# DoA-LF: A Location Fingerprint Positioning Algorithm with Millimeter-Wave

Zhiqing Wei, Yadong Zhao, Xinyi Liu, Zhiyong Feng The final version of this
paper is published in IEEE Access. Zhiqing Wei, Yadong Zhao, Xinyi Liu and
Zhiyong Feng are with Beijing University of Posts and Telecommunications,
email: {weizhiqing, zhaoyadong, xinyi<EMAIL_ADDRESS>
###### Abstract

Location fingerprint (LF) has been widely applied in indoor positioning.
However, the existing studies on LF mostly focus on the fingerprint of WiFi
below 6 GHz, bluetooth, ultra wideband (UWB), etc. The LF with millimeter-wave
(mmWave) was rarely addressed. Since mmWave has the characteristics of narrow
beam, fast signal attenuation and wide bandwidth, etc., the positioning error
can be reduced. In this paper, an LF positioning method with mmWave is
proposed, which is named as DoA-LF. Besides received signal strength indicator
(RSSI) of access points (APs), the fingerprint database contains direction of
arrival (DoA) information of APs, which is obtained via DoA estimation. Then
the impact of the number of APs, the interval of reference points (RPs), the
channel model of mmWave and the error of DoA estimation algorithm on
positioning error is analyzed with Cramer-Rao lower bound (CRLB). Finally, the
proposed DoA-LF algorithm with mmWave is verified through simulations. The
simulation results have proved that mmWave can reduce the positioning error
due to the fact that mmWave has larger path loss exponent and smaller variance
of shadow fading compared with low frequency signals. Besides, accurate DoA
estimation can reduce the positioning error.

###### Index Terms:

Millimeter-Wave, Location Fingerprint, Direction of Arrival Estimation,
Cramer-Rao Lower Bound

##  I Introduction

With the development of mobile Internet, the location based services and
applications such as mobile social networks, online maps and online-to-offline
(O2O) services are developing rapidly, which makes the positioning
technologies critical for modern life. The global navigation satellite system
(GNSS) consisting of GPS, GLONASS, GALILEO, BDS, etc. is efficient in outdoor
environment. However, in indoor environment, because satellite signal is
blocked by roofs and walls of buildings, GNSS does not work, which triggers
the studies on indoor positioning.

Figure 1: The scenario of LF positioning

LF positioning plays an important role in complex indoor environment because
it does not require line-of-sight (LOS) measurement [1]. The scenario of LF
positioning is illustrated in Fig. 1. There are five APs in a specific indoor
environment. The features of received signals from all APs are measures at
reference points (RPs), which constitute LF database. When a device appears at
a spot, it measures the features of all APs to find the RPs with similar
features. Then the estimated location can be the mean of the locations of $K$
RPs with most similar features, which is $K$ nearest node (KNN) method.

Among LF positioning systems, WiFi fingerprint systems are most widely studied
and applied in indoor positioning. Except of WiFi fingerprint, UWB fingerprint
systems are also studied [2][3]. Besides, with the development of bluetooth
low energy (BLE) subsystem, which is “designed for machine type communication”
[4], BLE devices will be more and more dense in buildings. Hence bluetooth LF
based positioning is also applied in indoor positioning [5].

The schemes of LF based indoor positioning focus on the construction of radio
map [6][7], the contour-based LF method [8], the gradient-based fingerprinting
[9], the CSI-based indoor positioning [10][11][12] and machine learning based
LF schemes [13][14], etc. As to the construction of radio map in LF
positioning, Jiang _et al._ in [6] constructed radio map via crowdsourcing
collection. Jung _et al._ in [7] evaluated the performance of radio map
construction methods in various environments. To provide accurate LF based
indoor positioning, He _et al._ in [8] developed the contour-based
trilateration to combine the advantages of trilateration and fingerprinting
[8]. Shu _et al._ in [9] proposed the gradient fingerprinting method to reduce
the impact of varying RSSI on the accuracy of LF positioning. LF positioning
schemes generally adopt RSSI information, however, the channel state
information (CSI) contains more information, which can be exploited to improve
the accuracy of indoor positioning. Wu _et al._ in [10] designed the
fingerprinting system exploiting CSI to construct the propagation model to
improve positioning accuracy. Wang _et al._ in [11] and [12] designed the CSI-
based fingerprinting schemes via deep learning approach. The positioning
problem is mathematically a regression problem. Besides, LF positioning
methods have large amount of data in the offline database. Thus machine
learning approaches can be adopted to solve the positioning problem. Mahfouz
_et al._ in [13] proposed a kernel-based machine learning for LF positioning.
Zou _et al._ applied extreme learning machine to solve indoor positioning
problem [14].

Besides the design of positioning schemes, the analysis of positioning error
is also an active area. In [15], Li proposed RSSI based positioning scheme
without path loss model and the Cramer-Rao lower bound (CRLB) was derived to
characterize the positioning error. Naik _et al._ in [16] studied the impact
of the height of AP on the positioning accuracy and the CRLB is
correspondingly derived. Jin _et al._ in [17] have studied the performance of
RSSI based LF positioning scheme using CRLB. And Hossain _et al._ in [18]
analyzed the CRLB of signal strength difference based LF positioning scheme.

Overall, LF positioning schemes are mostly focused on the fingerprint of WiFi,
Bluetooth, UWB, etc. To our best knowledge, there are rarely studies on LF
positioning with mmWave. Because mmWave has the characteristics of narrow
beam, fast signal attenuation and wide bandwidth, etc., mmWave signals can
provide centimeter level ranging accuracy [19][20][21]. Besides, with the
development of fifth generation mobile communication (5G), dense mmWave small
cells will be widely deployed [22], which makes LF positioning with mmWave to
be realizable. In the standard of IEEE 802.11ad, 60 GHz mmWave for multi-
gigabit-per-second WiFi is developed [23]. Besides, IEEE has also started the
standardization of IEEE 802.11aj, which is a WLAN system operating at 45 GHz
mmWave band [24]. Thus in the near future, LF positioning with mmWave is
practical.

In this paper, an LF positioning method with mmWave is proposed, which may be
promising in the era of 5G when APs with mmWave are widely deployed. We
utilize the characteristics of narrow beam and fast signal attenuation of
mmWave to reduce positioning error. The proposed LF positioning scheme is
called DoA-LF, because the fingerprint database contains DoA information of
APs besides RSSI information, which is obtained via DoA estimation. Then we
select $K$ candidate RPs with the most similar features and calculate their
weighted mean through weighted $K$ nearest neighbor (WKNN) algorithm, which is
the final estimated location. It is noted that the value of $K$ has an impact
on the positioning error and there exists an optimal $K$ that can minimize the
positioning error. With DoA information of APs operating on mmWave spectrum
band, the positioning error is significantly reduced compared with LF
positioning with signal below 6 GHz. Then the impact of the number of APs, the
interval of RPs, the variance of DoA estimation, the path loss exponent and
the shadow fading of mmWave on the positioning error is analyzed via Cramer-
Rao Lower Bound (CRLB). Finally, the analysis results are verified by
simulation results. The key parameters and notations in this paper are listed
in Table I.

The rest of this paper is organized as follows. In Section II, system model is
introduced. Section III presents the proposed DoA-LF algorithm. In Section IV,
CRLB is achieved, which yields the impact of various parameters on the
positioning error. Simulation results are provided in Section V to verify our
scheme and analysis results. Finally, Section VI summarizes this paper.

TABLE I: Key Parameters and Notations Symbol | Description
---|---
${PL({d})}$ | Path loss at a distance $d$
$n$ | Path loss exponent
${X_{\sigma}}$ | Shadow fading factor
$(\hat{x},\hat{y})$ | Estimated location coordinates
$({x_{i}},{y_{i}})$ | Location coordinates of $i$th RP
${{\bm{s}}}$ | Signal strength vector
${{\bm{\varphi}}}$ | Angle vector
$Q$ | Number of APs
$M$ | Number of RPs
$A$ | Test area
$\theta$ | Location of the test point
$r$ | Location of the estimated point
$\Pr(r|\theta)$ | Conditional probability
${d_{r\theta}}$ | Euclidean distance between $r$ and $\theta$
${d_{ir}}$ | Euclidean distance between $r$ and $i$th AP
${d_{i\theta}}$ | Euclidean distance between $\theta$ and $i$th AP
$\bm{s}_{i\theta}$ | Actual signal strength of $i$th AP at the test point
$\bm{s}_{ir}$ | Received signal strength of the $i$th AP at point $r$
${\varphi_{i\theta}}$ | DoA of $i$th AP at the test point
${\varphi_{ir}}$ | DoA of the $i$th AP at point $r$
${\sigma_{s}^{2}}$ | Variance of received signal strength
${\sigma_{\varphi}^{2}}$ | Variance of DoA estimation
$J(\hat{\theta})$ | Fisher Information Matrix
${C_{\hat{\theta}}}$ | Cramer-Rao Lower Bound
AP | Access point
RP | Reference point
TP | Test point
LF | Location fingerpint
RSSI | Received signal strength indicator

##  II System Model

The 60 GHz mmWave is adopted for multi-gigabit-per-second WiFi [23]. Hence 60
GHz mmWave is adopted in LF positioning. In this section, the channel model of
60 GHz mmWave and the process of LF positioning are presented.

###  II-A 60 GHz mmWave Channel Model

The channel model of 60 GHz mmWave is [25][26]

$PL(d)[dB]=PL({d_{0}})+10n\log(\frac{d}{{{d_{0}}}})+{X_{\sigma}},d>{d_{0}},$
(1)

where $PL({d_{0}})=20\log(\frac{{4\pi{d_{0}}}}{\lambda})$ is free-space path
loss at a reference distance ${d_{0}}$, which is generally set as
${d_{0}}=1\kern 2.0pt\rm{m}$. The path loss exponent is $n$. ${X_{\sigma}}$ is
the shadow fading factor modeled by a Gaussian random variable with mean zero
and variance $\sigma_{s}^{2}$.

In DOA-LF algorithm, besides RSSI information, DoA information is also needed,
which can be obtained via multiple signal classification (MUSIC) algorithm.
Since MUSIC algorithm is well known in the area of array signal processing, we
do not introduce MUSIC algorithm in this paper. Readers can refer to [31] for
details. The MUSIC algorithm has lower computational complexity compared with
other DoA algorithms. However, other DoA algorithms can also be adopted
besides MUSIC algorithm. If a DoA algorithm has small error, the performance
of positioning algorithm can be correspondingly improved.

###  II-B DoA-LF Positioning Algorithm

LF positioning plays an important role in complex indoor environment. LF
positioning algorithm consists of two steps, namely, offline database
construction and online matching. Weighted $K$ nearest neighbor (WKNN)
algorithm is widely applied in LF positioning [32][33][34], where the weighted
mean of the coordinates of $K$ nearest RPs is calculated as the estimated
coordinates. The weight coefficient $w_{i}$ is generally inversely
proportional to the Euclidean distance between the estimated point and $i$th
RP [32][33] in the feature space spanning by the vectors containing RSSI and
DoA information. Hence the estimated coordinates are as follows [32][33].

$\displaystyle{w_{i}}=\frac{\gamma}{{{d_{i}}+\varepsilon}},$ (2)
$\displaystyle(\hat{x},\hat{y})=\frac{{\sum\limits_{i=1}^{K}{{w_{i}}({x_{i}},{y_{i}})}}}{{\sum\limits_{i=1}^{K}{{w_{i}}}}},$

where ${d_{i}}$ is the Euclidean distance in the feature space between the
measured point and the $i$th RP. $(\hat{x},\hat{y})$ are the estimated
coordinates and $({x_{i}},{y_{i}})$ are the coordinates of $i$th RP. $\gamma$
is normalized parameter and $\varepsilon$ is a small positive number in order
to prevent the denominator being zero.

##  III The Proposed DoA-LF Algorithm

Figure 2: The proposed DoA-LF algorithm.

In this section, the characteristics of mmWave, such as narrow beam and fast
attenuation, are exploited to construct an LF positioning algorithm called
DoA-LF algorithm. The DoA information of APs obtained by MUSIC algorithm is
combined with the RSSI of APs to construct a new offline database for online
matching. Then WKNN algorithm is adopted to calculate the estimated
coordinates. Fig. 2 illustrates the process of proposed DoA-LF algorithm.

Step 1: Obtain RSSI and DoA information from APs to each RP.

$Q$ APs and $M$ RPs are deployed in a specific area, which is illustrated in
Fig. 3. The RSSIs of APs at each RP are measured, which are saved in a vector
${{\bm{s}_{i}}}=[{s_{1i}},{s_{2i}},\cdots,{s_{Qi}}]$ with
${s_{ji}},j=1,\cdots,Q$ denoting the RSSI of $j$th AP at $i$th RP.

Meanwhile, the MUSIC algorithm is adopted for DoA estimation. The directions
of APs at each RP are measured, which are saved in a vector
${{\bm{\varphi}_{i}}}=[{\varphi_{1i}},{\varphi_{2i}},\cdots,{\varphi_{Qi}}]$
with ${\varphi_{ji}},j=1,\cdots,Q$ denoting the DoA of $j$th AP at $i$th RP.
It is noted that the RSSI and DoA information is obtained simultaneously.

Step 2: Construct offline database.

Combining $Q$-dimensional RSSI vector and $Q$-dimensional direction vector, we
obtain $2Q$-dimensional vector $[\bm{s}_{i},\bm{\varphi}_{i}]$ representing
the features of $i$th RP. The vectors of all RPs span a feature space and
construct the offline database for DoA-LF positioning with mmWave.

Step 3: Online matching.

Comparing RSSI and direction information of test point with the data in
offline database, we can find out $K$ nearest RPs whose features are closest
to test point. Then we calculate the estimated coordinates via the weighted
mean of $K$ nearest RPs using (2).

Figure 3: A scenario of proposed DoA-LF positioning.

For example, in Fig. 3, there are $4$ APs and $77$ RPs in an area. Firstly, an
offline database is established, which consists of RSSI and DoA information of
all RPs. Secondly, $K=4$ candidate RPs with the most similar features are
selected, which are denoted by the squares in Fig. 3. Finally, the estimated
coordinates can be obtained via calculating the weighted mean of the selected
$K=4$ candidate RPs.

##  IV Analysis of Positioning Error

In the above section, the DoA-LF positioning algorithm with mmWave is
proposed. In DoA-LF positioning algorithm, the number of APs, the interval of
RPs, the channel model of mmWave and the error of DoA estimation have an
impact on the positioning error. In this section, we analyze the impact of
these parameters on positioning error, which can provide guideline for the
selection of appropriate parameters in DoA-LF algorithm with mmWave.

As illustrated in Fig. 3, there are $Q$ APs and $M$ RPs. APs are distributed
on the edge of a specific area denoted as area $A$. RPs are uniform
distributed in area $A$. The location of test point is denoted as $\theta$.
DoA-LF algorithm generates the estimated point $r$ with conditional
probability $\Pr(r|\theta)$. The Euclidean distance between $r$ and $\theta$
is ${d_{r\theta}}$. The conditional probability $\Pr(r|\theta)$ and the
distance ${d_{r\theta}}$ can represent positioning error. Besides,
$\Pr(r|\theta)$ is related with ${d_{r\theta}}$. In LF positioning, one-to-one
mapping can be established between $2Q$-dimensional feature space and two-
dimensional geo-location space. Thus the expression of conditional probability
$\Pr(r|\theta)$ in two-dimensional geo-location space equals to the
conditional probability
$\Pr({\bm{s}_{r}},{\bm{\varphi}_{r}}|{\bm{s}_{\theta}},{\bm{\varphi}_{\theta}})$
in $2Q$-dimensional feature space, where ${\bm{s}_{\theta}}$,
${\bm{\varphi}_{\theta}}$, ${\bm{s}_{r}}$ and ${\bm{\varphi}_{r}}$ are the
RSSI and DoA information of all APs at location $\theta$ and $r$ respectively.
Therefore the conditional probability $\Pr(r|\theta)$ is as follows.

$\displaystyle\Pr(r|\theta)$
$\displaystyle=\Pr({\bm{s}_{r}},{\bm{\varphi}_{r}}|{\bm{s}_{\theta}},{\bm{\varphi}_{\theta}})$
(3)
$\displaystyle=\prod\limits_{i=1}^{Q}{\Pr({\bm{s}_{ir}},{\bm{\varphi}_{ir}}|{\bm{s}_{i\theta}},{\bm{\varphi}_{i\theta}})}$
$\displaystyle=\prod\limits_{i=1}^{Q}{\Pr({\bm{s}_{ir}}|{\bm{s}_{i\theta}})}\prod\limits_{i=1}^{Q}{\Pr({\bm{\varphi}_{ir}}|{\bm{\varphi}_{i\theta}})}.$

The conditional probability $\Pr({\bm{s}_{ir}}|{\bm{s}_{i\theta}})$ denotes
that the actual signal strength of $i$th AP at the test point is
$\bm{s}_{i\theta}$ while the received signal strength of $i$th AP is
$\bm{s}_{ir}$, which is the signal strength at location $r$.
$\Pr({\bm{s}_{ir}}|{\bm{s}_{i\theta}})$ follows Gaussian distribution with
mean $\bm{s}_{i\theta}$ and variance ${\sigma_{s}^{2}}$ [18][37]. Meanwhile,
$\Pr({\bm{\varphi}_{ir}}|{\bm{\varphi}_{i\theta}})$ denotes the conditional
probability that the DoA of $i$th AP at test point is ${\varphi_{i\theta}}$
while the estimated DoA of $i$th AP is ${\varphi_{ir}}$, which is the DoA of
$i$th AP at location $r$. $\Pr({\bm{\varphi}_{ir}}|{\bm{\varphi}_{i\theta}})$
also follows Gaussian distribution with mean $\bm{\varphi}_{i\theta}$ and
variance ${\sigma_{\varphi}^{2}}$ [38][39].

$\displaystyle{\frac{{{\partial^{2}}\log f(r;\theta)}}{{\partial
x^{2}}}{\rm{=}}\sum\limits_{i=1}^{Q}{\left({\frac{{-2{\eta}{{\cos}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}+{\eta}\log(\frac{{d_{ir}^{2}}}{{d_{i\theta}^{2}}})\frac{{1-2{{\cos}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}-\frac{{si{n^{2}}{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}}}\right)}},$
(4) $\displaystyle{\frac{{{\partial^{2}}\log
f(r;\theta)}}{{\partial{x}\partial{y}}}{\rm{=}}\frac{{{\partial^{2}}\log
f(r,\theta)}}{{\partial{y}\partial{x}}}{\rm{=}}\sum\limits_{i=1}^{Q}{\left({-\frac{{{\eta}sin2{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}-{\eta}\log(\frac{{d_{ir}^{2}}}{{d_{i\theta}^{2}}})\frac{{{\rm{sin}}2{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}+\frac{{({\varphi_{ir}}-{\varphi_{i\theta}})cos2{\varphi_{i\theta}}-cos{\varphi_{i\theta}}\sin{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}}}\right)}},$
$\displaystyle{\frac{{{\partial^{2}}\log f(r,\theta)}}{{\partial
y^{2}}}{\rm{=}}\sum\limits_{i=1}^{Q}{\left({\frac{{-2{\eta}si{n^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}+{\eta}\log(\frac{{d_{ir}^{2}}}{{d_{i\theta}^{2}}})\frac{{1-2{{\sin}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}-\frac{{co{s^{2}}{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}}}\right)}}.$

According to the path loss model of mmWave in (1), assuming that the transmit
power of each AP is $P_{t}$, the average signal strength at $\theta$ from
$i$th AP is given by

${s_{i\theta}}=P_{t}-PL({d_{i\theta}})=P_{t}-PL({d_{0}})-10n\log(\frac{{{d_{i\theta}}}}{{{d_{0}}}})-{X_{\sigma}},$
(5)

where $X_{\sigma}$ is a Gaussian random variable with mean zero and variance
$\sigma_{s}^{2}$. $d_{i\theta}$ is the distance between $i$th AP and the
location $\theta$.

CRLB is an effective tool to estimate the minimum variance of parameter
estimation error [40][41][42]. In this subsection, the lower bound of the
variance of DoA-LF algorithm error is analyzed by CRLB. Assuming that the test
point is $\theta=(x,y)^{T}$, the unbiased estimation value of $\theta$ is
$\hat{\theta}=(\hat{x},\hat{y})^{T}$, which is the weighted mean of $K$
nearest RPs selected by DoA-LF algorithm. The covariance matrix of
$\hat{\theta}$ is

$\displaystyle{\mathop{\rm
cov}}\left(\hat{\theta}\right)=E\left({(\hat{\theta}-\theta){{(\hat{\theta}-\theta)}^{T}}}\right)$
(6) $\displaystyle=\left({\begin{array}[]{*{20}{c}}{{\mathop{\rm
var}}(\hat{x}-x)}&{{\mathop{\rm
cov}}\left({(\hat{x}-x),(\hat{y}-y)}\right)}\\\ {{\mathop{\rm
cov}}\left({(\hat{y}-y),(\hat{x}-x)}\right)}&{{\mathop{\rm
var}}(\hat{y}-y)}\end{array}}\right),$

where $E(\alpha)$ is the expectation of random variable $\alpha$.
${\mathop{\rm var}}(\alpha)$ is the variance of $\alpha$.
$\rm{cov}(\alpha,\beta)$ is the covariance of random variables $\alpha$ and
$\beta$. According to CRLB inequality, the covariance of $\hat{\theta}$
satisfies the following inequality [43][44].

${\mathop{\rm cov}}(\hat{\theta})\geq\left(J(\hat{\theta})\right)^{-1},$ (7)

where $J(\hat{\theta})$ is fisher information matrix (FIM), which is defined
as follows [41][44].

$J(\hat{\theta})={E}\left(-\frac{{{\partial^{2}}\ln
f(r;\theta)}}{{\partial\theta\partial{\theta^{T}}}}\right),$ (8)

where $r$ is the corresponding coordinates of $2Q$-dimensional observation
features and $f(r;\theta)$ is denoted as

$f(r;\theta)=f(r|\theta)f(\theta).$ (9)

According to one-to-one mapping relation between $2Q$-dimensional feature
space and two-dimensional geo-location space, the value of $f(r|\theta)$ is
provided in (3). Besides, $\theta$ is uniformly distributed in area $A$, hence
$f(\theta)=\frac{1}{{\left|A\right|}}$ where $\left|A\right|$ denotes the area
of $A$. Thus the value of $f(r;\theta)$ is

$\displaystyle
f(r;\theta)=\prod\limits_{i=1}^{Q}{{\kappa}\exp\left({-\frac{{{{({s_{ir}}-{s_{i\theta}})}^{2}}}}{{2\sigma_{s}^{2}}}-\frac{{{{({\varphi_{ir}}-{\varphi_{i\theta}})}^{2}}}}{{2\sigma_{\varphi}^{2}}}}\right)},$
(10)
$\displaystyle=\prod\limits_{i=1}^{Q}{{\kappa}\exp\left({-\frac{{{{\left({10n\lg(\frac{{{d_{ir}}}}{{{d_{i\theta}}}})}\right)}^{2}}}}{{2\sigma_{s}^{2}}}-\frac{{{{({\varphi_{ir}}-{\varphi_{i\theta}})}^{2}}}}{{2\sigma_{\varphi}^{2}}}}\right)},$

where $\kappa=\frac{1}{{2\pi|A|{\sigma_{s}}{\sigma_{\varphi}}}}$. $d_{ir}$ is
the distance between $i$th AP and the location $r$. $\lg(*)$ is a logarithmic
function with base $10$. Therefore the function $\log f(r;\theta)$ is as
follows.

$\displaystyle\log f(r;\theta)$ (11)
$\displaystyle=\sum\limits_{i=1}^{Q}{\log\left({{\kappa}\exp\left(-\frac{{{{(10n\lg(\frac{{{d_{ir}}}}{{{d_{i\theta}}}}))}^{2}}}}{{2\sigma_{s}^{2}}}-\frac{{{{({\varphi_{ir}}-{\varphi_{i\theta}})}^{2}}}}{{2\sigma_{\varphi}^{2}}}\right)}\right)}$
$\displaystyle=\sum\limits_{i=1}^{Q}{\left({{\rm{log}}{\kappa}-{\eta}{{\left(\log(\frac{{{d_{ir}}}}{{{d_{i\theta}}}})\right)}^{2}}-\frac{{{{({\varphi_{ir}}-{\varphi_{i\theta}})}^{2}}}}{{2\sigma_{\varphi}^{2}}}}\right)},$

where $\log(*)$ is a logarithmic function with base $e$. The value of ${\eta}$
is

${\eta}={(\frac{{10n}}{{\sqrt{2}{\sigma_{s}}\log 10}})^{2}}.$ (12)

It is noted that ${d_{i\theta}}$ and ${\varphi_{i\theta}}$ are functions of
$x$ and $y$, whose expressions are

$\displaystyle{d_{i\theta}}{\rm{=}}\sqrt{{{({x}-{x_{i}})}^{2}}+{{({y}-{y_{i}})}^{2}}},$
(13)
$\displaystyle{\varphi_{i\theta}}=\arccos(\frac{{{x_{i}}-{x}}}{{{d_{i\theta}}}})=\arcsin(\frac{{{y_{i}}-{y}}}{{{d_{i\theta}}}}),$

which are illustrated by Fig. 4.

In order to obtain (8), we first derive the first order partial derivatives as
follows.

$\displaystyle\frac{{\partial\log
f(r;\theta)}}{{\partial{x}}}=\sum\limits_{i=1}^{Q}{\eta\log(\frac{{d_{ir}^{2}}}{{d_{i\theta}^{2}}})\frac{{{x}-{x_{i}}}}{{d_{i\theta}^{2}}}+\frac{{({\varphi_{ir}}-{\varphi_{i\theta}})\sin{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}{d_{i\theta}}}}},$
(14) $\displaystyle\frac{{\partial\log
f(r;\theta)}}{{\partial{y}}}=\sum\limits_{i=1}^{Q}{\eta\log(\frac{{d_{ir}^{2}}}{{d_{i\theta}^{2}}})\frac{{{y}-{y_{i}}}}{{d_{i\theta}^{2}}}+\frac{{({\varphi_{ir}}-{\varphi_{i\theta}})\cos{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}{d_{i\theta}}}}}.$

Then the second order derivatives can be derived, which are shown in (4).

For DoA-LF algorithm, ${d_{ir}}$ approximately equals ${d_{i\theta}}$ and
${\varphi_{ir}}$ approximately equals ${\varphi_{i\theta}}$. Hence the values
of $\log(\frac{{d_{ir}^{2}}}{{d_{i\theta}^{2}}})$ and
${\varphi_{ir}}-{\varphi_{i\theta}}$ approximate zero. The entries of (4)
become as follows.

$\displaystyle\frac{{{\partial^{2}}\log f(r;\theta)}}{{\partial
x^{2}}}=\sum\limits_{i=1}^{Q}{\left({-\frac{{2\eta{{\cos}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}-\frac{{{{\sin}^{2}}{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}}}\right)},$
(15) $\displaystyle\frac{{{\partial^{2}}\log
f(r;\theta)}}{{\partial{x}\partial{y}}}=\frac{{{\partial^{2}}\log
f(r;\theta)}}{{\partial{y}\partial{x}}}$ $\displaystyle{\kern
57.0pt}=\sum\limits_{i=1}^{Q}{(-\frac{{{\eta}\sin
2{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}-\frac{{\sin
2{\varphi_{i\theta}}}}{{2\sigma_{\varphi}^{2}d_{i\theta}^{2}}})},$
$\displaystyle\frac{{{\partial^{2}}\log f(r;\theta)}}{{\partial
y^{2}}}=\sum\limits_{i=1}^{Q}{(-\frac{{2{\eta}{{\sin}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}-\frac{{{{\cos}^{2}}{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}})}.$

Then the FIM $J(\hat{\theta})$ can be derived as follows.

$J(\hat{\theta})=\left({\begin{array}[]{*{20}{c}}{{J_{xx}}(\hat{\theta})}&{{J_{xy}}(\hat{\theta})}\\\
{{J_{yx}}(\hat{\theta})}&{{J_{yy}}(\hat{\theta})}\end{array}}\right),$ (16)

where the entries of (16) are

$\displaystyle{J_{{x}{x}}}(\hat{\theta})=-\frac{{{\partial^{2}}\log
f(r;\theta)}}{{\partial
x^{2}}}=\sum\limits_{i=1}^{Q}{\frac{{2\eta{{\cos}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}+\frac{{{{\sin}^{2}}{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}}},$
(17)
$\displaystyle{J_{xy}}(\hat{\theta})={J_{yx}}(\hat{\theta})=\sum\limits_{i=1}^{Q}{\frac{{{\eta}\sin
2{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}+\frac{{\sin
2{\varphi_{i\theta}}}}{{2\sigma_{\varphi}^{2}d_{i\theta}^{2}}}},$
$\displaystyle{J_{yy}}(\hat{\theta})=-\frac{{{\partial^{2}}\log
f(r;\theta)}}{{\partial
y^{2}}}=\sum\limits_{i=1}^{Q}{\frac{{2{\eta}{{\sin}^{2}}{\varphi_{i\theta}}}}{{d_{i\theta}^{2}}}+\frac{{{{\cos}^{2}}{\varphi_{i\theta}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{2}}}}.$

Therefore ${\left({J(\hat{\theta})}\right)^{-1}}$ is

${\left({J(\hat{\theta})}\right)^{-1}}=\frac{1}{{\left|{J(\hat{\theta})}\right|}}\left({\begin{array}[]{*{20}{c}}{{J_{yy}}(\hat{\theta})}&{-{J_{yx}}(\hat{\theta})}\\\
{-{J_{xy}}(\hat{\theta})}&{{J_{xx}}(\hat{\theta})}\end{array}}\right),$ (18)

where $\left|{J(\hat{\theta})}\right|$ is

$\displaystyle\left|{J(\hat{\theta})}\right|$
$\displaystyle={J_{xx}}(\hat{\theta}){J_{yy}}(\hat{\theta})-{J_{xy}}(\hat{\theta}){J_{yx}}(\hat{\theta})$
(19)
$\displaystyle{=\sum\limits_{i=1}^{Q}{\frac{{2{\eta}{{({{\cos}^{2}}{\varphi_{i\theta}}-{{\sin}^{2}}{\varphi_{i\theta}})}^{2}}}}{{\sigma_{\varphi}^{2}d_{i\theta}^{4}}}}}.$

Substituting (6) and (18) into (7), we have

$\displaystyle\left({\begin{array}[]{*{20}{c}}{{\mathop{\rm
var}}(\hat{x}-x)}&{{\mathop{\rm
cov}}\left({(\hat{x}-x),(\hat{y}-y)}\right)}\\\ {{\mathop{\rm
cov}}\left({(\hat{y}-y),(\hat{x}-x)}\right)}&{{\mathop{\rm
var}}(\hat{y}-y)}\end{array}}\right)$ (20)
$\displaystyle\geq\frac{1}{{|J(\hat{\theta})|}}\left({\begin{array}[]{*{20}{c}}{{J_{yy}}(\hat{\theta})}&{-{J_{yx}}(\hat{\theta})}\\\
{-{J_{xy}}(\hat{\theta})}&{{J_{xx}}(\hat{\theta})}\end{array}}\right).$

Hence the CRLB ${C_{\hat{\theta}}}$ of positioning error is

$\displaystyle{C_{\hat{\theta}}}$ $\displaystyle={{\mathop{\rm
var}}(\hat{x}-x)}+{{\mathop{\rm
var}}(\hat{y}-y)}\geq\frac{{{J_{xx}}(\hat{\theta})+{J_{yy}}(\hat{\theta})}}{{|J(\hat{\theta})|}}$
(21)
$\displaystyle=\sum\limits_{i=1}^{Q}{\frac{{d_{i\theta}^{2}(1+2{\eta}\sigma_{\varphi}^{2})}}{{2{\eta}{{({{\cos}^{2}}{\varphi_{i\theta}}-{{\sin}^{2}}{\varphi_{i\theta}})}^{2}}}}}.$

Figure 4: The distance and DoA of $\theta$ and $r$.

According to (21), CRLB is an increasing function of $\sigma_{\varphi}^{2}$.
Besides, CRLB is a decreasing function of ${\eta}$. Because of (12), we have
conclusions as follows.

  1. 1.

The variance of positioning error is increasing with the variance
$\sigma_{s}^{2}$, which means that the positioning error is increasing with
the increase of shadow fading factor.

  2. 2.

The variance of positioning error is decreasing with path loss exponent $n$,
which means that LF positioning error can be reduced when using the signal
with large path loss exponent, such as mmWave.

  3. 3.

In LF positioning with mmWave, when $\sigma_{\varphi}^{2}$ is small, namely,
the error of DoA estimation is small, the positioning error can be reduced.

Meanwhile, the variance of positioning error is also impacted by the interval
of RPs and the number of APs. With the decreasing of the interval of RPs, the
variance of positioning error is decreasing. However, when the interval is
extremely small, the computational complexity of DoA-LF algorithm will be
large. Besides, with the increasing of the number of APs, the positioning
error can be reduced. However, when the number of APs is extremely large, the
computational complexity of DoA-LF algorithm will be intolerable. Simulation
results in Section V will support our analysis.

##  V Simulation Results and Analysis

###  V-A DoA-LF Algorithm

Firstly, the LF positioning scheme with mmWave is compared with the
traditional LF positioning with 2.4 GHz WiFi. The path loss models of 60 GHz
mmWave and 2.4 GHz WiFi under the same indoor NLOS111The LF positioning is
mainly applied in the NLOS environment. environment are given as follows
[27][28].

$\displaystyle P{L_{mmw}}(d)[dB]=-75.3+16.8\log(d)+{\cal N}(0,2.45),$ (22)
$\displaystyle P{L_{2.4GHz}}(d)[dB]=-48.5+20.5\log(d)+{\cal N}(0,3.04).$

In order to create an offline database in the training phase, $100$ samples of
RSSI and DoA estimations at each RP are achieved. Then the average RSSI and
DoA estimations at each RP are recorded in the offline database. The size of
indoor environment is $100\rm{m}\times 100\rm{m}$ and RPs are deployed per $5$
meters. Besides, $4$ APs are distributed in the $4$ corners of the entire
region. The transmit power of each AP is $30$ mW. Fig. 5 illustrates the
intuitive results of DoA-LF algorithm, where the estimated points match the
test points very well.

Figure 5: A test of DoA-LF positioning algorithm.

In order to present the errors of different positioning methods, the
probability of successful positioning within the range of location error,
namely, the cumulative distribution function (CDF) of LF positioning is
simulated. Assume that the positioning error is $E$, which is a random
variable. For any positioning error $E=e$ in the horizontal axis, its
corresponding CDF, namely, the value in vertical axis is the probability of
$E<e$. Besides, the average positioning errors of different methods are
simulated.

Figure 6: Comparison of CDFs between 60 GHz mmWave and 2.4 GHz WiFi with
different algorithms, where $K=6$ for KNN and WKNN. Figure 7: Comparison of
average positioning errors between 60 GHz mmWave and 2.4 GHz WiFi with
different algorithms, where $K=6$ for KNN and WKNN.

Firstly, the impact of path loss models of 2.4 GHz WiFi and 60 GHz mmWave on
three different LF positioning algorithms, namely, the algorithms of NN, KNN
and WKNN, is illustrated in Fig. 6 and Fig. 7. In Fig. 6, the CDF curve of 60
GHz mmWave grows much faster than that of 2.4 GHz WiFi when using the same LF
positioning algorithm222The proposed DoA-LF algorithm also adopts WKNN
operations. However, in this section, the algorithms of nearest node (NN), $K$
nearest node (KNN) and weight $K$ nearest node (WKNN) denote the LF
positioning algorithms without DoA estimation, namely, they only use RSSI
information.. The underlying reason is that mmWave has the characteristics of
narrow beam and fast attenuation, which can increase the discrimination of
RSSI and DoA information of different RPs and reduce the positioning error.
Besides, the positioning error of 60 GHz mmWave is much higher than that of
2.4 GHz WiFi when using the same LF positioning algorithm. Moreover, WKNN
algorithm yields the lowest error and the fastest convergence speed among the
three LF algorithms for 60 GHz mmWave. Fig. 7 shows that 60 GHz mmWave yields
less positioning error compared with 2.4 GHz WiFi. The average positioning
error of 60 GHz mmWave is $68.9\%$ less than 2.4 GHz WiFi.

Figure 8: Comparison of RSSI between 60 GHz mmWave and 2.4 GHz WiFi with the
same transmit power of APs.

Fig. 8 illustrates the comparison of RSSI between 60 GHz mmWave and 2.4 GHz
WiFi. In Fig. 8, the RSSI of 2.4 GHz WiFi is higher than that of 60 GHz mmWave
at the same RP, which verifies the fact that the attenuation of 60 GHz mmWave
signal is faster than that of 2.4 GHz signal. Besides, the distribution of
RSSIs of mmWave signal is more concentrated compared with 2.4 GHz signal,
which can increase the discrimination of RSSI to reduce positioning error.

Figure 9: Comparison of CDFs between DoA-LF algorithm and other algorithms in
60 GHz mmWave, where $K=4$ for DoA-LF, KNN and WKNN. Figure 10: Comparison of
average positioning errors between DoA-LF algorithm and other algorithms in 60
GHz mmWave, where $K=4$ for DoA-LF, KNN and WKNN.

Then we analyze the performance of the proposed DoA-LF algorithm which uses
RSSI and DoA information for hybrid LF positioning with mmWave. The
performance is compared with other three LF algorithms, which is illustrated
in Fig. 9 and Fig. 10. Fig. 9 shows that the CDF curve of DoA-LF algorithm
grows faster than other algorithms without DoA estimation, which means that
the positioning error of DoA-LF algorithm is the lowest. Similarly, as
illustrated in Fig. 10, DoA-LF algorithm yields less positioning error
compared with other algorithms. The average positioning error of DoA-LF
algorithm is 1.32 meters, which is approximately $50\%$ less than WKNN and KNN
algorithm without DoA estimation.

Figure 11: Comparison of CDFs between DoA-LF with 60 GHz mmWave, WKNN with 60
GHz mmWave and WKNN with 2.4GHz WiFi, where $K=6$ for DoA-LF and WKNN. Figure
12: Comparison of average positioning errors between DoA-LF with 60 GHz
mmWave, WKNN with 60 GHz mmWave and WKNN with 2.4GHz WiFi, where $K=6$ for
DoA-LF and WKNN.

Finally, we compare the performance of DoA-LF algorithm with mmWave, WKNN
algorithm with 60 GHz mmWave and WKNN algorithm with 2.4 GHz WiFi. As
illustrated in Fig. 11, the positioning error of DoA-LF algorithm with 60 GHz
mmWave is much lower than WKNN algorithm with 60 GHz mmWave and 2.4 GHz WiFi.
As illustrated in Fig. 12, the average error distance of DoA-LF algorithm is
1.37 meters, which is $48.5\%$ less than WKNN with 60 GHz mmWave and $85.6\%$
less than WKNN with 2.4 GHz WiFi.

Figure 13: The relation between the average positioning error and $K$.

The relation between the average positioning error and $K$ is illustrated in
Fig. 13 to yield the optimal value of $K$ in WKNN algorithm. It is noted that
if $K$ is small, the randomness is large and the average positioning error is
large. On the contrary, if $K$ is large, the disturbance from other RPs is
correspondingly large and the average positioning error is also large. Thus
there exists an optimal $K$ to minimize the average positioning error, which
is illustrated in Fig. 13. Besides, the average positioning error for any $K$
with mmWave is still smaller than that with low frequency signal.

Overall, we have verified that the positioning error can be reduced
significantly when employing DoA-LF positioning algorithm with mmWave.

###  V-B CRLB and Analysis

In this subsection, the simulation results are shown to analyze the impact of
various parameters on the average positioning error of DoA-LF algorithm, which
include the number of APs, the interval of RPs, the path loss exponent, the
DoA estimation error and the variance of received signal strength. These
simulations and analysis can verify our analysis of CRLB in section IV.
Besides, they can provide a guideline for the selection of appropriate
parameters in the proposed DoA-LF positioning algorithm.

Figure 14: The impact of the interval of RPs on average positioning error,
where the number of APs varies from 3 to 6.

Firstly, we analyze the impact of the interval of RPs on average positioning
error. Fig. 14 shows that the average positioning error is increasing with the
increase of the interval of RPs. The curve $y=x$ is plotted in Fig. 14, where
$x$ denotes the interval of RPs and $y$ denotes the average positioning error.
It is noted that when the interval of RPs is too small, the average
positioning error is larger than the interval of RPs, which is due to the fact
that the CRLB has determined that the average positioning error cannot be as
small as possible. Hence the interval of RPs does not have to be very small
because the computational complexity of DoA-LF algorithm will be intolerable
in this situation and the average positioning error is lower bounded by the
CRLB. However, when the interval of RP is larger than a threshold, the average
positioning error is smaller than the interval of RPs.

Figure 15: The impact of the number of APs on average positioning error, where
the interval of RPs varies from 5 to 8 meters.

Then we analyze the impact of the number of APs on average positioning error.
As illustrated in Fig. 15, the positioning error is decreasing with the
increase of the number of APs. Since the increase of the number of APs will
enlarge the computational complexity, there is an optimal number of APs in
order to balance the computational complexity and positioning error.

Figure 16: The relation between positioning error and path loss exponent.
Figure 17: The relation between positioning error and received signal strength
variance. Figure 18: The relation between positioning error and DoA estimation
variance.

Finally, we analyze the impact of the variance of DoA estimation, the path
loss exponent and the variance of received signal strength on the average
positioning error of DoA-LF algorithm with mmWave. In Fig. 16, the positioning
error is decreasing with the increase of path loss exponent. In Fig. 17, the
positioning error is decreasing with the decrease of the variance of received
signal strength. In Fig. 18, the positioning error is decreasing with the
decrease of the variance of DoA estimation. These results support our analysis
results of CRLB in Section IV, which can be explained. When the path loss
exponent is large, the discrimination of RSSI is correspondingly large, which
can reduce positioning error. Besides, when the variance of signal intensity
or DoA estimation is small, the fluctuation of RSSI or DoA estimation is
correspondingly small, which can also reduce positioning error. However, the
path loss exponent, the variance of received signal strength and the variance
of DoA estimation are not controllable variables because they are impacted by
the electromagnetic and physical environment. Therefore when the environment
is not ideal, we can properly reduce the interval of RPs or increase the
number of APs to bound positioning error.

##  VI Conclusion

In this paper, the DoA-LF positioning algorithm is proposed, which
incorporates the features of RSSI and DoA information into LF positioning. The
characteristics of narrow beam and fast attenuation of mmWave are exploited to
reduce positioning error. The DoA information obtained from MUSIC algorithm is
combined with RSSI information to construct a joint offline database for
online matching. Then the $K$ candidate RPs with the most similar features are
selected and their weighted mean is calculated, which is the estimated
location. Moreover, the CRLB of DoA-LF positioning algorithm with mmWave is
derived. Simulation results show that the positioning error of DoA-LF
algorithm is much smaller than the algorithms without DoA estimation. Besides,
the positioning error with mmWave is much smaller than that with low frequency
signals. Moreover, we have shown that there exists an optimal $K$ to minimize
the positioning error via simulation. Finally, simulation results show that
the positioning error is an increasing function of the shadow fading factor,
the interval of RPs and the error of DoA estimation. And the positioning error
is a decreasing function of the path loss exponent and the number of APs. The
study of this paper may provide guideline for indoor positioning in the era of
5G with densely deployed mmWave small cells.

## Acknowledgment

The authors appreciate editor and anonymous reviewers for their precious time
and great effort in improving this paper.

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