This section provides a brief overview of importance sampling [15, 16] in the context of ray tracing for the simulation of radio wave propagation. We consider the channel gain $g$ (5). To account for a continuum of paths, such as those arising from diffuse reflections, we generalize the gain from a discrete sum to a path integral:

$$g={\int }_{𝒫}{\left|a(p)\right|}^2𝑑\mu (p).$$

(180)

Here, $𝒫$ represents the high-dimensional space of all possible propagation paths from the transmitter to the receiver. A single element $p\in 𝒫$ denotes a specific path. The term $a(p)$ is the path coefficient associated with that path, and $\mu (p)$ is the measure on the path space $𝒫$. For a more in-depth discussion of path integrals within ray tracing, see [24, Chapter 8].

We aim to estimate the channel gain $g$ using $N_S$ samples, by sampling paths following a distribution $q(p)$, such that $q(p)>0$ if ${\left|a(p)\right|}^2>0$ and ${\int }_{𝒫}q(p)𝑑\mu (p)=1$. For a ray-tracing algorithm, this full path density is induced jointly by the source-direction distribution, the conditional interaction-type probabilities, the conditional outgoing-direction distributions (e.g., for diffuse reflection and diffraction), the mapping from sampled directions to surface intersections, and the target-connection and candidate-selection rules. The estimate of the channel gain is then given by:

$$\widehat{g}=\frac{1}{N_S}\sum _{n=1}^{N_S}\frac{{\left|a(p_n)\right|}^2}{q(p_n)}.$$

(181)

An important consideration is that the estimate $\widehat{g}$ is unbiased, i.e., $𝔼[\widehat{g}]=g$ for any sampling distribution $q(\cdot )$. Therefore, we aim to choose the sampling distribution $q(\cdot )$ such that the variance of the estimate $\widehat{g}$ is minimized. Practically, this would result in estimating the channel gain with a smaller number of samples $N_S$, i.e., achieving a higher sample efficiency. The variance of the estimator $\widehat{g}$ is given by:

	$\text{Var}(\widehat{g})$	$=𝔼[{\widehat{g}}^2]-g^2$		(182)
		$={\displaystyle \frac{1}{N_S^2}}𝔼\left[{\displaystyle \sum _{n=1}^{N_S}}{\displaystyle \sum _{m=1}^{N_S}}{\displaystyle \frac{{\left|a(p_n)\right|}^2{\left|a(p_m)\right|}^2}{q(p_n)q(p_m)}}\right]-g^2$		(183)
		$={\displaystyle \frac{1}{N_S}}\left(𝔼\left[{\left({\displaystyle \frac{{\left|a(p)\right|}^2}{q(p)}}\right)}^2\right]-g^2\right).$		(184)

where the last equality holds because the samples $p_n$ are independent. Observe that

$$𝔼\left[{\left(\frac{{\left|a(p)\right|}^2}{q(p)}\right)}^2\right]={\int }_{𝒫}q(p){\left(\frac{{\left|a(p)\right|}^2}{q(p)}\right)}^2𝑑\mu (p)={\int }_{𝒫}\frac{{\left|a(p)\right|}^4}{q(p)}𝑑\mu (p).$$

(185)

If we choose the sampling distribution

$$q^{\ast }(p)≔\frac{{\left|a(p)\right|}^2}{g}$$

(186)

then

$$𝔼\left[{\left(\frac{{\left|a(p)\right|}^2}{q^{\ast }(p)}\right)}^2\right]={\int }_{𝒫}g{\left|a(p)\right|}^2𝑑\mu (p)=g^2.$$

(187)

This implies that the variance of the estimator $\widehat{g}$ is zero, i.e., $\text{Var}(\widehat{g})=0$, which is optimal. A zero-variance estimator would allow us to determine the channel gain exactly from a single sample. In practice, however, this is unattainable because evaluating the optimal distribution $q^{\ast }(\cdot )$ requires prior knowledge of the path coefficients $a(p)$ and the channel gain $g$, i.e., the quantity we aim to estimate. Nevertheless, this result provides useful guidance: an effective sampling distribution $q(\cdot )$ should give more importance to paths that contribute more to the channel gain $g$. The distribution $𝒬$, introduced in (20), is designed as a practical choice by selecting interaction types based on the squared magnitudes of the corresponding reflection and refraction coefficients. However, it does not account for the antenna pattern, the free-space propagation loss, or the scattering pattern from diffuse reflection, and is thus suboptimal. Developing effective, practical importance sampling strategies remains an open and challenging problem.