< -1
and at
0.5
and
0.42
if n -1.
By the Cauchy residue theorem, we have n )
. 5 “ ~
.4”
Z”
0.8 z 1.25)(z 6.25)
+
Hz(z’
n, =
4(z 0.25)(z 1.25)
From (6) one may be tempted to conclude that for n
2 ,
H ~ ( z ,
n) has poles a t 0 .25 , 1 .25, and 6 .2 5. In real i ty , there are only two
poles: 0.25 and 6.25, obtained by “i nterac tion” of causal poles
I . C O M M E N T S
N A N D
C O U N T E R E X A M P L E
O THE
O R D E R
S ELECTI ON ETHOD
We first
on
the
order selection
method presented in
exploit the claim that for a noncausal ARMA model with
p +
causal
poles and
p -
anticausal poles
(p
=
p + + p - ),
the maximum order
p2
of the polynomial
A 2 ( z - l ,
n) is
p 2 = p ( p +
1 ) / 2 , w h er e
A , ( z ,
with causal poles, and anticausal poles with anticausal Poles, re-
causal poles and anticausal poles: details are in [l] . We illustrate
this by examining (6) a bit further.
Section
IV
of the above paper, Throughout the section the authors
spectively. There are no poles produced by “interaction”
between
The denominator polynomial of H2(z ,
n)
s given by
n) s the denominator polynomial of H 2 ( z , n)
:
= ~3~ H ( z ) * H ( z ) ,
Manuscript received May
8,
1990; revised April
15,
1991.
The author is with the Department
of
Electrical Engineering, Auburn
From (6) and (7 ) , the numerator polynomial Of H2 Z,) s given by
University, Auburn, AL 36849.
IEEE Log Number 9106003.
N(z)
=
-5(4.0z)”z2(z 0.25)
(O.5)”z2(z
6.25)
‘G.
B. Giannakis and A. Swami, IEEE Trans. Acoust. , Speech, Signal
Processing,
vol. 38, no. 3, pp. 478-495, Mar. 1990.
=
- ( O .~ Z) ”Z’ [ ~ ( Z 0 .25 ) + ( 1 . 2 5 ~ - ’ ) ” ( ~ 6 .2 5 )l .
1053-587X/92$03.00 992 IEEE