Nonuniform linear antenna arrays for enhanced near

field source localization

Houcem Gazzah, Jean-Pierre Delmas

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Houcem Gazzah, Jean-Pierre Delmas. Nonuniform linear antenna arrays for enhanced near field source
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 NONUNIFORM LINEAR ANTENNA ARRAYS FOR ENHANCED NEAR FIELD SOURCE
LOCALIZATION

Houcem Gazzah Jean Pierre Delmas

Dept. of Elec. and Computer Engineering Telecom SudParis, UMR CNRS 5157
University of Sharjah, 27272, UAE 91011 Evry, France
hgazzah@sharjah.ac.ae jean-pierre.delmas@it-sudparis.eu

ABSTRACT some geometric parameters that characterize linear (but non-

Arrays of sensors freely located along an axis are considered uniform) arrays with better estimation than the ULA for near-
in this paper that are used to localize a near-field emitting field sources, while being equivalent for far-field sources.
source. Using Taylor expansion and a suitable coordinate sys- The paper is organized as follows. Sec. 2 presents the
tem, simple, yet rich to interpret, Cramer-Rao bounds relative near-field signal model and coordinate system. Sec. 3 is dedi-
to the direction and range parameters are derived. Our analy- cated to the derivation of simple and interpretable closed-form
sis allows in particular to unveil a family of non-uniform lin- expressions of the CRB on DOA and range. Then, Sec. 4 is
ear arrays with better near field estimation capabilities, com- focused on the special class of centro-symmetric arrays and
pared to the well-established uniform linear arrays. key-geometric parameters that determine their performance.
This ultimately leads to non-uniform centro-symmetric arrays
Index Terms— Direction of arrival, range, near-field, an- that achieve better accuracy than the ULA for localizing near-
tenna arrays, performance analysis, Cramer-Rao bound. field sources. A conclusion is given is Sec. 5.

1. INTRODUCTION 2. SIGNAL MODEL

Antenna arrays is a major field of statistical signal processing We consider a linear antenna array of P sensors C1 , · · · , CP
and direction finding is among the most investigated topics located along the x-axis, at coordinates x1 , · · · , xP , respec-
[1]. The complexity of the antenna array near-field propaga- tively, as shown on Fig. 1. The array centroid is assumed to be
tion model explains the limited research and the scarcity of at the origin O, which allows for a more compact expressions
results, the existing literature being mostly concerned with of the CRB, compared to [2, 6]. A source S emits a narrow-
antenna arrays far-field analysis and applications. A recent band signal s(t) (with wavelength λ) in the direction of the
interest in parameter estimation of near-field sources has antenna array. It is at a distance r from the origin O and forms
just emerged, focusing on the Cramer-Rao Bounds (CRB) of an angle θ with [O, y). At time t, the source originates at sen-
DOA and range estimated by means of the Uniform Linear sor p the snapshot yp (t) = exp (iτp ) s(t)+np (t), where np (t)
Array (ULA) [2, 3] and the Uniform Circular Array (UCA) represents the ambient additive noise, while τp =2π(SOˆ −
[4], respectively. Despite the importance and wide use of SCp )/λ is also equal to
the ULA and the UCA, these results are of limited interest,
especially if one thinks about the developing interest about r( √ ) xp x2p
irregular array geometries and methods to design advanta- τp = 2π 1 − βp , with βp =1
ˆ − 2 sin(θ) + 2 .
λ r r
geous (w.r.t. the ULA) non-uniform linear arrays [5]. This
paper addresses linear arrays with freely spaced sensors and
shows that there is a room to outperform the ULA from the
viewpoint of near-field source localization.
Our Taylor expansion of the CRB is based on the exact 
θ i Source
R r
time delay formula, similarly as [2, 3], but uses a different 
coordinate system. It judiciously leads to more compact and 
more interpretable CRB expressions that apply to arbitrary y y y y y y - x
linear arrays. For instance, we identify a class of centro- C1 C2 O Cp CP
symmetric linear arrays with better near-field estimation ca- xp -
pabilities. They include, but are not limited to, the ULA.
In fact, within centro-symmetric linear arrays, we identify Fig. 1. Source in the near-field of the arbitrary linear array.
 S S 2 −S 2 S +P S 2 −P S S
Based on N snapshots {yp (t)}p=1,...,P ;t=t1 ,...,tN , es- ˆ S3 (P S2 4 −S
where γ1 =4P 3 2 4 3 5
2 )(P S S −S 3 −P S 2 ) and
4
2 2 4 2 3
timates of both range r and DOA θ are obtained using a 3P S S S +S 3 S −P S 3 −2P S 2 S
2 3 4 2 3 2 5
variety of algorithms. Some are capable of achieving the γ2 =2
ˆ 3
S2 (P S2 S4 −S23 −P S32 )
.
CRB [7]. Indeed, we adopt the stochastic CRB as our In these CRB expressions, terms in 1/r vanish if S3 = 0
evaluation criterion, assuming the usual statistical proper- in (2) and if S3 = S5 = 0 in (3). This is not a marginal
ties about np (t) and s(t): (i) np (t) and s(t) are indepen- scenario as it concerns in particular the ULA. To cover these
dent, (ii) {np (t)}p=1,...,P ;t=t1 ,...,tN are independent, zero- cases, we prove that, for arrays with S3 = 0, we have
mean circular Gaussian distributed with variance σn2 , (iii) [ ( ) ]
2
{s(t)}t=t1 ,...,tN are assumed to be independent zero-mean 1 + 1 + 1 + P 4P S4
S4 −S2 2 sin (θ) S4 1
S2 r 2
CRB(θ) = c +o(ϵ2 ).
circular Gaussian distributed with variance σs2 (the so-called 2
cos (θ)P S2
unconditional or stochastic model). (4)
For those satisfying both S3 = 0 and S5 = 0, we have
γ3 (θ)
3. TAYLOR EXPANSION OF THE CRB CRB(r) 4c 1 + 2(P S4 −S22 )r2
= + o(ϵ2 ) (5)
r4 cos4 (θ) P S4 − S22
To derive the stochastic CRB on θ and r, we use the general
matrix expression of the stochastic CRB given for several pa- 18P 2 S42 +2S24 +3P S22 S4 −23P 2 S2 S6
where γ3 (θ)=
ˆ P S2 sin2 (θ)+3P S6 −
rameters per source [8] and adapted to a single source. This 3S2 S4 .
gives CRB(θ, r) = F−1 with

2N σs4 [ ] 4. CENTRO-SYMMETRIC ARRAYS

F= 2 2 2
P DH D − DH aaH D , (1)
σn (σn + P σs )
We define centro-symmetric arrays as those linear arrays with
iτ1 iτP T sensors disposed symmetrically w.r.t. the origin, ones with
where a = [e , ..., e ] is the steering vector and D the
coordinates of the form ±x(1) , ±x(2) , · · ·, with, possibly, a
derivative [ ∂a ∂a
∂θ , ∂r ]. A long proof (proofs can be found in [9]) sensor at the origin. Obviously, such an array verifies Si =
leads to the following expression of the 2 × 2 matrix F:
0 for any odd i. The opposite is not always true. Actually
2c S3 3P S4 − S22 we prove that1 an array is centro-symmetric iff Si = 0 for
[F]1,2 = P + sin(θ) + o(ϵ4 ) any odd i less or equal to P . We will rather focus on the
r cos3 (θ) r 3 r4
closed-form expressions (4)-(5) to analyze the impact of the
c S2 S3
[F]1,1 = P 2 + 2P sin(θ) 3 array geometry (in terms of the non-zero S2 , S4 and S6 ) on
r2 cos2 (θ) r r the array estimation performance for both DOA and range, of
[ ]
S4 P 4 sin (θ) − 1 − S22 sin2 (θ)
2
both far-field and near-field sources.
+ + o(ϵ4 )
r4
c 1 P S4 − S22 P S5 − S2 S3 4.1. Relations between far-field and near-field DOA per-
[F] 2,2 = + sin(θ)
4
cos (θ) 4 r 4 r5 formance
[ ] [ ]
P S6 23 sin2(θ)−3 −3S2 S4 5 sin2(θ)−1 +8S32 sin2(θ) We start by recalling the stochastic DOA CRB in the far-field
+ +o(ϵ6)
8r6 propagation model [11, rel. (5)] given for arbitrary linear ar-
∑P rays by
ˆ p=1 xkp (with in particular, S1 = 0) are purely
where Sk = c
CRBFF (θ) = . (6)
geometric, c= λ
ˆ 4π
2 σ 2 (σ 2 +P σ 2 )
n n s
ˆ 1r maxp |xp | and o(ϵk ) sat-
, ϵ= cos2 (θ)P S2
2 2N σs4
k k Compared to (4), we realize that arrays for which S3 = 0 (in-
isfying limϵ→0 o(ϵ )/ϵ = 0.
cluding centro-symmetric arrays) achieve CRBFF (θ) when
From [F]1,2 , the DOA is decoupled from the range to the
the source-to-array distance tends to infinity. At the same
second-order in ϵ iff S3 = 0. This class of arrays that will be
time, θ and r estimates are decoupled, as pointed out ear-
studied in details in Sec. 4 will have many advantages. For
lier. Other advantages of centro-symmetric arrays include
the moment, we give results for arbitrary values of S3 .
a larger domain of validity of our approximations as a re-
From the above, the following expressions of the CRB on
sult of the convergence in 1/r2 compared to 1/r for non
the DOA and range are obtained after tedious derivations.
centro-symmetric arrays. More unusual is the behavior of
non centro-symmetric arrays (and, in general, those with a
c 1 1 + γ1 sin(θ)
CRB(θ) = r
+ o(ϵ) (2) non-zero S3 ). Because 1 P S4 − S22 > 0, such arrays verify
P S2 − P S32 2 cos2 (θ)
P S4 −S 2 lim CRB(θ) > CRBFF (θ).
r→∞
CRB(r) 4c 1 + γ2 sin(θ)
r
= S32
+ o(ϵ)(3) 1 Proofs, detailed in [9], about S , S and S are based on the Newton-
2 4 6
r4 P S4 − S22 −P cos4 (θ) Girard Formula [10, pp. 69-74] that allows iterative calculation of Si .
S2
Now, w.r.t. [F]1,2 , θ and r are coupled to the second-order 25
θ=π/2
in ϵ. More precisely, the square of [F]1,2 and the term [F]2,2 θ=π/8
θ=π/16
tend to zero with the same speed. This is in contrast to the θ=0
20
case S3 = 0, for which the square of [F]1,2 tends to zero more
rapidly than [F]2,2 when r tends to ∞. Consequently, from
the practical point-of-view, as far as only the DOA parameter
15
is considered, the far-field propagation model, although ap-
proximative, may be preferable to the exact near-field model

fθ(κ)
for non centro-symmetric arrays with S3 ̸= 0.
10
To illustrate the different behavior of centro-symmetric
and non centro-symmetric arrays in the near-field region, we
test in Fig. 2 antenna arrays of P = 4 sensors. On the one 5
hand, the ULA with a constant inter-sensors spacing d and for
which S3 = S5 = 0. On the other hand, the minimum hole
and redundancy linear array (MHRLA) with inter-spacings d, 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
3d, 2d [12] and for which S3 ̸= 0. Thanks to a larger aper- κ

ture, the MHRLA exhibits a lower far-field CRB, for instance,

CRBMHRLA
FF (θ)/CRBULAFF (θ) ≈ 0.22. However, due to the Fig. 3. Near-field DOA estimation performance, as expressed by
coupling of θ and r in the the matrix F of the MHRLA, we fθ (κ), as function of the geometry of the centro-symmetric array, as
have limr→∞ CRB(θ) > CRBFF (θ) for this array. Further- expressed by parameter κ.
more, this figure confirms that the domain of validity of our
approximations is much larger for the centro-symmetric than range (near-field) CRBs. The following rewriting of (4-5) will
for non centro-symmetric arrays. be helpful for this purpose
[( ) ]
2.5
Exact
1
S2 + κ
1 4
1 + 1−κ sin2 (θ) + 1 P1r2
Approximate CRB(θ) = c
ULA
MHRLA
cos2 (θ)P
+o(ϵ2 ) (7)
2 [
CRB(r) 1 4 1 S2
= c 4 +
r4 cos (θ) S22 κ1 − 1 2P r2
Normalized DOA CRB

( ) 2

18 + 3κ + 2κ2 − 23
η sin2 (θ) + 3 κη −3κ
1.5
×  + o(ϵ2 ). (8)
2
(1 − κ)

where
1
S22 S3
κ=
ˆ and η = ˆ 22
P S4 P S6
appear as two key geometric parameters that determine the
0.5 near-field accuracy of the antenna array. They also can be
2 4 6 8 10 12
r / maxp(|xp|) proved to verify the following interesting properties 1

Fig. 2. Approximative [(from (2), (4)) ] and exact [deduced from η≤κ≤1
(1)] ratios CRB(θ)/CRBFF (θ) for 4-sensors ULA and MHRLA and
and a source at θ = 50◦ . 4
η≃ (3 ) for P ≫ 1.
κ − 2
5
5
While S2 determines the array far-field DOA estimation
4.2. Near-field performance performance, κ and η are all what matters for near-field DOA
and range estimation performance. In practice, only κ is im-
We have favored centro-symmetric arrays verifying S3 = 0
portant, in regard of the predominant terms in (7) and (8). It
mainly because they do not suffer degradation in the DOA
affects (8) through the increasing function 1/(1/κ − 1), and
CRB when source tends to be in the antenna far-field. Among
affects (7) through function fθ (κ)
those arrays with S3 = 0, those with identical S2 achieve the
[( ) ]
same far-field DOA CRB. Now, out from these ones, we will 1 4 2
fθ (κ)=
ˆ 1+ sin (θ) + 1 ,
be able to identify better ones, on the basis of the DOA and κ 1−κ
illustrated in Fig. 3. Considering DOA estimation, choosing can be seen as an indicator of improvement (over the ULA)
κ loosely close to 1/2 ensures limited degradation in all look whenever it is lower than one. For instance, if P ≫ 1,
directions. If κ is close to (but lower than) 1/2, then it also
4 1
ensures better estimation of the range parameter as well. RP (κ) = .
5 1
κ −1
4.3. Comparison with the ULA
This ratio is illustrated in Fig. 5 for the domain2 [0.3, 0.7] of κ
To better illustrate the impact of κ on the estimation per- outside which DOA near-field performance degrades severely
formance of both DOA and range, we compare the 6- (as clear from Fig. 3). It can be seen from Fig. 5 that the (far-
sensors ULA (with sensors placed at ±0.1195, ±0.3586 field) range CRB can be reduced by a much as 50% by an-
and ±0.5976) against a non-ULA array of 6 sensors located tenna arrays with a κ moderately lower than that of the ULA.
at ±0.1281, ±0.2396 and ±0.6528. Both arrays exhibit the
2
same S2 = 1 (and, hence, have identical far-field DOA esti- P=6
P=∞
mation CRBs). However, κ is equal to 0.577 for the ULA and 1.8 ULA
to 0.45 for the non-ULA. Non−ULA

In Fig. 4, we report the ratios 1.6

CRB(θ)|non−ULA CRB(r)|non−ULA 1.4

and ,
CRB(θ)|ULA CRB(r)|ULA 1.2

RP(κ)
calculated using the exact CRB expressions and the approx- 1

imate CRB expressions in (7) and (8). There, we can see

that while we obtain similar DOA performance (with a slight 0.8

degradation), the non-ULA array has better range estimation 0.6

capabilities. From (7), all centro-symmetric linear arrays
characterized by a given size P and a given value of S2 verify 0.4

CRB(θ)|non−ULA 0.2
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
lim = 1. κ
r→∞ CRB(θ)|ULA
However, from (8), the κ-dependent function Fig. 5. Centro-symmetric non-ULA versus ULA: Compared range
estimation performance of far-field sources.
CRB(r)|non−ULA
1
κULA −1
RP (κ) =
ˆ lim =
κ −
CRB(r)|ULA 1
r→∞ 1
5. CONCLUSION

Accurate, simple and interpretable closed-form expressions

of the CRB for both angle and range parameters of a near-field
1.1
narrow-band source have been obtained for arbitrary linear ar-
rays using the exact expression of the time delay parameter.
1 They show the exact geometric condition for the antenna ar-
ray to have an attractive behavior in its near-field: better preci-
Normalized CRB

0.9 sion and faster convergence to the far-field DOA CRB. Such a
DOA CRB
Range CRB
class of centro-symmetric arrays includes, but is not restricted
0.8 Exact to the ULAs. Furthermore, it is proved that appropriately de-
Approximate
signed centro-symmetric non-ULA can largely improve the
0.7 range estimates without deteriorating the DOA estimates un-
der near-field conditions.
0.6

0.5
0.5 1 1.5 2 2.5 3
Range r

Fig. 4. DOA and range CRBs of the non-ULA (κ = 0.45) normal- 2 In fact, extreme values of κ (i.e., 0 and 1) are achieved by impractically
ized to that of the equivalent ULA (κ = 0.577) for θ = 40◦ . Both co-localized sensors, either at the origin, or at the same distance (and on both
arrays are made of P = 6 sensors and are such that S2 = 1. sides) from the origin.
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