The considered propagation scenario is shown in Figure 1. The indoor propagation environment is defined by orthonormal vectors **x**, **y**, and **z** of a rectangular coordinate system. The human body is modeled as a finite lossy dielectric cylinder and is walking in the direction ABC or CBA crossing the line-of-sight (LOS) path between the transmitter (Tx) and the receiver (Rx) at position B. The positions of the transmitting and receiving antennas are fixed.

In addition to the reflection of the signal by the floor, ceiling and walls of the indoor environment (reflections from furniture and other objects are not considered), the signal is also scattered by moving human body at, e.g., position A and C (see Figure 1). As the person approaches position B, the signal is subject to both scattering and diffraction from the top and side of the human body. Note that the strength of the scattered and diffracted signals are position dependent and change as the person walks though the propagation environment. Furthermore, the random movement of people create time varying channel conditions. Characterization of the Doppler spectra is thus important for the determination of the time variance of the wireless channel. Other important parameters are the shadowing and inter-shadowing intervals which are needed when simulating the crossing of the LOS path by multiple bodies. In subsequent sections, we discuss each of the aforementioned propagation mechanisms for signal effect by moving human bodies.

### 2.1. Reflection and scattering

The strength of the signal reflected by the floor, ceiling and walls of the indoor environment depend on the link configuration and material properties of the reflecting surfaces. The reflection coefficient is given by [8]

{R}_{N}=\frac{\text{cos}{\theta}_{r}-\sqrt{\eta -{\text{sin}}^{2}{\theta}_{r}}}{\text{cos}{\theta}_{r}+\sqrt{\eta -{\text{sin}}^{2}{\theta}_{r}}}

(1)

{R}_{P}=\frac{\text{cos}{\theta}_{r}-\sqrt{\left(\eta -{\text{sin}}^{2}{\theta}_{r}\right)/{\eta}^{2}}}{\text{cos}{\theta}_{r}+\sqrt{\left(\eta -{\text{sin}}^{2}{\theta}_{r}\right)/{\eta}^{2}}}

(2)

where *R*_{
N
}and *R*_{
P
}are the reflection coefficients when the electric field component is normal and parallel to the reflection plane, respectively. Parameter *θ*_{
r
}is the angle between the incident ray and the normal to the reflecting surface. Parameter *η* is the complex permittivity of the reflecting material. The received reflected signal power can then be calculated using

P=\frac{{P}_{t}{G}_{t}{G}_{r}}{{L}_{p}{L}_{r}}

(3)

where *P*_{
t
}is the transmitted power, *G*_{
t
}and *G*_{
r
}are transmitting and receiving antenna gains, respectively. Parameter *L*_{
p
}is the path lose and *L*_{
r
}= |*R*|^{2} is the reflection lose with *R* = *R*_{
P
}, *R*_{
N
}.

The signal is also scattered by people moving in the propagation environment. An illustration of the human body model adopted in this study is shown in Figure 2. The human body is modeled as a vertically oriented cylindrical volume in a rectangular coordinate systems defined by the orthonormal vectors **x**, **y**, and **z**. The radios *R* and the height *H* describe the dimension of the volume. Incident on the human body is a linearly polarized wave with wavenumber *k* = 2*π*/*λ, λ* being the wavelength, and direction of propagation **k**_{
i
}. The transmitter is assumed to be located far away from the human body, so that the incident field amplitude can be approximated as being constant over the human body. Similarly, the scattered wave is received at a point located far away from the human body, in the direction **k**_{
s
}relative to the center of the human body. The propagation vectors of the incident **k**_{
i
}and scattered **k**_{
s
}fields are given by (see Figure 2)

{\mathbf{k}}_{i}=-\text{sin}{\theta}_{i}\text{cos}{\varphi}_{i}\mathbf{z}-\text{sin}{\theta}_{i}\text{sin}{\varphi}_{i}\mathbf{x}-\text{cos}{\theta}_{i}\mathbf{y}

(4)

{\mathbf{k}}_{s}=\text{sin}{\theta}_{s}\text{cos}{\varphi}_{s}\mathbf{z}-\text{sin}{\theta}_{s}\text{sin}{\varphi}_{s}\mathbf{x}-\text{cos}{\theta}_{s}\mathbf{y}

(5)

where angles *θ*_{
i
}and *ϕ*_{
i
}are the elevation and azimuth directions of the incident field. Angles *θ*_{
s
}and *ϕ*_{
s
}are the elevation and azimuth directions of the scattered field.

The scattering of an arbitrarily polarized wave incident on the cylinder can be described in terms of the scattering amplitude tensor, *F*(**k**_{
s
},**k**_{
i
}) [9]. If the field inside the cylinder is estimated as the field inside a similar, but infinite cylinder, then *F*(**k**_{
s
},**k**_{
i
}) can be expressed as in [9, Equation (47)]. The scattering cross-section can then be calculated using

{\sigma}_{\mathsf{\text{cs}}}=4\pi {\left|F\left({\mathbf{k}}_{s},{\mathbf{k}}_{i}\right)\right|}^{2}

(6)

For the propagation scenario shown in Figure 1, at each location of the human body, the incident and scattering angles can be calculated using geometrical relations, and these can be used further to determine the scattered signal utilizing the scattering cross-section defined in Equation (6).

The multipath (fast fading) component of the Rice channel in indoor environment might then be described by the sum of the standard deviations of the reflected and scattered fields, *σ*_{reflected} + *σ*_{scattered}. Together with the direct signal component (discussed in Section 2.2), these are used in Section 2.5 to generate the signal fading caused by moving human body.

### 2.2. Diffraction

As the person approaches position B in Figure 1, the signal is also subject to diffraction from the top and side of the human body. The diffracted signal can be calculated using Kirchhoff diffraction equation [10, 11] which gives the relationship between the aperture diffracted electric field at an observation point, *P*, and the free-space electric field

\frac{E\left(P\right)}{{E}_{\mathsf{\text{o}}}\left(P\right)}=\frac{j}{2}\int {\int}_{\Sigma \left(u,v\right)}\text{exp}\left(-j\frac{\pi}{2}\left({u}^{2}+{v}^{2}\right)\right)\phantom{\rule{2.77695pt}{0ex}}du\phantom{\rule{2.77695pt}{0ex}}dv

(7)

with

\left[\begin{array}{c}u\hfill \\ v\hfill \end{array}\right]=\frac{\sqrt{2}}{{R}_{1}}\left[\begin{array}{c}{x}_{\mathsf{\text{o}}}\hfill \\ {y}_{\mathsf{\text{o}}}\hfill \end{array}\right]

(8)

where *O* is the intersection point (with Cartesian coordinates *x*_{o} and *y*_{o}) of the Tx-Rx straight line with the obstacle (see Figure 3). Parameter Σ(*u,v*) is the surface of the aperture, and *R*_{1} is the radius of the first Fresnel zone given by

{R}_{1}=\sqrt{\frac{\lambda ab}{a+b}}

(9)

where *a* and *b* are the distances from the transmitter and the receiver to the aperture plane.

The total received diffracted field is then the combination of the three aperture diffracted fields whose surfaces are Σ_{1}, Σ_{2}, and Σ_{3} with assumed infinite depth. Unlike simple ray tracing and Markov process models, the transitions between LOS and non-line-of-sight (NLOS) conditions are not sharp and thus more realistic regarding the empirical results. Using this model, good estimation of the shadow region behind a human body might be obtained. The standard deviation of the diffracted field, *σ*_{diffracted}, is used to describe the direct (slow fading) component of the received signal affected by moving human body. Together with the multipath signal component (discussed in Section 2.1), *σ*_{diffracted} is used in Section 2.5 to generate the signal fading caused by moving human body.

### 2.3. Doppler spectrum

The random movement of people in the propagation environment create time varying channel conditions. Characterization of the Doppler spectra is thus important for the determination of the time variance of the wireless channel. The situation where the antenna is moving in a random environment leads to the classical Jakes spectrum (with bathtub like shape) for scatters uniformly distributed in azimuth [12]. For the case where the antenna is stationary, moving scatterers in the channel such as vehicles (in outdoor environment) or persons (in indoor environment) will lead to a different Doppler spectrum which peaks at 0 Hz and falls off rapidly [13, 14].

For electrically large scatterer such as the human body with scattering pattern peaking in the forward direction, the Doppler spectrum is given by [15]

{S}_{n}\left(f\right)={\left(-1\right)}^{n}{P}_{n-0.5}\left(\frac{{f}^{2}}{2}-1\right)

(10)

where *P*_{n- 0.5}is the *n* th order Legendre function. In [15], a strong correlation was found between the shape of the scattering pattern and the Doppler spectrum. Based on this, a simplified model which mimics the measured spectra caused by randomly moving people is proposed, and is given by [15]

S\left(f\right)=\frac{1}{f+e}

(11)

where *e* is a model constant with value equal to 0.02 chosen to much the measured Doppler spectrum [15].

### 2.4. Shadowing and inter-shadowing events

Knowledge on the shadowing and inter-shadowing events are needed to simulate the multiple crossing of the LOS path by randomly moving people. Usually, an arrival process is described by Poisson distribution [16]. As in [1], the number of shadowing events may thus be modeled using Poisson distribution, expressed as

p\left(N\right)=\frac{{\hat{N}}^{N}}{N!}\text{exp}\left(-\hat{N}\right)

(12)

where \hat{N} is the average number of bodies crossing the LOS path. Equation (12) gives the probability that *N* persons cross the LOS path within a given period of time.

In Poisson process, the Exponential distribution plays an important role in describing the inter-event durations [17]. For a Poisson process with parameter \hat{N}, the time between events are Exponentially distributed, defined as

p\left(T\right)=\hat{N}\text{exp}\left(-\hat{N}T\right)

(13)

where *T* is the time interval to the next shadowing event. Equation (13) gives the probability of *T* time interval for the next crossing of the LOS path. Parameter \hat{N} depends on the type of the environment, i.e., indoor (office, home, airport terminal, etc.) or outdoor (urban, suburban, rural, etc.), and should be characterized using measurements for the considered propagation environment.

### 2.5. The simulation model

The proposed physical-statistical channel model for simulating the signal effect by moving human body is shown in Figure 4. In the model, a complex white Gaussian processes with zero mean and unit variance is filtered by Equation (10) or Equation (11) for Doppler spectrum shaping. The resulting time series are multiplied by the sum of the standard deviations of the reflected and scattered fields, *σ*_{reflected} + *σ*_{scattered} (see Section 2.1) to produce the multipath (fast fading) component of the received signal. Then, the standard deviation of the diffracted field, *σ*_{diffracted} (see Section 2.2) representing the direct (slow fading) signal component, is added to produce the total complex signal envelope of the received signal affected by moving human body.

This results in physical-statistical modeling of the propagation channel where the received signal envelope is Rice distributed with its position dependent parameters calculated using electromagnetic based methods (i.e., reflection, scattering, and diffraction). Note that parameters *σ*_{scattered} and *σ*_{diffracted} are position dependent, and change as the person walks through the propagation environment. Furthermore, Poisson and Exponential distributions are used to generate the shadowing and inter-shadowing events which can be used to simulate the crossing of the LOS component by multiple moving human bodies (see Section 2.4).