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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



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



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.



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

1992 04 application od uniform linear array bearing estimation techniques to uniform circular...

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