In this section, we talk about how to generate the 2D correlated random shadowing based on the MR-FDPF model.

As is known, when expressed in dB, the instantaneous path loss can be expressed as the sum of the mean path loss, the shadow fading, and the small-scale fading as follows

$$ PL(d) = L(d) + {X_{\sigma}} + F $$

((4))

where *P*
*L*(*d*), *L*(*d*), *X*
_{
σ
}, and *F* denote the instantaneous path loss, the mean path loss, the shadow fading, and the small-scale fading, respectively. The mean path loss *L*(*d*) and the shadow fading *X*
_{
σ
} characterize the signal variations over large distances, so they are usually called the large-scale propagation characteristics. On the contrary, the *F* is called the small-scale fading since it characterizes the rapid signal fluctuations over very short distances, e.g., over several wavelengths.

### 3.1 Extraction of the 2D deterministic shadowing from the MR-FDPF model

In (4), the term *L*(*d*) is considered to be deterministic and thus it can be described in a deterministic manner. Typically, the mean path loss *L*(*d*) is log dependent on the Tx-Rx separation distance *d* as follows

$$ L(d) = {L_{0}} + 10n \cdot {\log_{10}}(d) $$

((5))

where *L*
_{0} is a constant which accounts for system losses and *n* is the path loss exponent depending on the specific propagation environment. For instance, *n*=2 for the free space propagation. The mean path loss is the main factor which determines the coverage area of a transmitter.

The shadow fading *X*
_{
σ
} is normally a Gaussian distributed random variable (in dB) with zero mean and standard deviation *σ*
_{
X
} [2].

At last, the small-scale fading *F* in linear scale is typically either a Rayleigh random variable (for NLOS propagation) or a Rician random variable (for LOS propagation). However, for real propagation scenarios, it is sometimes very difficult to tell whether it is a pure NLOS or a LOS propagation. Thus, here we would like to model the small-scale fading by the Nakagami-*m* fading which includes the Rayleigh fading and Rice fading as special cases [24]. Moreover, recently the Nakagami-*m* fading has received more and more attention because it gives the best fit to many measurement data, such as land-mobile and indoor-mobile multipath propagation [2, 25].

As stated above, the small-scale fading represents the rapid signal fluctuations over short distances, so they can be removed by averaging over local areas. After removing the small-scale fading, we obtain the local mean path loss. The local mean path loss includes the mean path loss and the shadow fading as shown in Fig. 4. Since we already know that the mean path loss is log dependent on the Tx-Rx separation distance *d* according to (5), we can obtain the mean path loss by using the Matlab curve fitting tool. And finally, the shadow fading can be easily obtained by subtracting the mean path loss from the local mean path loss. The above procedures can be applied to the simulation results of the MR-FDPF model in order to obtain the large-scale propagation characteristics, which is one of our previous works published in [26].

Since the simulation results provided by the MR-FDPF model are deterministic, the extracted shadow fading from the MR-FDPF model is also deterministic.

### 3.2 Generation of the 2D correlated random shadowing

Since the MR-FDPF model is a 2D deterministic radio propagation model and it takes the specific propagation environment into account, the extracted shadow fading from the MR-FDPF model is a 2D correlated deterministic shadow fading. However, for real systems, shadow fading should only be modeled statistically due to the difficulty in modeling the randomly moving people and moving objects present in real environments. Thus, a random shadow fading is considered to be more realistic and more accurate than a deterministic shadow fading. In fact, in modern mobile telecommunications, shadow fading has to be modeled to be 2D correlated random shadowing since shadow fading may present both cross-correlation and spatial correlation due to the presence of similar obstacles during the propagation. For instance, nearby receivers are probable to experience very similar shadow fadings, i.e., their shadow fadings are correlated.

Now, we detail how to generate the 2D correlated random shadowing model based on the 2D correlated deterministic shadow fading provided by the MR-FDPF model. It is based on the method of Fraile et al. in [8]. Assume that there are totally *I* transmitters in the simulated scenario. The MR-FDPF model can provide a deterministic shadow fading map for each of these transmitters, e.g., shadow fading map *Φ*
_{
i
} for transmitter *i*. For each point (*x*,*y*) (i.e., where the virtual receiver is) in the simulated scenario, the shadow fading experienced by signals transmitted from *I* transmitters can be modeled by *I*+1 independent Gaussian random variables {*G*
_{0},*G*
_{1}⋯*G*
_{
I
}} which have zero mean and the same standard deviation *σ*
_{
X
}. To make sure that the generated shadow fading exhibits the same cross-correlation as presented in real systems, the shadow fading can be generated as follows:

$$ X_{\sigma}^{ij} = \sqrt {{\rho_{ij}}} \cdot {G_{0}} + \sqrt {1 - {\rho_{ij}}} \cdot {G_{i}} $$

((6))

where \(X_{\sigma }^{ij}\) is the generated shadow fading for transmitter *i* while taking into account its cross-correlation from transmitter *j*, with *i*,*j*∈{1,2⋯*I*}.

From the above, it is easy to know that

$$ {\mathbb{E}}\left({X_{\sigma}^{ij}} \right) = 0 $$

((7))

$$ \mathbb{S}\left({X_{\sigma}^{ij}} \right) = {\sigma_{X}} $$

((8))

where \({\mathbb {E}}\left (\cdot \right)\) and \(\mathbb {S}\left (\cdot \right)\) denote the expectation and the standard deviation. Thus, it guarantees that the generated shadow fading is still a Gaussian random variable with zero mean and standard deviation equal to *σ*
_{
X
}. Meanwhile, it also guarantees that the cross-correlation of shadow fadings between any pair of transmitters (*i*,*j*) is equal to

$$ {R_{ij}}\left(0 \right) = \frac{{\mathbb{E}\left[ {X_{\sigma}^{ij} \cdot X_{\sigma}^{ji}} \right]}}{{\sqrt {\mathbb{E}\left[ {{{\left({X_{\sigma}^{ij}} \right)}^{2}}} \right] \cdot \mathbb{E}\left[ {{{\left({X_{\sigma}^{ji}} \right)}^{2}}} \right]} }} = {\rho_{ij}} $$

((9))

Thus, in this approach, the common component *G*
_{0} is used to model the receiver-position-dependent cross-correlation of shadow fadings from different transmitters.

Since the same procedure above can be repeated at each point (*x*,*y*) in the simulated scenario to generate the cross-correlated shadow fading, the generated 2D cross-correlated shadow fading map can be rewritten as:

$$ {}X_{\sigma}^{ij}\left({x,y} \right) = \sqrt {{\rho_{ij}}\left({x,y} \right)} \cdot {G_{0}}\left({x,y} \right) + \sqrt {1 - {\rho_{ij}}\left({x,y} \right)} \cdot {G_{i}}\left({x,y} \right) $$

((10))

Although the above generated 2D shadow fadings are cross-correlated, there is not any spatial correlation inside (the correlation of the shadow fadings is zero when their positions are different). In order to generate the spatial correlation, a 2D filter can be applied to the 2D cross-correlated shadow fading map.

The impulse response of the 2D filter is denoted by *h*(*x*,*y*). The input of the 2D filter is supposed to be the above generated 2D cross-correlated shadow fading map, i.e., \(a(x,y) = X_{\sigma }^{ij}\left ({x,y} \right)\). The output *b*(*x*,*y*) is the expected shadow fading map which presents both the cross-correlation and the spatial correlation. As we know, if the impulse response *h*(*x*,*y*) of the 2D filter is known, the output *b*(*x*,*y*) can be easily obtained by a 2D convolution between the input *a*(*x*,*y*) and the impulse response *h*(*x*,*y*). Thus, the main task we should do here is to try to obtain the impulse response *h*(*x*,*y*) of the 2D filter.

According to the theory of random processes and linear systems, the power spectral density of *b*(*x*,*y*) is related to the power spectral density of *a*(*x*,*y*) according to

$$ {S_{bb}}({f_{x}},{f_{y}}) = {S_{aa}}({f_{x}},{f_{y}}) \cdot {\left| {H({f_{x}},{f_{y}})} \right|^{2}} $$

((11))

where *S*
_{
bb
}(*f*
_{
x
},*f*
_{
y
}), *S*
_{
aa
}(*f*
_{
x
},*f*
_{
y
}) are the power spectral density of *b*(*x*,*y*) and that of *a*(*x*,*y*), respectively. *H*(*f*
_{
x
},*f*
_{
y
}) is the system transfer function of the 2D filter. Since the input \(a(x,y) = X_{\sigma }^{ij}\left ({x,y} \right)\) is a white shadow fading map, its autocorrelation function *R*
_{
aa
}(*Δ*
*x*,*Δ*
*y*) is non-zero only at the position (0,0). Thus, its power spectral density is flat

$$ {S_{aa}}\left({{f_{x}},{f_{y}}} \right) = {\sigma_{a}^{2}} $$

((12))

After a simple mathematical derivation, we can find \({\sigma _{a}^{2}} = {\sigma _{X}^{2}}\). Therefore, the system transfer function can be obtained by

$$ H\left({{f_{x}},{f_{y}}} \right) = \sqrt {{{{S_{bb}}\left({{f_{x}},{f_{y}}} \right)} \left/\right. {{\sigma_{a}^{2}}}}} $$

((13))

Then, the impulse response *h*(*x*,*y*) can be easily obtained by performing a 2D inverse Fourier transform to the system transfer function.

When provided with the shadow fading map extracted from the MR-FDPF model, the power spectral density *S*
_{
bb
}(*f*
_{
x
},*f*
_{
y
}) can be obtained directly by applying a 2D Fourier transform to the autocorrelation function *R*
_{
bb
}(*Δ*
*x*,*Δ*
*y*). Here, the deterministic shadow fading map provided by the MR-FDPF model is considered to be one realization of the random process of *b*(*x*,*y*).