Array Design for Superresolution Direction-Finding ... - Manikas A

Array Design for Superresolution Direction-Finding Algorithms Naushad Hussein Dowlut BEng, ACGI, AMIEE Athanassios Manikas PhD, DIC, AMIEE, MIEEE Department of Electrical and Electronic Engineering Imperial College of Science, Technology and Medicine Exhibition Road, London SW7 2BT, U.K.

Abstract. It is well-known that the sensor characteristics and their physical arrangement place fundamental limitations on the ultimate capabilities of an array system. In this paper, the design of linear arrays, tailored to the so-called superresolution direction-finding (DF) algorithms, is considered. The proposed design approach is based on the differential geometry of the array manifold. The novel concept of a sensor locator polynomial is presented and its properties are investigated. Then it is shown how this concept can be applied to the design of a linear array with prespecified Cramer-Rao bounds on the azimuth estimates, without recourse to optimisation.

1. Introduction Direction-finding array systems are mainly concerned with the estimation of the directions-of-arrival (DOAs) of multiple emitters, present in the array environment, based on the statistics of the received signals. Currently, the most powerful DF algorithms are the socalled algorithms, which signal subspace asymptotically (i.e. infinite data or infinite Signal-toNoise Ratio) exhibit infinite resolution capabilities. However, under non-asymptotic conditions, their performance is influenced by both the behaviour of the particular algorithm employed and the array geometry. It is therefore clear that careful array design is fundamental for the proper operation of superresolution DF systems. Array design for conventional beamforming techniques, based on achieving a gain pattern with a narrow mainlobe and low sidelobes, is a well-researched topic (see for instance [1]). Sensor placement strategies tailored to the Maximum Entropy and Maximum Likelihood Methods have also been proposed [2]-[3]. Array design for the powerful signal subspace algorithms, on the other hand, has not been thoroughly investigated. The only recalled attempt is

one due to Spielman et al. [4], who showed that in the single-target scene, the design procedure for the MuSIC algorithm is equivalent to conventional beampattern design. Note that all the design procedures reviewed involve some kind of nonlinear optimisation. Signal subspace estimation techniques consist essentially of finding the intersections between the signal subspace and the array manifold, which characterises the array response over the entire parameter space, e.g. azimuths  to  for the linear array case. The array manifold is actually a vector continuum lying in the complex 5 -dimensional space ‘ 5 , where 5 is the number of sensors, and is determined by the sensor positions and characteristics. In this paper, a new array design methodology based on the differential geometry of the array manifold is proposed. In Section 2, the notational conventions used throughout this paper are introduced and the signal model for the narrowband far-field passive array signal processing problem is formulated. A formal definition of the array manifold is also presented. Section 3 contains some results of the application of differential geometry to the array manifold of a linear array of sensors ([5], [6]). The new concept of a sensor locator polynomial is presented in Section 4 and it is shown that if the length and curvatures of a manifold are known, the corresponding array can be synthesised from the roots of this polynomial. The sensor locator polynomial is then examined for various array structures in Section 5, with special attention paid to the fully symmetric and fully asymmetric linear array cases. In Section 6, the well-known Cartan Matrix in differential geometry is shown to have a special meaning to the array synthesis problem. The practical problem of designing an array when the number of sensors, and the desired performance levels are prespecified is examined in Section 7. Finally, the paper is concluded in Section 8.

Address for correspondence : Dr A. Manikas, Department of Electrical & Electronic Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BT, U.K. e-mail : [email protected]

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length in multidimensional space, is more appropriate. The arc length is formally defined as :  ~  4   or alternatively,  ~ ~  ! 5 

2. Notation and Problem Formulation

!  j !j

The following notation is used throughout the paper : 

[

; fix ! ** ++  À!  —  

]

scalar matrix transpose integer part absolute value Euclidian norm is a function of sensor locations normalised  arc length ! curvature Cartan matrix

 / exp !  sum !

[

4 5 3

 5 ‘5 "

vector hermitian transpose elemental exponential elemental ! power sum of elements ! power of matrix number of sources number of sensors number of snapshots 5 -dimensional real space 5 -complex space ! coordinate vector

!

!

À!

5

#

! % !! ~ [ ! !! b  !!

!

1

!

where  !  ‘ 4 is the vector of complex signal envelopes and [  ‘ 5 d4 is the 5 d 4 matrix defined as :

!

[

!

! ~ " !Á ÄÁ  !# z



4

2

!

with    ‘ 5 denoting the complex array response to a unit amplitude wavefront from direction  . For a linear array of omnidirectional sensors with positions given by the real vector    N (in units of halfwavelengths), the response vector is given by :

!

where exp À exponential.

² ³ ~ exp c    cos

!

3

!

denotes the element by element

Definition 1 The array manifold is defined as the locus of all response vectors  ¢     and it describes a curve in ‘ 5 .

$ !

+À +

!

where . denotes differentiation with respect to parameter . In the case of a linear array of isotropic sensors with locations  (in units of half-wavelengths), it can be shown from Equations 3, 4 and 5 that

The 5 d  complex signal vector % !  ‘ observed at the output of an array of 5 sensors operating in the presence of 4 far-field narrowband emitters impinging from the azimuth-directions ~  Á ÄÁ 4 ; with respect to the array axis, and additive noise  !  ‘ 5 can be modelled as :

"

À !

!

%

The array manifold completely characterises an array's spatial response and therefore reflects the intrinsic capabilities of the array independent of the type of processing employed. In this paper, the proposed design approach determines the sensor locations of a linear array given certain performance requirements which can be translated into desired characteristics of the array manifold.

! ~ ++  c cos ! 6! À ! ~ ++sin 7! where the initial condition ! ~  is assumed. The À rate of change of arc length, !, is a local property of

and

the curve and plays a crucial role in dictating the detection and resolution capabilities of the array [6]. Another important parameter of the array manifold curve is its total length,

 ~



! ~ ++

8

!

which increases in proportion to the sensor spacings. This parameter is important in the identification of manifold ambiguities of a linear array; if the manifold length is greater than the length of one winding (5 odd) or one half-winding (5 even) [5], then spurious intersections between the array manifold and the signal subspace is possible. Furthermore, at every point along the manifold curve, a set of  unit coordinate vectors o ~ " Ä"  ‘ 5 d and  c  curvatures  Á ÄÁ c can be defined, where   5 is the dimensionality of the manifold. The coordinate vectors and curvatures are related according to

!

! " ! oZ

!

!# $ !

!%

! ~ o !] !

9

!

where À Z denotes differentiation with respect to parameter and ] is the Cartan matrix which is a real skew-symmetric matrix of the curvatures defined as follows :

!

xz   ! c  s! c  s! ÄÄ  {} ! z~ zzz   s!  Ä  }}} Å Å Å y Å Å s!  |   

]



10

!

11

!





c

3. Differential Geometry of Array Manifold The array manifold is conventionally parameterised in terms of the azimuth angle ; however for the purposes of studying the geometry of the curve, parameterisation in terms of its arc length, which is the actual physical

with

International Symposium on Digital Signal Processing, ISDSP-96, London.

J !! ~~ +" ! !+ " 

z

z

Z

Z 

39

From Equation 9 and bearing in mind that the coordinate vectors are of unit length, the following expressions for the first three manifold curvatures, for example, may be derived for a linear array of isotropic sensors :

‚… h … ! ƒ … … 4 5 i„

++ h h + + h + + i —  

 ~ "Z ~

 —  c  —    — ~   c  b  —    

 ~ "Z b  " ~  ~ "Z b  "

12

while in general, it can be shown that the ! manifold curvature can be calculated according to the following recursive equation [5] :  ~

   Äc

‡‡‡  ‡‡

 %²°³b

c

~

~ — ~ where   £  € sum²³ ~  ( z

² c ³c Á —  cb



+ +

‡‡‡ ‡‡

13

!

i.e. phase reference ~ array centroid)

In Equation 13, the coefficients Á are given by :

 ~

Á

cb

 c

  Ä   Ä ,   c  ~  ~ b c ~c b

€

! 15!

14 or recursively,  Á ~ cÁ b c cÁc ,  € 

~  €

h h

with the initial conditions :    ~ — 16 Á ~ Á  ‚   Á ~  Note that the curvatures of a linear array of isotropic sensors depend on the relative rather than the absolute sensor spacings and are independent of the arc length parameter. This implies that the manifold curve has the shape of a circular hyperhelix lying on a complex 5 dimensional sphere of radius 5 in ‘ 5 .

!

l

4. The Sensor Locator Polynomial

It is known that for a curve embedded in a dimensional space, where   5 , only  c  manifold curvatures exist, i.e. the ! curvature vanishes and higher order curvatures are not defined. In the case of a linear array of 5 sensors, with  symmetrical sensors about the centroid, it can be shown that : 5 c  c  if E a sensor at array centroid 17 ~ otherwise 5 c  Note that a sensor at the centroid counts as a symmetrical sensor, i.e.  ~ , and therefore  is always even, and hence the zero curvature is always of even order.

!

H

Based on Equation 13, and in conjunction with the fact that  ~ , it can be shown that the roots of the polynomial :  —  ~—   c Á—  c b Á—  c c ÄbÁ °!b 18 are the normalised sensor positions. This polynomial named the “Sensor Locator Polynomial” (SLP) has coefficients Á given by Equation 14 where, for example,

!

(normalised sensor positions)

c

cb

Before proceeding further, linear arrays will be divided into three categories : 1) Fully symmetric All the sensors occur in symmetrical pairs about the centroid (e.g. Figure 1). 2) Partially symmetric At least one sensor has a symmetrical counterpart about the centroid (e.g. Figure 2). 3) Fully asymmetric No sensor has a symmetrical counterpart about the centroid (e.g. Figure 3). In the fully symmetric case, for example,  in Equation 17 is equal to 5 , the number of sensors.

!

!

‚… … ƒ … ~  ! … „

Á ~ Á

 c

 

~ c c

 

19



!

~ ~b

The roots of the sensor locator polynomial will always

occur in pairs of opposite signs (since the SLP consist  ). Then, in the case of of only the even powers of — fully asymmetric and partially symmetric arrays, the roots will actually represent two arrays which are mirror images of each other, but whose manifold curvatures are identical. It is obvious in the fully symmetric case that the roots will form a single array since the array and its mirror image are identical. In the fully asymmetric case, the set of roots will consist of two disjoint subsets corresponding to the two mirror arrays. In the partially symmetric case, however, the two subsets will overlap at the symmetric sensors. Examples 1, 2 and 3 at the end of this section, help to clarify these points. The framework and results of the previous discussion can be expressed in the following theorem :

!

$

%

Theorem 1 Given all the  c  curvatures  Á ÄÁ c and the length  of a manifold, then the locations of the elements of the array (and its mirror image) can be estimated from the following expression  20 array/mirror array/mirror ~ 

!

where  array and  mirror are two subsets of the set of the roots  of the following polymomial

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40

!

 —  ~—   c —  c b —  c c Äb °!b where

 

cb

 ~

cb

Ä

 ~

 ~ b

array

mirror

 c

array

""

21

!

mirror

Example 1 : Fully Symmetric Linear Array The normalised sensor positions of the fully symmetric linear array of Figure 1 are : —  ~ c À  Á c À Á , À , À  and its manifold curvatures, using Equation 13, are :  ~ À Á À Á À The SLP calculated from the above curvatures is given by :

"

"

#

#

 —  ! ~—   c À —   b À

with roots :

"

#

array ~ c À  Á c À Á À Á À  which are identical to the normalised sensor positions of the array. Note that the sensor at the centroid does not appear as a root since it does not affect the curvatures. However, its absence might give rise to ambiguities in the array manifold. Example 2 : Partially Symmetric Linear Array Figure 2 depicts a partially symmetric linear array, whose normalised sensor locations and curvatures are respectively given by : —  ~ " c À  Á c À Á À Á À Á À# and  ~ "À Á À Á À Á À Á À Á À Á À# The SLP is given by :

 —  ! ~—  c À  —  b À —   c À —   b À 

"

with roots :

~

##

Note that in this case, the subsets are disjoint.

mirror

array

#

f ÀÁ f À  Á f À Á f À 

array ~ c ÀÁ c À  Á À Á À  and mirror ~ c À  Á c À Á À  Á À

    Ä   c

mirror

array

~

and the subsets satisfying the conditions of Equation 21 are :

c ~c b

~  r  ~   ~ c with   + + ~ +  + ~  € sum  ! ~ sum  ! ~ 

"

with roots :

#

f À Á f À Á f À Á f À 

The subsets of the roots, which satisfy the conditions given by Equation 21 are :

array ~ " c À Á c À Á À Á À Á À # and mirror ~ " c À Á c À , c À Á À Á À #

Clearly, the two subsets overlap. Note that the minor discrepancies between the roots of the SLP and the actual normalised sensor locations are due to numerical error caused by rounding off the curvatures and the polynomial coefficients. Example 3 : Fully Asymmetric Linear Array The fully asymmetric linear array of Figure 3 has normalised sensor positions and curvatures given by : —  ~ " c ÀÁ c À  Á À Á À # and  ~ "À Á À Á ÀÁ À Á ÀÁ À Á À # respectively. The SLP corresponding to the above curvatures is :  —  ! ~—  c—  b À—   c À—   b À

The computation of the coefficients Á of the sensor locator polynomial using Equation 14 requires all the curvatures. However, in many situations only a limited set of curvatures may be known or can be estimated from the problem specifications. For instance, we will see in Section 7 that, sometimes, only the first curvature can be estimated from the problem specifications. Therefore, to allow the use of the sensor locator polynomial to situations where a subset of the curvatures is known, new expressions for the coefficients of the sensor locator polynomial, accommodating the incomplete knowledge of all the curvatures, need to be derived. This is possible only if the designs are restricted to particular configurations and a priori knowledge about the sensor positions is then exploited to derive simpler expressions. In the next section, it is shown that for the special cases of fully symmetric and fully asymmetric arrays, it is possible to compute the sensor locator polynomial coefficients with a small subset of the curvatures.

5. Fully Symmetric and Fully Asymmetric Arrays Consider the SLP ¢

!

 —  ~—   b —  c b —  c b Äb °! where

!

 ~ c 

!



Áb ~   Á  Á ÄÁ c

!

22

!

23

!

and  À is interpreted as ‘is a function of ’. The problem is to evaluate the coefficients  when not all of the curvatures are known. A useful theorem by Newton [7], states that : “The sums of the similar powers of the roots of an equation can be expressed rationally in terms of the coefficients.” Using the converse of the above theorem, it can be shown that the ! coefficient of the SLP is given by :  c 24  ~ c    ~ ²c³  z ~



~   z~  €

!



where

International Symposium on Digital Signal Processing, ISDSP-96, London.

°



~

 

~  sum



!

25

!

41

i.e.  is the sum of the ! power of all the positive or negative roots (since  is always even) of the SLP. Note that

 ~ 

 Á  Á ÄÁ 

!

26

!

!

Interestingly, the sums of similar powers of the normalised sensor locations, sum —   , can be expressed in terms of the manifold curvatures, as can be realised by expanding the explicit expressions in Equation 12 :

4 5 4 5 4 5

  ~  sum — sum —  ~   b   ~     b   b  sum —

4

!

5

4

‚… ƒ 5… „

! >D]4

!

and it can be proved that, in general, sum —   ~ c

°!b

27



 Á ÄÁ  °!c

5E ?

!

c

Á!

!

28 This implies that the sum of the ! power of the normalised sensor locations can be calculated as the first term of the  c  ! power of a matrix with identical structure to the Cartan matrix (Equation 10) but containing only the first ° c  manifold curvatures. It is worth noting that

!

!

! 4

sum —   ~   Á  Á ÄÁ  °!c

5

29

!

The next step is to find out the relationship between  and sum —   . As previously mentioned, the roots of the SLP represent a normalised array together with its mirror image about the centroid, and hence it can be deduced that for the fully symmetric and fully asymmetric arrays :

!

! I !

sym ~  sum —  asym ~ sum — 

30

!

31

!

and therefore, from Equations 26 and 29,

 ~   Á  Á ÄÁ c

!

i.e. the ! coefficient of the SLP of a fully symmetric or fully asymmetic array is a function of only the first c manifold curvatures. In particular,  °! ~   Á  Á ÄÁ  °!c and hence :

!

4

5

Theorem 2 The SLP of a fully symmetric or fully asymmetric array can be formed using only the first ° c  of the  c  manifold curvatures.

!

!

However, in the partially symmetric case, there exists no such relationship as Equation 30 between psym and   , because of the overlap of the roots, and sum — therefore, Equation 24 is not applicable. Considering the specific example of  , defined as :

!



++ it can be easily verified that, ‚…  ~ ƒ  ~ …   „ z  ~

°

~

 

~

 



sym    asym  psym   

psym

Note that the exact value of determined.

32

!

33

!

cannot be

Using Equations 24, 27 and 30, the first three coefficients of the SLP for a fully symmetric array are given by :  sym ~ c  sym ~ c    sym sym  ~ c   b  sym ~  c   34  sym sym sym  ~ c   b   b  sym   ~ c   b  c  b  

‚… … … … ! 4 5… ƒ ! ! … … … … 5 5 … „

4 4

6. Sensor Locations and the Cartan Matrix The Cartan matrix is a real skew-symmetric matrix of the manifold curvatures, with a special structure as illustrated in Equation 10. The Cartan matrix therefore contains all the information about the local behaviour of the manifold, and in the case of a linear array of isotropic sensors, it actually describes the whole manifold-hyperhelix, since the curvatures are constant and independent of the arc length parameter. Since the array manifold is in turn determined by the sensor positions and characteristics, it can be inferred that the Cartan matrix should also contain this information. The question is how it can be easily retrieved. It can be proved that there exists a direct relationship between the eigenvalues of the Cartan matrix and the normalised sensor positions, as stated in the following theorem :

!

$

%

Theorem 3 Given all the  c  curvatures  Á ÄÁ c of a manifold, the roots  of the corresponding sensor locator polynomial are simply given by :  ~  d eig ] 35 where eig ] represents the eigenvalues of the Cartan matrix.

!

!

!

The roots can then be partitioned into the normalised array-mirror pair and scaled to the specified manifold length by using Equations 21 and 20 respectively. In summary, if all the manifold curvatures are known, then it is more convenient to use Theorem 3, rather than the SLP, to design the array. However, if only a limited number of curvatures is available, as illustrated by the example in the next section, then the design should be based on the SLP concept.

International Symposium on Digital Signal Processing, ISDSP-96, London.

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7. Array Design Based on the CR Bound The most popular bound in array processing is the Cramer-Rao Bound (CRB), which represents the minimum estimation error variance achievable by any unbiased estimator. In other words, the CRB is estimator-independent, and it can be shown to depend on the number of snaphots 3, the Signal-to-Noise ratio, SNR, and the sensor placement and characteristics. Its use for the design of arrays for superresolution algorithms is justified by the fact that it is asymptotically attainable by the MuSIC algorithm [8]. The explicit expressions for the CRB in terms of the array manifold characteristics, and hence, the array geometry, derived in [9], will be usedÀ Assuming the phase reference to be taken at the array centroid, the differential geometric versions of the one-source and two-source CRB's are respectively given by :  CRB ~ 36 LSNR   and CRB  ~ 37 3 d SNR  "   c 5

!

!

+ +*

"

where

À !

À ! !4

~   cos  c cos 

*

! 5 ! 38!

Note that Equation 37 is an excellent approximation of the exact CRB expression proposed in [8] for two close emitters at  and  ~  b " respectively. Next, consider the following design example : Design a symmetrical -sensor linear array which exhibits a one-source CRB of À ° on the standard deviation of the DOA estimate of an emitter at  °, and a two-source CRB of À ° on the standard deviation of the DOA estimate of the same emitter, in the presence of a second emitter at  °, using a number of snapshots and SNR product, 3 d SNR ~  ²e.g. 3 ~  and SNR ~ ³. Note that under the same conditions, a standard sensor uniform linear array, would exhibit CRB ~ À ° and CRB ~ À °. From the design specifications and in conjunction with Equations 36 and 37, the following properties of the array to be designed can be calculated :

l

l

H ++ ~~ À  À 

39



!

and it is clear that in this case only the first curvature is available. The sensor locator polynomial, computed using Equations 34, is given by the following expression : —  b À

40  —  ~—   c À  with roots  ~ fÀ Á fÀ   , which, together with a sensor at the origin, actually constitute the normalised version of the desired array. After scaling, the designed array is :  ~ c À Á c À

Á Á À

Á À ;

!

"

"

!

#

#

As a check of the validity of the design procedure, the CRBs of the proposed array are computed using the exact expressions derived in [8] : CRB ~ À ° and CRB ~ À ° and are found to match the specifications very closely.

l

l

8. Conclusion The important problem of linear array design tailored to superresolution DF algorithms has been addressed. The design approach is based on the properties of the array manifold, which reflects the intrinsic capabilities of the array. The proposed concepts were supported by a number of examples and the connection between the sensor locator polynomial and the Cartan matrix was established. Current work is focussed on extending the concepts presented in this paper and deriving new superresolution design rules for general array geometries.

References [1]

[2]

[3]

[4]

[5]

[6]

[7] [8]

[9]

Y. T. Lo, “Aperiodic Arrays,” in Antenna Handbook, Theory, Applications, and Design, Y. T. Lo and S. W. Lee, Editors, Van Nostrand, New York, 1988. S. W. Lang, G. L. Duckworth and J. H. McClellan, “Array Design for MEM and MLM Array Processing,” IEEE ICASSP-81 Proceedings, Vol. 1, pp. 145-148, March 1981. X. Huang, J. P. Reilly and M. Wong, “Optimal Design of Linear Array of Sensors,” IEEE ICASSP-91 Proceedings, pp.1405-1408, May 1991. D. Spielman, A. Paulraj and T. Kailath, “Performance Analysis of the MUSIC Algorithm,” IEEE ICASSP-86 Proceedings, pp. 1909-1912, April 1986. I. Dacos and A. Manikas, “The Use of Differential Geometry in Estimating the Manifold Parameters of a One-Dimensional Array of Sensors,” Journal of the Franklin Institute, Engineering and Applied Mathematics, Vol. 332B, No. 3, pp. 307-332, 1995. A. Manikas, H. R. Karimi, I. Dacos, “Study of the Detection and Resolution Capabilities of a One-Dimensional Array of Sensors by Using Differential Geometry,” IEE Proceedings on Radar, Sonar and Navigation, Vol. 141, No. 2, pp. 83-92, Apr. 1994. W. S. Burnside and A. W. Panton, Theory of Equations, Vols I & II, Dover Publications Inc., New York, 1960. P. Stoica and A. Nehorai, “MUSIC, Maximum Likelihood and Cramer-Rao Bound,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 37, No. 5, pp. 720-741, May 1989. H. R. Karimi and A. Manikas, “The Manifold of a Planar Array and its Effects on the Accuracy of Direction-Finding Systems,” accepted for publication in IEE Proceedings on Radar, Sonar and Navigation, Apr. 1996.

Figure 1. Fully Symmetric Linear Array

Figure 2. Partially Symmetric Linear Array

Figure 3. Fully Asymmetric Linear Array

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