# How Much Data is Needed for Channel Knowledge Map Construction?

Xiaoli Xu, _Member, IEEE,_ Yong Zeng, _Senior Member, IEEE,_ X. Xu and Y. Zeng
(Corresponding author) are with the National Mobile Communications Research
Laboratory, School of Information Science and Engineering, Southeast
University, Nanjing 210096, China (email: xiaolixu, yong_zeng@seu.edu.cn).
This work was supported by the National Natural Science Foundation of China
with grant number 6504009712 and 62071114.

###### Abstract

Channel knowledge map (CKM) has been recently proposed to enable environment-
aware communications by utilizing historical or simulation generated wireless
channel data. This paper studies the construction of one particular type of
CKM, namely channel gain map (CGM), by using a finite number of measurements
or simulation-generated data, with model-based spatial channel prediction. We
try to answer the following question: How much data is sufficient for CKM
construction? To this end, we first derive the average mean square error
(AMSE) of the channel gain prediction as a function of the sample density of
data collection for offline CGM construction, as well as the number of data
points used for online spatial channel gain prediction. To model the spatial
variation of the wireless environment even within each cell, we divide the CGM
into subregions and estimate the channel parameters from the local data within
each subregion. The parameter estimation error and the channel prediction
error based on estimated channel parameters are derived as functions of the
number of data points within the subregion. The analytical results provide
useful guide for CGM construction and utilization by determining the required
spatial sample density for offline data collection and number of data points
to be used for online channel prediction, so that the desired level of channel
prediction accuracy is guaranteed.

###### Index Terms:

Channel gain map (CGM), environment-aware communication, spatial channel
prediction, parameter estimation, average mean square error, map construction.

##  I Introduction

With the ever-increasing node density and channel dimension in wireless
communication networks, the acquisition of real-time channel state information
(CSI) purely relying on the conventional pilot based channel estimation
becomes costly, and even infeasible in rate-hungry and delay-stringent
applications. On the other hand, the abundant location-specific channel data
and powerful data-mining capability of wireless networks make it possible to
shift from the conventional environment-unaware communication to future
environment-aware communication, which facilitates CSI acquisition [1]. One
promising technique for realizing environment-aware communication is by
leveraging channel knowledge map (CKM), which makes use of the available
channel knowledge learnt from the physical environment and/or the wireless
measurements [2]. As shown in Fig. 1, CKM can be constructed by using
location-tagged data from the physical-map assisted simulation or the actual
channel measurements by network devices. The site-specific CKM can be stored
and managed at the base station (BS). During the CKM utilization stage, the
user’s physical or virtual location is used to query the CKM for obtaining a
priori channel knowledge between the BS and the user. With such a priori
channel knowledge, the required online training overhead for real-time CSI
acquisition can be greatly reduced [3]. In addition, the training results
could be feedback to the construction algorithm for updating the CKM, so as to
reflect the environment dynamics.

Figure 1: An illustration of CKM concept.

This paper focuses on the construction of a specific type of CKM, termed
channel gain map (CGM), which provides channel gain knowledge at arbitrary
location within each cell. The most straightforward approach for channel gain
prediction is by utilizing stochastic channel models, such as the distance-
dependent path loss model. However, such models only use the very high-level
environment attributes, such as the environment type: urban, suburban or rural
[4]. As a result, the channel gain is only coarsely related to the user
location via its distance from the BS, without considering the specific
environment. On the other hand, CGM is a site-specific database constructed
based on the actual local wireless environment that is learnt from the
physical map or the actual channel measurements. The spatial and temporal
correlation of channel gains make it possible to construct CGM by using a
finite number of data samples.

Channel gains of wireless links are typically modelled by three components,
i.e., the path loss, large-scale shadowing and small-scale multi-path fading.
The path loss and shadowing are mainly affected by the propagation
environment, e.g., the blockage due to building, terrain and plants, which can
be considered as relatively static. On the other hand, the multi-path fading
de-correlates fast in both the temporal and spatial domain, which is hardly to
be predicted in complex environment. Hence, this paper focuses on the
_spatial_ prediction for CGM construction and aims to determine the required
spatial sample density for achieving the desired level of accuracy in channel
prediction. Intuitively, the larger the spatial sample density is, the more
likely to find the data points that are close and hence highly correlated with
the channel at the target location. However, larger spatial sample density
implies high construction and maintenance cost of the CGM. Besides, the number
of neighboring data points used for online channel prediction also affects the
prediction error. The effect of measurement error and small-scale multi-path
fading can be mitigated by including more samples. However, including more
samples not only incurs larger computational cost, but also introduce bias if
uncorrelated samples are included.

The spatial prediction of channel gain can be achieved in different ways,
e.g., the model-based prediction [5][6][7], data-based prediction [8][9][10]
and the hybrid model and data-based prediction [11]. For model-based
prediction, the large-scale path loss at the target location is modeled as a
deterministic function of the distance between the target location and the BS.
Then, the shadowing at the target location is estimated from the shadowing
experienced by the neighboring locations in CGM, which is obtained by
subtracting their respective path loss from the measured channel gain. Given
the path loss parameters and the spatial correlation model [12], the minimum
mean square error (MMSE) prediction has been proposed in [5]. The authors in
[5] also investigate the maximum likelihood (ML) and least square (LS)
estimation of the channel parameters from the sample measurements. The
extension of the spatial prediction algorithms to the spatial-temporal
prediction are discussed in [13] and [14], where the temporal predictions are
achieved by nonlinear filtering and autoregressive prediction, respectively.
Since the path loss and spatial correlation between channel gains are modelled
as functions of distances, the accuracy of the model-based spatial prediction
is sensitive to localization error [15]. For data-based prediction, the
channel gain is viewed as spatially distributed data and the geostatical tools
[16] are adopted to perform the spatial interpolation, such as the $k$-nearest
neighbors (KNN), inverse-distance weight (IDW) and the Kriging Algorithms [8].
Besides, the collected data points can also be used to train a deep neural
network (DNN), which is then used to predict the channel gain at the target
location [17]. The performance of the data-based prediction can be improved if
the environment information is properly used, e.g., the 3D physical map is
used to assist the DNN training process in [18]. The authors in [11] considers
the hybrid model and data based channel prediction, where the collected
channel data is first divided into different groups based on the expectation
maximization (EM) algorithm, and then model-based spatial prediction is
applied within each data group. However, such data division is only effective
when the environment is simple so that the boundary of each group can be
clearly identified.

In general, the model-based prediction algorithms exploit the expert knowledge
and enable the estimation under limited data set. However, the performance of
model-based prediction algorithms relies on the accuracy of the channel gain
model. On the other hand, the data-driven methods rely on the amount of data
available. Compared to model-based prediction, data-based methods are usually
more flexible and may achieve better performance in complex environment if
sufficient data is available. However, data-driven methods are usually not as
tractable as the model-based method. To get some insights on the question:
“How much data is needed for CKM construction?”, we adopt the classical model-
based prediction in this paper and analyze channel prediction performance as a
function of the spatial sample density for offline CGM construction, as well
as the number of data points used for online channel prediction. Specifically,
we consider two types of spatial sample distributions, i.e., the random
distribution modelled by homogeneous Poisson point process (PPP) and the grid
distribution with certain separation between adjacent samples. When the
channel modeling parameters are known, we first derive the average mean square
error (AMSE) of the channel gain prediction as a function of the spatial
sample density, when only the nearest data point is used for online channel
prediction. Then, the results are extended to the channel prediction by using
any number of data points in CKM. The analytical results show that the
marginal improvement for AMSE decreases when more samples are included, which
implies that a limited number of data points should be sufficient for CKM
construction.

One of the main challenges for model-based channel prediction is that the
wireless environment may vary significantly even within each site, e.g., the
users may experience frequent shift between line-of-sight (LoS) link and non-
LoS (NLoS) link as they move in urban environment. It was shown in [19] that
even for a relatively small area, different channel gain models need to be
used for users in different streets. Hence, it is desired to establish the
local channel gain models for CGM construction. Given the sample density, the
expected number of samples within the local region reduces with the region
size. The limited number of samples may lead to large error in channel
parameter estimation. Since the channel prediction is more sensitive to the
path loss parameters, we derive the estimation error of the path loss exponent
and path loss intercept as functions of the number of samples within the
region. We also point out that the estimation error in channel parameters may
not lead to error in channel prediction, since the correlated shadowing may be
counted in the path loss intercept in a small region. Finally, the analytical
results are verified by extensive simulations and a case study with actual CGM
construction from sparse samples is presented to validate the proposed CGM
construction method.

The rest of this paper is organized as follows. The CGM construction problem
is formulated in Section II. Section III presents the CGM-based spatial
channel prediction methods and analyzes the achievable AMSE when the model
parameters are known. Section IV considers the channel parameter estimation
and spatial prediction based on local measurements within a small region. The
analytical expressions are verified by the numerical results in Section V, and
finally Section VI draws the conclusion of this paper and summarizes the key
results from this study.

##  II System Model

We consider the BS of a macro cell that wants to construct a CGM to enable
environment-aware communications. The CGM stores the channel gains at some
limited sample locations, which are used to infer the channel gains at any
location within the cell, based on the spatial predictability of the channel.
According to [2], the CGM can be constructed either by channel measurements
returned by network users at random locations, or by dedicated channel
measurement devices at designed locations. Therefore, we consider the
following two distributions of the data measurement locations:

  * •

_Random Sampling_ : Data is collected at random locations, modeled by
homogeneous PPP with density $\lambda$.

  * •

_Grid Sampling_ : Data is collected at grid points separated with a certain
distance $d$.

(a) Random Sampling

(b) Grid Sampling

Figure 2: Distribution of data collecting locations for CGM construction in a
square area with side length $D=1$km. The red star is the location of the BS.

Fig. 2 presents an illustration example for the random and grid sampling of
the data collecting locations. The CGM constructed by the above two methods
are referred to as _random CGM_ and _grid CGM_ , and denoted by
$\mathcal{M}_{r}$ and $\mathcal{M}_{g}$, respectively. The location of the BS
is denoted as the origin and the cell is assumed to be a square area, denoted
by $\mathcal{A}$, with side length $D$. The number of data points to construct
the CGM is $\lambda D^{2}$ and $(\frac{D}{d})^{2}$ in random and grid CGM,
respectively. According to the spatial predictability of the channel, the
channel gain at any arbitrary location $\mathbf{q}\in\mathcal{A}$, denoted by
${\Upsilon}_{\mathrm{dB}}(\mathbf{q})$, can be estimated from the collected
data stored in the CGM. The prediction function is written as

$\displaystyle\tilde{\Upsilon}_{\mathrm{dB}}(\mathbf{q})=f(\mathbf{q},\mathcal{M}_{c}),c\in\\{r,g\\}.$
(1)

The AMSE of the spatial prediction over all target locations within the area
is defined as

$\displaystyle
AMSE=\frac{1}{|\mathcal{A}|}\int_{\mathbf{q}\in\mathcal{A}}\left(\Upsilon_{\mathrm{dB}}(\mathbf{q})-\tilde{\Upsilon}_{\mathrm{dB}}(\mathbf{q})\right)^{2}d\mathcal{A},$
(2)

where $|\mathcal{A}|$ is the size of the area.

Intuitively, the accuracy of the spatial channel prediction improves with the
density of the data collecting locations, represented by $\lambda$ and $d$ in
$\mathcal{M}_{r}$ and $\mathcal{M}_{g}$, respectively. Besides, the complexity
of the estimation algorithm also plays an important role in the spatial
prediction, which can be reflected by the number of neighboring data points in
CGM used for online channel prediction. In order to answer the question “how
much data is needed for CKM construction?”, we derive the relationship between
the AMSE with the data collection density and data used for online prediction.

##  III CGM-based Channel Prediction with Known Parameters

In the wireless communication literature, it is well established that the
channel gain constitutes of three major components, i.e., the path loss,
shadowing and multipath fading. Denote by $\Upsilon(\mathbf{q})$ the channel
gain between the BS located at the origin and a receiver at location
$\mathbf{q}$. We have
$\Upsilon(\mathbf{q})=\Upsilon_{\mathrm{PL}}(\mathbf{q})\Upsilon_{\mathrm{SH}}(\mathbf{q})\Upsilon_{\mathrm{MP}}(\mathbf{q})$,
where $\Upsilon_{\mathrm{PL}}(\mathbf{q})$,
$\Upsilon_{\mathrm{SH}}(\mathbf{q})$ and $\Upsilon_{\mathrm{ML}}(\mathbf{q})$
represent the impact of path loss, shadowing and multipath fading,
respectively. With
$\Upsilon_{\mathrm{dB}}(\mathbf{q})=10\log_{10}(\Upsilon(\mathbf{q}))$ being
the channel gain in dB, we have

$\displaystyle\Upsilon_{\mathrm{dB}}(\mathbf{q})=K_{\mathrm{dB}}-10n_{pL}\log_{10}(\|\mathbf{q}\|)+v(\mathbf{q})+\omega(\mathbf{q}),$
(3)

where $K_{\mathrm{dB}}$ accounts the path loss intercept, $n_{PL}$ is the path
loss exponent, $v(\mathbf{q})=10\log_{10}(\Upsilon_{\mathrm{SH}}(\mathbf{q}))$
is a random variable that models the effect of shadowing, and
$\omega(\mathbf{q})=10\log_{10}(\Upsilon_{\mathrm{ML}}(\mathbf{q}))$ models
the effect of multipath fading. Generally, $v(\mathbf{q})$ can be modelled as
a zero-mean Gaussian random variable with variance $\alpha$ and spatial
correlation function

$\displaystyle\varepsilon(h)=\alpha\exp\left(-\frac{h}{\beta}\right),$ (4)

where $h$ is the distance between the pair of data points and $\beta$ is known
as the correlation distance of shadowing [12]. In other words, we have
$\mathbb{E}\\{v(\mathbf{q}_{i})v(\mathbf{q}_{j})\\}=\varepsilon(\|\mathbf{q}_{i}-\mathbf{q}_{j}\|)$.
The multipath effect $\omega(\mathbf{q})$ is also modeled as a zero-mean
Gaussian random variable, with variance $\sigma^{2}$ and it spatially de-
correlates fast as compared with the shadowing.

###  III-A Spatial Prediction of Channel Gain

We first assume that the channel modeling parameters in (3), i.e.,
$\\{n_{\mathrm{PL}},K_{\mathrm{dB}},\alpha,\beta,\sigma^{2}\\}$, are fixed and
known. Then, the estimation of channel gain at location $\mathbf{q}$ is
equivalent to estimating the shadowing effect $v(\mathbf{q})$ since the path
loss can be computed from the distance to the BS and the multipath fading can
be hardly predicted, and hence treated as noise.

Ideally, all data points in the CGM should be used for estimating
$\Upsilon_{\mathrm{dB}}(\mathbf{q})$. However, since the channel correlation
diminishes with the separation distance and for the purpose of reducing the
computational complexity, we assume that only $k$ nearest data collecting
locations are included for online channel prediction. Denote the locations of
$k$ nearest data point in CGM by
$\mathcal{Q}_{k}=\\{\mathbf{q}_{1},...,\mathbf{q}_{k}\\}$. According to [5],
the MMSE estimation of $v(\mathbf{q})$ is given by

$\displaystyle\tilde{v}(\mathbf{q})=\phi_{Q}^{T}(\mathbf{q})(\mathbf{R}_{Q}+\sigma^{2}\mathbf{I}_{k\times
k})^{-1}(\mathbf{y}_{Q}-K_{\mathrm{dB}}-n_{\mathrm{PL}}\mathbf{h}),$ (5)

where
$\phi_{Q}(\mathbf{q})=\left[\varepsilon(\|\mathbf{q}-\mathbf{q}_{1}\|),\cdots,\varepsilon(\|\mathbf{q}-\mathbf{q}_{k}\|)\right]^{T}$
captures the correlation between the shadowing at the target location
$\mathbf{q}$ and that at all the data collecting locations $\mathcal{Q}_{k}$,
$\mathbf{R}_{Q}$ is the correlation matrix, with the $(i,j)$th component being
$[\mathbf{R}_{Q}]_{ij}=\varepsilon(\|\mathbf{q}_{i}-\mathbf{q}_{j}\|)$,
$\mathbf{I}_{k\times k}$ is the identity matrix,
$\mathbf{y}_{Q}=\left[\Upsilon_{\mathrm{dB}}(\mathbf{q}_{1}),\cdots,\Upsilon_{\mathrm{dB}}(\mathbf{q}_{k})\right]^{T}$
is the measurements stored in the CGM, and
$\mathbf{h}=[-10\log_{10}(\|\mathbf{q}_{1}\|),\cdots,-10\log_{10}(\|\mathbf{q}_{k}\|)]^{T}$
is the distance vector.

The estimated channel gain $\tilde{\Upsilon}_{\mathrm{dB}}(\mathbf{q})$ is the
summation of the path loss and the estimated shadowing
$\tilde{v}(\mathbf{q})$, i.e.,

$\displaystyle\tilde{\Upsilon}_{\mathrm{dB}}(\mathbf{q})=K_{\mathrm{dB}}-10n_{pL}\log_{10}(\|\mathbf{q}\|)+\tilde{v}(\mathbf{q}).$
(6)

The complexity of the online channel prediction according to (6) mainly
attributes to the inversion of a $k\times k$ matrix in (5), which can be
reduced by considering less number of neighboring data points, i.e., using a
smaller $k$. Besides, the MSE of the estimation in (6) is given by

$\displaystyle\xi_{\mathrm{dB}}(\mathbf{q})$
$\displaystyle=\left(\Upsilon_{\mathrm{dB}}(\mathbf{q})-\tilde{\Upsilon}_{\mathrm{dB}}(\mathbf{q})\right)^{2}$
$\displaystyle=\alpha+\sigma^{2}-\phi_{Q}^{T}(\mathbf{q})(\mathbf{R}_{Q}+\sigma^{2}\mathbf{I}_{k\times
k})^{-1}\phi_{Q}(\mathbf{q}).$ (7)

By substituting (7) into (2) and taking average with respect to the random
location $\mathbf{q}\in\mathcal{A}$, we can get the AMSE for the CGM
constructed by random and grid sampling, respectively, as elaborated below.

###  III-B AMSE of Random CGM

We start by considering the special case when only the nearest data point is
used, i.e., $k=1$. Denote by $\mathbf{q}_{1}$ the location of the nearest data
point stored in CGM and let $d_{\min}=\|\mathbf{q}-\mathbf{q}_{1}\|$. Then,
the correlation matrix is simplified to $\mathbf{R}_{Q}=1$ and the estimated
shadowing loss in (5) reduces to

$\displaystyle\tilde{v}(\mathbf{q})=\frac{\alpha}{1+\sigma^{2}}\exp\left(-\frac{d_{\min}}{\beta}\right)v(\mathbf{q}_{1}),$
(8)

where
$v(\mathbf{q}_{1})=\Upsilon_{\mathrm{dB}}(\mathbf{q}_{1})-K_{\mathrm{dB}}+10n_{\mathrm{PL}}\log_{10}(\|\mathbf{q}_{1}\|)$
is the shadowing loss at $\mathbf{q}_{1}$. Furthermore, the MMSE in (7) is
simplified to

$\displaystyle\xi_{\mathrm{dB}}(\mathbf{q})|_{(k=1)}=\alpha+\sigma^{2}-\frac{\alpha^{2}}{\alpha+\sigma^{2}}\exp\left(-\frac{2d_{\min}}{\beta}\right).$
(9)

When the data points are distributed randomly, according to the homogeneous
PPP with density $\lambda$, the distribution of $d_{\min}$ is given by the
probability density function (p.d.f.) $P_{r}(x)$ [20] with

$\displaystyle P_{r}(x)=2\pi\lambda x\exp\left(-\pi\lambda x^{2}\right).$ (10)

By substituting (9) and (10) into (2), we can obtain the AMSE as a function of
the density $\lambda$, which is presented in Lemma 1.

######  Lemma 1.

If the CGM is constructed by data collection at random locations according to
the homogeneous PPP with density $\lambda$ and only the nearest data point is
used for online channel prediction, the AMSE is given by

$\displaystyle
AMSE=\mathbb{E}[\xi_{\mathrm{dB}}(\mathbf{q})|_{(k=1)}]=\alpha+\sigma^{2}-\frac{\alpha^{2}}{\alpha+\sigma^{2}}\zeta_{r}(\lambda),$
(11)

where

$\displaystyle\zeta_{r}(\lambda)$
$\displaystyle=\int_{0}^{\infty}\exp\left(-\frac{2x}{\beta}\right)P_{r}(x)dx$
$\displaystyle=2\pi\lambda\int_{0}^{\infty}x\exp\left(-\pi\lambda
x^{2}-\frac{2x}{\beta}\right)dx$
$\displaystyle=1-\frac{1}{\beta\sqrt{\lambda}}\exp\left(\frac{1}{\pi\lambda\beta^{2}}\right)\left(1-\Phi\left(\frac{1}{\beta\sqrt{\pi\lambda}}\right)\right)$

and $\Phi(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$ is the Gauss error
function.

###### Proof.

Please refer to Appendix A. ∎

When $\lambda\rightarrow 0$, the AMSE in (11) reduces to
$(\alpha+\sigma^{2})$, which corresponds to the variance of the channel gain
prediction using path loss only. When $\lambda\rightarrow\infty$,
$\zeta_{r}(\lambda)\rightarrow 1$ and (11) reduces to
$\left(\sigma^{2}+\frac{\alpha\sigma^{2}}{\alpha+\sigma^{2}}\right)$,
representing the limit of channel gain spatial prediction. In general,
$\zeta_{r}(\lambda)$ is an increasing function with $\lambda$ and hence the
AMSE decreases with the spatial sample density $\lambda$. Besides, the spatial
prediction is more effective, i.e., with a larger decaying slope with
$\lambda$, when the shadowing variance is larger than the fading power.

Next, we derive the AMSE for general value of $k$, which is rather challenging
since it involves the inverse of a random matrix $\mathbf{R}_{Q}$. To gain
some insights, we first consider the asymptotical behaviors. When the CGM
density is sufficiently large, i.e., $\lambda\rightarrow\infty$, or the
spatial correlation vanishes, i.e., $\beta\rightarrow\infty$, we have
$\phi_{Q}\rightarrow\alpha\mathbf{1}_{k\times 1}$ and
$R_{Q}\rightarrow\mathbf{1}_{k\times k}$, where $\mathbf{1}_{m\times n}$ is a
matrix of size $m\times n$ with all the components being 1. Then, the inverse
of the matrix can be obtained as

$\displaystyle(\mathbf{R}_{Q}+\sigma^{2}\mathbf{I}_{k\times k})^{-1}$
$\displaystyle=(\alpha\mathbf{1}_{k\times k}+\sigma^{2}\mathbf{I}_{k\times
k})^{-1}$ $\displaystyle=\frac{1}{\sigma^{2}}\mathbf{I}_{k\times
k}+\frac{\alpha}{\sigma^{2}(k\alpha+\sigma^{2})}\mathbf{1}_{k\times k}.$ (12)

By substituting (12) and $\Phi_{Q}$ into (9), we have

$\displaystyle\lim_{\lambda\rightarrow\infty}\xi_{\mathrm{dB}}(\mathbf{q})=\lim_{\beta\rightarrow\infty}\xi_{\mathrm{dB}}(\mathbf{q})=\alpha+\sigma^{2}-\frac{k\alpha^{2}}{k\alpha+\sigma^{2}}.$
(13)

Since the limiting behavior of $\xi_{\mathrm{dB}}(\mathbf{q})$ no longer
depends on the target location, we can obtain AMSE directly. Specifically, for
CGM with general density $\lambda$ and prediction algorithm including $k$
neighboring data points, we can obtain the approximate AMSE by assuming that
all the $k$ measurements have the minimum distance with the target location,
i.e., setting
$\phi_{Q}(\mathbf{q})\approx\alpha\exp\left(-\frac{d_{\min}}{\beta}\right),\forall\mathbf{q}$,
and assuming that the measurements are close to each other, i.e.,
$\mathbf{R}_{Q}\approx\mathbf{1}_{k\times k}$. The obtained AMSE is presented
in Lemma 2.

######  Lemma 2.

If the CGM is constructed by data collection at random locations according to
the homogeneous PPP with density $\lambda$ and $k$ nearest data points are
used for online channel prediction, the achieved AMSE is approximated by

$\displaystyle
AMSE\approx\alpha+\sigma^{2}-\frac{k\alpha^{2}}{k\alpha+\sigma^{2}}\zeta_{r}(\lambda),$
(14)

where the approximation is tight when $k=1$ or $\lambda\rightarrow\infty$.

Similar to (11), when $\lambda\rightarrow 0$, the sample locations are too far
to be correlated, and hence the AMSE is $(\alpha+\sigma^{2})$, which is
independent of $k$. When $\lambda$ is sufficiently large, the shadowing can be
completely known from $k\rightarrow\infty$ samples, and hence the resultant
AMSE only contains the fading power $\sigma^{2}$. In general, for given
spatial sample density, the AMSE decays with $k$ with the slope

$\displaystyle\frac{\partial{AMSE}}{\partial
k}=-\frac{\alpha^{2}\sigma^{2}}{(k\alpha+\sigma^{2})^{2}}\zeta_{r}(\lambda),$
(15)

whose absolute value decays with $k$ and increases with $\lambda$. This
implies that the marginal improvement of AMSE decreases when $k$ is
sufficiently large, and including more data points is effective only when the
spatial sample density is large.

###  III-C AMSE of Grid CGM

When the data collecting locations in CGM are placed in the grid with
separation $d$, the distance between a random location $\mathbf{q}$ with the
nearest measurement is distritbuted according to $P_{g}(x)$ with

$\displaystyle P_{g}(x)=\begin{cases}\frac{2\pi x}{d^{2}},&0\leq
x\leq\frac{d}{2}\\\
\frac{4x}{d^{2}}\left(\frac{\pi}{2}-2\arccos\left(\frac{d}{2x}\right)\right),&\frac{d}{2}\leq
x\leq\frac{\sqrt{2}d}{2}\end{cases}$ (16)

The derivation of (16) is presented in Appendix B. Then, following similar
analysis as for random CGM, we can obtain the AMSE for grid CGM with $k=1$, as
presented in Lemma 3.

######  Lemma 3.

If the CGM is constructed by data collected at grid points with separation $d$
and only the nearest data point is used for online channel gain prediction,
the achieved AMSE is given by

$\displaystyle
AMSE=\mathbb{E}[\xi_{\mathrm{dB}}(\mathbf{q})|_{(k=1)}]=\alpha+\sigma^{2}-\frac{\alpha^{2}}{\alpha+\sigma^{2}}\zeta_{g}(d)$
(17)

where

$\displaystyle\zeta_{g}(d)$ $\displaystyle=\int_{0}^{\frac{d}{2}}\frac{2\pi
x}{d^{2}}e^{-\frac{2x}{\beta}}dx+\int_{\frac{d}{2}}^{\frac{\sqrt{2}d}{2}}\frac{8x}{d^{2}}e^{-\frac{2x}{\beta}}\arccos\left(\frac{d}{2x}\right)dx$
$\displaystyle=\left[\frac{\pi\beta^{2}}{2d^{2}}\left(1-\left(1+\frac{\sqrt{2}d}{\beta}\right)e^{-\sqrt{2}d/\beta}\right)-2\Psi\left(\frac{d}{\beta}\right)\right]$

and
$\Psi(x)=\int_{1}^{\sqrt{(}2)}ue^{-ux}\left(\arccos\left(\frac{1}{u}\right)\right)du\approx(0.2854-0.0725x+0.0108x^{2})e^{-x}$.

Following similar analysis as that for random CGM, we can obtain the same
asymptotical MSE in (13) for grid CGM when $d\rightarrow 0$. Furthermore, by
using similar approximation, we have

######  Lemma 4.

If the CGM is constructed by data collected at grid points with separation $d$
separation $d$ and $k$ nearest data points are used for online channel gain
prediction, the achieved AMSE is given by

$\displaystyle
AMSE\approx\alpha+\sigma^{2}-\frac{k\alpha^{2}}{k\alpha+\sigma^{2}}\zeta_{g}(d),$
(18)

where the approximation is tight when $k=1$ or $d\rightarrow 0$.

Comparing (18) with (14), it is observed that the channel prediction
performance in random CGM and grid CGM have the same limiting performance at
the extremely large or small spatial sample density, i.e., with
$\lim_{d\rightarrow 0}\zeta_{g}(d)=1$ and
$\lim_{d\rightarrow\infty}\zeta_{g}(d)=0$. In general, we have
$\zeta_{g}(d)>\zeta_{r}(\lambda)$ when they have the same effective density,
i.e., with $\lambda=1/d^{2}$.

The analytical results presented in Lemma 2 and Lemma 4 may be used to guide
the CGM construction and utilization procedures, by determining the required
sample density for offline CKM construction and the number of data points to
be used for online channel prediction, so that the desired level of channel
prediction accuracy is guaranteed.

##  IV CGM-based Channel Prediction with Unknown Parameters and Spatial
Variations

In practice, the channel modeling parameters in the path loss model (3) are
usually unknown and they vary across different regions [19]. An illustration
example is shown in Fig. 3, where Fig. 3(a) shows a city map and Fig. 3(b)
shows the corresponding CGM, generated by the commercial ray tracing software
Wireless Insite111https://www.remcom.com/wireless-insite-em-propagation-
software. It is clearly noted that the regions bounded by the three red boxes
suffer from different levels of path loss, due to the different levels of LoS
link blockage. Hence, different channel gain models should be used for channel
spatial prediction. To this end, we divide the area into small regions and
achieve the spatial prediction in two steps:

  * •

Build the local channel gain model based on the data points within each small
region, which is either determined by the distance threshold or building
borders.

  * •

For target location within each region, estimate the channel gain based on the
local channel model and the data points in proximity of the target location.

(a) City map

(b) Channel gain map

Figure 3: An illustration of the channel model variation within a site.

In the following, we assume that the region division has been completed and
consider the channel modeling parameter estimation for a local region with $N$
measurements in the CGM, located at $\\{\mathbf{q}_{1},...,\mathbf{q}_{N}\\}$.
Furthermore, we analyze the impact of spatial sample density and region size
on the channel parameter estimation and the channel prediction.

###  IV-A Estimation of Local Channel Modeling Parameters

Since the effects of shadowing and multipath fading are modeled as the zero-
mean random variables, we can obtain the least square (LS) estimation of the
path loss intercept and path loss exponent as

$\displaystyle\left[\begin{matrix}\hat{K}_{\mathrm{dB}}&\hat{n}_{\mathrm{PL}}\end{matrix}\right]=\arg\min\|\mathbf{y}-K_{\mathrm{dB}}-n_{\mathrm{PL}}\mathbf{h}\|^{2},$
(19)

where $\mathbf{y}$ is a vector containing the collected channel gain at all
the $N$ sample locations within the region,
$\mathbf{h}=\left[-10\log_{10}(\|\mathbf{q}_{1}\|),-10\log_{10}(\|\mathbf{q}_{2}\|),\cdots-10\log_{10}(\|\mathbf{q}_{N}\|)\right]$
is the vector containing the distance of the $N$ data points to the BS. Let
$\mathbf{H}=[\mathbf{1}_{N\times 1},\mathbf{h}]$, and the LS estimation of the
path loss intercept and path loss exponents is given by

$\displaystyle\left[\begin{matrix}\hat{K}_{\mathrm{dB}}\\\
\hat{n}_{\mathrm{PL}}\end{matrix}\right]=\left(\mathbf{H}^{T}\mathbf{H}\right)^{-1}\mathbf{H}^{T}\mathbf{y}.$
(20)

The covariance matrix of the estimation error vector is

$\displaystyle\mathbf{C}_{LS}$
$\displaystyle=(\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{H}^{T}\left(\mathbf{R}_{Q}+\sigma^{2}\mathbf{I}_{N\times
N}\right)\mathbf{H}(\mathbf{H}^{T}\mathbf{H})^{-1},$ (21)

where $\mathbf{R}_{Q}$ is the shadowing correlation matrix for $N$ data
points. The estimation error for path loss intercept $K_{\mathrm{dB}}$ and
path loss exponent $n_{\mathrm{PL}}$ are given by

$\displaystyle\sigma^{2}_{\hat{K}_{\mathrm{dB}}}=\mathbf{C}_{LS}(1,1),\sigma^{2}_{\hat{n}_{\mathrm{PL}}}=\mathbf{C}_{LS}(2,2).$
(22)

In general, the estimation errors in (22) depend on the number of data points
and their locations. To gain some insights on the impact of region size and
sample density, we assume that the distance of the data collection location to
the BS is a random number uniformly distributed within the range
$[\delta_{\min},\delta_{\max}]$, i.e.,
$\|\mathbf{q}\|\sim\mathcal{U}(\delta_{\min},\delta_{\max})$. Then, we can
approximate the channel parameter estimation error as a function of the number
of data points $N$, and their distribution parameters
$(\delta_{\min},\delta_{\max})$, as stated in Lemma 5.

######  Lemma 5.

Consider a typical region containing $N$ data points in the CGM with random
point density $\lambda$ or grid point separation $d$. If the distances of the
measurement locations to the BS are uniformly distributed within
$[\delta_{\min},\delta_{\max}]$, the LS estimation error of the path loss
exponent and path loss intercept in (21) can be approximated by

$\displaystyle\sigma^{2}_{\hat{K}_{\mathrm{dB}}}$
$\displaystyle=\frac{\alpha+\sigma^{2}/c}{N/c}\cdot\frac{\mu^{2}+\chi}{\chi}$
(23) $\displaystyle\sigma^{2}_{\hat{n}_{\mathrm{PL}}}$
$\displaystyle=\frac{\alpha+\sigma^{2}/c}{N/c}\cdot\frac{1}{\chi}$ (24)

where

$\displaystyle c$
$\displaystyle=\begin{cases}\max\\{1,\pi\lambda\beta^{2}\\},&\textnormal{for
random CGM}\\\ \max\\{1,\pi\beta^{2}/d^{2}\\},&\textnormal{for grid CGM}\\\
\end{cases}$ $\displaystyle\mu$
$\displaystyle=\frac{10\delta_{\max}\log_{10}(\delta_{\max})-10\delta_{\min}\log_{10}(\delta_{\min})}{\delta_{\max}-\delta_{\min}}-\frac{10}{\ln
10}$ $\displaystyle\chi$ $\displaystyle=\frac{100}{(\ln
10)^{2}}-\frac{100\delta_{\max}\delta_{\min}\log_{10}(\delta_{\max}/\delta_{\min})}{(\delta_{\max}-\delta_{\min})^{2}}$

###### Proof.

Please refer to Appendix C. ∎

When the shadowing correlation distance $\beta$ is small, we have $c=1$. Lemma
5 indicates that the estimation error is inversely proportional with the
number of data points within the region, which implies that we can improve the
parameter estimation accuracy by using a denser CGM. However, when $\beta$ is
large, increasing the CGM density will not change the value $N/c$ and hence
the improvement over the channel parameter estimation is rather limited.
Besides, the estimation error also depends on the region size, measured by
$(\delta_{\max}-\delta_{\min})$, within which the channel parameters are
consistent.

###  IV-B Estimation of Shadowing and Fading Parameters

The shadowing and fading parameters can be estimated using LS method or the
variogram fitting algorithms discussed in [21]. This paper adopts the LS
estimation for its simplicity and analytical tractability. Specifically, after
obtaining the estimated path loss parameters in (20), the shadowing and fading
at each data collecting location can be obtained by subtracting the path loss,
i.e.,

$\displaystyle\mathbf{s}=\mathbf{y}-\hat{K}_{\mathrm{dB}}-\hat{n}_{\mathrm{PL}}\mathbf{h},$
(25)

where $\hat{K}_{\mathrm{dB}}$ and $\hat{n}_{\mathrm{PL}}$ are given in (20).

Consider a pair of data points at the locations $\mathbf{q}_{i}$ and
$\mathbf{q}_{j}$, with residual channel gain $\mathbf{s}(i)$ and
$\mathbf{s}(j)$, respectively, after subtracting the estimated path loss. The
correlation between these residual channel gains is calculated as
$\mathbf{s}(i)\mathbf{s}(j)$, which can be used to estimate the shadowing
variance and correlation distance in (4). Specifically, let
$\mathcal{D}=\\{d_{1},...,d_{|\mathcal{D}|}\\}$ denote the set of all possible
distances between a pair sample locations. Further denote by
$\mathcal{I}_{u}=\\{(i,j):\|\mathbf{q}_{i}-\mathbf{q}_{j}\|=d_{u}\\}$ the set
of pairs with distance $d_{u}$ for $u=1,...,|\mathcal{D}|$. Then, we can
estimate the value of the correlation function at $d_{u}$ as

$\displaystyle\hat{\varepsilon}(d_{u})=\frac{1}{|\mathcal{I}_{u}|}\sum_{(i,j)\in\mathcal{I}_{u}}\mathbf{s}(i)\mathbf{s}(j).$
(26)

To reduce complexity and ensure meaningful estimation results, the cardinality
of distance vector $|\mathcal{D}|$ is selected such that
$\hat{\varepsilon}(d_{u})>0$ for $u\leq|\mathcal{D}|$. Then, the estimation of
shadowing parameters can be formulated as

$\displaystyle[\hat{\alpha},\hat{\beta}]=\arg\min\sum_{d_{u}\in\mathcal{D}}|\mathcal{I}_{u}|\left(\ln(\alpha
e^{-d_{u}/\beta})-\ln(\hat{\varepsilon}(d_{u}))\right),$ (27)

where $|\mathcal{I}_{u}|$ is set as the weight measuring the reliability of
the experimental value $\hat{\varepsilon}(d_{u})$. Problem (27) can be solved
easily with the solution given by [5]

$\displaystyle\left[\begin{matrix}\ln(\hat{\alpha})\\\
\frac{1}{\beta}\end{matrix}\right]=\left(\mathbf{M}^{T}\mathbf{W}\mathbf{M}\right)^{-1}\mathbf{M}^{T}\mathbf{W}\left[\begin{matrix}\ln(\hat{\varepsilon}(d_{1}))\\\
\vdots\\\ \ln(\hat{\varepsilon}(d_{|\mathcal{D}|}))\end{matrix}\right],$ (28)

where $\mathbf{M}=\left[\begin{matrix}1,&\cdots&1\\\
-d_{1}&\cdots&-d_{|\mathcal{D}|}\end{matrix}\right]^{T}$,
$\mathbf{W}=\mathrm{diag}(|\mathcal{I}_{1}|,\cdots,|\mathcal{I}_{\mathcal{D}}|)$

With $\hat{\alpha}$ obtained in (28), the estimated multipath fading variance
is given by

$\displaystyle\hat{\sigma}^{2}=\max\\{\mathbf{s}^{T}\mathbf{s}-\hat{\alpha},0\\}.$
(29)

###  IV-C AMSE of Spatial Channel Prediction

Based on the estimated channel modeling parameters within the local region, we
can perform the spatial prediction of the channel gain within the region as in
(6), by substituting the corresponding estimated channel parameters
$\\{\hat{n}_{\mathrm{PL}},\hat{K}_{\mathrm{dB}},\hat{\alpha},\hat{\beta},\hat{\sigma}^{2}\\}$.
Since the AMSE of channel gain spatial prediction is not sensitive to the
estimation error of the correlation distance and shadowing power. Hence, we
focus on analyzing the impact of path loss estimation error on the AMSE.

Under the special case when the correlation distance $\beta$ is smaller than
the separation between samples, the channel gain estimation reduces to the
estimation of path loss, and hence the AMSE is proportional to the path loss
parameter estimation error, given in Lemma 5. Specifically, the channel gain
estimation at $\mathbf{q}$ is given by

$\displaystyle\tilde{\Upsilon}_{\mathrm{dB}}(\mathbf{q})=\hat{K}_{\mathrm{dB}}-10\hat{n}_{\mathrm{PL}}\log_{10}(\|\mathbf{q}\|))$
(30)

The MSE of the path loss estimation is given by

$\displaystyle\xi_{\mathrm{dB}}(\mathbf{q})$
$\displaystyle=\mathbb{E}[(\tilde{\Upsilon}_{\mathrm{dB}}-{\Upsilon}_{\mathrm{dB}})^{2}]$
$\displaystyle=\alpha+\sigma^{2}+\sigma_{\hat{K}_{\mathrm{dB}}}^{2}+(10\log_{10}(\|\mathbf{q}\|)))^{2}\sigma_{\hat{n}_{\mathrm{PL}}}^{2}$
$\displaystyle-10\log_{10}(\|\mathbf{q}\|))(\mathbf{C}_{\mathrm{LS}}(2,1)+\mathbf{C}_{\mathrm{LS}}(1,2))$
(31)

where $\sigma_{\hat{K}_{\mathrm{dB}}}^{2}$ and
$\sigma_{\hat{n}_{\mathrm{PL}}}^{2}$ are given by (23) and (24), respectively.
Further assuming $\|\mathbf{q}\|\sim\mathcal{U}(\delta_{\min},\delta_{\max})$,
we get the AMSE as

$\displaystyle
AMSE=\mathbb{E}[\xi_{\mathrm{dB}}(\mathbf{q})]=(\alpha+\sigma^{2})\left(1+\frac{2}{N}\right).$
(32)

which decreases monotonically with the number of data points $N$ within the
region.

On the other hand, when $\beta$ is large, the path loss at the target location
is calculated from the estimated parameters and the shadowing is estimated by
the shadowing experience by the neighboring measurements within the same
region. In this case, the AMSE does not necessarily increase with the
parameter estimation error, especially when the region is small. This is
because the correlated shadowing loss can also be counted into the path loss
intercept $\hat{K}_{\mathrm{dB}}$, without affecting the accuracy of channel
gain prediction. Hence, deriving the AMSE is challenging. For most of the
practical scenarios, we can approximate the AMSE by the case with known
channel parameters, presented in Lemma 1 and Lemma 2.

##  V Numerical Results

The analysis presented in the preceding sections is verified by the numerical
results in this section. We consider the CGM construction in a $D\times D$
square area. Unless otherwise stated, the channel gain is generated according
to the model in (3) with channel parameters $n_{\mathrm{PL}}=2.2$,
$K_{\mathrm{dB}}=-80$, $\alpha=8$, $\beta=30$ m and $\sigma^{2}=2$.

First, we verify the analytical AMSE expressions presented in Lemma 1 and
Lemma 3 for $k=1$ and various CGM densities in Fig. 4. It is observed that the
analysis matches very well with the simulation results, even though an
approximation is adopted to obtain the closed-form expression for
$\zeta_{g}(d)$ in Lemma 3. Besides, the AMSE monotonically decreases with the
CGM density. The price paid for denser CGM is the higher construction and
storage cost. With the same density, grid CGM slightly outperforms the random
CGM since the distance between the random target locations and measurements
are guaranteed to be no greater than $d/2$.

Figure 4: The AMSE of channel gain prediction with various CGM densities. For
fair comparison, the separation distance in Grid CGM is set as
$d=1/\sqrt{\lambda}$.

Under the fixed CGM density, the prediction accuracy can be improved by
considering more neighboring data points in CGM for online channel prediction,
i.e., with a larger $k$, as indicated by Lemma 2 and Lemma 4. The analytical
results in (14) and (18) are verified in Fig. 5(a) and Fig. 5(b),
respectively. As expected, the analytical results are tight when $k=1$ and
rather accurate when CGM density is large. It is observed that the nearest
data point provides the most information for channel prediction at the target
location. With further increase of $k$, the performance improvement is
limited, and it becomes almost flat far before $k$ reaches the number of
neighboring measurements within the correlation distance, i.e.,
$\pi\beta^{2}\lambda$ in random CGM and $\left(\frac{\beta}{d}\right)^{2}$ in
grid CGM. This justifies for the low complexity online channel estimation
algorithms that only utilize the limited number of measurements in proximity.

(a) Random CGM

(b) Grid CGM

Figure 5: The AMSE of channel gain prediction using various number of data
points.

Next, we examine the accuracy of the path loss parameter estimation for the
scenario with unknown channel modeling parameters. Assume that the channel
parameters within a square area of side length $\bar{D}$ is constant, and the
border lines of the area is known, so that all the channel gain measurements
within that region can be used for channel parameter estimation. Fig. 6
compares the analytical estimation error of $n_{\mathrm{PL}}$ and
$K_{\mathrm{dB}}$, given in (23) and $\eqref{eq:nPLerror}$, respectively, with
the simulation results. The estimation error decreases with the region size
within which the channel parameters remain constant. When the shadowing
correlation distance is small, i.e., $\beta=1$, the estimation error can be
reduced by increasing the map density, as shown in Fig. 6(a) and Fig. 6(b).
However, when $\beta=30$, the reduction of the estimation error with the
increase of the data collection density is negligible, as shown in Fig. 6(c)
and Fig. 6(d). The simulation results are consistent with our expectation, as
revealed from the analytical results. Besides, similar parameter estimation
accuracy is observed for random and grid CGM when they have the same density.

(a) $\beta=1$

(b) $\beta=1$

(c) $\beta=30$

(d) $\beta=30$

Figure 6: The estimation error of channel parameters $n_{\mathrm{PL}}$ and
$K_{\mathrm{dB}}$, with $\alpha=8$ and $\sigma^{2}=2$.

Further consider the AMSE of channel prediction with the estimated parameters
within a local model-consistent region. When $\beta$ is small, the AMSE is
mainly caused by the error in path loss parameters and it changes with the
number of collected data points within the region. Fig. 7(a) shows that the
simulation results match well with the analytical expression shown in (32).
The gap between the AMSE under estimated channel parameters and that under
known parameters vanishes when the number of measurements within the region is
large. Fig. 7(b) shows the AMSE of channel prediction when $\beta=30$ in a
small region with $N=20$ measurement samples. When the region is small, the
correlated shadowing may be considered as part of the path loss. This explains
why the AMSE of the channel prediction with estimated parameters is lower than
that of the prediction with true channel parameters when $k\leq 2$. When $k$
gets large, the shadowing loss can be properly calculated and hence the AMSE
performance converges. The performance degradation caused by parameter
estimation error is almost negligible, and hence the performance can still be
approximated by the analytical results given in Lemma 4.

(a) $\beta=1,k=0$

(b) $\beta=30,N=20$

Figure 7: The AMSE of channel prediction within the model-consistent region
based on grid CGM with $d=25$.

In practical environment, finding the correct channel gain model becomes even
more challenging. We consider a case study by reconstructing the CGM shown in
Fig. 3(b) via the grid samples shown in Fig. 8(a) with $d=20$m, where the
building areas are excluded. For simplicity, the whole area is only divided
into two regions based on whether the LoS link is blocked by buildings or not.
Following the procedures in Section IV, the channel parameters for each region
are estimated using the LS methods. For LoS region, the estimated channel
parameters are $\hat{n}_{\mathrm{PL}}=1.3442$,
$\hat{K}_{\mathrm{dB}}=-77.9267$, $\hat{\alpha}=0$, $\hat{\beta}=0$ and
$\hat{\sigma}^{2}=9.4255$. For NLoS region, the estimated channel parameters
are $\hat{n}_{\mathrm{PL}}=4.92$, $\hat{K}_{\mathrm{dB}}=-31.6323$,
$\hat{\alpha}=339.5941$, $\hat{\beta}=33.9911$ and $\hat{\sigma}^{2}=0$. For
LoS region, only the path loss need to be calculated and hence the neighboring
measurements are not used for online channel prediction, i.e., with $k=0$. For
NLoS region, based on the analytical results, $k=3$ is selected for balancing
the accuracy and complexity for online channel prediction. The reconstructed
CGM based on the estimated parameters is shown in Fig. 8(b) with the AMSE
being 109.9952.

(a) Sample Locations

(b) Reconstructed CGM

Figure 8: The AMSE of channel prediction based on grid CGM with $d=20$ m with
region division based on LoS link status.

Further consider the channel prediction in a small region highlighted in Fig.
8, which contains 27 measurement samples. The estimated channel model from the
samples within this region has the parameters $\hat{n}_{\mathrm{PL}}=0$,
$\hat{K}_{\mathrm{dB}}=-120.8396$, $\hat{\alpha}=148.0263$,
$\hat{\beta}=7.5880$ and $\hat{\sigma}^{2}=0$, and the AMSE is 67.4203, which
is much smaller than the AMSE if only the LoS/NLoS regions are considered.
This justifies the necessity for using different models for different regions.
However, if we further divide the this region into even smaller regions, as
shown in Fig. 9(a), the AMSE increases since the number of data points within
each region reduces. From Fig. 9(b), it is observed that if this region is
further divided into two regions (separated by the $y$-axis in Fig. 9(a)), the
AMSE slightly increases to 71.2606. If this region is divided into 4 small
regions, the average AMSE increases 91.1041 and it varies in different sub-
regions.

(a) Sample Locations

(b) Reconstructed CGM

Figure 9: The AMSE of channel prediction within small regions.

##  VI Conclusion

This paper investigates the CGM construction and utilization problems towards
environment-aware communications. For both the random and grid distribution of
data collection locations, we derive the AMSE of channel prediction as
functions of the data collection density and the number of data points used
for online prediction. To model the spatial variation of the wireless
environment, we divide the area into small regions and estimate the local
channel modeling parameters based on data points within the region. The
estimation errors of the path loss parameters are derived as functions of the
number of samples within the region. In general, the channel prediction error
reduces with the sample density and the number of measurements used for online
channel prediction, so does the AMSE. Some important results from this study
are summarized below:

  * •

Given the channel modeling parameters, the number of measurements need to be
used for online channel estimation is far below the total number of samples
within the shadowing distance, i.e., a small $k$ is sufficient in most of the
scenarios.

  * •

When the correlation is strong, increasing the sample density within the
region does not necessarily lead to more accurate estimation of the channel
parameters, but it improves the channel prediction performance.

  * •

For small $k$ and large $\beta$, a larger parameter estimation error may not
lead to larger channel prediction error since it is unnecessary to distinguish
the correlated shadowing and path loss intercept.

Through this study, we have noticed that the proper region division is
essential for model-based channel prediction. In the future, the physical map
assisted region division will be considered. Besides, the data driven CGM
construction is also promising in the complex urban environment, since it does
not rely on the accuracy of channel model.

##  Appendix A Proof of Lemma 1

Since the MSE of the estimation is only related with the distance between the
target location $\mathbf{q}$ and its nearest data point, we can obtain the
average MSE by taking the average with respect to the distribution of
$d_{\min}$ in (10), which renders

$\displaystyle\mathbb{E}[\xi_{\mathrm{dB}}(\mathbf{q})|_{(k=1)}]$
$\displaystyle=\alpha+\sigma^{2}-\frac{\alpha^{2}}{\alpha+\sigma^{2}}\int_{0}^{\infty}e^{-\frac{2x}{\beta}}2\pi\lambda
xe^{-\pi\lambda x^{2}}dx$
$\displaystyle=\alpha+\sigma^{2}-\frac{2\pi\lambda\alpha^{2}}{\alpha+\sigma^{2}}\underbrace{\int_{0}^{\infty}xe^{-\pi\lambda
x^{2}-\frac{2x}{\beta}}dx}_{\Xi(\lambda)}.$ (33)

To further simplify the above equation, we have

$\displaystyle\Xi(\lambda)=\int_{0}^{\infty}x\exp\left(-\pi\lambda\left(x+\frac{1}{\pi\lambda\beta}\right)^{2}+\frac{1}{\pi\lambda\beta^{2}}\right)dx$
$\displaystyle=\exp\left(\frac{1}{\pi\lambda\beta^{2}}\right)\left(\int_{\frac{1}{\pi\lambda\beta}}^{\infty}xe^{-\pi\lambda
x^{2}}dx-\frac{1}{\pi\lambda\beta}\int_{\frac{1}{\pi\lambda\beta}}^{\infty}e^{-\pi\lambda
x^{2}}dx\right)$
$\displaystyle\overset{(a)}{=}\frac{1}{2\pi\lambda}\left(1-\frac{1}{\beta\sqrt{\lambda}}\exp\left(\frac{1}{\pi\lambda\beta^{2}}\right)\right)\left(1-\Phi\left(\frac{1}{\beta\sqrt{\pi\lambda}}\right)\right)$
(34)

where $(a)$ follows from the integration formulas in [22].

##  Appendix B Derivation of $d_{\mathrm{min}}$ Distribution in Grid CGM

As shown in Fig. 10(a), the middle data point is the closest neighbor for all
the target locations within the red box. Hence, finding the distribution of
$d_{\min}$ in the grid CGM, i.e., $P_{g}(x)=\Pr(d_{\min}=x)$, is equivalent to
finding the probability that a random location within the box has a distance
$x$ to the center.

(a)

(b)

Figure 10: The distribution of $d_{\min}$ in Grid CGM.

From Fig. 10(b), for $x\leq\frac{d}{2}$, the random location has distance $x$
with the center if it falls into the ring with inner and outer radius $x$ and
$x+\epsilon$, respectively, where $\epsilon\rightarrow 0$. Hence, we have

$\displaystyle P_{g}(x)=\lim_{\epsilon\rightarrow
0}\frac{\pi(x+\epsilon)^{2}-\pi x^{2}}{\epsilon d^{2}}=\frac{2\pi x}{d^{2}}.$
(35)

Similarly, for $\frac{d}{2}<x<\frac{\sqrt{2}d}{2}$, the random location has
distance $x$ from the center if it falls into the overlapped area of the ring
from $x$ and $x+\epsilon$ and the red square. Hence, we have

$\displaystyle P_{g}(x)$ $\displaystyle=\lim_{\epsilon\rightarrow
0}\frac{4x\epsilon\left(\frac{\pi}{2}-2\arccos(\frac{d}{2x})\right)}{\epsilon
d^{2}}$
$\displaystyle=\frac{4x}{d^{2}}\left(\frac{\pi}{2}-2\arccos\left(\frac{d}{2x}\right)\right).$
(36)

##  Appendix C Proof of Lemma 5

The channel parameter estimation error is given by the diagonal elements of
the error covariance matrix $C_{LS}$ in (21). Deriving the explicit expression
of $C_{LS}$ is challenging due to the inverse of a $N\times N$ matrix
$(R_{Q}+\sigma^{2}\mathbf{I}_{N\times N})$. The shadowing correlation among
the data points introduces the bias error on the path loss parameter
estimations. To get some insights on the parameter estimation error with CGM
construction and utilization, we propose to group the data points that are
within the correlation distance $\beta$ and treat each group as an effective
data point, as shown in Fig. 11(a). The shadowing correlation among the
effective data points shown in Fig. 11(b) can be neglected, i.e., with
$R_{Q}=\alpha\mathbf{I}_{N\times N}$.

(a)

(b)

Figure 11: An illustration diagram for deriving the parameter estimation
error.

Consider the random CGM with density $\lambda$ and grid CGM with separation
$d$. The above procedures will lead to a sparse map by a factor
$\pi\beta^{2}$, i.e., on average the number of measurements grouped together
is

$\displaystyle c=\begin{cases}\max\\{1,\pi\lambda\beta^{2}\\},&\textnormal{
Random CGM}\\\ \max\\{1,\pi\beta^{2}/d^{2}\\},&\textnormal{ Grid CGM}\\\
\end{cases}$ (37)

Since the multipath fading at data collection locations is assumed to be
independent, the grouped data points have an effective multipath variance
$\sigma^{2}/c_{r}$. Now, we can obtain an approximation on the parameter
estimation error by considering the estimation over the effective data points
after grouping, which renders

$\displaystyle
C_{LS}\approx(\alpha+\sigma^{2}/c)(\tilde{H}^{T}\tilde{H})^{-1}$
$\displaystyle=\left(\alpha+\frac{\sigma^{2}}{c}\right)\left[\begin{matrix}\tilde{N}&-10\log_{10}\prod_{i=1}^{\tilde{N}}d_{i}\\\
-10\log_{10}\prod_{i=1}^{\tilde{N}}d_{i}&\sum_{n=1}^{\tilde{N}}(10\log_{10}d_{i})^{2}\end{matrix}\right]^{-1},$
(38)

where $\tilde{H}$ contains the distance information of the effective data
points, $\tilde{N}=N/c$ is the number of effective data points,
$d_{i}=\|\mathbf{q}_{i}\|$ is the distance between the $i$th effective data
location to the BS at the origin. We have assumed that $d_{i}$ is a random
variable uniformly distributed within the range
$[\delta_{\min},\delta_{\max}]$, then its value in dB scale, denoted by
$y_{i}=10\log_{10}d_{i}$, is distributed according to

$\displaystyle
f(y)=\Pr(y_{i}=y)=\frac{\ln(10)}{(\delta_{\max}-\delta_{\min})}10^{\left(\frac{y}{10}-1\right)}.$
(39)

Denote by $\mu=\frac{1}{\tilde{N}}\sum_{i=1}^{\tilde{N}}y_{i}$. When
$\tilde{N}$ is sufficiently large, $\mu$ will approach the expectation of
$y_{i}$, which is derived as

$\displaystyle\mu$
$\displaystyle=\int_{10\log_{10}\delta_{\min}}^{10\log_{10}\delta_{\max}}yf(y)dy$
$\displaystyle=\frac{10\delta_{\max}\log_{10}(\delta_{\max})-10\delta_{\min}\log_{10}(\delta_{\min})}{\delta_{\max}-\delta_{\min}}-\frac{10}{\ln
10}$ (40)

Denoted by $\chi$ the variance of the random variable $y_{i}$, we have

$\displaystyle\chi$
$\displaystyle=\int_{10\log_{10}\delta_{\min}}^{10\log_{10}\delta_{\max}}(y-\mu)^{2}f(y)dy$
$\displaystyle=\frac{100}{(\ln
10)^{2}}-\frac{100\delta_{\max}\delta_{\min}\log_{10}(\delta_{\max}/\delta_{\min})}{(\delta_{\max}-\delta_{\min})^{2}}.$
(41)

When $\tilde{N}$ is sufficiently large, we have
$\sum_{n=1}^{N}(10\log_{10}d_{i})^{2}=\sum_{n=1}^{\tilde{N}}y_{i}^{2}=\tilde{N}(\chi+\mu^{2})$.
Hence the error covariance matrix can be in (38) can be written as

$\displaystyle C_{LS}$
$\displaystyle=(\alpha+\sigma^{2}/c)\left[\begin{matrix}\tilde{N}&-\tilde{N}\mu\\\
-\tilde{N}\mu&\tilde{N}(\chi+\mu^{2})\end{matrix}\right]^{-1}$
$\displaystyle=\frac{\alpha+\sigma^{2}/c}{\tilde{N}\chi}\left[\begin{matrix}\chi+\mu^{2}&\mu\\\
\mu&1\end{matrix}\right]$ (42)

Together with (22), it completes the proof of Lemma 5.

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