1
Large Intelligent Surface for Positioning in
Millimeter Wave MIMO Systems
Jiguang He
†
, Henk Wymeersch
?
, Long Kong
∗
, Olli Silv
´
en
‡
, Markku Juntti
†
†
Centre for Wireless Communications, FI-90014, University of Oulu, Finland
?
Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden
∗
Interdisciplinary Centre for Security Reliablility and Trust (SnT), University of Luxembourg, Luxembourg
‡
Center for Machine Vision and Signal Analysis (CMVS), FI-90014, University of Oulu, Finland
Abstract—Millimeter-wave (mmWave) multiple-input multiple-
output (MIMO) system for the fifth generation (5G) cellular
communications can also enable single-anchor positioning and
object tracking due to its large bandwidth and inherently high
angular resolution. In this paper, we introduce the newly invented
concept, large intelligent surface (LIS), to mmWave positioning
systems, study the theoretical performance bounds (i.e., Cram
´
er-
Rao lower bounds) for positioning, and evaluate the impact of
the number of LIS elements and the value of phase shifters on
the position estimation accuracy compared to the conventional
scheme with one direct link and one non-line-of-sight path. It
is verified that better performance can be achieved with a LIS
from the theoretical analyses and numerical study.
I. INTRODUCTION
Conventionally, indoor and outdoor positioning is carried
out by using received signal strength (RSS), time-difference-
of-arrival (TDoA) [1] or fingerprinting-based approaches [2],
[3]. Recently, efforts on positioning has been made by leverag-
ing millimeter-wave (mmWave) multiple-input multiple-output
(MIMO) systems and the geometric relationship between the
base station (BS) and the mobile station (MS) [4]–[8]. It
was shown that even a single BS can achieve promising
positioning accuracy. Extensions to multiple-carrier and multi-
user scenarios were studied in [7].
Practical positioning algorithms for the single-anchor
mmWave MIMO system can be classified into two major cate-
gories, i.e., direct positioning [6] and two-stage positioning [4],
[5], [7], [8]. The direct positioning aims at estimating the coor-
dinates of the MS from the received signal directly, while the
two-stage positioning first estimates the instantaneous channel
realization: the channel gains, the angle of departure (AoD),
the angle of arrival (AoA), and the time of arrival (ToA). In
the second stage, the user location is calculated based on the
channel estimates and the environmental geometry.
Large intelligent surfaces (LIS), also known as reconfig-
urable intelligent surfaces (RIS), can effectively control the
propagation wave, e.g., phase and even amplitude, without
any need of baseband processing units [9], [10]. The LIS has
been proposed to be used as a Tx/Rx antenna like in hybrid
This work has been performed in the framework of the IIoT Connectivity for
Mechanical Systems (ICONICAL), funded by the Academy of Finland. This
work is also partially supported by the Academy of Finland 6Genesis Flagship
(grant 318927) and Swedish Research Council (grant no. 2018-03701).
Base station
LIS
Mobile station
…
…
Fig. 1: Positioning system with the aid of a large intelligent surface
and multi-carrier mmWave OFDM signals. The coordinates and ori-
entation of MS, (m
x
, m
y
) and α, are unknown and to be estimated.
beamforming for positioning [11] and relay type reflector for
communications [9].
In this paper, we study the positioning with the assistance
of a LIS based reflector and multiple subcarriers at mmWave
frequency bands. First, the performance bounds (i.e., Cram
´
er-
Rao lower bound) are evaluated based on the equivalent Fisher
information matrix (FIM). The impact of the LIS, e.g., phase
shifter value and the number of LIS elements, is studied
on the estimation of channel parameters, positioning error
bound (PEB), and orientation error bound (OEB). Numerical
results show the superiority of the LIS aided mmWave MIMO
positioning system over its conventional counterpart without
incorporating a LIS.
II. SYSTEM MODEL
The positioning system is presented in Fig. 1, which consists
of one multiple-antenna BS, one multiple-antenna MS and
one LIS. We consider a two-dimensional (2D) scenario with
uniform linear arrays (ULAs) for both the antenna elements
and LIS elements (i.e., analog phase shifters). The numbers
of antenna elements at the BS and MS are N
B
and N
M
,
respectively, while the number of LIS elements is N
L
. No
rotation is assumed for the BS and the LIS while α-rad
rotation is assumed for the MS. The objective of the system
is to localize the MS and estimate its orientation by using
the received signals at the MS with N mmWave orthogonal
frequency division multiplexing (OFDM) subcarriers.
2
The propagation channel is composed of one direct path and
one reflection path via the LIS. The direct channel between
the BS and MS for the n-th subcarrier is expressed as
H
B,M
[n] = ρ
B,M
e
−j2πτ
B,M
nB
N
α
r
(φ
B,M
)α
H
t
(θ
B,M
),
for n = −(N − 1)/2, ··· , (N − 1)/2, (1)
where α
r
(φ
B,M
) ∈ C
N
M
×1
and α
t
(θ
B,M
) ∈ C
N
B
×1
are
the antenna array response and steering vectors at the
MS and BS, respectively. The i-th entry of α
r
(φ
B,M
) and
α
t
(θ
B,M
) are [α
r
(φ
B,M
)]
i
= e
j2π(i−1)
d
λ
sin(φ
B,M
)
, [α
t
(θ
B,M
)]
i
=
e
j2π(i−1)
d
λ
sin(θ
B,M
)
with d being the antenna element spacing
1
,
λ being the wavelength of the signal, and θ
B,M
and φ
B,M
being
the AoD and AoA, respectively. j =
√
−1, τ
B,M
is the ToA, B
is the overall bandwidth for all the subcarriers, and B f
c
2
,
where f
c
is the center frequency. ρ
B,M
∈ R
+
is the free-space
path loss occurred in the direct link for all the subcarriers, and
(·)
H
denotes the conjugate transpose operation.
The two tandem channels (H
B,L
[n] ∈ C
N
L
×N
B
for the first
hop and H
L,M
[n] ∈ C
N
M
×N
L
for the second hop) for the n-th
subcarrier, which connect the BS to the MS via the LIS, are
defined as
H
B,L
[n] = ρ
B,L
e
−j2πτ
B,L
nB
N
α
r
(φ
B,L
)α
H
t
(θ
B,L
), (2)
and
H
L,M
[n] = ρ
L,M
e
−j2πτ
L,M
nB
N
α
r
(φ
L,M
)α
H
t
(θ
L,M
), (3)
where the notations α
t
(θ
B,L
), α
r
(φ
B,L
), α
t
(θ
L,M
), α
r
(φ
L,M
),
ρ
B,L
, ρ
L,M
, τ
B,L
, and τ
L,M
are defined in the same way as those
in (1).
The entire channel, including both the line-of-sight (LoS)
path and the non-line-of-sight (NLoS) path (i.e., the reflection
path via the LIS), between the BS and the MS for the n-th
subcarrier can be formulated as
H[n] = H
B,M
[n] + H
L,M
[n]ΩH
B,L
[n], (4)
where Ω = diag(exp{jω
1
}, ··· , exp{jω
N
L
}) ∈ C
N
L
×N
L
is
the phase control matrix at the LIS. It is a diagonal matrix
with constant-modulus entries in the diagonal.
Assuming that precoding F is exploited at the BS and the
positioning reference signal (PRS) x[n] is transmitted over the
n-th subcarrier, the downlink received signal is in the form of
y[n] =
√
P H[n]Fx[n] + n[n], (5)
where each entry in the additive white noise n[n]
follows circularly-symmetric complex normal distribution
CN(0, 2σ
2
), and P is the transmit power of the PRS.
1
With notation reuse, d also denotes element spacing in the LIS.
2
All the wavelengths λ’s of the subcarriers are nearly the same because of
B f
c
.
The geometric relationship among the BS, LIS, and MS is
formulated as
τ
B,M
= kb − mk
2
/c,
τ
B,L
+ τ
L,M
= kb − lk
2
/c + km − lk
2
/c,
θ
B,M
= arccos((m
x
− b
x
)/kb − mk
2
),
θ
B,L
= arccos((l
x
− b
x
)/kl − bk
2
),
θ
L,M
= −arccos((m
x
− l
x
)/kl − mk
2
),
φ
B,M
= π + arccos((m
x
− b
x
)/kb − mk
2
) − α
= π + θ
B,M
− α,
φ
B,L
= −π + arccos((l
x
− b
x
)/kb − lk
2
) = −π + θ
B,L
,
φ
L,M
= π − arccos((m
x
− l
x
)/kl − mk
2
) − α
= π + θ
L,M
− α,
ρ
B,M
= (kb − mk
2
)
−µ/2
,
ρ
B,L
= (kb − lk
2
)
−µ/2
,
ρ
L,M
= (kl − mk
2
)
−µ/2
, (6)
where b = [b
x
b
y
]
T
, l = [l
x
l
y
]
T
, and m = [m
x
m
y
]
T
are the centers of the BS, LIS, and MS, respectively, α is
the orientation of the MS, µ is the path loss exponent, c
is the speed of light, and k · k
2
stands for the Euclidean
norm. Based on Fig. 1, we can further impose the following
constraints on the channel angular parameters: 1) θ
B,M
, θ
B,L
∈
(0, π/2), 2) θ
L,M
∈ (−π/2, 0), 3) φ
B,L
∈ (−π, −π/2), 4)
φ
B,M
, φ
L,M
∈ (π/2, π), and 5) α ∈ (0, π/2). It should be
noted that we consider far-field communications. Therefore,
additional constraints are imposed to the number of LIS
elements:
2(N
L
d)
2
λ
< kb−lk and
2(N
L
d)
2
λ
< kl−mk, which can
be summarized as N
L
<
√
λ
√
2d
· min{
p
kb − lk,
p
kl − mk}.
Under the condition that the positions of the BS and the LIS
are known a priori, the system can be virtually regarded as a
two-LoS aided positioning system. Intuitively, better position
estimation accuracy is expected compared to the scenario,
which is a mixture of one LoS path and one NLoS path [4].
III. PROBLEM FORMULATION
We use the two-stage approach to estimate the
user coordinate and orientation. In the first stage,
we estimate the channel parameters, defined as
η = [τ
B,M
, θ
B,M
, φ
B,M
, ρ
B,M
, τ
L,M
, φ
L,M
, ρ
L,M
]
T
with (·)
T
denoting the transpose operation. The estimate of η can be
presented in a general form as
ˆ
η = η + w, (7)
where w ∼ CN(0, Σ) denotes the estimation error.
Relying on the estimate
ˆ
η, we further obtain the MS’s
coordinate
ˆ
m and orientation ˆα via
[
ˆ
m, ˆα] = argmax
[m,α]
p(
ˆ
η|η(m, α)) (8)
= argmin
[m,α]
(
ˆ
η − η(m, α))
T
Σ
−1
(
ˆ
η − η(m, α)), (9)
where η(m, α) is a function of m and α, building the
relationship among η, m, and α detailed in (6).
3
In practice, the estimate of η can be done via compres-
sive sensing techniques, e.g., orthogonal matching pursuit
(OMP) [12], basis pursuit (BP) [12], or approximate message
passing (AMP) [13] due to the inherent sparsity property of
the mmWave MIMO channels [14].
The goal of the LIS aided mmWave MIMO positioning
system is to minimize the average distortion of the position
estimation with Euclidean distance measure, i.e.,
var(
ˆ
m) = E[(m
x
− ˆm
x
)
2
] + E[(m
y
− ˆm
y
)
2
], (10)
and that of orientation estimation, i.e.,
var(ˆα) = E[(α − ˆα)
2
], (11)
where E[·] is the expectation operator.
IV. CRAM
´
ER RAO LOWER BOUNDS
In general, we aim at calculating the Cram
´
er Rao lower
bounds for the vector-valued unknown parameter ζ =
[m
x
m
y
α]
T
. However, it is not straightforward to obtain
them. We first calculate the Fisher information matrix (FIM)
of η for the n-th subcarrier, defined as
¯
J[n] ∈ R
7×7
with
[
¯
J[n]]
i,j
= Ψ
n
(η
i
, η
j
) =
P
σ
2
<{
∂µ
H
[n]
∂η
i
∂µ[n]
∂η
j
}, where µ[n] =
√
P H[n]Fx[n]. The details on all the elements in
¯
J
n
are
described in Appendix A.
Observation 1: According to (18), (25), (31), and (36), the
estimate of channel parameters in the direct link is independent
from the NLoS via the LIS. The estimation performance
depends on the design of precoding matrix F and PRS x[n].
Observation 2: According to (40), (43), and (45), the esti-
mate of channel parameters in the indirect link is independent
from the LoS. The estimation performance depends on β[n],
which is a function of F, PRS x[n] and Ω, in the form of
β[n] = α
H
t
(θ
L,M
)Ωα
r
(φ
B,L
)α
H
t
(θ
B,L
)Fx[n]
= [α
t
(θ
L,M
) α
∗
r
(φ
B,L
)]
H
ωα
H
t
(θ
B,L
)Fx[n], (12)
where Ω = diag(ω) and denotes element-wise prod-
uct. |β[n]| ≤ N
L
|α
H
t
(θ
B,L
)Fx[n]|. When ω
i
= 2π(i −
1)
d
λ
[sin(θ
L,M
) − sin(φ
B,L
)], |β[n]| = N
L
|α
H
t
(θ
B,L
)Fx[n]|. In
other words, when F and x[n] are fixed, we can get the optimal
estimate of channel parameters in the indirect link when the
phase control matrix at LIS satisfies the following condition:
ω
i
= 2π(i − 1)
d
λ
[sin(θ
L,M
) − sin(φ
B,L
)].
Then, we derive the Jacobian matrix T
1
with [T
1
]
i,j
=
∂η
i
/∂ζ
j
, detailed in the below:
∂τ
B,M
/∂m
x
=
cos(θ
B,M
)
c
,
∂θ
B,M
/∂m
x
= ∂φ
B,M
/∂m
x
= −
sin(θ
B,M
)
kb − mk
2
,
∂ρ
B,M
/∂m
x
= −µ/2kb − mk
−µ/2−1
2
cos(θ
B,M
),
∂τ
L,M
/∂m
x
=
cos(θ
l,u
)
c
, ∂φ
L,M
/∂u
x
= −
sin(θ
L,M
)
kl − mk
2
,
∂ρ
L,M
/∂m
x
= −µ/2kl − mk
−µ/2−1
2
cos(θ
L,M
),
∂τ
B,M
/∂m
y
=
sin(θ
B,M
)
c
,
∂θ
B,M
/∂m
y
= ∂φ
B,M
/∂m
y
=
cos(θ
B,M
)
kb − mk
2
,
∂ρ
B,M
/∂m
y
= −µ/2kb − mk
−µ/2−1
2
sin(θ
B,M
),
∂τ
L,M
/∂m
y
=
sin(θ
L,M
)
c
, ∂φ
L,M
/∂m
y
=
cos(θ
L,M
)
kl − mk
2
,
∂ρ
L,M
/∂m
y
= −µ/2kl − mk
−µ/2−1
2
sin(θ
L,M
),
∂τ
B,M
/∂α = 0,
∂θ
B,M
/∂α = −1, ∂φ
B,M
/∂α = 1,
∂ρ
B,M
/∂α = 0, ∂τ
L,M
/∂α = 0,
∂φ
L,M
/∂α = −1, ∂ρ
L,M
/∂α = 0. (13)
The FIM of ζ for the n-th subcarrier is
˜
J[n] = T
1
¯
J[n]T
T
1
, (14)
and by summing up all the contributions from the N subcar-
riers, the FIM
˜
J of ζ is in the form of
˜
J =
(N−1)/2
X
n=−(N−1)/2
˜
J[n]. (15)
The objective is to find the minimal theoretically achievable
value for the standard deviation of positioning estimation error
and orientation estimation error, which is the Cram
´
er Rao
lower bound, written as
PEB =
q
tr{[
˜
J
−1
]
1:2,1:2
} ≤
p
var(
ˆ
m), (16)
and
OEB =
q
[
˜
J
−1
]
3,3
≤
p
var(ˆα). (17)
V. SIMULATION RESULTS
The parameters are set up as follows: (b
x
, b
y
) = (0, 0),
(l
x
, l
y
) = (160/3, 80), (m
x
, m
y
) = (80, 40), α = π/10, µ =
2.08 (path loss exponent), N
B
= 128, N
M
= 32, N = 31,
B = 100 MHz, f
c
= 60 GHz, and α = π/10. According to
the far field constraints, N
L
≤ 138. The signal-to-noise ratio
(SNR) is defined as
P
σ
2
. For the purpose of comparison, we
introduce a benchmark scenario [4] with one LoS and one
NLoS path with the scatter located at (160/3, 80). For the
calculation of theoretical performance limits, the location of
the scatter is deterministic but unknown to the MS.
A. Impact of the LIS: Phase Shifter
In this experiment, we set each element of f = Fx[n]
as e
jν
with ν ∼ U(0 2π], ∀n. It shown in (12) that if
ω
i
= 2π(i − 1)
d
λ
[sin(θ
L,M
) − sin(φ
B,L
)], the diagonal entries
in FIM, e.g., (40), (43), and (45), can achieve the maximum
value, i.e., α
H
t
(θ
L,M
)Ωα
r
(φ
B,L
) in β satisfies the following
condition α
H
t
(θ
L,M
)Ωα
r
(φ
B,L
) = N
L
when ω
i
= 2π(i −
1)
d
λ
[sin(θ
L,M
) −sin(φ
B,L
)] (labeled as “Incremental phase” in
Fig. 2). This in turn provides a better estimation performance
of NLoS channel parameters from the theoretical perspective,
which is verified in Fig. 2. “LIS” in the legend stands for the
4
-20 -15 -10 -5 0 5 10 15 20
SNR [dB]
10
-4
10
-3
10
-2
10
-1
10
0
10
1
CRB of Standard Deviation
Fig. 2: The impact of phases on CRB of standard deviation of channel
parameters in the reflection path with N
L
= 100.
studied LIS aided mmWave MIMO positioning system. Note
that all the simulation curves are obtained under the same
randomly-generated f and other parameters but only different
phase conditions (benchmark scheme with random ω
i
in Ω,
labeled as “Random phase” in Fig. 2). The incremental phase
significantly outperforms the random phase (roughly 10 times
better in the current experiment). The optimal design of f will
bring better performance, and be left as our future work.
B. Impact of the LIS: Number of Elements
In this experiment, we set ω
i
= 2π(i − 1)
d
λ
[sin(θ
L,M
) −
sin(φ
B,L
)] and SNR to 5 dB. The number of LIS elements
plays a critical role on the estimation performance of channel
parameters in the reflection path. The CRB of normalized
standard deviation of τ
L,M
, φ
L,M
, and ρ
L,M
are inversely
proportional to the number of elements in the LIS, e.g., N
L
.
As verified in Fig. 3, the larger the number of elements in
the LIS, the better the estimation performance of the channel
parameters. With around 100 LIS elements, the estimation
performance can be improved around 100 times from the
results shown in Fig. 3. In the legends, ρ
B,S,M
, φ
B,S,M
, and
τ
B,S,M
denote the path loss, AoA, and ToA of the NLoS path
via the scatter, respectively, for the benchmark scheme. Due to
the fixed number of scatter, the estimation performance of the
benchmark scheme stays unchanged. Note that the increase
of elements does not have any great impact on the estimation
accuracy of channel parameters of the direct path, which is
not difficult to understand based on the FIM
¯
J[n].
C. PEB and OEB
In this subsection, we evaluate the impact of the number
of LIS elements on both the PEB and the OEB while fixing
ω
i
= 2π(i − 1)
d
λ
[sin(θ
L,M
) − sin(φ
B,L
)]. As shown in Figs. 4
and 5, the increase of elements improves the positioning
performance. Even with the help of a 40-element LIS, around
3 dB gain can be achieved when the OEB is at the level of
10
−2
. It should be noted that the performance enhancement
mainly come from the improvement of the NLoS via the LIS.
0 10 20 30 40 50 60 70 80 90
10
-3
10
-2
10
-1
10
0
CRB of Normalized Standard Deviation
Fig. 3: CRB of normalized standard deviation of channel parameters
versus the number of LIS elements.
-20 -15 -10 -5 0 5 10 15 20
SNR [dB]
10
-1
10
0
10
1
CRB of Standard Deviation
Fig. 4: PEB versus the SNR.
VI. CONCLUSIONS
We have studied the fundamental limits of mmWave MIMO
positioning with the aid of a LIS. The impact of the number
of LIS elements and the phases of LIS elements on the
estimation of channel parameters has been investigated. This
in turn, helps the analyses on the positioning and orientation
error bounds. The comparison has been made between the
positioning system with and without the assistance of LIS to
show the potential benefits brought by the introduction of LIS
even with passive elements. Since the beamformer design at
BS and phase shifter design at LIS play a critical role in the
positioning, the joint consideration of them will be left as our
future investigation.
APPENDIX A
DERIVATION OF FIM
The derivation of the FIM on all the channel parameters is
shown as follows:
Ψ
n
(τ
B,M
, τ
B,M
) =
P N
M
ρ
2
B,M
σ
2
(2πnB)
2
N
2
kα
H
t
(θ
B,M
)Fx[n]k
2
2
, (18)
Ψ
n
(τ
B,M
, θ
B,M
) =
P ρ
2
B,M
σ
2
<{j2π
nB
N
(x[n])
H
F
H
α
t
(θ
B,M
)
˙
α
H
t
(θ
B,M
)Fx[n]}, (19)
Ψ
n
(τ
B,M
, φ
B,M
) =
P ρ
2
B,M
σ
2
× <{j2π
nB
N
(x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)
˙
α
r
(φ
B,M
)α
H
t
(θ
B,M
)Fx[n]},
(20)
Ψ
n
(τ
B,M
, ρ
B,M
) =
P ρ
B,M
σ
2
kα
H
t
(θ
B,M
)Fx[n]k
2
2
<{j2π
nB
N
} = 0, (21)
5
-20 -15 -10 -5 0 5 10 15 20
SNR [dB]
10
-3
10
-2
10
-1
CRB of Standard Deviation
Fig. 5: OEB versus the SNR.
Ψ
n
(τ
B,M
, τ
L,M
) =
P ρ
B,M
ρ
B,L
ρ
L,M
σ
2
(2πnB)
2
N
2
× <{β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (22)
Ψ
n
(τ
B,M
, φ
L,M
) =
P ρ
B,M
ρ
B,L
ρ
L,M
σ
2
× <{j2π
nB
N
β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)
˙
α
r
(φ
L,M
)}, (23)
Ψ
n
(τ
B,M
, ρ
L,M
) =
P ρ
B,M
ρ
B,L
σ
2
× <{j2π
nB
N
β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (24)
Ψ
n
(θ
B,M
, θ
B,M
) =
P N
M
ρ
2
B,M
σ
2
k
˙
α
H
t
(θ
B,M
)Fx[n]k
2
2
, (25)
Ψ
n
(θ
B,M
, φ
B,M
) =
P ρ
2
B,M
σ
2
× <{(x[n])
H
F
H
˙
α
t
(θ
B,M
)α
H
r
(φ
B,M
)
˙
α
r
(φ
B,M
)α
H
t
(θ
B,M
)Fx[n]}, (26)
Ψ
n
(θ
B,M
, ρ
B,M
) =
P N
M
ρ
B,M
σ
2
<{(x[n])
H
F
H
˙
α
t
(θ
B,M
)α
H
t
(θ
B,M
)Fx[n]}, (27)
Ψ
n
(θ
B,M
, τ
L,M
) =
P ρ
B,M
ρ
B,L
ρ
L,M
σ
2
× <{−j2π
nB
N
β[n]ξ[n](x[n])
H
F
H
˙
α
t
(θ
B,M
)α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (28)
Ψ
n
(θ
B,M
, φ
L,M
) =
P ρ
B,M
ρ
B,L
ρ
L,M
σ
2
× <{β[n]ξ[n](x[n])
H
F
H
˙
α
t
(θ
B,M
)α
H
r
(φ
B,M
)
˙
α
r
(φ
L,M
)}, (29)
Ψ
n
(θ
B,M
, ρ
L,M
) =
P ρ
B,M
ρ
B,L
σ
2
× <{β[n]ξ[n](x[n])
H
F
H
˙
α
t
(θ
B,M
)α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (30)
Ψ
n
(φ
B,M
, φ
B,M
) =
P ρ
2
B,M
σ
2
k
˙
α
r
(φ
B,M
)α
H
t
(θ
B,M
)Fx[n]k
2
2
, (31)
Ψ
n
(φ
B,M
, ρ
B,M
) =
P ρ
B,M
σ
2
× <{(x[n])
H
F
H
α
t
(θ
B,M
)
˙
α
H
r
(φ
B,M
)α
r
(φ
B,M
)α
H
t
(θ
B,M
)Fx[n]}, (32)
Ψ
n
(φ
B,M
, τ
L,M
) =
P ρ
B,M
ρ
B,L
ρ
L,M
σ
2
× <{−j2π
nB
N
β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)
˙
α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (33)
Ψ
n
(φ
B,M
, φ
L,M
) =
P ρ
B,M
ρ
B,L
ρ
L,M
σ
2
× <{β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)
˙
α
H
r
(φ
B,M
)
˙
α
r
(φ
L,M
)}, (34)
Ψ
n
(φ
B,M
, ρ
L,M
) =
P ρ
B,M
ρ
B,L
σ
2
× <{β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)
˙
α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (35)
Ψ
n
(ρ
B,M
, ρ
B,M
) =
P N
M
σ
2
kα
H
t
(θ
B,M
)Fx[n]k
2
2
, (36)
Ψ
n
(ρ
B,M
, τ
L,M
) =
P ρ
B,L
ρ
L,M
σ
2
× <{−j2π
nB
N
β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (37)
Ψ
n
(ρ
B,M
, φ
L,M
) =
P ρ
B,L
ρ
L,M
σ
2
× <{β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)
˙
α
r
(φ
L,M
)}, (38)
Ψ
n
(ρ
B,M
, ρ
L,M
) =
P ρ
B,L
σ
2
× <{β[n]ξ[n](x[n])
H
F
H
α
t
(θ
B,M
)α
H
r
(φ
B,M
)α
r
(φ
L,M
)}, (39)
Ψ
n
(τ
L,M
, τ
L,M
) =
P N
M
ρ
2
B,L
ρ
2
L,M
σ
2
(2πnB)
2
|β[n]|
2
N
2
, (40)
Ψ
n
(τ
L,M
, φ
L,M
) =
P ρ
2
B,L
ρ
2
L,M
σ
2
× <{j2π
nB
N
(β[n])
∗
α
H
r
(φ
L,M
)
˙
α
r
(φ
L,M
)Ωα
r
(φ
B,L
)α
H
t
(θ
B,L
)Fx[n]}, (41)
Ψ
n
(τ
L,M
, ρ
L,M
) =
P ρ
2
B,L
ρ
L,M
|β[n]|
2
σ
2
<{j2π
nB
N
} = 0, (42)
Ψ
n
(φ
L,M
, φ
L,M
) =
P ρ
2
B,L
ρ
2
L,M
σ
2
kβ[n]
˙
α
r
(φ
L,M
)k
2
2
, (43)
Ψ
n
(φ
L,M
, ρ
L,M
) =
P ρ
2
B,L
ρ
L,M
σ
2
× <{(β[n])
∗
α
H
r
(φ
L,M
)
˙
α
r
(φ
L,M
)Ωα
r
(φ
B,L
)α
H
t
(θ
B,L
)Fx[n]}, (44)
Ψ
n
(ρ
L,M
, ρ
L,M
) =
P N
M
ρ
2
B,L
|β[n]|
2
σ
2
, (45)
where
˙
α
t
(θ
B,M
) = ∂α
t
(θ
B,M
)/∂θ
B,M
= D
t
(θ
B,M
)α
t
(θ
B,M
),
˙
α
r
(φ
B,M
) = ∂α
r
(φ
B,M
)/∂φ
B,M
= D
r
(φ
B,M
)α
r
(φ
B,M
),
˙
α
r
(φ
L,M
) = ∂α
r
(φ
L,M
)/∂φ
L,M
= D
r
(φ
L,M
)α
r
(φ
L,M
) with
D
t
(θ
B,M
) = j2π
d
λ
cos(θ
B,M
)diag(0, 1, ··· , i, ··· , N
B
− 1),
D
r
(φ
B,M
) = j2π
d
λ
cos(φ
B,M
)diag(0, 1, ··· , i, ··· , N
M
− 1),
D
r
(φ
L,M
) = j2π
d
λ
cos(φ
L,M
)diag(0, 1, ··· , i, ··· , N
M
− 1),
ξ[n] = e
−j[2π(τ
B,L
+τ
L,M
−τ
B,M
)
nB
N
]
, and β[n] =
α
H
t
(θ
L,M
)Ωα
r
(φ
B,L
)α
H
t
(θ
B,L
)Fx[n].
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