Comparison of the ultimate direction-finding capabilities of a number of planar array geometries

A.Manikas, PhD, DIC A.Alexiou, Dip. EE H.R. Karimi, PhD, DIC

Indexing terms: Planar arrays, Direction-finding systems

Abstract: Although the properties of direction- finding (DF) algorithms have heen investigated extensively, the fundamental effects of the array configuration on the performance of DF systems remain unknown. Furthermore, it is often overlooked that there are some theoretical lower limits on the DF performance which are imposed by the array geometry itself. In the paper eight diverse array geometries of elevated feed monopoles, which are used in a number of experimental sites in the UK, are investigated and compared using the ultimate detection, resolution and accuracy thresholds as figures of merit.

List of symbols

A , a scalar A, a vector A, A matrix

transpose conjugate transpose element by element square absolute value of a scalar Euclidian norm of vector projection operator orthogonal projection operator real N-dimensional space complex N-dimensional space general bearing parameter azimuth bearing elevation bearing arc length parameter rate of change of arc length of p-curve bearing separation [PI -p21 arc length separation Is1 - s21

0 IEE, 1997 IEE Proceedings online no. 19971444 Paper first received 9th July 1996 and in revised form 14th May 1997 A. Manikas and A. Alexiou are with the Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2BT, UK H.R. Karimi was with Imperial College of Science, Technology and Medicine, and is now with Motorola, GSM Products Division, 16 Euroway, Blagrove, Swindon SN5 SYQ, UK

a aalap a’ a d a s

ith curvature N number of sensors M number of sources L P signal power 0’ noise power exp{ A element by element exponential sum(a) sum of the elements of vector a L{a, b} subspace spanned by vectors a and b DF direction finding DOA direction of arrival CRB Cramer-Rao bound SNR signal-to-noise ratio RMS root-mean square

number of snapshots (observation interval)

1 Introduction

Consider a planar array of N sensors receiving A4 narrowband plane waves. The response of the array to a signal incident from azimuth 8 E [0, 360’1 and elevation $ E [0, 90’1 is described by the source position vector (or manifold vector), which is defined as

40,4) = e.P{-jW, d4} = exp(-j7r(rzcos8 +rYsin8)cosq3} (1)

where r = [rx, ry, r,] E RNx3 is the matrix of sensor locations (in units of half-wavelengths) and k(8, @) = n(cos8 cos$, sin8 cos$, sin@IT E R3x1 denotes the wavenumber vector. The array manifold is then defined as the locus of the vector a(8, $) V8, 4.

The overall performance of a DF system is a func- tion of both the array geometrykharacteristics and the D F algorithm employed, since a particular algorithm behaves differently when used in conjunction with dif- ferent array structures and, similarly, a certain array generates different results when its output is applied to different algorithms.

The effect of the array structure on the system per- formance may be assessed quantitatively by determin- ing the shape and orientation of the array manifold through the study of the manifold’s differential geome- try [14].

In the case of a linear array employed in a 8 direc- tion-finding system (where $ = 0 and ry = rz = 0) the array manifold is a single curve shaped in the form of a

321 IEE Proc-Radar. Sonar Nuvig., Vol. 144, No. 6, December 1997

Table 1: Differential geometry features of planar array manifold

Parameter curves @curves $curves

Rate of change at arc length S ( $ ) = K sin$ IRI S(8) = cos$ lRel

hyperhelix [Note 11, [l], and, therefore, can be fully characterised by its length and curvatures. However, in a (0, 4) direction-finding system which employs a pla- nar array, the manifold is a two-parameter surface lying on a hypersphere of radius dN embedded in CN and is shaped in the form of a conoid. The parameters of this conoid have been estimated in [2]. Alternatively, the surface of this conoid can be considered to consist of two families of constant-parameter curves defined as follows:

(&curves of constant elevation $z) 1 ($-curves of constant azimuth 0,) where each family can be used to fully describe the array manifold surface. The properties of these curves have been investigated in [3] and are summarised next: (i) $-parameter curves According to eqn. 1, the manifold surface of a planar array can be generated by a family of @parameter curves which meet at the apex of the manifold for $ = 90” and can be described as

with 8 = constant

&parameter curves a(B1$z)

$-parameter curves a($lQz)

a($) = exp { -j.R(Q) cos $} (2)

where R(0) = rx cos 8 + ry sin 0. Eqn. 2 can be seen as the manifold of an equivalent linear array with sensor locations given by the vector R(8). This implies that the +parameter curves of a planar array are shaped as complex hyperhelices [ 1, 31. (ii) @-parameter curves The manifold surface can also be generated by a family of @-parameter curves

$ = constant (3) These curves are not hyperhelical and only limited information, regarding their complete shape, is available. A hyperhelical curve a@) such as a @-curve is analytically ‘convenient’ in the sense that all its curvatures are independent of the parameter p , and hence the procedure for their calculation is identical to that of linear arrays [1]. However, for a non-helical curve, such as a &curve, this is not the case and an analytical approach for the calculation of all curvatures is impractical. Fortunately, the first curvature is quite adequate for describing a curve’s shape at a local level.

The essential properties of both 8 and 4 curves, which will be used in this study, are: (a) the arc length s (the most basic feature of a curve),

a(0) = exp { -j.R(O) cos $}

(b) the rate of change of arc length S,

(c) and the first curvature ~ ~ ( p ) ,

(4)

(5)

Note 1: A hyperheh is a curve whose curvatures remain constant at all points along its length.

where p is a generic parameter representing both 8 and 4-

Table 1 provides analytical expressions for the first curvature and the rate of change of arc length of both @ and 4 curves for a planar array. Note that the differ- ential geometry of the @curves varies as a function of azimuth with a periodicity of at least 180”.

In this paper eight array geometries are compared according to three fundamental limits which are imposed on the performance of a superresolution D F system by the array structure employed. These are (a) the detection threshold (b) the resolution threshold (c) the estimation accuracy lower bound where each is a function of the differential geometry of manifold parameter curves.

These limits will be used as figures of merit for the different array structures. Since no DF algorithm can exceed these performance levels, they also provide a benchmark against which any DF algorithm can be compared.

2 Performance criteria

The most essential feature of signal subspace type tech- niques is their ability to eliminate the effects of noise from the estimation process and thus to provide exact estimates of the signal subspace spanned by the true manifold vectors. This feature is achieved asymptoti- cally over an infinite number of snapshots (the obser- vation interval). Unfortunately, in practice the availability of only a limited number of snapshots, L, prevents the full elimination of the noise and can result in poor direction-finding performance. This ‘uncer- tainty’ due to the remaining noise has been modelled in [4] based on the processing of L snapshots at the out- put of an N-element array which receives M signals in the presence of noise of power d. The conclusion of the investigation in [4] is that the ‘uncertainty’ due to the remaining noise in the estimation process of the ith signal after L snapshots can be represented as an N- dimensional hypersphere centred at the point a, whose radius oe, represents the RMS value of the remaining noise. These spheres are known as ‘uncertainty spheres’ and they ‘shrink’ as a function of the observation inter- val according to the model

I -2 i = l , . . . , M

= q& where Pi denotes the power of the ith source. The fac- tor C is a positive real number smaller than or equal to 1 (i.e. 0 < C < 1) which models the additional uncer- tainties introduced by the employment of a specific practical D F algorithm. The value C = 1 corresponds to a theoretical limit achieved by an ‘ideal’ DF-algo- rithm which does not introduce extra uncertainties and eliminates any dependency which may exist between the received signals (for instance decorrelating any cor- related signals, etc).

IEE ProccRadar, Sonar Navig., Vol. 144, No. 6, December 1997 322

Consider the case of two closely spaced emitters with manifold vectors al and a2. The thresholds of detection and resolution can be identified in terms of the relative positions of the uncertainty spheres centred at points al and a2. Then, according to [4],

‘Two sources are detected if and only if the uncer- tainty spheres do not make contact’,

with the implication that the detection threshold occurs when the two uncertainty spheres just make contact. It should be emphasised that detecting the presence of two sources does not necessarily mean that their bear- ings have been resolved. In fact, for a sufficiently low SNR or a sufficiently small number of snapshots, a typical high-resolution algorithm will always provide a spectrum with a single null, even if it has been given the true number of sources a priori.

If a similar line of arguments is followed for the resolution, then

‘Two sources corresponding to points s1 and s2 on the manifold are resolved if and only if the uncer- tainty spheres do not make contact with the sub- space W = L{a(so), uI(so)} at any point so between s1 and s2, where u1 = da(so)/ds’.

In this case, the resolution threshold occurs when the uncertainty spheres just make contact with the worst case signal subspace W.

According to the model of uncertainty spheres and the above definitions of detection and resolution thresholds, the products (SNRI x L) required for a source of power PI to be detected and resolved in the presence of a source of power Pz, when the two sources have a bearing separation Ap, are, respectively,

(6) and

( 1 + ~ ) 4

where [Note 21

(7)

and p is a parameter representing 4 exmessions it is assumed that the

when p = q5

when p = 0 (8)

or 8. In the above reference point is

taken at the array centroid. Bearing p corresponds to arc length (sl + s2)/2 on the manifold, which, to a first order approximation, also corresponds to bearing

+ p2)/2. Note also that the arc length separation is related to the bearing separation with the expression As = Ap x S@). For C = 1 the above expressions become ‘lower bounds’ and form a benchmark against which any practical DF algorithm can be compared.

Having presented the detection and resolution crite- ria, the discussion will be confined to the estimation accuracy. The estimation accuracy can be expressed in terms of the error variance of the estimation of the parameter p 1 for a source of power PI in the presence Note 2: ri, is the first curvature of the circular approximation to the man- ifold, i.e. it takes into account the orientation of the manifold in the N- dimensional observation space.

IEE Proc-Radur. Sonar Nuvig.. Vol. 144. No. 6, December 1997

of a source of power P2. If the two sources have a bearing separation Ap, then this estimation accuracy is bounded by the CramCr-Rao bound [3, 51

2SNR1 x L CRB[p,] =

0 1 L

N - SNRl x L [& x S(p)12S2(p1)[R3p) - 1”

(9) where al is the source position vector corresponding to the source at p1 and SNRl is the signal-to-noise ratio of the signal impinging from bearing p l .

Furthermore, if the CRB is evaluated at the thresh- old level (SNR1 x L),,, then the estimation error (accu- racy of the estimate) at the resolution threshold is given as follows:

N C P P x w2 - (If ‘ p ) 4 S”(P1)

It is important to note that for two equipowered sources the above expression is simplified (for C = 1) to

(RMS estimation error at resolution threshold)

which is an expected result indicating the generality and significance of the resolution threshold.

The above discussion was carried out under the assumption that the planar array consists of antennas that are isotropic (with gain of unity) in both azimuth 8 and elevation 4. This assumption might seem unreal- istic since many practical antennas (like the elevated- feed monopoles which will be investigated in this study) are nonisotropic and exhibit a complex gain response g E C’ as a function of one or both bearing parameters. In this case the array manifold is given by

By using ag(p) instead of a@), where p = (8, @), eqns. 6 , 7 and 9 can be transformed for directional sensors to the expressions presented in eqns. 11-13.

ado, 4) = d e , 4)a(O, 4) (10)

3

It is clear from these equations that the directional pattern g @ ) of a directional sensor behaves simply as a ‘voltage gain’ term boosting or deteriorating the

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tive signal-to-noise ratio at the output of the array. Consequently, it can be statcd that directional sensors does not affect the r one array geometry over the other, although, of course, performance is affected in absolute terms.

ance as a function of elevation entia1 geometry reveals that the varia-

performance with elevation $ is independ- ent of array geometry and consequently the merits of

uration as compared with another a t different elevations, i.e. perform- need only be carried out at a single choice of q0 is arbitrary and can be

made according to the particular application. Here we consider an elevation of q50 = 30" at which the elevated feed monopoles, which will be used in this study, exhibit maximum gain.

Another characteristic feature of planar arrays of isotropic sensors is that while azimuth-estimation performance is greatest at low elevations, elevation- estimation performance is greatest at high elevations. Naturally this characteristic is masked by the directional pattern of the array elements.

2.2 Performance as a function of azimuth Manifold differential geometry also reveals that the 4- and 8-estimation capabilities of a planar array are dependent on azimuth 8 through vector R(8) and its derivative R&8) = R(8 + go"), respectively. Re(@] are both direct functions of the array tion and are 90" out of phase.

Fig. 1 uzmuth (3 Phase reference is assumed to be at array centroid (I) IR(0)l = large

(U) lR(6')l = small

Variations in parameter estimation performance as a function of

* @eshmation = GOOD &estimation = POOR

3 @estimation = POOR + &estimation = GOOD

IR(0 i 90")l = small

lR(0 + 90")l = large

It is evident from eqns. 6, 7, and 9 that an array is a more powerful direction finder along those azimuths 8 which correspond to larger values of IR(6')l (for 4- estimation) and larger values of iR(8 -t 90")l (for 8- estimation), where R(8) equals the sensor locations when the array is projected along the azimuthal direction 6' (Fig. 1). This rule is particularly easy to apply in cases where the sensors are predominantly distributed along one direction. In such cases it is easy to judge the variations of iR(@)i with 8, by a simple observation of the array geometry. For more complicated array structures the variations of iR(f3)I are not so obvious and actual computation is necessary. In the following Section the variations of performance with azimuth will be examined in de of different array configurations.

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3 Array geometries for study

The eight array geometries investigated in this paper consist of vertical elevated-feed monopoles which are directional in elevation but isotropic in azimuth. These array structures are used in a number of experimental sites in the UK operating as DF systems in the HF frequency band (3-30MHz) and employ 12m-high monopoles. Fig. 2 shows the elevation gain pattern of these monopole antennas. The pattern consists of a single beam pointing at an elevation of approximately $ = 30" with nulls at $ = 90" (due to the vertical orien- tation of the monopole) and $I = 0" (due to the reflec- tivity of conducting ground). Such antennas are unsuitable for the reception of ground waves or high elevation signals but are often employed in the over- the-horizon reception of long-range sky waves.

O Y 0 10 20 30 40 50 60 70 80 90

elevation, deg

Fig. 2 Monopole gain pattern

The eight array geometries under investigation are: (a) linear array of 20 antennas (sensors) consisting of monopoles uniformly spaced at 27.58m, with the exception of the end-sensors which are at double spac- ing (b) 24-element uniform circular array of 75m radius (e) Y, X, L and t (cross) shaped arrays with each branch of these shapes having eight monopoles with locations (8m, 22m, 38m, 57m, 79m, 105m, 136m, 170m) (d) two %element arrays with dual-ring and dual-spiral geometries, respectively. Figs. 3-6 illustrate the above described array geometries, in metres. Note that the array centroids are used as reference points (0 m, 0 m).

Finally, we have seen in Section 2 that the variations in @estimation performance and $-estimation perform- ance, as a function of 8, are essentially identical apart from a phase of 90" (due to dependence on IR(0 + 90")l and IR(8)1, respectively) and a scaling factor (due to dependence on cos $ and sin$, respectively). Therefore, only the @estimation performances for q50 = 30" will be investigated and compared in the following Section. Various array geometries and values of 8 from 0" to 180" are considered, while the results are repeated for 6' from 180" to 360".

Radar, Sonar Navig , VoL 144, Nu 6, December 1997

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4 Comparison based on detection, resolution and accuracy criteria

The theoretical lower limits on the accuracy, detection and resolution thresholds imposed on a DF system by the array geometry itself were evaluated for the array geometries described in the preceding Section. The results are presented in Figs. 7-9 for an operating fre- quency of 1 SMHz. A discussion associated with the performance of each geometry is presented below.

0

(iii)

0 20 40 60 80 100 120 140 160 180

azimuth, deg Fig.7 for two equipowered emitters separated by A0 = 0.5" about 0

@ = 30" (i) Cross; (ii) circular; (iii) dual ring; (iv) dual spiral; (v) linear

Detection threshold (SNR x L) as a function of azimuth angle

35

3o I--\----.- (iii) - - - - - . - - -

0 1 I 0 20 40 60 80 100 120 140 160 180

azimuth, deg Fig. 8 for two equipowered emitters separated by A 0 = 0.5" about 0 @ = 30" (i) Cross; (ii) circular; (iii) dual ring; (iv) dual spiral; (v) linear

Resolution threshold ( S N R x L) as a function of azimuth angle

4. I Linear array The behaviour of a linear array can be readily pre- dicted by noting that R(8) = r, cos8 and that Re(@ = R(8 + 90") = -rx sin@ Hence one could predict that a linear array would have great difficulty in estimating azimuth bearings near 0" or 180" (end-fire), where IR(8)i = 0. This is demonstrated in Figs. 7-9, where the three criteria (SNR x L),,,, (SNR x L),es,e and CRB, tend to infinity at the above angles. One could also pre- dict that a linear array would have the highest azimuth estimation capability for azimuths near 90" (broadside), where lR(8)i is maximum. Again this is confirmed in Figs. 7-9, where the functions (SNR x L)det,8, (SNR x ~5),,,,~ and CRB, are smallest at 8 = 90".

326

1 O-a

0 20 40 60 80 100 120 140 160 180

azimuth, deg Error standard deviation dCRB, as a function of azimuth angle Fig.9

for two uncorrelated emitters separated by A0 = 0.5" about 0 L = 100, SNR = 20dB, @ = 30" (i) Cross; (ii) circular; (iii) dual ring; (iv) dual spiral; (v) linear

4.2 Circular array It can be shown that the uniform circular array config- uration is the only planar array geometry for which both IR(8)i and lR2(8)l are independent of 8. This implies that, in the case of a circular array, all the per- formance criteria investigated herein exhibit no depend- ence on azimuth 8. Such a geometry seems to be the obvious choice for applications where uniform per- formance at all azimuths is a necessity.

4.3 Y-shaped and X-shaped arrays Both Y-shaped and X-shaped arrays have a balanced- symmetric [Note 31 geometry [3], and consequently their detection thresholds (SNR x L)det,e are independ- ent of 8. On the other hand both (SNR x L)yes,8 and CRBe vary periodically as a function of 8 with a period of 60" for the Y-array and of 90" for the X-array. Note that the best performance of the Y and X arrays occurs at even multiples of 30" and 45", respectively, with worst performances occurring at odd multiples of 30" and 45", respectively.

4.4 L-shaped and cross-shaped arrays These two geometries are not balanced-symmetric and so the detection thresholds (SNR x L),,, exhibit signif- icant variations with best performance at 4.5" and worst performance at 135" for the L-array and at 0"/180" and 90", respectively, for the cross-array. Note also that the resolution threshold (SNR x L),es,e and CRB, indicate best and worst performances at similar azimuths.

4.5 Dual-ring array The dual-ring array is approximately balanced-symmet- ric and also resembles a circular-type structure. Conse- quently an almost uniform performance is provided at all azimuths. The detection (SNR x L)det,, indicates best performance at 77" and worst performance at 167" while the resolution threshold function (SNR x L),es,e and CRB, indicate best and worst performances at 68" and 126", respectively.

4.6 Dual-spiral array The dual-spiral array is an example of an array geome- try which has sensors clearly distributed in one pre- dominant direction and so it is easier to observe the variations of IR(8)l as a function of 8. The detection (SNR x L)det,, indicates best performance at 153" and

Note 3: Balanced symmetric array: rxTry = 0 and [r,i = lryl

IEE Proc.-Radar, Sonar Nuvig., Vol. 144, No. 6, December 1997

worst performance at 63", while both (SNR x L)res,B and CRBe indicate best and worst performances at 162" and 54", respectively.

A comparison of the array structures reveals that at, and only at, 8 = 90" (broadside) the linear array has the best performance but that there is a monotonic degradation in performance as emitters approach the end-fire position. In terms of average performance over all azimuths, the X-shaped array outperforms the oth- ers, with the Y-shaped array taking second position. The worst performance is obtained with the dual-ring array, which clearly suffers due to its small dimensions.

5 Performance comparison and array normalisation

It is well known that the aperture of an array (i.e. the maximum separation between any two sensors) plays an important role in the ability of an array to resolve two sources close together. Therefore it may be argued that, to perform a fair comparison, the DF capabilities should be investigated with the array geometries nor- malised with respect to aperture. To perform such a comparison the exercise of Section 4 is repeated with the sensor locations scaled such that each array has the same aperture of 57.918 half-wavelengths.

-4

-8

m -12

A-

a 5 -16

0

X

-20

-24

(ii)

0 20 40 60 80 100 120 140 160 180 azimuth, deg

Fig. 10 for two equipowered emitters separated by AB = 0.5" about B 0 = 30" The array geometries have been normalised with respect to aperture (i) Cross: (ii) circular; (iii) dual ring; (iv) dual spiral: (v) linear

Detection threshold (SNR x L) as a f i c t i o n of azimuth angle

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30

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m 2o

x 15 a * 10

5

0

-5

TJ i

z

i 20 40 60 80 100 120 140 160 '

azimuth. d w Fig. 11 for two equipowered emitters separated by AB = d5' about B 0 = 30" The array geometries have been normalised with respect to aperture (i) Cross: (ii) circular: (iii) dual ring; (iv) dual spiral; (v) linear

IEE Proc-Radar. Sonar Navig.. Vol. 144, No. 6, December 1997

Resolution threshold (SNR x L) &a unction of azimuth angle

1 0.'

0) 0) 0 L- e 5

a

m a

cn z lo2 C

0

t 1

I I 0 20 40 60 80 100 120 140 160 180

azimuth, deg Fig. 12 for two uncorrelated emitters separated by AB = 0.5" about B L = 100, SNR = 20dB, 8 = 30" The array geometries have been normalised with respect to aperture (i) Cross: (ii) circular: (iii) dual ring; (iv) dual spiral: (v) linear

Error standard deviation &RBO as a function of azimuth angle

From the results shown in Figs. 10-12, it is apparent that now the circular array exhibits the best perform- ance which is independent of azimuth. The perform- ance of the linear array approaches that of the circular array only at 8 = 90" (broadside) but rapidly shows a degradation as emitters approach the end-fire position. In terms of performance averaged over all azimuths, the X-shaped array and the Y-shaped array take sec- ond position. The worst performance is obtained with the dual-spiral array.

As expected, the variations of array performance with azimuth remain unchanged, apart from a scaling factor, when compared to the unnormalised case. This is because all array shapes are preserved through the normalisation process. The absolute performances of all the arrays are slightly improved as a result of an increase in array dimensions.

-8

-1 2 m

J-

5 -16 TJ

X

*

-20

-24

(iii) \ - -

(ii)

0 20 40 60 80 100 120 140 160 180 azimuth, deg

Fig. 13 Detection threshold (SNR x L ) as a function of azimuth angle for two equipowered emitters separated by AB = 0.5" about 0 0 = 30" The arrays have been normalised with respect to aperture and number of sensors (i) Cross; (ii) circular; (iii) linear

It is known that the resolution is also influenced by the length of the manifold curve (the longer the mani- fold curve the better the resolution). For a given aper- ture the length of the manifold curve is a function of the number of sensors. Therefore it may also be argued

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that a fairer performance comparison would be achieved with arrays normalised with respect to both aperture and number of sensors. In Figs. 13-15 the results of this normalisation are shown with all array geometries having 24 elements. The normalisation was not possible for the dual ring and the dual spiral arrays, due to insufficient information regarding the rules defining their original sensor locations.

30 1 I

0 20 40 60 80 100 120 140 160 180 azimuth, deg

Fig. 14 Resolution threshold (SNR x L) as a fmction of azimuth angle for two equpowered emitters separated by A8 = 0.5" about 8

= 30" The arrays have been normalised with resuect to auerture and number of sensors (i) Cross; (ii) circular; (iii) linear

10.'

a d e b

i

v) z 2 cc 10.~

C

0

I 0-4

(ii)

0 20 40 60 80 100 120 140 160

azimuth, deg 0

Fig. 15 Error standard deviation dCRB, as a function of azimuth angle for two uncorrelated emitters separated by A8 = 0.5" about 8 L = 100, SNR = 20dB, q5 = 30" The arrays have been normalised with respect to aperture and number of sensors (i) Cross; (ii) circular; (iii) linear

An important feature of an array geometry is its abil- ity to provide a uniform DF performance over all azi- muth directions. This property can be quantified by the following differential comparison rule:

dzjferential(SNR x L ) = dB{(SNR x L)best}

- dB{(SNR x L ) w o r s t }

where a small diffeerentiaZ(SNR x L) corresponds to more consistent performance and therefore to a better array geometry.

m

linear Y circular X L cross dual dual nng spiral

Fig. 16 Dlf erence between best and worst detection and resolution threshold p e ~ m a n c e s for unnormalised and normalised array structures (1) detection threshold (ii) detection threshold (normalised wrt aperture) (in) detection threshold (normalised wrt aperture and number of sensors) (iv) resolution threshold (v) resolution threshold (normalised wrt aperture) (vi) resolution threshold (normalised wrt aperture and number of sensors)

To demonstrate the significance of this differential rule, it has been used to compare the original array geometries as well as their two normalised counter- parts. The results are shown in Fig. 16 in a bar-chart format. It is clear from this bar-chart that the differ- ence is not affected by the normalisation and is a func- tion only of the array geometry itself. Thus, using the above differential rule, the array geometries may be ordered as shown in Table 2.

6 Conclusions

In this paper the ultimate direction-finding capabilities of a number of planar arrays have been investigated and compared. Criteria relating to the DF performance in terms of the minimum achievable bearing estimation error, and the detection and resolution thresholds, have been determined using the differential geometry proper- ties of the array manifold.

The arrays were initially unnormalised and then nor- malised with respect to aperture and with respect to aperture and number of sensors. From the results it was concluded that the difference between the best and

Table 2: Arrays order with respect to difference between best and worst performance

Detection threshold Resolution threshold

Array Unnormalised Normalised Array Unnormalised Normalised

aperture aperture, N aperture aperture, N

Circular 1st 1st 1st Circular 1st 1 st 1 st Y 1 st 1st 1st Dual ring 2nd 2nd - X 1 st 1st 1st Y 3rd 3rd 2nd

X 4th 4th 3rd Dual ring 2nd 2nd -

Dual spiral 3rd 3rd - L 5th 5th 4th Cross 4th 4th 2nd Dual spiral 6th 6th - L 5th 5th 3rd Cross 6th 7th 5th Linear 6th 6th 4th Linear 7th 8th 6th

328 IEE Proc.-Radar, Sonar Navig., Vol. 144, No. 6, December 1997

the worst detection and resolution performance is inde- pendent of the normalisation procedure being employed and is a characteristic only of the array geometry itself. However, this should not be confused with the absolute performance of an array, which is, of course, a function of both aperture and number of sen- sors. A comprehensive set of results has been presented indicating the theoretical limits on accuracy, detection and resolution thresholds.

Thus these limits can be used as figures of merit to provide the performance level against which any exist- ing or new algorithm can be compared. That is, for a given array, the closer a DF algorithm comes, perform- ance-wise, to these theoretical limits the better.

7 References

1 DACOS, I., and MANIKAS, A.: ‘Estimating the manifold parameters of a one-dimensional array of sensors’, J. Franklin Inst. Eng. Appl. Math. , 1995, 332B, (3), pp. 307-332

2

5

6

MANIKAS, A., and DACOS, I.: ‘Investigating the manifold parameters of a non linear array of omnidirectional sensors. Part 11: Specialization to planar arrays of arbitrary geometry’. Unclas- sified-Reseach report: AM-91-5, Department of Electrical and Electronic Engineering, Imperial College London, 1991 KARIMI, H.R., and MANIKAS, A.: ‘The manifold of a planar array and its effects on the accuracy of direction-finding systems’, IEE Proc.-Radar, Sonar Navig., 1996, 143, (6), pp. 349-357 MANIKAS, A., KARIMI, H.R., and DACOS, I . : ‘Study of the detection and resolution capabilities of one-dimensional array of sensors by using differential geometry’, IEE Proc. -Radar, Sonar Navig., 1994, 141, (2), pp. 83-92 STOICA, P., and NEHORAI, A.: ‘MUSIC, maximum likelihood and Cramer-Rao bound’, IEEE Trans. Acoust. Speech Signal Process., 1989, 37, pp. 720-741 SCHMIDT, R.O.: ‘Multiple emitter location and signal parame- ter estimation’, IEEE Trans. Antennas Propag., 1986, 34, (3), pp. 276-280 GUGGENHEIMER, H.W.: ‘Differential geometry’ (McGraw- Hill, 1973) LIPSCHUTZ, M.M.: ‘Differential geometry’ (McGraw-Hill, 1969) VAN TREES, H.L.: ‘Detection, estimation and modulation the- ory - part I’ (Wiley, 1968)

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