We define radio maps as path integrals, similarly to how measurements are defined in computer graphics [23, 24, Section 3.7]. Consider a point source $𝐬$ and a measurement surface $ℳ$, divided into $N_M$ cells $M_i$, for $i=1,\mathrm{\dots },N_M$, each of area $\left|M_i\right|$. For clarity, we ignore diffraction in this subsection. Diffraction is considered in Section 4.2.

In the propagation model implemented by Sionna RT, diffuse reflections spawn new ray tubes, whereas specular reflections and refractions merely deviate incident ray tubes, as illustrated in Figure 26.

Ignoring diffuse reflections for now, $𝐬$ is the only source of ray tubes, and every path therefore consists of a single ray tube. The radio map value for the $i$-th cell due to specular reflections and refractions is

$$C_i^{(ℛ,𝒯)}=\frac{1}{\left|M_i\right|}{\int }_{M_i}{e}^{(ℛ,𝒯)}(𝐭)𝑑A(𝐭),$$

(39)

where $dA(𝐭)$ is an infinitesimal surface element of $M_i$, and

$$e^{(ℛ,𝒯)}(𝐭)=\sum _{p\in {𝒫}^{(ℛ,𝒯)}(𝐬\to 𝐭)}e(p)𝒱\left(p\right)$$

(40)

is the channel gain at $𝐭$ due to specular reflections and refractions. Here, ${𝒫}^{(ℛ,𝒯)}(𝐬\to 𝐭)$ denotes the countable set of paths $p=(𝐬,{𝐯}^{(1)},\mathrm{\dots },{𝐯}^{(\mathrm{\ell })},𝐭)$ connecting $𝐬$ to $𝐭$ through specular reflections and refractions. The visibility function $𝒱\left(p\right)$ equals $1$ if $p$ is not occluded and $0$ otherwise. The channel gain associated with $p$ is

$$e(p)={\parallel \frac{{\lambda }_0}{4\pi }{\mathrm{𝐓𝐂}}_{\mathrm{T}}({\theta }_{\mathrm{T}},{\phi }_{\mathrm{T}})\parallel }_2^2,$$

(41)

where ${\lambda }_0$ is the free-space wavelength, ${𝐂}_{\mathrm{T}}$ is the two-dimensional complex transmit antenna pattern evaluated at the departure angles $({\theta }_{\mathrm{T}},{\phi }_{\mathrm{T}})$ of $p$, and $𝐓$ is a $2\times 2$ complex matrix modeling the effects of the interactions and free-space propagation along $p$. Unlike in (4), the receive antenna pattern is not applied. Instead, the squared norm of the electric-field amplitude is used, corresponding to an isotropic receive antenna whose polarization is matched to that of the incident wave. Because the squared amplitudes of the individual paths are summed, the radio map is non-coherent and does not capture interference between paths.

We now generalize the definition to diffuse reflections. Let $𝒜$ denote the union of all diffuse reflecting surfaces in the scene. These surfaces can be regarded as a continuum of point sources. As detailed in Appendix A.9, the squared amplitude of the diffuse scattered field is proportional to the area of the impinging ray-tube footprint on the scattering surface. For a path $p$ containing a diffuse reflection at ${𝐯}^{(𝒮)}\in 𝒜$, its channel gain can therefore be written as

$$e^{(𝒮)}(p)={\tilde{e}}^{(𝒮)}(p)dA({𝐯}^{(𝒮)})$$

(42)

where ${\tilde{e}}^{(𝒮)}(p)$ is the channel gain normalized by the area of the impinging ray-tube footprint. A path containing one diffuse reflection consists of two ray tubes: one originating at $𝐬$ and one originating at ${𝐯}^{(𝒮)}$. The path may additionally contain specular reflections and refractions before or after ${𝐯}^{(𝒮)}$. The contribution of paths containing exactly one diffuse reflection is

$$C_i^{(𝒮,1)}=\frac{1}{\left|M_i\right|}{\int }_{M_i}{\int }_{𝒜}{\tilde{e}}^{(𝒮,1)}({𝐯}^{(𝒮)},𝐭)𝑑A({𝐯}^{(𝒮)})𝑑A(𝐭)$$

(43)

where

$${\tilde{e}}^{(𝒮,1)}({𝐯}^{(𝒮)},𝐭)=\sum _{p\in {𝒫}^{(𝒮,1)}(𝐬\to {𝐯}^{(𝒮)}\to 𝐭)}{\tilde{e}}^{(𝒮)}(p)𝒱\left(p\right).$$

(44)

The diffuse-reflection point ${𝐯}^{(𝒮)}$ is held fixed when defining ${𝒫}^{(𝒮,1)}(𝐬\to {𝐯}^{(𝒮)}\to 𝐭)$. Consequently, this path set is countable, although the subsequent integration over ${𝐯}^{(𝒮)}\in 𝒜$ is continuous.

For an arbitrary number $n_{𝒮}$ of diffuse reflections, define

$${𝒜}^{n_{𝒮}}=\underset{n_{𝒮}\text{ times}}{\underbrace{𝒜\times \mathrm{\cdots }\times 𝒜}}$$

(45)

as the set of ordered sequences ${𝐯}_{𝒮}^{(n_{𝒮})}=({𝐯}^{(𝒮,1)},\mathrm{\dots },{𝐯}^{(𝒮,n_{𝒮})})$ of diffuse-reflection points, and define the product surface element

$$d{A}^{n_{𝒮}}\left({𝐯}_{𝒮}^{(n_{𝒮})}\right)=\prod _{i=1}^{n_{𝒮}}dA\left({𝐯}^{(𝒮,i)}\right).$$

(46)

Repeated application of (42) gives

$$e^{(𝒮,n_{𝒮})}(p)={\tilde{e}}^{(𝒮,n_{𝒮})}(p)d{A}^{n_{𝒮}}\left({𝐯}_{𝒮}^{(n_{𝒮})}\right),$$

(47)

where for fixed diffuse-reflection points ${𝐯}_{𝒮}^{(n_{𝒮})}$, we define

$${\tilde{e}}^{(𝒮,n_{𝒮})}({𝐯}_{𝒮}^{(n_{𝒮})},𝐭)=\sum _{p\in {𝒫}^{(𝒮,n_{𝒮})}\left(𝐬\to {𝐯}^{(𝒮,1)}\to \mathrm{\cdots }\to {𝐯}^{(𝒮,n_{𝒮})}\to 𝐭\right)}{\tilde{e}}^{(𝒮,n_{𝒮})}(p)𝒱\left(p\right).$$

(48)

As in the single-reflection case, the path set in this expression is countable because all diffuse-reflection points are held fixed. The radio map value is obtained by summing over all possible numbers of diffuse reflections:

$$C_i=\frac{1}{\left|M_i\right|}{\int }_{M_i}\sum _{n_{𝒮}=0}^{\mathrm{\infty }}{\int }_{{𝒜}^{n_{𝒮}}}{\tilde{e}}^{(𝒮,n_{𝒮})}({𝐯}_{𝒮}^{(n_{𝒮})},𝐭)𝑑A^{n_{𝒮}}\left({𝐯}_{𝒮}^{(n_{𝒮})}\right)𝑑A(𝐭).$$

(49)

For $n_{𝒮}=0$, the inner surface integral is omitted and ${\tilde{e}}^{(𝒮,0)}(𝐭)=e^{(ℛ,𝒯)}(𝐭)$, so that the corresponding term is precisely (39).

SBR is not applicable to paths that include diffraction: as explained in Section 3.1, the probability of a ray intersecting an edge is zero. To address this, Sionna RT computes the radio map in two steps, as illustrated in Figure 27. First, the radio map is computed for all interaction types except diffraction as described previously. Wedges near the source are detected and stored during this step. Second, only diffracted paths are computed. Currently, Sionna RT supports only first-order diffraction for radio map computation, i.e., paths connecting the source to the measurement surface through a single diffracting wedge and no other interactions with the scene. The two radio maps are then summed to obtain the final result. The following describes the computation of radio map due to diffraction.

For clarity, we consider diffraction on a single edge $ℰ$, defined by an origin $𝐨$, an edge vector $\widehat{𝐞}$, and length $L_{ℰ}$. Sionna RT processes all such wedges found near the source in parallel. Let $𝐯\in ℰ$ be a diffraction point, which is parametrized by the distance along the edge $x$ and the Keller cone azimuth $\varphi$, i.e., $𝐯=𝐨+x\widehat{𝐞}$. Then for a measurement point $𝐭\in M_i$, define:

$$w(𝐭,𝐯)=\underset{\cos {\beta }_0^{\prime }}{\underbrace{\frac{{\left(𝐯-𝐬\right)}^{𝖳}\widehat{𝐞}}{{\parallel 𝐯-𝐬\parallel }_2}}}-\underset{\cos {\beta }_0}{\underbrace{\frac{{\left(𝐭-𝐯\right)}^{𝖳}\widehat{𝐞}}{{\parallel 𝐭-𝐯\parallel }_2}}}$$

(56)

where ${\beta }_0^{\prime }$ and ${\beta }_0$ are the angles between the edge and the incident and diffracted wave, respectively (see Figure 5(b)). The Keller cone condition is $w(𝐭,𝐯)=0$, which can be shown to be satisfied only for a single diffraction point on the edge, denoted by ${𝐯}_{𝒟}(𝐭)$. For a single diffraction wedge, as the diffraction point is unique, the radio map definition is similar to (39):

$$C_i^{(𝒟)}=\frac{1}{\left|M_i\right|}{\int }_{M_i}{e}^{(𝒟)}\left(p\right)𝒱\left(p\right)𝑑A(𝐭)$$

(57)

where $p=(𝐬,{𝐯}_{𝒟}(𝐭),𝐭)$ is the path from the source to the measurement point $𝐭$ via the diffraction point ${𝐯}_{𝒟}(𝐭)$.

Direct integration over the measurement surface is inefficient, as it would require connecting surface points to the source (for example, by placing targets within each cell and running the path solver). To overcome this, consider an orthogonal basis $({\widehat{𝐭}}_0,{\widehat{𝐧}}_0,\widehat{𝐞})$, where ${\widehat{𝐧}}_0$ is the normal to a wedge face and ${\widehat{𝐭}}_0\times {\widehat{𝐧}}_0=\widehat{𝐞}$, as illustrated in Figure 28. For a given angle of incidence ${\beta }_0^{\prime }$, any point on $M_i$ that can be reached by a diffracted ray can be written as:

$$𝐭=\underset{{𝐯}_{𝒟}(𝐭)}{\underbrace{𝐨+x\widehat{𝐞}}}+\gamma \underset{{\widehat{𝐤}}_s}{\underbrace{\left(\sin {\beta }_0^{\prime }\cos \varphi {\widehat{𝐭}}_0+\sin {\beta }_0^{\prime }\sin \varphi {\widehat{𝐧}}_0+\cos {\beta }_0^{\prime }\widehat{𝐞}\right)}}.$$

(58)

Here, ${\widehat{𝐤}}_s$ is the direction of the diffracted ray (with $\varphi$ the Keller cone azimuth), and $\gamma >0$ is the distance for which the ray from ${𝐯}_{𝒟}(𝐭)$ in direction ${\widehat{𝐤}}_s$ intersects the measurement cell $M_i$ (see Figure 28). Since a ray and a plane intersect at most once, $𝐭$ is uniquely determined by $x$ and $\varphi$, and thus $𝐭(x,\varphi )$ forms a reparametrization of the subset of $M_i$ that can be reached by a diffracted ray when the angle of incidence is ${\beta }_0^{\prime }$.

With this reparametrization, (57) becomes:

$$C_i^{(𝒟)}=\frac{1}{\left|M_i\right|}{\int }_0^{L_{ℰ}}{\int }_0^{n\pi }e\left(p\right)𝒱\left(p\right){\parallel \frac{\partial 𝐭}{\partial x}\times \frac{\partial 𝐭}{\partial \varphi }\parallel }_2𝑑x𝑑\varphi .$$

(59)

To estimate (59), $N_S^{\prime }$ samples are drawn on the wedge: $x$ uniformly in $[0,L_{ℰ}]$ and $\varphi$ uniformly in $[0,n\pi ]$. This yields the Monte Carlo estimator:

$${\widehat{C}}_i^{(𝒟)}=\frac{1}{\left|M_i\right|}\frac{L_{ℰ}n\pi }{N_S^{\prime }}\sum _{n=1}^{N_S^{\prime }}e\left(p_n\right)𝕀\left(p_n\right){\parallel \frac{\partial 𝐭}{\partial x}\times \frac{\partial 𝐭}{\partial \varphi }\parallel }_2$$

(60)

where $p_n$ is the $n$-th path, $𝕀\left(p_n\right)$ is 1 if $p_n$ intersects the measurement surface and is not occluded by other scatterers and 0 otherwise, and the weighting factor ${\parallel \frac{\partial 𝐭}{\partial x}\times \frac{\partial 𝐭}{\partial \varphi }\parallel }_2$ is computed as described in Appendix D.

The proposed definition of radio maps involves a non-coherent summation of path gains (41). As a result, it does not capture fast fading effects, which result from the constructive and destructive interference of multiple paths. Figure 29 demonstrates this by comparing three scenarios: a radio map generated using the radio map solver, a radio map created by non-coherently summing the path coefficients (4) calculated with the path solver (Section 3) by placing a single target at the center of each measurement cell, and a radio map produced by coherently summing the path coefficients (4) calculated with the path solver, also by placing a single target at the center of each measurement cell. The radio map obtained using the radio map solver aligns with the non-coherent summation of path coefficients calculated using the path solver, as anticipated. However, it does not account for fast fading effects, which require the coherent summation of path coefficients. Also, because the diffracted contributions are accumulated non-coherently with the specular and LoS contributions, field-level continuity at incident and reflection shadow boundaries is not preserved.

The Sionna RT radio map solver supports antenna arrays of any size at the transmitter and allows for user-defined precoding vectors. However, it only supports synthetic arrays. In this setup, paths are generated from a single source located at the center of the array, and the channel response for each individual antenna is calculated by applying the appropriate phase shifts synthetically. This approximation requires plan wave approximation, which is valid only when $D^2\ll R{\lambda }_0$, where $D$ is the array size and $R$ is the distance from the array center to the first interaction point or measurement plane.

Using synthetic arrays enables efficient scaling with respect to the number of antennas. To see this, let $N_A^{\text{tx}}$ denote the number of antennas at the transmitter, and let ${𝐮}_P\in {ℂ}^{N_A^{\text{tx}}}$ be the precoding vector. For a given path connecting the source to the measurement plane, let $𝐄\in {ℂ}^2$ represent the electric field observed at the measurement plane. By employing a synthetic array, the corresponding electric fields for each antenna element are obtained by applying the array response vector (36), denoted as ${𝐮}_A^{\text{tx}}$, leading to

$${𝐄}_A=𝐄{\left({𝐮}_A^{\text{tx}}\right)}^{𝖳}.$$

(61)

Here, ${𝐄}_A$ is a matrix of size $2\times N_A^{\text{tx}}$, where each column represents the electric fields radiated by each transmit antenna. The contribution of this path to the radio map is given by

$${\parallel {𝐄}_A{𝐮}_P\parallel }_2^2.$$

(62)

This expression allows the path’s contribution to the radio map to be represented as

$${\parallel {𝐄}_A{𝐮}_P\parallel }_2^2={\parallel \left(𝐄{\left({𝐮}_A^{\text{tx}}\right)}^{𝖳}{𝐮}_P\right)\parallel }_2^2={\parallel 𝐄\alpha \parallel }_2^2$$

(63)

where $\alpha ={\left({𝐮}_A^{\text{tx}}\right)}^{𝖳}{𝐮}_P$ is a scalar, which depends on the departure direction of the path from the transmitter and thus varies for paths originating from the source. By precomputing $\alpha$ for each path, the effect of the transmitter array can be efficiently simulated by multiplying the electric field of each path by $\alpha$ whenever the path intersects the measurement surface. This operation does not depend on the size of the array.

Figure 30(b) illustrates the computation time needed to generate a radio map for a single source as the number of samples $N_S$ and the number of transmit antennas $N_A^{\text{tx}}$ are varied. The setup, depicted in Figure 30(a), employs the “simple street canyon” scene built into Sionna RT. The red ball indicates the source, positioned 20 meters above the ground. The radio map in Figure 30(a) is calculated using ${10}^8$ samples and a linear array of 512 transmit antennas. Notably, the radio map solver can compute radio maps with ${10}^9$ samples on a single NVIDIA RTX 4090 GPU. The computation time increases with both the number of samples and antennas, exceeding one second only for the most demanding computations. It is important to note that while the computation time increases with the number of antennas due to the precomputation of the array weighting factor $\alpha$, it does not increase the memory usage of the radio map solver.