1

Exact Conditional and Unconditional Cramèr-Rao
Bounds for Near Field Localization

arXiv:1303.1725v2 [] 8 Mar 2013

Youcef Begriche1 , Messaoud Thameri1 and Karim Abed-Meraim2

Abstract
This paper considers the Cramer-Rao lower Bound (CRB) for the source localization problem in the near field.
More specifically, we use the exact expression of the delay parameter for the CRB derivation and show how this
‘exact CRB’ can be significantly different from the one given in the literature and based on an approximate time
delay expression (usually considered in the Fresnel region). This CRB derivation is then generalized by considering
the exact expression of the received power profile (i.e., variable gain case) which, to our best knowledge, has
been ignored in the literature. Finally, we exploit the CRB expression to introduce the new concept of Near Field
Localization (NFL) region for a target localization performance associated to the application at hand. We illustrate
the usefulness of the proposed CRB derivation and its developments as well as the NFL region concept through
numerical simulations in different scenarios.

Index Terms
Cramèr-Rao Bound, Near field, Source localization, Fresnel versus Localization region.

1

TELECOM ParisTech, TSI Department, France. e-mail:Youcef.Begriche@ieee.org

1

TELECOM ParisTech, TSI Department, France. e-mail:thameri@telecom-paristech.fr

2

PRISME Laboratory, Polytech’Orléans, Orléans University, France. e-mail: karim.abed-meraim@univ-orleans.fr

2

I. I NTRODUCTION
Sources localization problem has been extensively studied in the literature but most of the research works are
dedicated to the far field case, e.g., [1], [2].
In this paper, we focus on the situation where the sources are in a near field region which occurs when the source
ranges to the array are not ‘sufficiently large’ compared with the aperture of the array system [3], [4]. Indeed,
this particular case has several practical applications including speaker localization and robot navigation [5], [6],
underwater source localization [7], near field antenna measurements [8], [9] and certain biomedical applications,
e.g., [10]. Recently, some works considered both far filed and near filed localization [11], [12] where the authors
propose different methods to achieve a better localization performance when the source moves from far field to
near field and vice versa.
More specifically, this paper is dedicated to the derivation of the Cramer Rao Bound expressions for different
signal models and their use for the better understanding of this particular localization problem. CRB derivation
for the near field case has already been considered in the literature [3], [13], [14], [15]. In [3], [13], the exact
expression of the time delay has been used to derive the unconditional CRB in matrix form, i.e., expressed and
computed numerically as the inverse of the Fisher Information Matrix (FIM). In [14], the conditional CRB based
on an approximate model (i.e., approximate time delay expression as shown in Section II-A) is provided.
Recently, EL Korso et al. derived analytical expressions of the conditional and unconditional CRB for near field
localization based on an approximate model [15]. In the latter work both conditional and unconditional CRB of
the angle parameter are found independent from the range value and are equal to those of far field region.
In our work, we propose first, to use the exact time delay expression for the derivation of the conditional and
unconditional CRB and provide closed form formula that are compared to those given in [15]. The development
(i.e., Taylor expansion) of the exact CRB allows us to highlight many interesting features including: (i) a more
accurate approximate CRB for the exact model as compared to the CRB based on an approximate model, (ii) and
a detailed analysis of the source range parameter effect on its angle estimation performance1 .
Secondly, we take into consideration the spherical form of the wavefront into the power profile. Indeed, in the
1

Part of this work has been published in [16]

3

near field case, the received power is variable from sensor to sensor which should be taken into account in the
data model. By considering such variable gain model, we generalize the previous CRB analysis and investigate the
impact of the gain variation onto the localization performance limit.
Finally, we propose to exploit the exact CRB expression to specify the ’near field localization region’ based on
a desired localization performance. In that case, the ’near field localization region’ is shown to depend not only on
the source range parameter and array aperture but also on the sources SNR and observation sample size.
The paper is organized as follows: Section II introduces the data model and formulates the main paper objectives.
Section III, the exact conditional and unconditional CRB derivations and their Taylor expansions are provided for
the equal gain case. Section IV generalizes the previous analysis to the variable gain model and highlights the
differences between equal and variable gain cases. In Section V, we introduce the concept of near field localization
region and illustrate its usefulness through specific examples. Section VI is dedicated to simulation experiments
while Section VII is for the concluding remarks.

II. P ROBLEM

FORMULATION

A. Data model
In this paper, we consider a uniform linear array with N sensors receiving a signal, emitted from one source
located in the near field region, and corrupted by circular white Gaussian noise vn of covariance matrix σ 2 IN . The
nth array output, n = 0, · · · , N − 1, is expressed as
xn (t) = s(t)ejτn + vn (t)

t = 1, · · · , T

where s(t) is the emitted signal. The exact expression of the time delay2 τn is given by
!
r
2πr
n2 d2 2nd sin θ
τn =
−1
1+ 2 −
λ
r
r

(1)

(2)

where d is the inter-element spacing, λ is the propagation wavelength and (r, θ ) are the polar coordinates of the
source as shown in Fig. 1. In the literature, the near field region (also called Fresnel region) is given by [17]
0.62
2



d3 (N − 1)3
λ

The first sensor, n = 0, is considered for the time reference.

 21

<r<2

d2 (N − 1)2
λ

(3)

4

Fig. 1.

Near field source model

Most existing works on near field source localization consider the following approximation of the time delay
expression to derive simple localization algorithms as well as CRB expressions, e.g., [18] [15]
dn
d2 n2 1
τn = −2π
sin(θ) + π
cos2 (θ) + o
λ
λ r

 
d
r

(4)

In the sequel, the source signal will be treated either as deterministic (conditional model) or stochastic (unconditional model). Indeed, in the array processing, both models can be found:
1) Conditional model in which the source signal is assumed deterministic but its parameters are unknown.
2) Unconditional model in which we assume that the source signal is random. In our case, we will assume s(t)
to be a complex circular Gaussian process with zero mean and unknown variance σs2 .

B. Objectives
1) Based on the exact expression of the time delay τn in equation (2), we aim to derive the exact conditional and
unconditional CRB and compare them with the existing CRB given in [15] and derived from the approximate
model in (4).
2) The distance from the source to the nth sensor dn is a function of the sensor position (i.e., sensor index)
according to
p

r 2 − 2ndr sin(θ) + n2 d2
s
 2
nd
nd
= r 1−2
sin(θ) +
r
r

dn =

(5)

5

where r is the distance from the source to the reference sensor (i.e., for n = 0, d0 = r ). Hence, the
received power profile is variable from sensor to sensor. In the far field case, this variation is negligible but
not necessarily in the near field context. To the best of our knowledge, this point has not been taken into
consideration in the existing literature. We would like to investigate the impact of such gain variation into
the localization performance limit.
3) The near field region has been so far assimilated to the Fresnel region which depends on the antenna size
and the signal wavelength only. However, the localization performance depends on other system parameters
(SNR, sample size, angle position, · · · ) for which reason we introduce the concept of near field localization
region where the localization error is upper bounded by a desired threshold value depending on the considered
application.

III. C ONDITIONAL

AND UNCONDITIONAL

CRB

DERIVATION WITH

E QUAL G AIN (EG)

Here, we do not consider the variation of the power profile which is equivalent to assuming that all sensors
have equal gain as shown in equation (1). The goal of this Section is to derive the conditional and unconditional
CRB for the angle and range parameter estimation of near field source and to make comparison with the results
presented in [15].

A. Exact conditional CRB with EG
In this Section, we consider the source signal as deterministic according to the model

s(t) = α(t)ej(2πf0 t+ψ(t))

where f0 is the known carrier frequency while α(t) and ψ(t) are the unknown amplitude and phase parameters of
the source signal. Under the data model assumption of Section II-A, we derive next the exact deterministic (i.e.,
conditional) CRB for the location source parameter estimation.
1) Exact conditional CRB derivation: In this deterministic case, the log-likelihood function of the observations
is given by
l(ξ) = −N T ln π − N T ln σ 2 −

1
kx − µk2
σ2

6

where
x = [xT (1) · · · , · · · , xT (T )]T
xT (t) = [x0 (t), · · · , xN −1 (t)]T
µ = [s(1)aT (θ, r), · · · , s(T )aT (θ, r)]T
a(θ, r) = [1, ejτ1 (θ,r) , · · · , ejτN −1 (θ,r) ]T
ξ = [θ, r, ΨT , αT , σ 2 ]T
Ψ = [ψ(1), · · · , ψ(T )]T
α = [α(1), · · · , α(T )]T

where k.k refers to the Frobenius norm and [.]T is the transpose operator.
The CRB is equal to the inverse of the Fisher Information Matrix (FIM) defined by
[FIM(ξ)]i,j = E



∂l(ξ) ∂l(ξ)
∂ξi ∂ξj



(6)

the latter is given in our particular case by
[FIM(ξ)]i,j =

2
∂µH ∂µ
N L ∂σ 2 ∂σ 2
+ 2 Re{
}
4
σ ∂ξi ∂ξj
σ
∂ξi ∂ξj

(7)

where Re{.} refers to the real part of a complex valued entity. The FIM is given by the block diagonal matrix as


 Q


FIM = 



0

2N
σ 2 IT

0 







NT
σ4

Indeed, it is shown in [15], that the FIM of the desired location parameters is decoupled from the noise variance
σ 2 and the sources magnitude parameters α. Consequently, the CRB of the range and angle parameters is equal to

the 2 × 2 top left sub-matrix of the inverse matrix

Q−1

 fθθ fθr


=
 frθ frr


fψθ fψr

−1

fθψ 


frψ 



Fψψ

7

where
fθθ = 2T DSNR kτ̇ θ k2
frr = 2T DSNR kτ̇ r k2
frθ = fθr = 2T DSNR (τ̇ H
θ τ̇ r )
kαk2
(deterministic SNR)
T σ2


∂τ0
∂τN −1 T
,...,
=
∂θ
∂θ


∂τ0
∂τN −1 T
=
,...,
∂r
∂r

DSNR =
τ̇ θ
τ̇ r

Moreover, the vectors of size T × 1, fψθ , fTθψ , fψr and fTrψ are given by
2 T
(1 τ̇ θ )(α ⊙ α)
σ2 N
2
= 2 (1TN τ̇ r )(α ⊙ α)
σ

fψθ = fTθψ =
fψr = fTrψ

where ⊙ represents the Hadamard product and 1N is the all-one vector of size N × 1. Finally, matrix Fψψ of size
T × T , is given by

Fψψ =

2N
diag(α ⊙ α)
σ2

which is the diagonal matrix formed by vector α ⊙ α.
By using the block structure of the FIM and the Schur’s matrix inversion lemma [19], we obtain the closed form
expression of the CRB given by the following lemma:
Lemma 1: The non-matrix expressions of the exact conditional CRB in the equal gain case for a source in the
near field, for N ≥ 3 and θ 6= ± π2 , are given by


Eeg (θ)
1
c
(8)
CRBeg (r) = 2T D
Eeg (θ)Eeg (r)−Eeg (r,θ)2
SNR


Eeg (r)
CRBceg (θ) = 2T D1
(9)
Eeg (θ)Eeg (r)−Eeg (r,θ)2
SNR


Eeg (r,θ)
(10)
CRBceg (r, θ) = 2T D1
Eeg (θ)Eeg (r)−Eeg (r,θ)2
SNR
where CRBceg (r, θ) is the non diagonal entry of the considered 2×2 CRB matrix (it represents the coupling between

8

the 2 parameters) and
1 T
(1N τ̇ θ )2
N
1
Eeg (r) = kτ̇ r k2 − (1TN τ̇ r )2
N
1 T
(1N τ̇ θ )(1TN τ̇ r )
Eeg (r, θ) = τ̇ H
θ τ̇ r −
N
Eeg (θ) = kτ̇ θ k2 −

Proof: The proof of this lemma can be deduced directly from lemma 5 proof given in appendix A.
2) Taylor Expansion (TE) of the CRB: This section aims to highlight the effects of some system parameters on
the localization performance and to better compare our exact CRB with the CRB given in [15]. For that, we propose
to use a Taylor expansion of the expressions of lemma 1. For simplicity, we omit the details of the cumbersome
(but straightforward) derivations and present only the final results in the following lemma:
Lemma 2: The Taylor expansions of the exact CRB expressions of lemma 1 lead to
3λ2
×
2T DSNR d2 π 2 cos2 (θ)p3 (N )

d
p2 (N )−6(N −1)(6N 2 −15N +11) sin(θ)
r
d2
1
+ (2N − 1)(384N 3 − 1353N 2 + 1379N − 368) 2
70
r

1
d2
+ (186N 4 − 1590N 3 + 5351N 2 − 6795N + 2890) sin2 (θ) 2
14
r
2
2

6r λ
× 15r 2 − 60(N − 1) sin(θ)dr
CRBceg (r) ≈
4
2
4
T DSNR d π cos (θ)p3 (N )
1  2
+
sin (θ)(1061N 2 − 2625N + 2911)
14

+225N 2 − 315N − 135 d2

CRBceg (θ) ≈

(11)

(12)

where p2 (N ) = (8N − 11)(2N − 1) and p3 (N ) = N (N 2 − 1)(N 2 − 4).
These expressions are useful to compare our CRB to the one in [15] but also, they allow us to reveal how the
different system parameters affect the localization performance. For example, one can see in particular that:
•

The CRB decreases linearly w.r.t. the observation time as well as the deterministic SNR.

•

The CRB goes to infinity when θ → ± π2 and the best localization results are obtained in the central direction,
i.e., for θ = 0.

9

•

Asymptotically (for large antenna sizes), the first term of the TE decreases in N 3 , i.e., if we double the number
of sensors, the CRB will be decreases approximately by a factor of 8.

3) Comparison with the ’CRB of the approximate model’: Let us first recall the CRB expressions given in [15]
and based on the approximate model in (4):
Lemma 3: The non-matrix expressions of the approximate conditional CRB [15] for a source in the near field,
for N ≥ 3 and θ 6= ± π2 , are given by
CRBeg (θ) =
CRBeg (r) =

3λ2
p2 (N )
2T DSNR d2 π 2 cos2 (θ)p3 (N )
6r 2 λ2
T DSNR d4 π 2 cos4 (θ)p3 (N )


15r 2 + 30dr(N − 1) sin(θ) + d2 p2 (N ) sin2 (θ)

(13)

(14)

Comparing lemma 2 to lemma 3, one can make the following observations:
•

First, we note that the first and main term of equation (11) (resp. of equation (12)) is equal to the first term
of equation (13) (resp. of equation (14)).

•

We note also, that the Taylor expansion (of the time delay) followed by CRB derivation leads to different
results as compared to CRB derivation followed by Taylor expansion (of the CRB). The latter expansion being
more accurate than the former as illustrated by our simulation results.

•

The approxim.ate CRB of the angle estimate shown in lemma 3 is independent of the range parameter (it is
the same as the far field CRB) while the expression of CRB(θ) in lemma 2 reveals how it is affected by the
range parameter. In particular, at the first order, one can see that the CRB decreases (resp. increases) as a
function of

d
r

for θ ∈ [0, π2 [ (resp. for θ ∈] − π2 , 0]).

B. Exact unconditional CRB with EG
For unconditional CRB, the signal is assumed Gaussian complex circular with zero mean and variance σs2 .
Assuming that ξ = [θ, r, σs2 , σ 2 ]T is the vector of the unknown parameters, the log-likelihood function of the
observed data is given by
ln(ξ) = −N T ln π − ln det(Σ) − T tr(Σ−1 R̂)

10

where Σ = σs2 a(θ, r)a(θ, r)H + σ 2 IN (theoretical covariance matrix), R̂ =

1
T

PT

t=1 x(t)x

H (t)

(sample estimate

covariance matrix), tr(.) refers to the matrix trace and (.)H is the transpose conjugate operator.
The Fisher Information matrix in this Gaussian case is given by


−1 ∂Σ

[FIM(ξ)]i,j = T tr Σ

∂ξi

Σ

−1 ∂Σ

∂ξj



(15)

Since the CRB is equal to the inverse of the Fisher information matrix and using the results of [20] for the two
desired localization parameters, we have the following lemma:
Lemma 4: The non-matrix expressions of the unconditional CRB in the equal gain case for a source in the near
field, for N ≥ 3 and θ 6= ± π2 are given by

(1 + (SNR) N )DSNR
=
CRBceg (θ)
(SNR)2 N


(1 + (SNR) N )DSNR
u
CRBeg (r) =
CRBceg (r)
(SNR)2 N


(1 + (SNR) N )DSNR
u
CRBeg (r, θ) =
CRBceg (r, θ)
(SNR)2 N
CRBueg (θ)

where SNR =

σs2
σ2



(16)
(17)
(18)

and DSNR ×CRBceg represents the normalized CRBceg (see lemma 1) depending on the localization

parameters, the array geometry, and the sample size only.
Proof: The proof of this lemma can be deduced directly from lemma 6 proof given in appendix (B).
This result translates the fact that the unconditional CRB varies in a similar way as the conditional CRB w.r.t.
variables θ , r and T .
However, concerning the SNR parameter and the number of sensors, it is interesting to observe that:
•

At low SNRs (i.e., if SNR × N << 1), the CRB decreases quadratically (instead of linearly in the conditional
case) w.r.t. the SNR and in N 4 (instead of N 3 ) in terms of the number of sensors.

•

However, for large SNRs (i.e., if SNR × N >> 1), the unconditional CRB behaves similarly as the conditional
CRB w.r.t. parameters SNR and N (i.e., it decreases linearly with the SNR and in N 3 w.r.t. the number of
sensors).

11

IV. C ONDITIONAL

AND UNCONDITIONAL

CRB

DERIVATION WITH

VARIABLE G AIN (VG)

In this case, we consider the same model as that described in Section II-A except that the received power is
variable from sensor to sensor. The nth output array is expressed as:
xn (t) = γn (r, θ)s(t)ejτn (r,θ) + vn (t) = s(t)(γn (r, θ)ejτn (r,θ) ) + vn (t)

t = 1, · · · , T

(19)

where γn (r, θ) represents the power profile of the nth sensor and is given by [17]
γn (r, θ) =

1
= q
dn
r 1−

1
2nd
r

sin θ +

dn being the distance between the source signal and the nth sensor.


nd 2
r

(20)

As we can see, in this model both the time delay profile and the power profile carry information on the desired
source location (r, θ). Our objective here, is to investigate the roles of both profiles in the performance limit given
by the CRB.

A. Exact conditional CRB with VG
Similarly to Section III-A, the conditional CRB is given by equation (7) which reduces in the variable gain case
to



 Q̃
FIM = 

0



0 



(21)

NT
σ4

which translates the fact that the FIM of the desired localization parameters is decoupled from the noise variance
σ 2 but not from the source magnitude parameter as in the equal gain case. Hence the CRB matrix of the range and

angle parameters is equal to the 2 × 2 top left sub-matrix of the following inverse matrix

−1
Q̃

−1

 fθθ


 f
 rθ
=


 fψθ


fψα

fθr

fθψ

frr

frψ

fψr

Fψψ

fαr

Fαψ

fθα 


frα 




Fψα 


Fαα

12

where frθ = fθr and
fθθ = 2T D̃SNR (kγ̇ θ k2 + (γ ⊙ γ)T (τ̇ θ ⊙ τ̇ θ ))

(22)

frr = 2T D̃SNR (kγ̇ r k2 + (γ ⊙ γ)T (τ̇ r ⊙ τ̇ r ))

(23)

T
frθ = 2T D̃SNR ((γ̇ H
θ γ̇ r ) + (γ ⊙ γ) (τ̇ θ ⊙ τ̇ r ))

(24)

Note that D̃SNR is defined here similarly to DSNR (D̃SNR =

kαk2
T σ2 )

but it is implicitly scaled by the square of

the range value which has been extracted from the received signal amplitude in (19) in order to explicit the role of
the power profile.
Consequently, D̃SNR unit is square meter (while DSNR is a constant without unit). Also, this way of normalizing
the signals leads to CRB expressions in the variable gain case that are scaled (multiplied) by a factor of r 2 as
compared to the equal gain case3. For this reason, when comparing the equal gain scenario to the variable gain
one in Section VI, we normalize the received signal power by dividing kαk2 by r 2 in the former case.
Moreover, the vectors of size T × 1, fψθ , fTθψ , fψr and fTrψ are given by
2
(γ ⊙ γ)T τ̇ θ
σ2
2
= 2 (γ ⊙ γ)T τ̇ r
σ
2
= 2 (γ T γ̇ θ )α
σ
2
= 2 (γ T γ̇ r )α
σ

fψθ = fTθψ =
fψr = fTrψ
fαθ = fTθα
fαr = fTrα
where

γ = [γ0 (r, θ), ..., γN −1 (r, θ)]T


∂γ0
∂γN −1 T
γ̇ θ =
, ...,
∂θ
∂θ


∂γ0
∂γN −1 T
γ̇ r =
, ...,
∂r
∂r

Finally, matrices Fψψ , Fαα , Fψα of size T × T , are given by
Fψψ =
3

2
2
kγk2 diag(α ⊙ α), Fψψ = 2 kγk2 IT and Fψα = 0.
σ2
σ

This can be seen as if in the equal gain case, the received power is set equal to a constant value while in the variable gain case, it is the

transmit power that is set equal to a constant.

13

Again by using the Schur’s matrix inversion lemma, one can obtain:
Lemma 5: The non-matrix expressions of the conditional CRB in the variable gain case for a source in the near
field, for N ≥ 3 and θ 6= ± π2 , are given by
CRBcvg (r) =

1
2T D̃SNR

!

Evg (θ)
Evg (θ)Evg (r) − Evg (r, θ)2

(25)

CRBcvg (θ) =

1
2T D̃SNR

!

Evg (r)
Evg (θ)Evg (r) − Evg (r, θ)2

(26)

CRBcvg (r, θ) =

1
2T D̃SNR

!

Evg (r, θ)
Evg (θ)Evg (r) − Evg (r, θ)2

(27)

where

1 
2
T
2
T
2 ((γ ⊙ γ) τ̇ θ ) + (γ γ̇ θ )
kγk

1 
2
T
2
T
Evg (r) = kγ̇ r k2 + (γ ⊙ γ)T (τ̇ r ⊙ τ̇ r ) −
γ̇
)
((γ
⊙
γ)
τ̇
)
+
(γ
r
r
kγk2

1 
T
T
T
T
Evg (r, θ) = γ̇ Tθ γ̇ r + (γ ⊙ γ)T (τ̇ θ ⊙ τ̇ r ) −
2 (γ ⊙ γ) τ̇ θ (γ ⊙ γ) τ̇ r + (γ γ̇ r )(γ γ̇ θ )
kγk
Evg (θ) = kγ̇ θ k2 + (γ ⊙ γ)T (τ̇ θ ⊙ τ̇ θ ) −

(28)
(29)
(30)

Proof: See appendix A
One can observe, that the equal gain case (i.e., lemma 1) is a particular situation of lemma 5 where γ = 1N
and DSNR is divided by r 2 .

B. Exact unconditional CRB with VG
Under the stochastic Gaussian model of Section III-B, the unconditional CRB with variable gain can be expressed
as stated by the following lemma:
Lemma 6: The non-matrix expressions of the exact unconditional CRB in the variable case for a source in the
near field, for N ≥ 3 and θ 6= ± π2 , are given by
CRBuvg (θ) =

˜
(1+SNRkγk2 )D̃
SNR
˜ )2 kγk2
(SNR

CRBuvg (r)

=

˜
(1+SNRkγk2 )D̃
SNR
˜ )2 kγk2
(SNR

=

˜
(1+SNRkγk2 )D̃
SNR
˜ )2 kγk2
(SNR

CRBuvg (r, θ)
Proof: See appendix B

!
!
!

CRBcvg (θ)

(31)

CRBcvg (r)

(32)

CRBcvg (r, θ)

(33)

14

Similarly to the equal gain case, D̃SNR × CRBceg represents a normalized CRB independent from the SNR value.
˜ is defined in the same way as the SNR in lemma 4 except that it has been implicitly scaled by the
Also, SNR
˜ unit is now square meter).
square of the range parameter (SNR

C. Comparison between VG and EG cases
By considering the extra-information given by the received power profile, one is able to achieve better localization
performance as will be shown later in Section VI. To better compare the CRB expressions in the constant and variable
gain cases, we provide here the first term of their Taylor expansion
CRBceg (θ) ≈
CRBcvg (θ)

≈

CRBceg (r) ≈
CRBcvg (r) ≈
where f1 (θ) =

15 sin2 (θ)
π 2 d2 cos4 (θ)p2 (N )

3λ2 p2 (N )
2π 2 T DSNR d2 cos2 (θ)p3 (N )


3λ2 r 2 p2 (N ) 1 + λ2 f1 (θ)

2π 2 T D̃SNR d2 cos2 (θ)p3 (N ) (1 + λ2 f2 (θ))
90λ2 r 4
π 2 T DSNR d4 cos4 (θ)p3 (N )
90λ2 r 6
π 2 T D̃SNR d4 cos4 (θ)p3 (N ) (1 + λ2 f2 (θ))

and f2 (θ) =

15 sin2 (θ)
π 2 d2 cos4 (θ)(N 2 −4) .

From these expressions, one can see that the two

CRBs are quite similar for sources located in the central direction (i.e., for small θ values) while at lateral directions
(i.e., |θ| close to

π
2)

the variable gain CRB is much lower than the equal gain one. This translates the fact that

when the source location information contained in the time delay profile becomes weak, it is somehow partially
compensated by the location information contained in the received power profile especially for the estimation of
the range value. This observation is illustrated by the simulation experiments (cf. Fig. 5, Fig. 6 and Fig. 7) given
in Section VI.

V. N EAR FIELD

LOCALIZATION REGION

(NFLR)

The radiating near field or Fresnel region is the region between the near and far fields [17] corresponding to the
space region defined by equation (3). The latter depends on the source-antenna range, the signal wavelength, and
the antenna aperture.
Note that the lower bound of Fresnel region is related to the fact that in the immediate vicinity of the antenna,
the fields are predominately reactive fields meaning that the E and H fields are orthogonal [17]. Therefore the

15

space region given by r ≤ 0.62



d3 (N −1)3
λ

1
2

should be kept out of the localization region. However, the upper

bound of the Fresnel region as given in equation (3) does not take into consideration the localization performance
limit. For this reason, we suggest to define the near field localization region (NFLR) based on a target estimation
performance relative to the application at hand. More precisely, if Stdmax > 0 is the maximum standard deviation
of the localization error that is tolerated by the considered application, i.e.,
s 

E

d̂ − d

2

≤ Stdmax

(34)

where d = (x, y)T (resp. d̂) refers to the location vector (resp. its estimate), then the NFL region can be defined as
the one for which the minimum standard deviation (given by the square root of the CRB) satisfies condition (34).


2
Since E d̂ − d
= E((x̂ − x)2 ) + E((ŷ − y)2 ), the previous condition on the minimum MSE can be
expressed as
p

CRB(x) + CRB(y) ≤ Stdmax

(35)

Now, the source coordinates can be rewritten according to
x = r sin(θ) = gx (r, θ)
y = r cos(θ) = gy (r, θ)

and hence, by using the delta method in [21], one can express
CRB(x) + CRB(y) = ∇gxT (r, θ)C∇gx (r, θ) + ∇gyT (r, θ)C∇gy (r, θ)

(36)

where
∇gx (r, θ) =





∂gx ∂gx
∂r ∂θ

∂gy
∇gy (r, θ) =
∂r


∂gy
∂θ

T

T

= [cos(θ)

r sin(θ)]T

= [sin(θ)

r cos(θ)]T


 CRB(θ) CRB(r, θ) 

C = 


CRB(θ, r) CRB(r)

A straightforward derivation of (36) leads to

CRB(x) + CRB(y) = r 2 CRB(θ) + CRB(r)

(37)

16

and therefore the NFL region is defined as the one corresponding to
p

r 2 CRB(θ) + CRB(r) ≤ Stdmax

(38)

An alternative approach would be to use a maximum tolerance value on the relative location error, i.e., a threshold
value ǫ such that

v 

u
2
u
u E d̂ − d
t
≤ǫ
kdk2

(39)

which corresponds to
r

CRB(θ) +

CRB(r)
≤ǫ
r2

(40)

For example, in the conditional case, equation (40) becomes
1
GN (r, θ) ≤ ǫ2
2T D̃SNR

(41)

where
GN (r, θ) =

Evg (θ) + Evg (r)/r 2
Evg (θ)Evg (r) − Evg (r, θ)2

From a practical point of view, equation (41) can be used to tune the system parameters in order to achieve a
desired localization performance. Different scenarii can be considered, according to the parameter, we can (or wish
to) tune.

Scenario 1: One can define the minimum observation time to achieve a desired localization performance at a given
location and a given SNR value as
Tmin (r, θ) =

GN (r, θ)
2ǫ2 D̃SNR

(42)

Similarly, one can also define the minimum SNR value for a target localization quality as
D̃SNRmin (r, θ) =

GN (r, θ)
2ǫ2 T

(43)

In Section VI, we provide simulation examples to illustrate the variation of these two parameters w.r.t. the source
location.

17

Scenario 2: The previous parameters can be also be defined for a desired localization region Rd (instead of a single
location point (r, θ)) as
Tmin (Rd ) =
DSNRmin (Rd ) =

max Tmin (r, θ)

(44)

max DSNRmin (r, θ)

(45)

(r,θ)∈Rd

(r,θ)∈Rd

For example, if we are interested into the surveillance of a space sector limited by rmin < r < rmax and
−θmax < θ < θmax , then
Tmin =
≈
D̃SNRmin

=
≈

1
2ǫ2 D̃

max GN (r, θ)

SNR Rd
1
GN (rmax , θmax )
2ǫ2 D̃SNR
1
max GN (r, θ)
2ǫ2 T Rd
1
GN (rmax , θmax )
2ǫ2 T

(46)

(47)

where the second equality holds from the observation that, away from the origin, GN is a decreasing function w.r.t.
the angular and range parameters4 .

Scenario 3: One can also wish to optimize the number of sensors with respect to a desired localization region Rd
and for a target localization quality ǫ. In that case, the minimum number of sensors needed to achieve the target
quality can be calculated as
n
o
Nmin = arg min N ∈ N∗ GN (r, θ) ≤ 2ǫ2 T D̃SNR ∀(r, θ) ∈ Rd
N

VI. S IMULATION
In this Section, three experimental sets are considered. The first one is to compare the provided exact CRB
expressions of lemma 1 with the CRB expressions given in [15]. Also, in that first experiment, we used the
maximum likelihood approach (with the exact model in (1) and (2)) to validate our exact CRB derivation and to
illustrate the gain in location estimation accuracy when using (2) instead of (4) in the data model. In the second
4

This is not an exact and proven statement but just an approximation that expresses the fact that the localization accuracy decreases when

the source moves away from the antenna or towards its lateral directions.

18

experiment, we investigate the effect of considering the variable gain model instead of the constant gain model for
near field source localization. Finally, the third experiment is to illustrate the usefulness of the near field localization
region as compared to the standard Fresnel region.
In all our simulations, we consider a uniform linear antenna with N = 15 sensors and iter-element spacing
d=

λ
2

(λ = 0.5m) receiving signals from one near field source located at (r, θ). The sample size is T = 90 (unless

stated otherwise) and the observed signal is corrupted by a white Gaussian circular noise of variance σ 2 . In the
conditional case, the source signal is of unit amplitude (i.e., α(t) = 1 ∀t).
The dotted vertical plots in all figures represent the upper and lower range limits of the Fresnel region given by
(3).

A. Experiment 1: Comparison with existing work
In Fig. 2, we compare the three CRB expressions for the source location parameter estimates versus the range
values in the interval [0 50m] and versus angle values5 in the interval [−90o 90o ]. The noise level is set to
σ 2 = 0.001 which corresponds to SNR = 30 dB . A similar comparison leading to similar results is given in Fig.

3 for a noise level set equal to σ 2 = 0.5 which corresponds to SNR ≈ 3 dB . The source angle is θ = 45o in the
comparison versus range values and r = 20λ in the comparison versus angle values.
From these figures, one can observe a non negligeable difference between the exact CRB and the proposed one
in [15] especially at low range values: i.e., the given CRB in [15] can be up to 30 times larger than the exact CRB.
Also, from Fig. 2.(c) and Fig. 3.(c), contrary to the given CRB in [15], the exact one varies with the range value
with a relative difference varying from approximately 60% for small ranges to 0 when r goes to infinity.
From Fig. 2.(b) - Fig. 3.(b), one can observe that the lowest CRB is obtained in the central directions, this
observation can be seen from the TE given in lemma 2 where the factor

1
cos(θ)

is minimum for this directions and

goes to infinity when |θ| → π2 .
Note that the provided Taylor expansion of the exact CRB is more accurate than the one obtained by expanding
the time delay expression before CRB derivation i.e., the one in [15].
5

Note that the curves w.r.t. the angle parameter (i.e., Fig. 2, Fig. 3, Fig. 7, Fig. 10 and Fig. 11) are not symmetrical around θ = 0 because

we have chosen the first sensor for the time reference as shown in Fig. 1.

19
σ2 = 0.001

(a): θ = 45°

N = 15

T = 90

(b): r = 20λ

−2

CRB(r)

CRB(r)

−4

10

10

0

10

20

30

40

50

−100

N = 15

T = 90

−50

Range r
σ2 = 0.001

(c): θ = 45°

(d):

r = 20λ

−4

−7.3

CRB(θ)

10

−7.5

10

0

Angle θ
σ2 = 0.001

50

N = 15

T = 90

Exact CRB in (9)
TE of CRB in (11)
CRB in [15]

10

CRB(θ)

T = 90

Exact CRB in (8)
TE of CRB in (12)
CRB in [15]

−6

10

−6

10

Exact CRB in (9)
TE of CRB in (11)
CRB in [15]

−7.7

10

10

20

30

40

50

−100

−50

Range r

0

Angle θ

50

100

CRB comparison: Exact conditional CRB versus approximate CRB in [15] in high SNR case
2

(a): θ = 45°

σ = 0.5

N = 15

(b): r = 20λ

T = 90

2

10

10

−2

Exact CRB in (8)
TE of CRB in (12)
CRB in [15]

10

10

T = 90

20

30

40

0

10

50

−100

−50

Range r
σ2 = 0.5

(c): θ = 45°

N = 15

T = 90

−1

−4.5

(d):

r = 20λ

10

10

−2

CRB(θ)

−4.7

10

Exact CRB in (9)
TE of CRB in (11)
CRB in [15]

−4.9

10

10

20

30

Range r

40

0

Angle θ
σ2 = 0.5

50

N = 15

100

T = 90

Exact CRB in (9)
TE of CRB in (11)
CRB in [15]

10

CRB(θ)

N = 15

CRB(r)

CRB(r)

0

σ2 = 0.5

Exact CRB in (8)
TE of CRB in (12)
CRB in [15]

5

10

Fig. 3.

N = 15

Exact CRB in (8)
TE of CRB in (12)
CRB in [15]

10

Fig. 2.

σ2 = 0.001

−3

10

−4

10

−5

10
−100

−50

0

Angle θ

50

100

CRB comparison: Exact conditional CRB versus approximate CRB in [15] in low SNR case

To validate our exact CRB expression (see Fig. 4), we have considered the Maximum likelihood (ML) algorithm.
The ML function is computed for the equal gain conditional model, and its maximization is ensured by using
Newton-Raphson method (6000 Monte Carlo runs are considered for the ML estimator MSE plot in Fig. 4). One
can see that these CRB expressions can be reached and that we can gain up to 60% of MSE reduction by avoiding
the standard model approximation in (4).

20

CRB(r)

(a): θ = 45°

σ2 = 0.001

N = 15

T = 90

Exact CRB in (8)
TE of CRB in (12)
CRB in [15]
ML estimator

−5

10

5

10

15

20

25

30

35

40

45

50

45

50

Range r
(c): θ = 45°

σ2 = 0.001

N = 15

T = 90

−7.3

CRB(θ)

10

−7.5

10

Exact CRB in (8)
TE of CRB in (12)
CRB in [15]
ML estimator

−7.7

10

5

10

15

20

25

30

35

40

Range r

Fig. 4.

Exact conditional CRB validation

B. Experiment 2: EG versus VG cases
To this end, we have to ensure first that the received power in the two cases (i.e., constant and variable gain
cases) is the same for the reference sensor (i.e., dividing the power of constant gain case per the square of the
range as explained in Section IV-A). To better compare CRB expressions, we consider two contexts where θ = 0o
for the first context (central direction) and θ = 85o for the second context (lateral direction). One can observe from
Fig. 5 and Fig. 6 that, for small |θ| values, the constant gain CRB is quite similar to the variable gain one while
at lateral direction (i.e., |θ| close to

π
2)

the variable gain CRB is much lower than the equal gain one, due to the

extra information brought by the considered gain profile.
This can be seen again from Fig. 7 where we can observe the large CRB difference for high |θ| values.

C. Experiment 3: Near Field Localization Region
The plots in Fig. 8 represent the upper limit of the NFL region for different tolerance values. From this figure,
one can observe that the Fresnel region is not appropriate to characterize the localization performance. Indeed,
depending on the target quality, one can have space locations (i.e., sub-regions) in the Fresnel region that are out
of the NFLR. Inversely, we have space locations not part of the Fresnel region that are attainable, i.e., they belong
to the NFLR.

21

(a): θ = 0°

2

σ = 0.001

N = 15

(b): θ = 85°

T = 90

2

σ = 0.001

N = 15

T = 90

6

10

2

10

1

10

4

10
0

10

2

CRB(r)

CRB(r)

10
−1

10

−2

0

10

10

−3

10

−2

10
−4

10

CRB with equal Gain
CRB with variable Gain

−5

10

10

20

30

CRB with equal Gain
CRB with variable Gain

−4

10

40

50

10

20

Range r

Fig. 5.

30

40

50

N = 15

T = 90

Range r

CRB comparison: Equal Gain versus Variable Gain cases for range estimation
(a): θ = 0°

2

σ = 0.001

N = 15

(b): θ = 85°

T = 90

2

σ = 0.001

−2

10

−4

10

−3

10

−5

10

−4

CRB(θ)

CRB(θ)

10

−5

10

−6

10

−6

10

CRB with equal Gain
CRB with variable Gain

CRB with equal Gain
CRB with variable Gain
−7

10
10

20

30

40

Range r

Fig. 6.

50

10

20

30

40

50

Range r

CRB comparison: Equal Gain versus Variable Gain cases for angle estimation

Fig. 9 compares the NFL region in the variable gain and equal gain cases with SNR = 30 dB . One can observe
that in the lateral directions the NFLR associated to the variable gain model is much larger than its counterpart
associated to the standard equal gain model. Also, in the short observation time context (i.e., Fig. 9.(a)) the NFLR
is included in the Fresnel region while for large observation time (i.e., Fig. 9.(b)) the NFLR region is much more
expanded and contains most of the Fresnel region6 .
6

Except for the extreme lateral directions where the target quality can never be met since the CRB goes to infinity for |θ| →

π
.
2

22

4

2

σ = 0.001

(a): r = 20λ

N = 15

T = 90

−2

10

(b): r = 20λ

σ2 = 0.001

N = 15

T = 90

10
CRB with equal Gain
CRB with variable Gain

CRB with equal Gain
CRB with variable Gain

3

10

−3

10

2

10

1

CRB(θ)

CRB(r)

10

0

−4

10

10

−1

10

−5

10
−2

10

−3

−6

10
−100

Fig. 7.

−50

0

Angle θ

50

10
−100

100

−50

0

50

Angle θ

100

CRB comparison of the Equal Gain and Variable Gain cases versus angle value
2

σ = 0.001 N = 15 T = 90

(a):
120

(b):

Stdmax = 10

Std

= 10

Stdmax = 20

Std

= 20

max
max

45

Stdmax = 50
Std

max

100

σ2 = 0.5 N = 15 T = 90

50

Stdmax = 50
Stdmax = 100

= 100

Fresnel region upper limit

Fresnel region upper limit

40

Frsenel region lower limit

Frsenel region lower limit

Source coordinate y

Source coordinate y

35
80

60

40

30

25

20

15

10
20
5

0
−100

−50

0

50

Source coordinate x

Fig. 8.

100

0
−50

0

50

Source coordinate x

Near field localization regions for different values of the target quality: Stdmax

In Fig. 10 - Fig. 11, we illustrate the variation of the two parameters Tmin and D̃SNRmin w.r.t. the source location
parameters and for a relative tolerance error equal to ǫ = 10%. From these figures, one can observe that D̃SNRmin
and Tmin increase significantly for sources that are located far from the antenna or in the lateral directions.

VII. C ONCLUSION
In this paper, three important results are proposed, discussed, and assessed through simulation experiments: (i)
Exact conditional and unconditional CRB derivation for near field source localization and its development in non

23

(a):Std

max

= 10 σ2 = 0.001 N = 15 T = 90

50

(b):Std

max

= 10 σ2 = 0.001 N = 15 T = 3000

90

Variable gain

Variable gain
Equal gain

Equal gain

Fresnel region upper limit

Fresnel region upper limit

45

Frsenel region lower limit

Frsenel region lower limit

80

40

70

Source coordinate y

Source coordinate y

35

30

25

20

60

50

40

30

15
20

10

10

5

0
−50

0

0
−50

50

0

Source coordinate x

Fig. 9.

100

Comparison of NFL regions in equal and variable gain cases
ε = 10%

DSNR = 30dB

ε = 10%

, N = 15

6

DSNR = 30dB , N = 15
range = 10 m
range = 25 m
range = 45 m

6

10

10

5

5

10

10

4

4

10

min

T

Tmin

10

3

3

10

10

2

2

10

10

θ = −30°
θ = 0°
θ = 30°
θ = 85°

1

10

1

10

0

0

10

10

20

30

Range r

Fig. 10.

50

Source coordinate x

40

50

10
−100

−50

0

Angle θ

50

Variation of the minimum observation time versus the source location parameters

matrix form. The latter reveals interesting features and interpretations not shown by the CRB given in the literature
based on an approximate model (i.e., approximate time delay). (ii) This CRB derivation is generalized to the variable
gain case where the exact expression of the received power profile is taken into account. This generalization allows
us to investigate the importance of the power profile information in ’adverse’ localization contexts (i.e., for lateral
lookup directions).
(iii) Based on the previous CRB derivation, a new concept of ’localization region’ is introduced to better define the

24
ε = 10%

N = 15,

ε = 10%

T = 3000

6

N = 15,

T = 3000

range = 10 m
range = 25 m
range = 45 m

6

10

10

5

10

5

10
4

10

4

10

3

min

DSNR

2

10

~

D~SNR

min

10

3

10

1

10

2

10

0

10

θ = −30°
θ = 0°
θ = 30°
θ = 85°

−1

10

1

10

−2

10

10

20

30

40

50

−50

Range r

Fig. 11.

0

Angle θ

50

100

Variation of the minimum deterministic SNR versus the source location parameters

space region where the localization quality can meet a target value or otherwise to better tune the system parameters
to achieve the target localization quality for a given location region.

A PPENDIX A
P ROOF

OF

L EMMA 5

A direct calculation of matrix Q̃ in (21) using equation (7) leads to


 Q1
Q̃ = 

QT2

where



Q2 


Q3



 fθθ fθr 

Q1 = 


frθ frr


 v1 v1′ 

Q2 = 


v2 v2′




 v3 diag(α ⊙ α) 0T 

Q3 = 


0T
v3 IT

the entries of Q1 are given by (22), (23) and (24), ⊗ represents the Kronecker product, v1 =
v1′ =

2 T
σ2 γ γ̇ θ ,

v2 =

2
σ2 (γ

⊙ γ)T τ̇ r ,

v2′ =

2 T
σ2 γ γ̇ r ,

2
σ2 (γ

⊙ γ)T τ̇ θ ,

25

and v3 =

2
σ2

kγk2 .

Because the CRB of the range and the angle parameters is equal to the 2 × 2top left sub-matrix
of the inverse


matrix Q̃

−1

, Schur lemma [19] can be used and the results will be as Q̃

T
Q2 .Q−1
3 .Q2

After a straightforward computation, one obtain


Qc = 


fθθ −

frθ −

kαk
v3

2

(v12 + v1′2 )

kαk2
v3 (v1 v2

x 
 Q−1
c
 where Qc = Q1 −
=


x
x

2



(v1 v2 + v1′ v2′ ) 


kαk2 2
frr − v3 (v2 + v2′2 )

fθr −

+ v1′ v2′ )

kαk
v3

−1

Now, by comparing this expression of Qc to the expressions in (28), (29) and (30), one can rewrite


leading finally to


Qc = 2T D̃SNR 


CRBcvg (r) =
=

CRBcvg (θ) =
=

CRBcvg (r, θ) =
=

Evg (r)

−Evg (r, θ) 


−Evg (r, θ)
Evg (θ)

Evg (r)
det(Qc )


Evg (θ)
1
2T DSNR Evg (θ)Evg (r) − Evg (r, θ)2
Evg (θ)
det(Qc )


Evg (r)
1
2T DSNR Evg (θ)Evg (r) − Evg (r, θ)2
Evg (r, θ)
det(Qc )


1
Evg (r, θ)
2T DSNR Evg (θ)Evg (r) − Evg (r, θ)2

26

A PPENDIX B
P ROOF

OF

L EMMA 6

2 T
2
For the unconditional case, the considered unknown
 parameter
 vector is ξ = (θ, r, σs , σ ) which leads to the

 F1
following 4 × 4 Fisher Information matrix FIM = 

FT2


F2 
 where the 2 × 2 matrices Fi are given by

F3


 fθθ fθr 

F1 = 


frθ frr



 fθσs2 fθσ2 

F2 = 


frσs2 frσ2



 fσs2 σs2
F3 = 

fσ2 σs2

fσs2 σ2 


fσ 2 σ 2




By using Schur’s lemma for matrix inversion [19], one can obtain FIM−1 = 



L−1
GT





G 
 where L = F1 −

H

 u x 
T
. F3 and L are 2 × 2 matrices and their inverse can be computed easily as

F2 F−1
3 F2 = 

x v

 

L−1 =

where

u −x   CRB(r) CRB(θ, r) 
1 

=


 
det 
CRB(r, θ) CRB(θ)
−x v
1
(fθσs2 c1 + fθσ2 c2 )
det1
1
v = fθθ −
(frσs2 c1 + frσ2 c2 )
det1
1
x = frθ −
(frσs2 c3 + frσ2 c4 )
det1

u = frr −

det1 = fσ2 σ2 fσs2 θ − fσs2 σ2 fσ2 θ

(48)
(49)
(50)
(51)

27

c1 = fσ2 σ2 fσs2 θ − fσs2 σ2 fσ2 θ

(52)

c2 = fσs2 σs2 fσ2 θ − fσ2 σs2 fσs2 θ

(53)

c3 = fσs2 σs2 fσs2 r − fσs2 σ2 fσ2 r

(54)

c4 = fσs2 σs2 fσ2 r − fσ2 σs2 fσs2 r

(55)

det = uv − x2

(56)

Now, it remains only to compute the entries of the FIM by using equation (15) and taking into account that
the matrix Σ = σs2 b(θ, r)b(θ, r)H + σ 2 IN and its inverse is given as Σ−1 =

1
σ2 (IN

1
C = SNR
+ kγk2 and b(θ, r) = [γ0 , γ1 ejτ1 , · · · , γN −1 ejτN −1 ]T .

A straightforward (but cumbersome) computation leads to
fθθ

=

2T
2
(1 − SNR kγk )(γ T γ̇ θ )2
C2
2

−(1 + SNR kγk )((γ ⊙ γ)T τ̇ θ )2
2

2

+C SNR kγk (kγ̇ θ k + (γ ⊙ γ)T (τ̇ θ ⊙ τ̇ θ ))
frr

=

2T
2
(1 − SNR kγk )(γ T γ̇ r )2
C2
2

−(1 + SNR kγk )((γ ⊙ γ)T τ̇ r )2
2

2

+C SNR kγk (kγ̇ r k + (γ ⊙ γ)T (τ̇ r ⊙ τ̇ r ))
frθ

=

2T
2
(1 − SNR kγk )(γ T γ̇ θ )(γ T γ̇ r )
C2
2

−(1 + SNR kγk )((γ ⊙ γ)T τ̇ θ )((γ ⊙ γ)T τ̇ r )
2

+C SNR kγk (γ̇ Tθ γ̇ r + (γ ⊙ γ)T (τ̇ θ ⊙ τ̇ r ))
4

fσs2 σs2

=

fσ 2 σ 2

=

fσs2 σ2

=

fθσs2

=

fθσ2

=

frσs2

=

frσ2

=

T kγk
4
σ (C SNR)2
T
2
2
(N C 2 − kγk (2C − kγk ))
4
σ C2
4
T kγk
σ 4 (C SNR)2
2

2T kγk
(γ T γ̇ θ )
σ 2 C 2 SNR
2T
(γ T γ̇ θ )
σ 2 C 2 SNR
2
2T kγk
(γ T γ̇ r )
2
2
σ C SNR
2T
(γ T γ̇ r )
σ 2 C 2 SNR

−

1
H
C b(θ, r)b(θ, r) )

where

28

By replacing these entries in equations (48)-(56), we obtain
u =
v =
x =

2T SNR2 kγk2

Evg (r)
(1 + SNR kγk2 )
2T SNR2 kγk2
Evg (θ)
(1 + SNR kγk2 )
2T SNR2 kγk2
Evg (r, θ)
(1 + SNR kγk2 )

leading finally to Lemma 6 result
CRBuvg (θ) =
CRBuvg (r) =
CRBuvg (r, θ) =

Evg (r)
1 + SNR kγk2
2 E (θ)E (r) − E (r, θ)2
2
2T SNR kγk vg
vg
vg
2
Evg (θ)
1 + SNR kγk
2 E (θ)E (r) − E (r, θ)2
2
2T SNR kγk vg
vg
vg
Evg (r, θ)
1 + SNR kγk2
2 E (θ)E (r) − E (r, θ)2
2
2T SNR kγk vg
vg
vg

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