Higher rank ambiguity for star arrays

ARTICLE IN PRESS

Signal Processing 84 (2004) 2265–2269 www.elsevier.com/locate/sigpro

Higher rank ambiguity for star arrays Monika Agrawala,, Kah-Chye Tanb a CARE, Indian Institute of Technology, New Delhi 110016, India Addest Technovation, 20 Ayer Rajah Crescent #09-19, Singapore 139964

b

Received 22 November 2002; received in revised form 14 January 2004

Abstract In this paper, ambiguities associated with the geometry of star arrays are studied. The angles between the various axes of a star array are very crucial, in the sense that, if not chosen properly, they give rise to inherent ambiguities. Such inherent ambiguities pose serious problems—they exist regardless of the number of sensors and inter-sensor separations. Fortunately, such ambiguities can be removed by suitable choices of the angles between the axes. The inherent ambiguities associated with uniform circular arrays are also studied. r 2004 Elsevier B.V. All rights reserved. Keywords: Ambiguity; Star array; Direction of arrival; Array manifold; Steering vector

1. Introduction When finding the direction of arrival (DOA) of narrowband signals using an array of sensors, it is important to be certain that the problem has a unique solution. The linear independence of steering vectors is key to obtaining unique DOA estimates. If the array has identical response to two different sets of DOAs, then the ambiguity problem is said to arise. The first attempt to introduce a mathematical framework for dealing with the ambiguity problem was made by Schmidt [4] in the early 1980s. They

classified the ambiguities according to the ‘‘rank’’ of steering vectors. Around the same time Godara and Cantoni [1] identified the arrays which are free of up to rank-1 ambiguity1 for a more general scenario, i.e., when both the azimuth and the elevation are of concern. But still, till today, array ambiguity in general is an open problem. It is still unknown as to what sensor arrangement would lead to an array that is free of up to rank-n ambiguities, for nX3; when the signals arrive from any direction, i.e. from both sides of the plane. Some special cases of array ambiguity have known solutions. It is known that an uniform linear array (ULA) consisting of M sensors with

Corresponding author.

E-mail address: [email protected] (M. Agrawal).

1 An array is free of up to rank-n ambiguities if every ðn þ 1Þ steering vectors with distinct DOAs are linearly independent.

0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.06.022

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inter-sensor spacing less than l=2; where l is the wavelength of the signals of interest, is free of up to rank-ðM  1Þ ambiguities, if the signals are coming from one side of the ULA. In [6], the ambiguity problem was examined for the case of linear arrays, and a special class of linear arrays that are free of rank-1 ambiguity was identified. Lo and Marple [2] derived a theorem for characterizing rank-2 ambiguities. They had also proposed a conjecture providing a simple way of identifying whether a set of directions is rank-n ambiguous or not. However, in [5], this conjecture was shown, through a counter example, to be, in general, incorrect. Manikas and Proukakis [3] have studied the ambiguities of linear arrays using differential geometry and have suggested methods to find them. In [5], a specific class of arrays (cross arrays2) has been studied in detail. The authors have shown that if the angle between the linear arrays, that make up a cross array, is not chosen properly, the cross array suffers from inherent ambiguities that cannot be removed by increasing the number of sensors or by changing the inter-sensor separations. Motivated by the study of the ambiguous behavior of cross arrays [5], in this paper, we extend the results of [5] for star arrays3 (SA) by tailoring the lemmas and theorems of [5] to cater to the requirements of SA. It has been shown that if none of the angles between the linear arrays that make up a SA is chosen properly, the SA suffers from inherent ambiguities. Such inherent ambiguities pose a serious problem. They exist regardless of the number of sensors and the inter-sensor separations. The only way to get rid of them is to suitably re-choose the cross angles.4 Further, these results offer insight into the inherent ambiguities of an uniform circular array (UCA). We show that an UCA with an even number of sensors is always rank-2M  1 ambig2 Cross array is an array whose sensors lie on two non-parallel straight lines. 3 Star array is an array whose sensors lie on several nonparallel straight lines intersecting at a single point. 4 Cross angle is the angle between the linear arrays that make up a SA.

uous, where 2M is the number of sensors comprising the UCA, regardless of its radius.

2. Array manifold In the Cartesian coordinate system, consider P narrowband plane waves of wavelength l; impinging on an array of M omni-directional sensors. The signal vector received by the array is rðtÞ ¼

P X

aðyp Þsp ðtÞ þ nðtÞ;

(1)

p¼0

where rðtÞ ¼ ½ r1 ðtÞ

r2 ðtÞ



nðtÞ ¼ ½ n1 ðtÞ n2 ðtÞ   

rM ðtÞ T ; nM ðtÞ T ;

(2) (3)

sp ðtÞ is the amplitude of the pth plane wave arriving from the direction yp relative to the broad side of the array, rm ðtÞ and nm ðtÞ are, respectively, the sensor output and the noise at the mth sensor. Since we are concerned with the azimuth-only estimation, we shall assume that y 2 ½0; 2pÞ (measured clockwise from the x-axis). The steering vector aðyÞ is defined as, aðyÞ ¼ ½ a1 ðyÞ

a2 ðyÞ   

aM ðyÞ T ;

(4)

where am ðyÞ ¼ eð{2pðxm cos yþym sin yÞ=lÞ ; with ðxm ; ym Þ being the coordinate of the mth sensor. (ðx1 ; y1 Þ is often chosen to be (0,0), therefore a1 ðyÞ ¼ 1). The collection of all possible steering vectors is termed as array manifold.

3. Ambiguity in star arrays We define an SA as an array whose sensors lie on the non-parallel straight lines intersecting at a common point. This common intersecting point is termed as reference point whereas these straight lines are termed as axes. The smallest angle between any two axes is called reference cross angle and one of the axes among these two axes (axes spanning the reference cross angle) is called reference axis. All the other angles are measured

ARTICLE IN PRESS M. Agrawal, K.-C. Tan / Signal Processing 84 (2004) 2265–2269

from this reference axis. We need the following lemmas, Lemma 1. Let there be K sets, each having 2l vector elements, s.t. the length of the vectors in a set is constant but may differ from the lengths of vectors of other sets. These sets are denoted by f ðkÞ 1 2l K ðkÞ fvðkÞ Þ: Also, for each even i m gm¼1 k¼1 ðvm 2 C ðkÞ there exists an odd j s.t. vðkÞ for 1pi; jp2l: i ¼ vj Then the column vectors 2 ð1Þ 3 2 ð1Þ 3 v2l v1 6 . 7 6 . 7 6 . 76 . 7 4 . 5 4 . 5 vðKÞ 1

vðKÞ 2l

are linearly dependent. Proof. See [5].

Lemma 2. Let n 2 Z; a 2 ð0; 2p; b 2 ð0; p and s1 ¼ ½cos a; sin aT and the sequence s1 ; s2 ; . . . ; satisfy the following recursive conditions; s2k ¼ Rb FRb s2k1 ; s2kþ1 ¼ Fs2k  and

F¼

1 0

0 : 1

Then the sequence s1 ; s2 ; . . . ; s2n is unique but s2nþ1 ¼ s1 ; in general s2nþl ¼ sl l 2 Z: This is true if and only if b ¼ mp=n for some m 2 Z þ ; gcdðm; nÞ ¼ 1 and na=peZ: Proof. See [5].

On the other hand, if there exist distinct f1 ; . . . ; f2n 2 ½0; 2pÞ such that Eq. (5) is satisfied then b ¼ mp=n where m; n 2 Zþ with gcdðm; nÞ ¼ 1: Proof. See Appendix A. Now we are ready to establish a theorem on the inherent ambiguities of SAs. Theorem 1. If a SA consists of M sensors and i LAs (axes) with reference cross angle (angle between the reference axis and the next axis), b ¼ mp=n and all other cross angles are integer multiples of b where m; n 2 Z þ with gcdðm; nÞ ¼ 1; npM=2; mpn=i; then there exist 2n steering vectors with distinct DOAs that are linearly dependent. Proof. See Appendix B.

&

for k ¼ 1; 2; . . . ;  cos b  sin b Rb ¼ sin b cos b

2267

&

Using the above theorem it can be shown that if the cross angles are of the above form, then there exist linearly dependent steering vectors corresponding to distinct DOAs. This theorem can be made clear with the following example: Example 1. Consider a SA consisting of 3 LAs with a reference cross angle of p=3 (Fig. 1). One LA is along the x-axis (which shall be taken as the reference axis), and the other LAs are at an angle of p=3 and 2p=3 w.r.t. the x-axis. Each LA contains at least 2 sensors, independent of all other LAs. SA may (or may not) contain a sensor at the reference point (common point where all the LAs intersect). Therefore, the SA is comprised of a

Lemma 3. Let n 2 Zþ ; and b 2 ð0; p: If b ¼ mp=n where m 2 Z þ with gcdðm; nÞ ¼ 1 then there exist distinct f1 ; f2 ; . . . ; f2n 2 ½0; 2pÞ such that nf1 =peZ; f2kþ1 ¼ f1 þ 2kb ðmod 2pÞ f2k ¼ 2p  f2kþ1 f2n ¼ 2p  f1 ;

for k ¼ 1; . . . ; n  1; for k ¼ 1; . . . ; n  1;

π/3

(5) f2k þ b ¼ 2p  ðf2k1 þ bÞ

for k ¼ 1; . . . ; n;

f2k þ ib ¼ 2p  ðf2ðkiÞþ1 þ ibÞ for k ¼ 1; . . . ; n; i ¼ 1; 2; . . . ; n  1:

Fig. 1. A star array (SA) with cross angle of p=3:

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M. Agrawal, K.-C. Tan / Signal Processing 84 (2004) 2265–2269

minimum of six sensors. Consider the signals coming with DOAs as y ¼ ða; 2p  a; 2p=3 þ a; 4p=3 þ a; 4p=3  a; 2p=3  aÞ where a 2 ð0; p=3Þ: These DOAs are symmetric w.r.t. x- and y-axes; therefore the corresponding steering vectors aðy P1 Þ; aðy2 Þ; . . . ; aðy6 Þ are linearly dependent ð 6n¼1 aðyn Þ ¼ 0Þ: Since a can take any value, therefore, there are infinitely many sets of y that give rise to ambiguous sets of DOAs. Relating the above theorem with Theorem 1 of [5], it can be seen that the number of distinct DOAs resulting in linearly dependent steering vectors directly depends upon the cross angles. Like SA (studies in Example 1), cross array (of [5]) with b ¼ p=3 also have six distinct DOAs corresponding to linearly dependent steering vectors. This implies that the cross array as well as the star array having the same cross angle have utmost the same rank ambiguity. Therefore, the inherent ambiguities present in the cross array cannot be removed by adding another axis at an angle of b (or multiple of b) w.r.t. the existing axes i.e. generalizing the cross array to a star array. In line with the above discussion, we can also study the inherent ambiguities of uniform circular arrays (UCA) having an even number of sensors. An UCA of 2M sensors can also be viewed as a special SA with M LAs, with the angle between any two LA being p=M (reference angle is p=M), and each LA having two sensors. This view of UCAs allows us to apply Theorem 1, which is basically stated for SAs. Lemma 4 addresses inherent ambiguities associated with UCAs.

4. Ambiguity-free star arrays Designing an array free of rank-n ambiguities is one of the most challenging tasks. Indeed, very little is known about the array geometries that give rise to unambiguous steering vectors. In this paper, we are concerned with the SA, which can be thought of as a generalization of the cross array. A set of criteria for designing a cross array free of rank-n ambiguities is presented in [5]. For completeness, we present a set of criteria for ensuring that a SA is free of up to a certain rank of ambiguities, as follows: Lemma 5. Consider a SA comprising of i LA, s.t. the lth LA contains pðlÞ sensors 8l: If all the LAs intersecting at a common point, are free of up to rank-ðk  1Þ ambiguities (where 2pkp minðpð1Þ; . . . ; pðiÞÞÞ for the signals coming from one side of the array, and if any one of the cross angles do not fall in Bk ; where Bk ¼ fmp=n : m; n 2 Zþ ; npk=2; mpn=i; gcdðm; nÞ ¼ 1g; then SA is free of up to rank-ðk  1Þ ambiguities. Proof. It been shown in [5] that a SA with i ¼ 2 is free of up to rank-ðk  1Þ ambiguities. By adding more axes we do not disturb the entries of array steering vector corresponding to these two LAs; hence we preserve the rank of array steering vectors. So if any of the cross angles is different from Bk ; then we have a SA free of up to rankðk  1Þ ambiguities.

5. Conclusion Lemma 4. For an UCA with 2M sensors there exist 2M steering vectors with distinct DOAs that are linearly dependent regardless of the radius of the UCA. Proof. A direct application of Theorem 1.

&

Since UCA is a popular choice in many practical systems, we believe that this result will be very helpful in designing systems. Here, Example 1, studies the ambiguities associated with UCAs comprising of six sensors.

In this paper, the ambiguity issues associated with SAs are studied. If the angles between the various axes of a SA are not appropriately chosen, it suffers from inherent ambiguities that cannot be got rid of by increasing the number of sensors and/ or the inter-sensor separations. Similarly, an UCA, which has been adopted in many practical array systems, suffers from inherent ambiguities regardless of its radius. A set of criteria for ensuring that a SA is free of up to a certain rank of ambiguities has also been derived.

ARTICLE IN PRESS M. Agrawal, K.-C. Tan / Signal Processing 84 (2004) 2265–2269

2269

and

Appendix A. Proof of Lemma 3 This lemma is largely proved in [5] except the claim that f2k þ ib ¼ 2p  ðf2ðkiÞþ1 þ ibÞ for k ¼ 1; . . . ; n; i ¼ 1; 2; . . . ; n  1:

bi ðy þ ibÞ ¼ ½ ezi1 o cosðyþibÞ



i

T ezpðiÞ o cosðyþibÞ  ;

(8)

f2k þ ib ¼ 2p  f2kþ1 þ ib; f2k þ ib ¼ 2p  ðf1 þ 2kb  ibÞ;

here b ¼ mp=n; where gcdðm; nÞ ¼ 1 and npr=2: By Lemma 3, we can find f1 ; . . . ; f2n 2 ½0; 2pÞ (distinct DOAs) such that

f2k þ ib ¼ 2p  ðf1 þ ð2k  2iÞb þ ibÞ; f2k þ ib ¼ 2p  ðf2ðkiÞþ1 þ ibÞ:

b1 ðf2k Þ

¼ b1 ð2p  f2kþ1 Þ

¼ b1 ðf2kþ1 Þ;

b1 ðf2n Þ

¼ b1 ð2p  f1 Þ

¼ b1 ðf1 Þ;

bi ðf2k þ ibÞ ¼ bi ð2p  f2ðkiÞþ1 þ ibÞ ¼ bi ðf2ðkiÞþ1 þ ibÞ:

Appendix B. Proof of Theorem 1 Consider an r-sensor SA, consisting of i LAs s.t. r ¼ pð1Þ þ    þ pðiÞ: LAs (axes) are indexed as LA1 ; . . . ; LAi : Without loss of generality, we can assume that LA1 ; the reference axis, is the x-axis, and each of the other axes are arranged in the order of increasing cross angles, i.e. the angle between LA1 and LA2 is b and the other angles are integer multiples of b: Let the coordinates of the sensor lying on LA1 be ðx1 ; 0Þ; . . . ; ðxpð1Þ ; 0Þ and lying on LAi th axis be ðzi1 cos ib; zi1 sin ibÞ    ðzipðiÞ cos ib; zipðiÞ sin ibÞ: Now we can express the steering vector of the SA as, 2 3 b1 ðyÞ 6 7 .. 7; aðyÞ ¼ 6 (6) . 4 5 bi ðy þ ibÞ where b1 ðyÞ ¼ ½ ex1 o cos y



expð1Þ o cos y T

(7)

It directly follows from Lemma 1 that aðf1 Þ; . . . ; aðf2n Þ are linearly dependent.

References [1] L.C. Godara, A. Cantoni, Uniqueness and linear independence of steering vectors in array space, J. Acoust. Soc. America 70 (2) (1981) 467–475. [2] J.T.-H. Lo, S.L. Marple, Observability conditions for multiple signal direction finding and array sensor localization, IEEE Trans. Signal Process. 40 (1992) 2641–2650. [3] A. Manikas, C. Proukakis, Modeling and estimation of ambiguity in linear arrays, IEEE Trans. Signal Process. 46 (August 1998) 2166–2179. [4] R.O. Schmidt, A signal subspace approach to multiple emitter location and spectral estimation, Ph.D. Thesis, Stanford University, Stanford, CA, 1981. [5] K.-C. Tan, Z. Goh, A detailed derivation of arrays free of higher rank ambiguities, IEEE Trans. Signal Process. 44 (February 1996) 351–359. [6] K.-C. Tan, G.L. Oh, M.H. Er, A study of the uniqueness of steering vectors in array steering, Signal Processing 34 (1993) 245–256.