The Use of Differential Geometry in Array Signal Processing A. Manikas† PhD DIC CEng MIEE MIEEE N. H. Dowlut PhD DIC AMIEE Digital Communications Research Section Department of Electrical and Electronic Engineering Imperial College of Science, Technology and Medicine Exhibition Road, London SW7 2BT, U.K.
Abstract This paper reports new results obtained through the application of differential geometry to the array manifold of a direction finding (DF) sensor array. It emphasises the crucial but so far disregarded role of the array manifold in the performance of subspace-based direction finding (DF) algorithms and then proceeds to a rigorous mathematical analysis of the array manifold using the tools of differential geometry. The results thus obtained are used to quantify the effects of the array manifold properties on the performance of a DF system and to design superresolution sensor arrays.
1. Introduction During the past few decades, there has been significant research into source localisation algorithms for sensor array signal processing, culminating in the development of subspace-based methods, which asymptotically exhibit infinite resolution, and are hence also referred to as “superresolution” algorithms. The operation of subspace-based direction finding (DF) algorithms essentially involves partitioning the observation space into the mutually orthogonal signal and noise subspaces and searching the array manifold for the response vectors which lie in the signal subspace or which are, equivalently, orthogonal to the noise subspace. The †
Corresponding author:
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concept of the array manifold is fundamental to the subspace class of DF algorithms and is defined as the locus of all the response vectors of the array over the feasible set of source directions-of-arrival (DOAs). In the case of an azimuth-only DF system e.g. a linear array, the array manifold is a curve in the complex 5 -dimensional space 5 , where 5 is the number of sensors of the array, while for an azimuth-elevation DF system e.g. a planar array, the array manifold is a surface in 5 . The significance of the array manifold in the performance of superresolution DF array systems strongly supports the need for a rigorous analysis. The nature of the array manifold makes the techniques of differential geometry particularly appropriate for this task. For ease of understanding, an azimuth-only DF system will first be considered. In such a case, the array manifold is a curve embedded in 5 and signal sources may then be located using a simple search of this curve. It is clear that if the manifold curve crosses over or upon itself, then the two response or manifold vectors are indistinguishable with respect to a signal subspace, although they correspond to sources impinging from distinct directions. Another unresolvable situation arises when a manifold vector is a linear combination of two or more manifold vectors. This phenomenon is known as the “ambiguity problem” and is a consequence of anomalies in the manifold curve. It may be envisaged that the longer the manifold curve, the more prone it is to such anomalies; however, this also means that the manifold vectors are more widely spaced and hence it is easier to distinguish between close sources, that is, the corresponding array displays better detection, resolution and DOA estimation capabilities. From the previous discussion, it follows that the array manifold determines the detection, resolution, DOA estimation and ambiguity performance of the corresponding array. A quantitative analysis of these effects requires the application of differential geometry techniques to the array manifold. The paper is organised as follows. In Section 2, the notational conventions used throughout the paper are presented and the far-field passive narrowband array signal processing model is defined. In addition formal definitions of the linear and planar array manifolds are provided while, in Section 3, the results of the application of differential geometry to the array manifold are presented. In particular, expressions for the arc length, rate of change of arc length, coordinate vectors and curvatures of the manifold curve are derived in terms of the physical characteristics of the array. Based on these results, the detection, resolution, DOA estimation and ambiguity performance of an array of sensors are examined in Sections 4 and 5. Finally, an array design approach tailored to
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subspace-based methods and based on the differential geometry of the array manifold is presented in Section 6. The paper is concluded in Section 7.
2 Notation and Preliminaries The following notation is used throughout the paper:
scalar vector matrix transpose complex conjugate transpose integer part
[
! !
À; À/ fix .
! ! !
exp sum
++
elemental exponentials elemental ! powers elemental sum Euclidian norm
3 4 5 7
number of snapshots number of sources number of sensors signal power noise power SNR signal-to-noise ratio , azimuth, elevation generic bearing parameter % sensor locations along %-axis % normalised sensor locations equivalent linear array , array response vector
!
"
-dimensional complex space arc length parameter ! coordinate vector ! curvature
3
manifold length matrix of coordinate vectors Cartan matrix of curvatures
o ]
!
!
The 5 d complex signal vector % ! 5 observed at the output of an array of 5 sensors operating in the presence of 4 far-field narrowband emitters impinging from bearings ~ Á Ä Á 4 ; and additive noise ! 5 can be c modelled as:
"
#
!
!
! !
!
% ! ~[ ! b ! c
!
!
1
!
where ! 4 is the vector of complex signal envelopes and [ 5d4 is c the response matrix defined as: [
c
!
! ~z " ! Á Ä Á ! #
!
2
4
with 5 , known as the “array response vector”, denoting the complex array response to a unit amplitude wavefront from direction . Note that for an azimuthonly DF system, the bearing represents the azimuth direction , and for an azimuthelevation system the bearing represents the azimuth-elevation direction Á .
!
!
For an azimuth-only ~ DF system consisting of a linear array of isotropic sensors, conventionally taken as lying along the %-axis and whose locations are given z ´ Á Á ÄÁ 5 µ; in units of half-wavelengths, the response vector by the vector % ~ is given by:
!
² ³ ~ exp c % cos
!
!
3
where exp À denotes the vector of elemental exponentials; whereas the array response vector of an azimuth-elevation ~ Á planar array DF system is given by:
!
!!
!
!
Á ~ exp c cos
4
z % cos b & sin ~
5
where
"
#
!
!
with % Á & denoting the %Á & coordinates of the sensors in half-wavelength units. Note that represents the projections of the sensors of the planar array onto the line of azimuth . Furthermore, the similarity between Equations 3 and 4 leads us to define as the “Equivalent Linear Array (ELA)” of the planar array along the direction .
4
The array manifold is defined as the locus of all response vectors over the parameter space +,
z$
!
7~ ¢ +
%
!
6
and it describes a geometric object in the complex 5 -dimensional space 5 . In the case of an azimuth-only DF system, ~ and + ~ ´ °Á °³ and the singleparameter array manifold traces out a curve in 5 , whereas for an azimuth-elevation DF system, ~ Á and + ~ ¸´ °Á ° ³Á ´ °Á °³ ¹ and the two-parameter array manifold traces out a surface in 5 . An alternative equivalent - parametrisation used in this paper is given by + ~ ¸´ °Á ° ³ Á ´ °Á °³ ¹ and as will be seen, this equivalent parametrisation is extremely useful in that it allows a unified framework for the analysis of the linear and planar array manifolds.
!
To study the influence of the array manifold on the system's DF performance, the array manifold can be analysed using the powerful mathematical tools of the differential geometry of curves and surfaces. However, to enable a common analysis of the linear and planar array manifolds, the latter can be treated as a family of curves which make up the surface. A particularly interesting family is the set of so-called “ curves” which is defined as the vector continuum described by the set of all response vectors over the whole elevation space at a particular azimuth k Á
z$
!
%
9 , k ~ k Á ¢ ´ °Á ° ³
7
!
The planar array manifold surface can then be alternatively treated as a family of curves spanning half the azimuth space:
$
%
7 ~ 9 , k ¢ k ´ °Á ° ³
!
8
From Equation 4, it is apparent that a -line is identical to the manifold of a linear array with sensor locations given by , which is hence designated as the “Equivalent Linear Array (ELA)”.
3 Differential geometry of the manifold curve The array manifold is conventionally parametrised in terms of the bearing ; however, for the purposes of studying the geometry of the linear array (LA) manifold curve or the planar array manifold -curve, parametrisation in terms of the arc length, which is the actual physical length of a manifold curve in 5 , is more suitable. The arc length is formally defined as: 5
! j ! j
z ~
!
9
and the rate of change of arc length is given by:
À ! ~z
+ À !+
!
~
10
where represents the azimuth for the LA manifold and the elevation for the curve of the planar array manifold. Using Equations 3 and 4, the following characteristics of the LA manifold and -curve can be easily derived:
! + + c cos ! I ´ °Á °³ ! ~ + +sin À !! ~~ ++ ++sin c cos ! I ´ °Á °³
À
LA
~ %
LA
%
11
!
12
!
The rate of change of arc length is a local property of the curve and it has been shown to strongly influence the DF performance of the corresponding array [1,2]. Another important characteristic of the curve is its total length which should intuitively have an impact on the ambiguity properties of the array, since ambiguities are caused by spurious intersections between the estimated signal subspace and the array manifold. From Equations 11 and 12, the total length of the LA manifold and -curve are respectively given by:
+ +I ++
!
LA ~ % ~
13
Furthermore, at every point along the manifold curve, a set of unit coordinate o ~ " Á ÄÁ " 5d c vectors and curvatures Á ÄÁ c can be defined according to:
! " ! !%
$ !
!#
oZ
!
!
! ~ o !] !
!
14
!
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 ! z ! ~z zzz Å z y
]
c
Å
!
!
c Å
!
Ä Ä Ä Æ Å Ä Ä c
Å
!
c c
{} }} }}} !|
!
15
6
with
z ! J ! z~~ +" ! !+
!
Z
"
16
Z
and is the dimensionality of the space in which the curve is embedded. It has been shown that in the case of a linear asymmetric array, which is defined as one in which no sensor has a symmetric counterpart with respect to the phase reference (taken at the array centroid), the corresponding manifold occupies the whole of 5 , that is,
~ 5 . If, on the other hand, the linear array is symmetric, that is, all the sensors occur in symmetrical pairs, then ~ 5 Â and in general, for an 5 sensor linear array, with sensors in symmetrical pairs, it can be shown that:
!
~ 5 c
17
Note that by analogy, the same results pertain to the equivalent linear array (ELA) of a planar array manifold. Equation 14 is a first order matrix differential equation whose solution can be easily derived as: o
!
! ~ o ! expm ]!
!
18
!
where o is the matrix of coordinate vectors at arc length and expm À denotes the matrix exponential. From Equation 14, and bearing in mind that the coordinate vectors are of unit length, the following expressions for the first three manifold curvatures, for instance, may be derived ¢
h h c
z h ~ +" b " + ~ c 4 b 5 h
z + + h h z ~ +"Z b " + ~ ~ "Z ~
Z
!
19
where
~
H
%
!
linear array planar array
20
In general, it can be shown that the ! curvature of a LA manifold or -curve is given by the following recursive equation [3]:
~
Äc
fix² ³b
~
² c ³c Á cb
!
21
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where
~ z~ (normalised sensor positions) h h ~£ sum² ³ ~ (i.e. phase reference ~ array centroid)
+ +
c
In Equation 21, the coefficients Á are given by:
cb
cb
~
~ b
Á ~
c
Ä
Ä Á c
Á
!
22
c ~c b
with
Á D
Á ~ Á ~
c
~
Á D
23
!
24
!
or recursively: Á ~ cÁ b c cÁc Á
with the initial conditions: ~
h h
Á ~ Á Á ~
Á
!
25
Note that the manifold curvatures 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 Performance Criteria Popular performance criteria of superresolution DF array systems include the detection, resolution and DOA estimation performance. The detection performance refers to the ability of the DF system to correctly enumerate the emitters present in the environment; this information is then used to partition the observation space into the signal and noise subspaces, which are subsequently employed in the DOA estimation
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process. The resolving power of the array is assessed by its ability to distinguish between closely spaced sources. For a spectrum-based DF algorithm such as MUSIC, two sources are said to be resolved if the spectrum exhibits two distinct peaks at, or near to, the two actual emitter directions. The DOA estimation performance on the other hand refers to the error incurred when estimating the DOAs of the emitters. This is usually assessed using the well-known Cramer-Rao bound (CRB) which represents the minimum achievable estimation error variance of any unbiased estimator. Recall that when the exact covariance matrix is available, the signal and noise subspaces can be accurately determined, and arbitrarily close emitters can be successfully detected and resolved, while their DOAs can be perfectly estimated. In practice, however, when only a finite amount of noisy data is available, it has been shown that the detection, resolution and estimation performances depend not only on the observation interval and SNR, but also on the physical characteristics of the array via the local structure of the array manifold [1,2]. For instance, the SNR thresholds of detection and resolution of an emitter of power 7 in the presence of a second close emitter of power 7 , over an observation interval of 3 snapshots, are respectively given by [1]:
SNR ! ~ 8 b n 9 3 " !
7 SNR ! ~ bn 9
8 7 V c 5 3 " ! 4
DET
RES
7 7
5
!
26
where " is the arc length separation of the two emitters, which from Equation 11 or 12, can be written as:
+ +*
" ~ cos c cos
n c
V ~ Àsin ~ À and
