# Outage Analysis and Beamwidth Optimization for Positioning-Assisted
Beamforming

Bingcheng Zhu, , Zaichen Zhang, , Julian Cheng, , Lei Wang, Jian Dang, , and
Liang Wu, , Bingcheng Zhu, Zaichen Zhang, Lei Wang, Jian Dang and Liang Wu are
with School of Information Science and Engineering, Southeast University,
Nanjing, Jiangsu, China (e-mail: {zbc, zczhang, wang_lei_seu, dangjian,
wuliang}@seu.edu.cn). The corresponding author of this work is Zaichen
Zhang.Julian Cheng is with School of Engineering, The University of British
Columbia, Kelowna, BC, Canada (e-mail<EMAIL_ADDRESS>
###### Abstract

Conventional beamforming is based on channel estimation, which can be
computationally intensive and inaccurate when the antenna array is large. In
this work, we study the outage probability of positioning-assisted beamforming
systems. Closed-form outage probability bounds are derived by considering
positioning error, link distance and beamwidth. Based on the analytical
result, we show that the beamwidth should be optimized with respect to the
link distance and the transmit power, and such optimization significantly
suppresses the outage probability.

###### Index Terms:

Beamforming, location-aware, outage probability, pointing error, positioning.

##  I Introduction

Massive multiple-input multiple output (MIMO) technique significantly narrows
radio-frequency beams, enabling more precise power assignment to receivers.
However, implementation of massive MIMO systems still have some challenges.
For example, even though the antenna array can be manufactured large, the
associated units for channel estimation and beamforming are expensive and
energy intensive [1, 2]. Another important issue is that the feedback overhead
surges with the number of antenna elements. Partial or reduced-dimensional
channel state information (CSI) [3] can be used to address this issue at the
expense of channel estimation error. Moreover, channel estimation error
increases with the link distance, resulting from the SNR deterioration and
nonnegligible quantization errors [4]. Furthermore, outdated CSI, multiuser
interferences and hareware impairments can also cause channel estimation error
[5, 6]. All these factors lead to pointing error and loss of beamforming
gains.

Positioning information can be exploited to resolve the practical difficulties
of beamforming. This idea originated from the fact that the desired antenna
phase shifts can be expressed as functions of receiver locations in line-of-
sight cases [7], and positioning outputs can provide prior knowledge on
conventional beamforming to help alleviate the complexity [8, 9]. It was shown
that the positioning information can be used to design beamforming in 5G
networks [10], and a subsequent work analyzed a beamformed radio link capacity
with positioning errors [11]. However, the work did not assume a certain
probability density function model for the positioning error, and thus the
resulting capacity expression contains multifold integrals. An elegant
performance expression is crucial especially when the system needs
instantaneous parameter adjustment, which is the case for the positioning-
assisted beamforming. In such systems, there exists a tradeoff between
beamwidth and directivity [12], but the existing analytical results were based
on the dichotomy of the beam’s coverage and the receiver’s movement is one-
dimensional, making it unable to model two-dimensional random walking cases.

In this work, we carry out an outage analysis for positioning-assisted
beamforming and optimize the beamwidth. Closed-form outage probability bounds
are derived and their asymptotic tightness is verified. Besides, the optimal
beamwidth is expressed as a closed-form function of the positioning error,
link distance and transmit power. The new analytical tool reveals insights
into the design of reliable beamforming systems by exploiting positioning
results, and lowers the computational complexity for parameter optimization.
We show that the conventional wisdom does not necessarily hold for the
narrowest beams to achieve the highest reliability.

##  II System Model

###  II-A Positioning Error

As shown in Fig. 1, we assume that the transmitter is located at
$\left(0,0\right)$ and the exact receiver position is
${\textbf{p}}_{u}=\left[0,d\right]^{T}$, where $d$ is the distance between the
transmitter and the receiver. The estimated user position is
$\hat{\textbf{p}}_{u}={\left[{\hat{x}_{u},\hat{y}_{u}}\right]^{T}}$. The joint
probability density function of $\hat{\textbf{p}}_{u}$ can be expressed as [9]

${f_{\hat{\textbf{p}}_{u}}}\left(\hat{\textbf{p}}_{u}\right)=\frac{1}{{2\pi\sqrt{\det\left({\textbf{R}}\right)}}}\exp\left({-\frac{{{{\left({\hat{\textbf{p}}_{u}-{\textbf{p}}_{u}}\right)}^{T}}{{\textbf{R}}^{-1}}\left({\hat{\textbf{p}}_{u}-{\textbf{p}}_{u}}\right)}}{2}}\right)$
(1)

where
${\textbf{R}}=E\left[{\left({\hat{\textbf{p}}_{u}-{\textbf{p}}_{u}}\right)\left({\hat{\textbf{p}}_{u}-{\textbf{p}}_{u}}\right)^{T}}\right]$
is the covariance matrix of the positioning error vector
$\hat{\textbf{p}}_{u}-{\textbf{p}}_{u}$, and $\det(\cdot)$ denotes the
determinant. Applying eigenvalue decomposition, we can express R as

${\textbf{R}}=\underbrace{\left[{\begin{array}[]{*{20}{c}}{\cos\varphi}&{-\sin\varphi}\\\
{\sin\varphi}&{\cos\varphi}\end{array}}\right]}_{\textbf{G}}\underbrace{\left[{\begin{array}[]{*{20}{c}}{\sigma_{1}^{2}}&0\\\
0&{\sigma_{2}^{2}}\end{array}}\right]}_{\bf{\Lambda}}\\\
\underbrace{\left[{\begin{array}[]{*{20}{c}}{\cos\varphi}&{\sin\varphi}\\\
{-\sin\varphi}&{\cos\varphi}\end{array}}\right]}_{{{\textbf{G}}^{T}}}$ (2)

where $\sigma_{1}>\sigma_{2}$, and $\varphi$ denotes the directional angle
along which the larger positioning error occurs, which has variance
$\sigma_{1}^{2}$, while $\pi/2-\varphi$ corresponds to the direction having
the smaller positioning error with variance $\sigma_{2}^{2}$. This model
complies with the conclusion that a Kalman filter outputs predictive Gaussian
distributions when the prior observations follow a Gaussian distribution [13].

Figure 1: The beam direction and the exact and estimated receiver positions.
The receiver is located at $\textbf{p}_{u}=(0,d)$, and the beam points to the
estimated receiver position $\hat{\textbf{p}}_{u}=(\hat{x}_{u},\hat{y}_{u})$.

###  II-B Channel Model

Assuming the antenna gains are unity, we can express the received power as

${P_{r}}={P_{\max}}{G_{\theta}}{G_{d}}$ (3)

where $P_{\max}$ is the maximum antenna gain at the boresight direction;
${G_{\theta}}$ and ${G_{d}}$ denote, respectively, the transmit field
radiation patterns induced power gain and the free-space propagation induced
power gain, which can be calculated through the Friis equation [14, eq. (2.7)]

${G_{d}}=\frac{{{\lambda^{2}}}}{{{{\left({4\pi d}\right)}^{2}}}}$ (4)

where $\lambda$ is the carrier wavelength, and we have [3, eq. (2)]

${G_{\theta}}=\max\left\\{{{{10}^{-1.2\frac{{{\theta^{2}}}}{{\theta_{3dB}^{2}}}}},{a_{m}}}\right\\}$
(5)

where $\theta\in(-\pi,\pi]$ denotes the angle of the receiver direction from
the boresight direction and $2\theta_{3dB}$ indicates the 3-dB beamwidth of
the antenna radiation pattern, and $a_{m}$ equals the side-lobe level
attenuation of the antenna pattern.

###  II-C Positioning-Assisted Beamforming

As shown in Fig. 1, the transmitter rotates the boresight direction of the
antenna array towards the estimated user position $\hat{\textbf{p}}_{u}$,
where the pointing error angle is $\theta$. Therefore, we have
$\theta=\textrm{atan2}(\hat{y}_{u},\hat{x}_{u})\in(-\pi,\pi]$, where
$\textrm{atan2}(\cdot,\cdot)$ is the four-quadrant inverse tangent function.
The outage probability of the positioning-assisted beamforming can be
expressed as

$\begin{split}&{P_{out}}\left({{\gamma_{th}}}\right)=\Pr\left({{P_{r}}\leq{\gamma_{th}}}\right)\\\
&=\Pr\left({{P_{\max}}{G_{\theta}}{G_{d}}\leq{\gamma_{th}}}\right)\\\
&=\Pr\left({\max\left\\{{{{10}^{-\frac{{1.2}}{{\theta_{3dB}^{2}}}{{\textrm{atan2}}^{2}}\left({{{{{\hat{y}}_{u}}}},{{{{\hat{x}}_{u}}}}}\right)}},{a_{m}}}\right\\}\frac{{{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}\leq\frac{{{\gamma_{th}}}}{{{P_{\max}}}}}\right)\\\
&=\left\\{\begin{array}[]{l}\Pr\left({\left|{{{\hat{x}}_{u}}}\right|\geq
k{{\hat{y}}_{u}}}\right),\frac{{{P_{\max}}{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}>{\gamma_{th}}>\frac{{{P_{\max}}{a_{m}}{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{{{P_{\max}}{a_{m}}{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}\geq{\gamma_{th}}\\\ 1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{{{P_{\max}}{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}\leq{\gamma_{th}}\end{array}\right.\end{split}$ (6)

where
$k=\tan\sqrt{\frac{{\theta_{3dB}^{2}}}{{1.2}}\lg\left({\frac{{{\lambda^{2}}{P_{\max}}}}{{{{\left({4\pi
d}\right)}^{2}}{\gamma_{th}}}}}\right)}>0$;
${P_{out}}\left({{\gamma_{th}}}\right)=0$ corresponds to the case where the
received lobe power surpasses the threshold $\gamma_{th}$ and
${P_{out}}\left({{\gamma_{th}}}\right)=1$ corresponds to the case where the
received power is below the threshold even with perfect pointing.

Figure 2: The beam direction and the exact and estimated receiver positions.
The receiver is located at $\textbf{p}_{u}=(0,d)$, and the beam points to the
estimated receiver position $\hat{\textbf{p}}_{u}=(\hat{x}_{u},\hat{y}_{u})$.

##  III Outage Probability

###  III-A Outage Probability Bounds

As shown in (1) and (6), when $\frac{{{P_{\max}}{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}>{\gamma_{th}}>\frac{{{P_{\max}}{a_{m}}{\lambda^{2}}}}{{{{\left({4\pi
d}\right)}^{2}}}}$, the outage probability can be expressed

${{\mathop{\rm
P}\nolimits}_{out}}\left({{\gamma_{th}}}\right)=\iint\limits_{\left|{{{\hat{x}}_{u}}}\right|\geq
k{{\hat{y}}_{u}}}{f_{\hat{\textbf{p}}_{u}}}\left(\hat{\textbf{p}}_{u}\right)d{\hat{\textbf{p}}_{u}}$
(7)

where $d\hat{\textbf{p}}_{u}:=d\hat{x}_{u}d\hat{y}_{u}$, and the integral
region can be defined as

${\Phi_{out}}=\left\\{{\left.{\left({{{\hat{x}}_{u}},{{\hat{y}}_{u}}}\right)}\right|\left|{{{\hat{x}}_{u}}}\right|\geq
k{{\hat{y}}_{u}}}\right\\}.$ (8)

It can be verified that

${\Phi_{out}}={\Phi_{L}}\cup{\Phi_{R}}$ (9)

where

$\begin{split}{\Phi_{R}}&=\left\\{{\left.{\left({{{\hat{x}}_{u}},{{\hat{y}}_{u}}}\right)}\right|{{\hat{x}}_{u}}\geq
k{{\hat{y}}_{u}}}\right\\},\\\
{\Phi_{L}}&=\left\\{{\left.{\left({{{\hat{x}}_{u}},{{\hat{y}}_{u}}}\right)}\right|{{\hat{x}}_{u}}\leq-k{{\hat{y}}_{u}}}\right\\}\end{split}$
(10)

which are shown in Fig. 2. Therefore, splitting the integral region in (7), we
have

${{\rm{P}}_{out}}\left({{\gamma_{th}}}\right)<\underbrace{\iint\limits_{{{\hat{x}}_{u}}\geq
k{{\hat{y}}_{u}}}{{f_{\hat{\textbf{p}}_{u}}}\left(\hat{\textbf{p}}_{u}\right)d{\hat{\textbf{p}}_{u}}}}_{{I_{R}}}+\underbrace{\iint\limits_{{{\hat{x}}_{u}}\leq-k{{\hat{y}}_{u}}}{{f_{\hat{\textbf{p}}_{u}}}\left({{\hat{\textbf{p}}_{u}}}\right)d\hat{\textbf{p}}_{u}}}_{{I_{L}}}.$
(11)

and after complicated simplification in Appendix A, we have

${I_{R}}=Q\left({-\frac{{\left[{1,-k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,-k}\right]\sqrt{\textbf{R}}}\right\|}}}\right)$
(12)

where $Q(x)=1/\sqrt{2\pi}\int_{x}^{\infty}\exp\\{-t^{2}/2\\}dt$ is the
Gaussian $Q$-function and

${I_{L}}=Q\left({\frac{{\left[{1,k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,k}\right]\sqrt{\textbf{R}}}\right\|}}}\right).$
(13)

According to Fig. 2, we can express the outage probability as

$\begin{split}{{\mathop{\rm
P}\nolimits}_{out}}\left({{\gamma_{th}}}\right)={I_{R}}+{I_{L}}-\int\limits_{{\Phi_{R}}\cap{\Phi_{R}}}{{f_{\hat{\textbf{p}}_{u}}}\left(\hat{\textbf{p}}_{u}\right)d}\hat{\textbf{p}}_{u}\end{split}$
(14)

where the integral region is bounded as

${\Phi_{L}}\cap{\Phi_{R}}\subset\left\\{{\left.{\left({{{\hat{x}}_{u}},{{\hat{y}}_{u}}}\right)}\right|{{\hat{y}}_{u}}\leq
0}\right\\}:={\Phi_{B}}$ (15)

because

$\left\\{\begin{array}[]{l}{{\hat{x}}_{u}}\geq k{{\hat{y}}_{u}}\\\
{{\hat{x}}_{u}}\leq-k{{\hat{y}}_{u}}\end{array}\right.\Rightarrow\left\\{\begin{array}[]{l}{{\hat{x}}_{u}}\geq
k{{\hat{y}}_{u}}\\\ -{{\hat{x}}_{u}}\geq
k{{\hat{y}}_{u}}\end{array}\right.\Rightarrow{{\hat{y}}_{u}}\leq 0.$ (16)

Using the bounding region in (15) to replace the integral region in (14), we
obtain

${{\mathop{\rm
P}\nolimits}_{out}}\left({{\gamma_{th}}}\right)>{I_{R}}+{I_{L}}-\underbrace{\int\limits_{{{\hat{y}}_{u}}\leq
0}{{f_{\hat{\textbf{p}}_{u}}}\left(\hat{\textbf{p}}_{u}\right)d}{\hat{\textbf{p}}_{u}}}_{{I_{B}}}$
(17)

where $I_{B}$ can be simplified according to Appendix A as

$I_{B}=Q\left({\frac{{\left[{0,1}\right]{{\textbf{p}}_{u}}}}{{\left\|{\sqrt{\textbf{R}}{{\left[{0,1}\right]}^{T}}}\right\|}}}\right).$
(18)

Substituting (12) and (13) into (11), and (12), (13), (18) into (17), we
obtain the upper and lower bounds as

$\begin{split}&\underbrace{Q\left({-\frac{{\left[{1,-k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,-k}\right]\sqrt{\textbf{R}}}\right\|}}}\right)}_{I_{R}}+\underbrace{Q\left({\frac{{\left[{1,k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,k}\right]\sqrt{\textbf{R}}}\right\|}}}\right)}_{I_{L}}>{{\mathop{\rm
P}\nolimits}_{out}}\left({{\gamma_{th}}}\right)\\\
&>\underbrace{Q\left({-\frac{{\left[{1,-k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,-k}\right]\sqrt{\textbf{R}}}\right\|}}}\right)}_{I_{R}}+\underbrace{Q\left({\frac{{\left[{1,k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,k}\right]\sqrt{\textbf{R}}}\right\|}}}\right)}_{I_{L}}\\\
&-\underbrace{Q\left({\frac{{\left[{0,1}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{0,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right)}_{I_{B}}.\end{split}$
(19)

###  III-B Convergence of the Bounds

To prove the tightness of the bounds, we need to develop an equality and two
inequalities. Reforming (12), (13) and (18), we obtain

$\begin{split}{I_{R}}&=Q\left({\frac{d}{{\left\|{\left[{-1/k,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right),\\\
{I_{L}}&=Q\left({\frac{d}{{\left\|{\left[{1/k,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right),\\\
{I_{B}}&=Q\left({\frac{d}{{\left\|{\left[{0,1}\right]\sqrt{\textbf{R}}}\right\|}}}\right)\end{split}$
(20)

where the denominators can be compared as

$\begin{split}&\left\|{\left({-\frac{1}{k},1}\right)\sqrt{\textbf{R}}}\right\|{\rm{=}}\left\|{-\frac{1}{k}{{\left({\sqrt{\textbf{R}}}\right)}_{1}}{\rm{+}}{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\right\|\\\
&=\sqrt{{{\left\|{\frac{1}{k}{{\left({\sqrt{\textbf{R}}}\right)}_{1}}}\right\|}^{2}}+{{\left\|{{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\right\|}^{2}}-\frac{2}{k}{{\left({\sqrt{\textbf{R}}}\right)}_{1}}\cdot{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\\\
&>\left\|{{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\right\|=\left\|{\left[{0,1}\right]\sqrt{\textbf{R}}}\right\|\end{split}$
(21)

when ${({\sqrt{\textbf{R}}})_{1}}\cdot{({\sqrt{\textbf{R}}})_{2}}<0$ where
$({\sqrt{\textbf{R}}})_{l}$ denotes the $l$th row vector of
$\sqrt{\textbf{R}}$, and

$\begin{split}&\left\|{\left({\frac{1}{k},1}\right)\sqrt{\textbf{R}}}\right\|{\rm{=}}\left\|{\frac{1}{k}{{\left({\sqrt{\textbf{R}}}\right)}_{1}}{\rm{+}}{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\right\|\\\
&=\sqrt{{{\left\|{\frac{1}{k}{{\left({\sqrt{\textbf{R}}}\right)}_{1}}}\right\|}^{2}}+{{\left\|{{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\right\|}^{2}}+\frac{2}{k}{{\left({\sqrt{\textbf{R}}}\right)}_{1}}\cdot{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\\\
&>\left\|{{{\left({\sqrt{\textbf{R}}}\right)}_{2}}}\right\|=\left\|{\left[{0,1}\right]\sqrt{\textbf{R}}}\right\|\end{split}$
(22)

when ${({\sqrt{\textbf{R}}})_{1}}\cdot{({\sqrt{\textbf{R}}})_{2}}\geq 0$.

The ratio of two Gaussian $Q$-functions satisfies

$\begin{split}&\mathop{\lim}\limits_{t\to\infty}\frac{{Q\left({\alpha
t}\right)}}{{Q\left(t\right)}}=\frac{{\int\limits_{\alpha
t}^{\infty}{\frac{1}{{\sqrt{2\pi}}}\exp\left({-\frac{{{x^{2}}}}{2}}\right)dx}}}{{\int\limits_{t}^{\infty}{\frac{1}{{\sqrt{2\pi}}}\exp\left({-\frac{{{x^{2}}}}{2}}\right)dx}}}\\\
&=\mathop{\lim}\limits_{t\to\infty}\frac{{\frac{1}{{\sqrt{2\pi}}}\exp\left({-\frac{{{{\left({\alpha
t}\right)}^{2}}}}{2}}\right)\alpha}}{{\frac{1}{{\sqrt{2\pi}}}\exp\left({-\frac{{{t^{2}}}}{2}}\right)}}\\\
&=\mathop{\lim}\limits_{t\to\infty}\alpha\exp\left({\frac{{{t^{2}}}}{2}\left({1-{\alpha^{2}}}\right)}\right){\rm{=}}\left\\{\begin{array}[]{l}\infty,0<\alpha<1\\\
0,\alpha>1\end{array}\right.\end{split}$ (23)

where the second equality is based on the L’Hospital’s rule.

The bounds in (19) are asymptotically tight in two cases.

####  III-B1 Case 1: $d\to\infty$

As shown in (21) and (22), the denominator in $I_{B}$ in (20) is never the
biggest among the three denominators. Therefore, applying (23) we can show

$\mathop{\lim}\limits_{d\to\infty}\frac{{{I_{B}}}}{{{I_{L}}+{I_{R}}}}=0$ (24)

which leads to the conclusion that the bounds in (19) are asymptotically tight
when $d\to\infty$.

####  III-B2 Case 2: ${{\rm{tr}}\left({\textbf{\bf{R}}}\right)}\to 0$

The covariance matrix R can be normalized as
${{\textbf{R}}_{N}}={{\textbf{R}}}/{{{\rm{tr}}\left({\textbf{R}}\right)}}$,
which is assumed to be fixed, thus (20) becomes

$\begin{split}{I_{R}}&=Q\left({\frac{{d/\sqrt{{\rm{tr}}\left({\textbf{R}}\right)}}}{{\left\|{\left[{-1/k,1}\right]\sqrt{{{\textbf{R}}_{N}}}}\right\|}}}\right),\\\
{I_{L}}&=Q\left({\frac{{d/\sqrt{{\rm{tr}}\left({\textbf{R}}\right)}}}{{\left\|{\left[{1/k,1}\right]\sqrt{{{\textbf{R}}_{N}}}}\right\|}}}\right),\\\
{I_{B}}&=Q\left({\frac{{d/\sqrt{{\rm{tr}}\left({\textbf{R}}\right)}}}{{\left\|{\left[{0,1}\right]\sqrt{{{\textbf{R}}_{N}}}}\right\|}}}\right).\end{split}$
(25)

Applying (21)-(23) to compare the equations in (25), we have

$\mathop{\lim}\limits_{{\rm{tr}}\left({\textbf{R}}\right)\to
0}\frac{{{I_{B}}}}{{{I_{L}}+{I_{R}}}}=0$ (26)

which leads to the conclusion that the bounds in (19) are asymptotically tight
when ${\rm{tr}}\left({\textbf{R}}\right)\to 0$.

Case 1 describes the case when the transmitter and the receiver are
sufficiently far away, and case 2 describes the case when the positioning
system is sufficiently accurate.

##  IV Optimization of the beamwidth

The beamwidth $\theta_{3dB}$ can be adjusted at the transmitter to minimize
the outage probability, and we assume the transmit power to be fixed, which
can be calculated as

$\begin{split}P_{t}=\int\limits_{-{\pi}}^{{\pi}}{{P_{\max}}{{10}^{-1.2\frac{{{\theta^{2}}}}{{\theta_{3dB}^{2}}}}}d\theta}={P_{\max}}\frac{{\sqrt{\pi}erf\left({{\pi}\sqrt{\frac{{1.2}}{{\theta_{3dB}^{2}}}\ln
10}}\right)}}{{\sqrt{\frac{{1.2}}{{\theta_{3dB}^{2}}}\ln 10}}}\end{split}$
(27)

where the side-lobe level is assumed to be negligible. When
$\theta_{3dB}\approx 0$, we have
$erf\left({{\pi}\sqrt{\frac{{1.2}}{{\theta_{3dB}^{2}}}\ln 10}}\right)\approx
1$, thus

$\begin{split}P_{t}\approx{P_{\max}}\frac{{{\theta_{3dB}}\sqrt{\pi}}}{{\sqrt{1.2\ln
10}}}\end{split}$ (28)

and

$\begin{split}{P_{\max}}\approx{P_{t}}\frac{{\sqrt{1.2\ln
10}}}{{{\theta_{3dB}}\sqrt{\pi}}}.\end{split}$ (29)

To minimize the outage probability in (7), we need to minimize the integral
region, which is equivalent to maximizing the factor $k$. The problem has the
same maximizer with the following problem

$\mathop{\max}\limits_{{\theta_{3dB}}}\theta_{3dB}^{2}\lg\left({\frac{{{\lambda^{2}}{P_{t}}\sqrt{1.2\ln
10}}}{{{\theta_{3dB}}\sqrt{\pi}{{\left({4\pi
d}\right)}^{2}}{\gamma_{th}}}}}\right)$ (30)

where $P_{\max}$ is replaced according to (29) and we focus on the case
${\theta_{3dB}}\leq\frac{{{\lambda^{2}}{P_{t}}\sqrt{1.2\ln
10}}}{{\sqrt{\pi}{{\left({4\pi d}\right)}^{2}}{\gamma_{th}}}}$ so that $k>0$.
Taking the derivative of (30) in terms of $\theta_{3dB}$ and making it zero,
we can obtain the maximizer of (30) as

$\theta_{3dB}^{*}=\frac{{{\lambda^{2}}{P_{t}}\sqrt{1.2\ln
10}}}{{\sqrt{\pi}{{\left({4\pi
d}\right)}^{2}}{\gamma_{th}}}}{10^{-\frac{1}{{2\ln 10}}}}$ (31)

which results in the optimal $k$ as

${k^{*}}=\tan\left({\frac{{{\lambda^{2}}{P_{t}}}}{{\sqrt{2\pi}{{\left({4\pi
d}\right)}^{2}}{\gamma_{th}}}}{{10}^{-\frac{1}{{2\ln 10}}}}}\right).$ (32)

Equation (31) indicates that the optimal beamwidth is a function of the
transmit power $P_{t}$, the link distance $d$ and the outage threshold
$\gamma_{th}$. When $d$ grows, the optimal beamwidth decreases to counter the
pathloss; when $P_{t}$ grows, the optimal beamwidth increases to cover a
larger angular range to counter the misalignment due to the positioning error.

##  V Numerical Results

Figure 3: Outage probability as a function of the distance $d$.
$P_{\max}=100$; $\gamma_{th}=10^{-7}$ W; $\theta_{3dB}=0.1$ rad;
$a_{m}=10^{-4}$; $\lambda=5.0$ cm.

Figure 3 compares the outage probabilities with different positioning error,
where $\theta_{3dB}$ is fixed to be $0.1$ rad ($5.73^{\circ}$). It is shown
that the bounds in (19) are sufficiently tight at the practical link
distances. It can also be observed that the outage probability drops within
the region $d\in(0,75]$, and then rises in $d\in(75,120]$. An explanation is
that when the receiver is too close to the transmitter, the pointing error
angle $\theta$ in Fig. 1 randomly varies in a large region, and using a fixed
beam is not able to cover such a large angular range, resulting in a high
outage probability. By comparing the black and blue curves, we can observe
that the higher positioning accuracy, quantified by $\sigma_{1}$ and
$\sigma_{2}$, can reduce the outage probability. Besides, the largest
positioning error direction in Fig. 1, quantified by $\varphi$, also has
significant impact on the outage probability. Specifically, when
$\varphi=\pi/2$, the outage probability is the lowest, this is because the
error along the $\varphi=\pi/2$ direction influences less on the pointing
error angle $\theta$. These results imply that the base station should be
selected based on the positioning error pattern characterized by R.

Figure 4: Outage probability as a function of the transmit power $P_{t}$.
$d=30$ m; $\gamma_{th}=10^{-7}$ W; $a_{m}=10^{-2}$; $\varphi=\pi/4$;
$\lambda=1.25$ cm. The fixed beamwidth systems have $\theta_{3dB}=0.1$ rad,
and the optimized beamwidths are calculated by (31).

Figure 4 compares the outage probability with a fixed beamwidth and an
optimized beamwidth. It can be observed again that the derived upper and lower
bounds are sufficiently tight. Besides, the diversity order of the system is
related to the positioning accuracy, which can be concluded by comparing the
two curves with a fixed $\theta_{3dB}$ but different
$(\sigma_{1},\sigma_{2})$. More importantly, by comparing the curves with the
fixed and the optimized $\theta_{3dB}$’s calculated by (31), we can see that
the optimized $\theta_{3dB}$ can lower the outage probability. More
importantly, the outage probability drops dramatically without approaching a
straight asymptote when $\theta_{3dB}$ is optimized. Another interesting
observation is that $P_{out}(\gamma_{th})<1$ for all $P_{t}$’s when
$\theta_{3dB}=\theta_{3dB}^{*}$, which is in contrast with the cases with
fixed $\theta_{3dB}$’s. An explanation is that even though the transmit power
is low, $\theta_{3dB}^{*}$ becomes small to main communication in a small
angular range, and thus the outage event will not always happen. The
explanation can be verified by the labels of $\theta_{3dB}$’s at different
$P_{t}$’s, showing that $\theta_{3dB}^{*}$ grows with $P_{t}$. The observation
agrees with the intuition that extra power can be allocated to the angular
range that was previously uncovered.

Figure 5: Outage probability as a function of the half-power beamwidth
$\theta_{3dB}$. $\gamma_{th}=10^{-7}$ W; $a_{m}=10^{-2}$; $\varphi=\pi/4$;
$\lambda=1.25$ cm; $P_{t}=24$ dB

Figure 5 shows how $\theta_{3dB}$ influences the outage probability. It is
shown that the bounds in (19) are tight for all $\theta_{3dB}$. More
importantly, the figure shows four outage probability minima when
$\theta_{3dB}$ changes, and they all agree with the analytical results in
(31). It is shown that $\theta_{3dB}^{*}$ is related to $d$, implying that the
beamwidth should be adjusted according to the receiver location; besides, the
positioning accuracy, quantified by $(\sigma_{1},\sigma_{2})$, does not
influence $\theta_{3dB}^{*}$. Another observation is that when $\theta_{3dB}$
is too large, we have $P_{out}(\gamma_{th})=1$, because the limited transmit
power has been too divergent and unable to counter the pathloss. On the other
hand, when the beam is too narrow, it is unlikely to cover the receiver when
positioning error exists, as is predicted near $\theta_{3dB}=0$.

##  VI Conclusions

We studied beamforming schemes based on positioning systems. Closed-form
outage probability bounds were developed, whose asymptotic tightness was
discussed. It was shown that the beamwidth should be optimized with respect to
the link distance and transmit power, and the optimal beamwidth can be
expressed in a closed form.

## Appendix A

Changing the dummy variables $\hat{\textbf{p}}_{u}$ to
${\bf{t}}=\sqrt{{{\textbf{R}}^{-1}}}\left({\hat{\textbf{p}}_{u}-{{\textbf{p}}_{u}}}\right)$,
we can express $I_{R}$ as

${I_{R}}=\int\limits_{\left[{1,-k}\right]\left({\sqrt{\textbf{R}}{\bf{t}}+{\bf{p}}}\right)\geq
0}{\frac{{\exp\left({-\frac{{{{\bf{t}}^{T}}{\bf{t}}}}{2}}\right)\det\left({\sqrt{\textbf{R}}}\right)}}{{2\pi\sqrt{\det\left({\textbf{R}}\right)}}}d}{\bf{t}}$
(33)

where the $\det({\sqrt{\textbf{R}}})$ is the Jacobian determinant. Then we
construct an orthogonal matrix

${\bf{W}}{\rm{=}}\left[{\underbrace{\frac{{\sqrt{\textbf{R}}{{\left[{1,-k}\right]}^{T}}}}{{\left\|{\left[{1,-k}\right]\sqrt{\textbf{R}}}\right\|}}}_{{{\bf{w}}_{1}}},{{\bf{w}}_{2}}}\right]$
(34)

where $\textbf{w}_{2}$ can be found through Schmidt orthogonalization so that
${\textbf{w}}_{1}^{T}{\textbf{w}}_{2}=0$ and $\|{\textbf{w}}_{2}\|=1$.
Changing the dummy variables t to ${\bf{l}}={{\bf{W}}^{T}\textbf{t}}$, we have

$\begin{split}{I_{R}}&=\int\limits_{\left[{1,-k}\right]\sqrt{\textbf{R}}{\textbf{Wl}}+\left[{1,-k}\right]{{\textbf{p}}_{u}}\geq
0}{\frac{{\exp\left({-\frac{{{{\textbf{l}}^{T}}{\textbf{l}}}}{2}}\right)}}{{2\pi}}d}{\textbf{l}}\\\
&=\int\limits_{\left({\left\|{\left[{1,-k}\right]\sqrt{\textbf{R}}}\right\|,0}\right){\textbf{l}}+\left[{1,-k}\right]{{\textbf{p}}_{u}}\geq
0}{\frac{{\exp\left({-\frac{{{{\textbf{l}}^{T}}{\textbf{l}}}}{2}}\right)}}{{2\pi}}d}{\textbf{l}}\\\
&=Q\left({-\frac{{\left[{1,-k}\right]{{\textbf{p}}_{u}}}}{{\left\|{\left[{1,-k}\right]\sqrt{\textbf{R}}}\right\|}}}\right)\end{split}$
(35)

which proves (12). Following similar procedures we can prove (13) and (18).

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