CRB analysis of near-field source localization using uniform circular arrays

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CRB analysis of near-field source localization using

uniform circular arrays

Jean-Pierre Delmas, Houcem Gazzah

To cite this version:

Jean-Pierre Delmas, Houcem Gazzah. CRB analysis of near-field source localization using uniform

circular arrays. ICASSP 2013 : 38th IEEE International Conference on Acoustics, Speech and Signal

Processing, May 2013, Vancouver, Canada. pp.3996 - 4000, �10.1109/ICASSP.2013.6638409�.

�hal-01277795�

CRB ANALYSIS OF NEAR-FIELD SOURCE LOCALIZATION USING UNIFORM CIRCULAR

ARRAYS

Jean-Pierre Delmas

Telecom SudParis, UMR CNRS 5157

91011 Evry, France

jean-pierre.delmas@it-sudparis.eu

Houcem Gazzah

Dept. of Elec. and Computer Engineering

University of Sharjah, 27272, UAE

hgazzah@sharjah.ac.ae

ABSTRACT

This paper is devoted to the Cramer Rao bound (CRB) on the azimuth, elevation and range of a narrow-band near-field source localized by means of a uniform circular array (UCA), using the exact expression of the time delay parameter. After proving that the conditional and unconditional CRB are gen-erally proportional for constant modulus steering vectors, we specify conditions of isotropy w.r.t. the distance and the num-ber of sensors. Then we derive very simple, yet very accurate non-matrix closed-form expressions of different approxima-tions of the CRBs.

Index Terms— Cramer Rao bound, near field source

lo-calization, uniform circular array, isotropy.

1. INTRODUCTION

A considerable literature has been dedicated to the CRB on the direction of arrival (DOA) of narrow-band sources. How-ever, most of it is limited the far-field case, meaning that a planar wavefront is impinging on the sensors array. The near-field assumption implies that the curvature of the waves has to be taken into account, resulting into more complicated mod-els parameterized by the source DOAs and range. Hence, we can identify only very few studies of this kind. Inspired by the subspace-based DOA algorithms, early ones were based on an approximate propagation model based on second-order Taylor expansion of the time delay parameter [1, 2]. Only lately has the exact time delay formula been used [3] for the only uniform linear arrays (ULA).

The UCA has been popular with many applications [4, 5, 6] and dedicated estimation algorithms have been proposed for far-field source localization [12, 13]. Isotropy is a com-pelling feature of UCA. Not only does it mean that the CRB on the azimuth and elevation of a single source is uniform for all azimuths [7], but also that the two estimates are uncorre-lated and that the CRB is not affected by the indeterminacy about the wave speed of propagation [8]. We are interested in analyzing the behavior of UCA for near-field sources, and in particular to what extend is isotropy maintained. Further-more, very simple non-matrix closed-form expressions of

ap-proximations of the CRB on the azimuth, elevation and range are given, where the sixth-order approximation of the CRB on the azimuth turns out to be very accurate.

The paper is organized as follows. Section 2 formulates the problem and specifies the data model. Section 3 is devoted to the CRB derivations. After proving that the so-called con-ditional and unconcon-ditional CRB are generally proportional for constant modulus steering vectors, we specify conditions of isotropy w.r.t. the range and the number of sensors. We derive very simple non-matrix closed-form expressions of different approximations of the CRB. Finally some numerical illustra-tions are given in Section 4.

2. DATA MODEL

A UCA is made of P identical and omni-directional

sen-sors C1, · · · , CP located in the (x, y) plane, at a distance

r0from the originO and an angle, respectively, θ1, · · · , θP

from[O, x) such that θp def= θ − 2π(p−1)_P . It is radiated by a

narrow-band source located in the antenna near-field, whose position is characterized by azimuthθ, elevation φ and range r as shown in Fig.1

z

φ

y

r

S(r, θ, φ)

C

C

C

x

C

θ

r

Fig.1 Uniform circular array and source DOAs.

Corrupted by an additive noise with complex envelope n(k),

the complex envelope of the signals collected by the array of sensors is modelled as

where a(α) is the so-called steering vector parameterized by α= [θ, φ, r]T. Referenced to the origin, its is given by

a_{(α) = [e}iτ1(α)

, ..., eiτp(α)_{, ..., e}iτP(α)_]T

whereτp(α) = 2πd_λp₀ withdp = SO − SCp andλ0 is the

propagation wavelength. In fact,τpis given by

τp(α) = 2π r λ0 1 − r 1 − 2r0 r cos θpsin φ + r2 0 r2 ! .

The usual statistical properties about nk andsk are the

following: (i)sk and nk are independent, (ii)(nk)k=1,...,K

are independent, zero-mean circular Gaussian distributed with covarianceσ2

nIP, (iii)(sk)k=1,...,K are assumed to be

either deterministic unknown parameters (the so-called con-ditional or deterministic model), or independent zero-mean circular Gaussian distributed with varianceσ2

s (the so-called

unconditional or stochastic model).

3. CRB DERIVATION 3.1. General expression of the CRB

General compact expressions of the CRB, concentrated on the DOA parameter alone have been derived for these two models of sources (see e.g. [9] for one parameter per source and [10, Appendix D] for several parameters per source). Specialized to a single source with one or several parameters per source, these expressionsCRBdet(α) and CRBsto(α) have all been

studied independently in the literature (see e.g. the recent pa-pers [11] and [3] which even concludes by ”extension of this work for stochastic sources is under consideration”).

In fact, it can be proved (proof is not given here by lack of space) that the expressionsCRBdet(α) and CRBsto(α) for

arbitrary parametrization α = [α1, ...., αL]T of the steering

vector a(α) related to array geometry or polarization such

thatka(α)k2 _{= P , are proportional}1_{, an issue previously}

overlooked. More precisely:

CRBsto(α) =  1 + σ 2 n P σ2 s  CRBdet(α) = [FIM(α)]−1 (1) whereσ2 s def = _K1 PK

k=1|s2k| for the deterministic model of

sources, with FIM(α) = cσ P DH(α)D(α) − DH(α)a(α)aH(α)D(α)
(2) wherecσ def= 2Kσ 4 s σ2 n(σ 2 n+P σ 2 s)and D(α) def = h∂a(α)_∂α1 , ...,∂a(α)_∂α L i . In the addressed problem, D(α) =h∂a(α)_∂θ ,∂a(α)_∂φ ,∂a(α)_∂r i= [a′θ, a

′

φ, a

′

r]. Thanks to (1), we only consider the stochastic

source model.

1_{Note that (1) is no longer valid for parametrizations for whichka(α)k}

depends on α.

3.2. Exact Fisher information matrix

The elements of the FIM (2) are, fori, j = 1, 2, 3, given by FIMi,j(α) = cσ[P (a ′_H αia ′ αj) − (a ′_H αia)(a ′_H αja) ∗_], where a′_αH_ia′ αj = PP p=1τ ′ p,αiτ ′ p,αj, a ′_H αia= PP p=1τ ′ p,αi, and τp,α′ i def = ∂τp(α) ∂αi . Consequently, we get FIM1,1(α) = c ′ σsin2φ  P P X p=1 u2p− P X p=1 up !2  FIM2,2(α) = c ′ σcos2φ  P P X p=1 v2 p− P X p=1 vp !2  FIM1,2(α) = c ′ σ sin(2φ) 2 " _P X p=1 up P X p=1 vp− P P X p=1 vpup # FIM3,3(α) = c ′′ σ  P P X p=1 (1 + wp)2− P X p=1 [1 + wp] !2  FIM1,3(α) = c ′′′ σ sin φ " _P X p=1 up P X p=1 wp− P P X p=1 upwp # FIM2,3(α) = c ′′′ σ cos φ " _P X p=1 vp P X p=1 wp− P P X p=1 vpwp #

where up def= sin θp/pβp, vp def= cos θp/pβp, wp def=

ǫvpsin φ − βp−1/2, βp def= 1 − 2ǫ cos θpsin φ + ǫ2 and

ǫdef= r0

r. There, we also define constantsc

′ σ def = cσ  2πr0 λ0 2 , c′′σ def = cσ  2π λ0 2 andc′′′σ def = cσ  2π λ0 2 r0.

Taylor series expansions of1/βp and1/pβpw.r.t. ǫ =

r0/r, followed by elementary trigonometric relations, show

that all elements of the FIM depend on the azimuthθ only

through the sumsPP

p=1cos kθp and

PP

p=1sin kθp fork =

1, 2, ..., which can be easily simplified thanks to

P X p=1 eikθp₌  P eikθ _{if k/P ∈ N} 0 otherwise (3)

It turns to be that while the FIM is periodic inθ of period 2π/P , as one may expect, it is not constant anymore, i.e. the

UCA is no longer strictly isotropic in the near-field.

3.3. Isotropy of the UCA

We analyze the way isotropy, guaranteed at the antenna far-field [7], is deteriorated as a result of a decreasing source range r or a decreasing number of sensors P . Careful

w.r.t.ǫ, where only the θ dependence is retained, yields to FIMi,j(α) = ∞ X k=0 P X p=1

g(i,j)_k (cos θp, sin θp)

!

ǫk, (4)

whereg_k(i,j) is a polynomial expression ofcos θp andsin θp

of degreek + 2, k + 1 or k for (i, j = 1, 2), (i = 1, 2, j = 3)

or(i = j = 3) respectively. By linearizing this polynomial,

we have for example fori, j = 1, 2:

g(i,j)_k (cos θp, sin θp) = k+2 X ℓ=0 c(i,j)_ℓ,k cos(ℓθp)+ k+2 X ℓ=1 s(i,j)_ℓ,k sin(ℓθp)

wherec(i,j)_0,k = 0 for odd degrees of g(i,j)_k . Then, using (3), fo-cusing onθ and carefully studying the first terms of the Taylor

expansion inǫ (4), we obtain FIMi,j(α) = ⌊(P −3)/2⌋ X k=0 bi,j_2kǫ2k+ ∞ X k=P −2 bi,j_k (θ)ǫk (5) FIMi,j(α) = ∞ X k=P −2 bi,j_k (θ)ǫk ₍₆₎ FIM2,3(α) = P −1 X k=3 bi,j_k ǫk₊ ∞ X k=P bi,j_k (θ)ǫk ₍₇₎ FIM3,3(α) = ⌊(P −1)/2⌋ X k=2 bi,j_2kǫ2k+ ∞ X k=P bi,j_k (θ)ǫk, (8) forP ≥ 4 and i = j = 1 or i = j = 2 (5), i = 1, j = 2, 3

(6), where the termsbi,jk andb i,j

2k do not depend onθ.

Note that zero-order terms of the block[FIM(α)](1;2,1:2)

derived from (5) and (6) give theFIM(θ, φ) of the far-field

case. Using the decoupling ofθ and φ of the far-field case

that we uncover by (6), we haveCRBFF(θ) = 1/b1,10 and

CRBFF(φ) = 1/b2,20 .

To illustrate the behavior of the FIM w.r.t. P , FIM1,1(α)

cσcr0sin2φ is given by the following expressions forP = 3, 4, 5, 6:

P

3 1 − ǫ2

sin φ cos 3θ + o(ǫ2

) 4 1 − ǫ2 (cos2 φ − sin2 φsin 4θ) + o(ǫ2 ) 5 _{1 − ǫ}2 cos2 φ − ǫ3 sin3 φcos 5θ + o(ǫ3 ) 6 1 − ǫ2 cos2 φ − ǫ4 (1 − 3 sin2 φ+ 2 sin4 φ − sin 4 φcos 6θ) + o(ǫ4 ) withcr0 def = 2πr0 λ0 2 P2

2 and where limǫ→0o(ǫ)/ǫ = 0.

Consequently, from (5)-(8), the following can be concluded: For a numberP of sensors, the FIM of the UCA does not

depend on the azimuth up to the orderP − 3 in r0/r. More

precisely, we see from (6), that the parametersθ and (φ, r)

are decoupled up to the orderP − 1 in r0/r. Consequently,

the azimuth CRB does not depend on the azimuth up to the

orderP − 1 in r0/r, in contrast to the elevation and range

CRBs that require an order smaller or equal toP − 3. So, for r or P fixed, isotropy increases when P or r increases,

re-spectively, and the azimuth CRB is much less sensitive to the azimuth than those of the elevation and range.

3.4. Closed-form expression of the CRB

To further improve our understanding of the near-field CRBs, we investigate different closed-form expressions of the ap-proximations of the CRB on α. Note that to obtain these approximations, we have to elaborate a little bit the neces-sary order of the Taylor expansions of each term ofFIM(α)

because the order inǫ of these terms may be different.

ForP > 8, we have proved that2

FIM1,1(α) cσcr0 = sin2φ1 − ǫ2_cos2_{φ + ǫ}4_g 1(sin2φ) +ǫ6g2(sin2φ) + o(ǫ7) FIM1,2(α) = o(ǫ7) FIM1,3(α) = o(ǫ7) FIM2,2(α) cσcr0 = cos2_φ  1 − ǫ2  1 − 5 2sin 2_φ_{+ o(ǫ}3₎ FIM3,3(α) cσcr0 = sin 2_φ 16r2 0 sin2_{φ + ǫ}2_g 3(sin2φ) ǫ4+ o(ǫ7) FIM2,3(α) cσcr0 = −sin φ cos φ r0  1−7 8sin 2_φ  ǫ3+ o(ǫ3), with g1(sin2φ) def = 1 − 3 sin2φ + 2 sin4φ g2(sin2φ) def

= −1 + 6 sin2φ − 10 sin4φ + 5 sin6φ g3(sin2φ)

def

= 16 − 33 sin2φ +35 2 sin

4_φ.

This expression of FIM(α) allows us to derive after some

manipulations ofFIM−1(α) the following closed-form

ex-pressions of the CRBs: CRB(θ) = CRBFF(θ)  1 + cos2φr 2 0 r2 + sin2_{φ cos}2_φr40 r4 + h1(sin 2_φ)r06 r6 + o r7 0 r7  (9) CRB(φ) = CRBFF(φ)  1+h2(sin2φ)r 2 0 r2 + o r2 0 r2  (10) CRB(r)= 16 cσcr0sin 4_φ r4 r2 0  1+h3(sin2φ) r2 0 r2 +o r2 0 r2  , (11) where CRBFF(θ) = 1/(cσcr0sin 2_φ) CRBFF(φ) = 1/(cσcr0cos 2_φ) 2_{ForP > 6, FIM} 1,1(α) = cσcr0sin 2 φ[1−ǫ2 cos2 φ+ǫ4 g1(sin2φ)+ o(ǫ5

) and FIM1,2(α) = FIM1,3(α) = o(ǫ5) and the other FIM expressions

denote the CRB on the azimuth and elevation in the far field case [7], and where we have introduced

h1(sin2φ) def

= − sin2φ + 3 sin4φ − 2 sin6φ h2(sin2φ) def= 16 sin2φ+ 39 4 sin 2_{φ − 27} h3(sin2φ) def= 5 − 21 4 sin 2_φ.

Using the decoupling between θ and (φ, r), note that the

second-order (resp. fourth-order) expansion in r0/r of

CRB(θ) in (9) is still valid for only P > 4 (resp. P > 6).

Interestingly, when the source is known to be in the(x, y)

plane, we deduce from theFIM(α) for φ = π/2 that CRB(θ) = CRBFF(θ)  1 + o r 7 0 r7  CRB(r) = 16 cσcr0 r4 r2 0  1 − r 2 0 2r2 + o  r2 0 r2  ,

where here CRBFF(θ) = 1/cσcr0. Note that the first or-der termr0/r vanishes in all the expressions of the CRB for

P > 6. Furthermore, for a source located in the plane (x, y),

the CRB on the azimuth is very insensitive to the range. This contrasts with the near field expression of the CRB for the ULA [3] for which, CRB(θ) includes a first order term in r0/r and thus is much more sensitive to the range than for

the UCA. Finally, note the simplicity of our closed-form ex-pressions w.r.t. the complicated non interpretable closed-form expressions obtained for the ULA [2, 3].

4. ILLUSTRATIONS

In order to characterize the level of isotropy, we introduce a non-isotropy criterion

ρ = sup

θ

|CRB(θ) − CRB(θ)| CRB(θ)

(whereCRB(θ) denotes the mean of CRB(θ) w.r.t. θ) that

we illustrate in Fig.2 for a UCA with half-wavelength inter-sensors spacing, and a source emitting at1 MHz3_{. This figure}

shows that the isotropy is much more sensitive toP than to r.

The validity of some approximate closed-form expres-sions of the CRB is illustrated for a source located with an azimuthθ = 70◦ _{and elevation φ = 70}◦_{. Fig.3 and 4}

compare the approximate ratios CRB(θ)/CRBFF(θ) and

CRB(φ)/CRBFF(φ) given by (9) and (10) to the exact ones

and Fig.5 compares the approximateCRB(r) (11) to the

ex-act one. We see from these three figures that the proposed approximations are very accurate from the ratior/r0 = 2 for

P = 10 and from r/r0= 4 for P = 7.

3_{Note that these characteristics impact onlyr}

0, but not the different ratios

of CRB. 2 3 4 5 6 7 8 9 10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 r/r 0 P=4 P=6 P=8 P=10

Fig.2 Non-isotropy criterionρ w.r.t. r and P .

2 3 4 5 6 7 8 9 10 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 1.045 1.05 r/r0 P=7. Exact CRB P=7. Approx. CRB P=10. Exact CRB P=10. Approx. CRB

Fig.3 Approximate and exact ratiosCRB(θ)/CRBFF(θ).

2 3 4 5 6 7 8 9 10 0.9 0.95 1 1.05 1.1 1.15 r/r0 P=7. Exact CRB P=7. Approx. CRB P=10. Exact CRB P=10. Approx. CRB

Fig.4 Approximate and exact ratiosCRB(φ)/CRBFF(φ).

2 3 4 5 6 7 8 9 10 0.8 0.85 0.9 0.95 1 1.05 r/r0 P=7 P=10

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