Wireless Performance Evaluation of Building Layouts: Closed-Form Computation
of Figures of Merit

It has been predicted that mobile traffic will increase by up to $1000$ times in the next decade, with over $80\%$ of mobile traffic taking place indoors [1]. 6G systems will meet the required mobile traffic through the combined effects of densification of the wireless network through small cells and the large-scale deployment of massive multiple-input multiple-output (MIMO) array antennas [2]. The large number of small cells are anticipated to enhance coverage and spectral efficiency [3, 4, 5, 6, 7, 8, 9], whereas in massive MIMO, array antennas are scaled up to increase reliability and spectral efficiency [11, 12, 13, 10]. On the other hand, wireless communication plays an important role in the realization of the “smart building” paradigm, or more broadly, in the “smart city” vision [14]. Different types of buildings have an intrinsic wireless performance which is independent of how densely small cells are deployed, or the massive MIMO array antenna deployed [15, 16]. The present work is the first step towards providing guidance, to both civil engineers and architects, on incorporating the wireless performance of buildings at the design stage.

Building design and construction has mainly focused on improving safety, visual and thermal comfort, and indoor air quality [17]. In the last decade, energy efficiency became an important metric in building design tools [18]. The operational energy required by a building can be evaluated by taking into account the lighting, the heating, the ventilation and the cooling as well as the provision of hot water [19]. At present, mobile operators are already among the top energy consumers [20], and an increasing trend of power consumption of wireless networks can be observed in different generations of wireless systems [21, 22]. Therefore, the energy saving is worthwhile to be considered in the design of green buildings. However, the indoor power consumption of wireless networks has not been considered.

One important factor impacting the power consumption in small-cell networks is the building-blockage. In [23, 24, 25], the LOS probability is analytically and empirically modelled. In [26], an analytic expression for the coverage rate is provided by approximating the line-of-sight (LOS) region as a fixed LOS ball. In [27], four random wall generation methods to place blockage objects in an indoor scenario were proposed and analyzed, where walls are either distributed randomly, semi-deterministically or heuristically. However, previous evaluations of the blockage effect proposed that buildings form a random process of shapes. The LOS probability was there assumed to be irrelevant for different directions of the link. This model seems to be promising in providing insights on the configuration of a wireless network in a homogeneous indoor scenario, where walls are assumed to be uniformly distributed in a building. However, in specific building designs, whether a wireless link is blocked by structures not only depends on the distance between the transmitter and the receiver, but also on their relative position and direction in the floor plan of the building. That is, the blockage condition of the wireless signal depends on the considered direction. Therefore, the homogeneous assumption is not suitable for evaluating specific building designs.

Rather than random blockages, ray-tracing/ray-launching techniques are employed to evaluate the indoor performance of wireless networks (i.e., for users located indoors) for specific building designs in [28, 29, 30, 31, 32, 33, 34]. However, results attained through this approach are only valid for a specific network deployment, and are not general enough to evaluate the wireless performance of buildings, where all possible positions of transmitters and receivers in the building under design are needed to be evaluated. In this case, ray-tracing/ray-launching techniques become computationally expensive, and therefore it is not practical to evaluate the wireless performance of a building layout in a reasonable time.

Moreover, wireless networks are traditionally designed and deployed when the buildings have been built. Before the networks are deployed, wireless communications engineers have to carry out ray-tracing/ray-launching based simulations for specific deployments of wireless networks. In other words, wireless communications engineers adapt a network to suit its deployment scenarios without modifying any property of the building layout. However, for an existing building with a specific intrinsic wireless performance determined by the building layout, the indoor wireless networks may not achieve an acceptable performance regardless how densely small cells or MIMO array antenna elements are deployed. The wireless performance of building layout has to be taken into account carefully in the building design stage, which means that building designers, both civil engineers and architects, requires guidance on how to predict and evaluate the wireless performance of a building under design.

Taking wireless performance of buildings into account will become indispensable for the design of future smart buildings in which appliances and many other devices and “things” will all be connected wirelessly. To address the challenges posed by the building evaluation in terms of wireless performance, ground-breaking works are needed. We are engaged in the collaborative project Build-Wise [35]. Supported by this project, figures of merits (FoMs) reflecting the intrinsic wireless performance of buildings have been defined considering the impact of blockage, bounce reflections, and waveguide effects introduced by indoor building structures employing the open space scenario as a benchmark for wireless network performance evaluation [15, 16]. The proposed FoMs bridge the gap between building design and wireless communications, laying the foundation to an improved wireless service indoors by optimising building layouts. However, the computation of the FoMs is based on Monte Carlo simulations, which is computationally expensive since an infinite number of transmit elements is assumed in the FoM definition. Therefore, an analytic framework is critically needed for fast computation of the FoMs.

The main contribution of this paper is a mathematical framework to evaluate the wireless performance of buildings. The proposed approach facilitates straightforward and more computationally efficient numerical evaluation of the proposed FoMs. Firstly, two FoMs, i.e., the interference gain (IG) and power gain (PG) are defined to capture the impact of the building layout on the power of interference signals and intended signals, respectively. Secondly, an approach to derive exact FoMs in closed form is proposed for polygonal-shaped rooms. By using this approach, the FoMs can be quickly and exactly computed given building layout tiled by polygonal rooms. Thirdly, the analytic approach is validated through Monte Carlo simulations for typical indoor path gain models. This proposed analytic framework provides a direct and practical quantification of the wireless performance of a building. Finally, the impact of the building layout, e.g., their sizes and the aspect ratios (AR) of rectangular rooms, on wireless performance FoMs is analyzed based on the proposed analytic approach.

The remainder of this paper is organized as follows. Section II summarizes the notation and main assumptions made in this paper. In Section III, two FoMs, i.e., the IG, and the PG are introduced to measure the wireless performance of a building. In Section IV, an approach to compute the closed-form expressions of the proposed FoMs is provided. In Section V, the approach is validated through Monte Carlo simulations. In Section VI, the impact of building layout on FoMs is analyzed. Conclusions are provided in Section VII.

In order to meaningfully evaluate the wireless performance of buildings, the impact of buildings on wireless networks has to be well understood. To help in the analysis we employ the open space scenario with path gain model (2) as a benchmark or reference. Then, the impact of buildings on the wireless networks can be evaluated by comparing the SINR of the signals received indoors with that in the open space scenario.

The expression of the SINR in the open space is given as follows. Using a polar-coordinate system $(R,\theta)$ by choosing the UE as the reference point and an arbitrary angle as the reference 0-angle in the open space, the power of the intended signal is given by the integral of the receive power from all transmit elements with a distance that is less than $R_{\mathrm{O}}$, i.e.,

Whereas, the power of the interference signal is given by

Thus, the SINR in an open space scenario is given by

Then, the SINR of a UE in the considered building is given as follows. For an arbitrary transmit element located at $(R,\theta)$, the transmission from the element to the UE could be either LOS or NLOS. If the transmission is LOS, the received signal is considered as an intended signal if $R<R_{\mathrm{L}}$. Otherwise, if $R\geq R_{\mathrm{L}}$, the received signal is considered as an interference signal. Similarly, if the transmission is NLOS, the intended signal satisfies $R<R_{\mathrm{N}}$, and the interference signal satisfies $R\geq R_{\mathrm{N}}$. For a given $(R,\theta)$, we use a Boolean variable $\mathcal{V}(R,\theta)$ to denote that “the transmission from the transmit element at $(R,\theta)$ to the UE is LOS”. Accordingly, $\lnot\mathcal{V}(R,\theta)$ denotes that “the transmission from the transmit element at $(R,\theta)$ to the UE is NLOS”. If a signal transmitted from $(R,\theta)$ is an intended signal, we have $[\mathcal{V}(R,\theta)\land(R<R_{L})]\lor[\lnot\mathcal{V}(R,\theta)\land(R<R_{N})]$. Then, the power of intended signals in the considered building is given by

where $P_{\mathrm{L}}$ and $P_{\mathrm{N}}$, respectively, denote receive power of intended signals from LOS and NLOS small cells/antenna elements, and can be computed separately since $\mathcal{V}(R,\theta)\land(R<R_{L})$ and $\lnot\mathcal{V}(R,\theta)\land(R<R_{N})$ are exclusive. Similarly, the power of the interference signal is given by

where $I_{\mathrm{L}}$ and $I_{\mathrm{N}}$, respectively, denote the LOS and the NLOS received powers of interference signals. Therefore, we have the expression of the indoor SINR as

Combining (17) and (20), we obtain

and we define two FoMs of a building’s wireless performance as [15, 16]

where $g_{\mathrm{I}}$ and $g_{\mathrm{P}}$ denote the IG and the PG, respectively. The variables $I_{\mathrm{O}}$ and $P_{\mathrm{O}}$ denote the power of interference signals and the power of intended signals in the open space scenario, respectively. In the LOS scenario, $I_{\mathrm{L}}$ and $P_{\mathrm{L}}$ denote the power of interference signals and the power of intended signals, respectively. In the NLOS scenario, $I_{\mathrm{N}}$ and $P_{\mathrm{N}}$ denote the power of interference signals and the power of intended signals from the transmit elements in the floor where the UE located, respectively. $\sigma^{2}$ denotes the power of thermal noise.

The proposed FoMs and their analytic computation approach, will provide civil engineers and architects guidance to evaluate the wireless friendliness of a building layout under their design. If FoMs are lower than expected, an acceptable wireless performance will not be achievable. For communication engineers, the FoM is a performance gain upper bound provided by the buildings under design. To achieve the upper bound, Assumptions 1) and 7) stated in Section II need to be approached.

In our model we consider an infinite number of small cells/antenna elements. Therefore, analytic expressions are comparatively more computationally efficient than performing Monte Carlo simulations to approximately compute the FoMs. Also, analytic results can provide accurate results and lead to a more straightforward comparison and optimization of the building layout. Therefore, we propose next an approach to derive $g_{\mathrm{I}}$ and $g_{\mathrm{P}}$ in closed form for evaluating the wireless performance of buildings.

Next we provide analytic equations for intended and interference signals starting with the open space scenario ($I_{\mathrm{O}}$ and $P_{\mathrm{O}}$) and later for the LOS and NLOS scenarios too (i.e., $I_{\mathrm{L}}$, $P_{\mathrm{L}}$, $I_{\mathrm{N}}$, and $P_{\mathrm{N}}$, respectively).

The parameters of the equations have been defined above in Section II and are omitted here for the sake of compactness.

The computation of $I_{\mathrm{L}}$, $P_{\mathrm{L}}$, $I_{\mathrm{N}}$, and $P_{\mathrm{N}}$ is not straightforward since the indoor scenarios are not homogeneous. Therefore, in order to facilitate their derivation, we propose a toy model (TM) as shown in Fig. 3. In the TM, we consider a circular sector with an infinite radius. A reference (or probing) UE is located at the vertex of the sector. A wall of finite length is modeled as a straight line segment across the sector with endpoints lying on the radii of the circular sector. The distance from the UE to the wall is denoted as $D_{0}$. A polar-coordinate system $(R,\theta)$ is introduced by choosing the UE as the reference point. The perpendicular direction to the wall is assumed to be the reference $0$-angle. $\theta_{\mathrm{l}}$ and $\theta_{\mathrm{r}}$ denote angles of the left and right end points of the wall, respectively. With these definitions, $-\frac{\pi}{2}<\theta_{\mathrm{l}}<\theta_{\mathrm{r}}<\frac{\pi}{2}$.

It is worthwhile to note that the proposed TM of a wall can be used to represent any polygonal-shaped room. Indeed, any room can be modelled by adding several TMs, i.e., one for each wall that can be seen by the reference UE in the room. Some examples are given in Fig. 3. For a specific direction, whether an area is LOS or NLOS is determined by the closest wall.

For a room consisting of $N_{\mathrm{TM}}$ TMs, we then have

For known closed-form expressions of $I_{\mathrm{L},n_{\mathrm{TM}}}$, $P_{\mathrm{L},n_{\mathrm{TM}}}$, $I_{\mathrm{N},n_{\mathrm{TM}}}$, and $P_{\mathrm{N},n_{\mathrm{TM}}}$, it is straightforward to compute $I_{\mathrm{L}}$, $P_{\mathrm{L}}$, $I_{\mathrm{N}}$, and $P_{\mathrm{N}}$ by (30-33). For an arbitrary $n_{\mathrm{TM}}$ (i.e., TM of a wall contribution to the computed signals) we have

where $\Omega_{\mathrm{PL}}$, $\Omega_{\mathrm{IL}}$, $\Omega_{\mathrm{PN}}$, and $\Omega_{\mathrm{IN}}$ denote the areas contributing to the LOS intended signals, the LOS interference signals, the NLOS intended signals, and the NLOS interference signals at the reference UE, respectively. Expressed in polar-coordinates, the straight line equation of the wall is given by $R=\frac{D_{0}}{\cos{\theta}}$. For any infinitesimal area at position $(R,\theta)$, the area is a LOS area if $R<\frac{D_{0}}{\cos{\theta}}.$ Else, the area is NLOS if $R\geq\frac{D_{0}}{\cos{\theta}}.$ Therefore, $\Omega_{\mathrm{IL}}$, $\Omega_{\mathrm{PL}}$, $\Omega_{\mathrm{IN}}$, and $\Omega_{\mathrm{PN}}$ are, respectively, defined as

In a TM with a given set of ($P_{\mathrm{T}}$, $P_{\mathrm{th}}$, $\theta_{\mathrm{r}}$, $\theta_{\mathrm{l}}$, $D_{0}$, $\lambda$), the integrals $P_{\mathrm{L},n_{\mathrm{TM}}}$, $I_{\mathrm{L},n_{\mathrm{TM}}}$, $P_{\mathrm{N},n_{\mathrm{TM}}}$ and $I_{\mathrm{N},n_{\mathrm{TM}}}$ are computed by following Theorems 2-5.

In this section, we compare the results obtained by the derived closed-form expressions and the Monte Carlo simulations. The analytic results used in the following Figs. 5-11 are summarized in TABLE V. In Sections V-VI, we use $h_{\mathrm{T}}=h_{\mathrm{R}}=1.2$ m without further specification.

In Fig. 5, we compare the results obtained by the derived closed-form expressions of $P_{\mathrm{O}}$ and $I_{\mathrm{O}}$, i.e., (26) and (29), with Monte Carlo simulations. The numerical results show an excellent agreement between the analytic, i.e., closed-form, results and the Monte Carlo simulations.

Let’s now consider two generic wireless systems operating at the $1$ GHz and the $28$ GHz bands, respectively. The parameters $P_{\mathrm{L}}$, $I_{\mathrm{L}}$, $P_{\mathrm{N}}$, and $I_{\mathrm{N}}$ are validated in Figs. 7-7, respectively, where analytic $P_{\mathrm{L}}$, $I_{\mathrm{L}}$, $P_{\mathrm{N}}$, and $I_{\mathrm{N}}$ are provided in TABLEs I-IV. The analytic results match Monte Carlo simulations very well. With an increasing $D_{0}$, more transmitted power is considered as LOS power, and therefore $P_{\mathrm{L}}$ and $I_{\mathrm{L}}$ increase, while $P_{\mathrm{N}}$ and $I_{\mathrm{N}}$ decrease.

Following Remark 2, the analytic $g_{\mathrm{I}}$ and $g_{\mathrm{P}}$ of a rectangular room of size $5\ \mathrm{m}\times 10\ \mathrm{m}$ are validated in Figs. 11 and 11 for the 1 GHz and the 28 GHz bands, respectively. Also, $g_{\mathrm{I}}$ and $g_{\mathrm{P}}$ of an L-shaped corner room are validated in Figs. 11 and 11. Floor plans of the considered rooms are given in Fig. 3. Numerical results show that the analytic $g_{\mathrm{I}}$ and $g_{\mathrm{P}}$ agree well with the Monte Carlo simulations.

In the following sections, we use only the analytic results to propose some examples of building wireless performance analysis since the computation of $g_{\mathrm{I}}$ and $g_{\mathrm{P}}$ by Monte Carlo simulation is computationally expensive.

In this section, we give examples of the evaluation of building wireless performance of two reference floor plans: a single rectangular room layout and an office building layout with rectangular rooms.