# Auralization of High-Order Directional Sources from First-Order RIR Measurements
## 2 Auralization of Arbitrary Source Directivity from Measurements
### Theory behind Source and Receiver Directional (SRD) Rits  
Based on the image source method [27], or more generally, the geometrical theory of diffraction [28], physically consistent room acoustic models including edge diffraction can be devised based on geometric sound propagation paths, see, e.g., in [29; 30; 31]. Accordingly, we may write any source-and-receiver-directional room impulse response (SRD RIR) as the sum of discrete propagation paths of the index \(i\)  
\[h\left(\theta_{\mathrm{R}},t,\theta_{\mathrm{S}}\right)=\sum_{i}\frac{\delta (\theta_{\mathrm{R}}-\theta_{\mathrm{R},i})\;\mathrm{a}_{i}\,\delta(t-\tau_{ i})\delta(\theta_{\mathrm{S}}-\theta_{\mathrm{S},i})}{\mathrm{r}_{i}}, \tag{1}\]  
where each path is characterized by a direction of arrival (DOA) and departure (DOD) denoted as \(\theta_{\mathrm{R},i}\) and \(\theta_{\mathrm{S},i}\), respectively; an arrival time \(\tau_{i}=\frac{\tau_{i}}{c}\); its geometric length \(\mathrm{r}_{i}\); and its attenuation \(\mathrm{a}_{i}\) through reflection and diffraction on rigid or sound-soft surfaces. For complex surface impedances, multiplication by \(\mathrm{a}_{i}\) theoretically becomes convolution by an impulse response \(\mathrm{a}_{i}\to\mathrm{a}_{i}(t)*\) or it can be expanded to additional paths in discrete-time processing, as preferred here. All vectors describing continuous or discrete directions \(\theta\) are denoted as unit direction vectors \(\theta=[\cos(\varphi)\sin(\vartheta),\)\(\sin(\varphi)\sin(\vartheta),\cos(\vartheta)]^{\mathrm{T}}\), with \(\varphi\) denoting the azimuth and \(\vartheta\) the zenith angle; labels \(\mathrm{S}\) and \(\mathrm{R}\) refer to source or receiver. We assume a signal \(x(t)\) that gets emitted by a source with the directivity \(g_{\mathrm{S}}(\theta_{\mathrm{S}})\) and gets picked up with the receiver directivity \(g_{\mathrm{R}}(\theta_{\mathrm{R}})\). The resulting signal \(y(t)\) is described by the convolution with the following impulse response,  
\[y(t)= x(t)*h(t)\qquad\qquad\text{with }h(t)=\int_{\mathrm{S}^{2}}\int_{ \mathrm{S}^{2}}g_{\mathrm{R}}(\theta_{\mathrm{R}})\;h\left(\theta_{\mathrm{R} },t,\theta_{\mathrm{S}}\right)g_{\mathrm{S}}(\theta_{\mathrm{S}})\;\mathrm{d} \theta_{\mathrm{R}}\;\mathrm{d}\theta_{\mathrm{S}}. \tag{2}\]  
The RIR \(h(t)\) is obtained by weighting the SRD RIR \(h\left(\theta_{\mathrm{R}},t,\theta_{\mathrm{S}}\right)\) with both the source and receiver directivities \(g_{\mathrm{S}}(\theta_{\mathrm{S}})\) and \(g_{\mathrm{R}}(\theta_{\mathrm{R}})\), assuming they are frequency-independent; for frequency-dependent  
Figure 1: Auralization scenario using the source and receiver directional (SRD) ARIR (in the Ambisonics domain). Here, \(x(t)\) is the source signal, \(h^{m^{\prime}m^{\prime}}_{nm}(t)\) is the SRD ARIR, and \(\gamma^{\mathrm{S}}_{nm}(t)\) and \(\gamma^{\mathrm{R}}_{nm^{\prime}}(t)\) are the directional impulse responses of the source and receiver, e.g., ear directivity, respectively. \(\beta_{\rm S}\)-directivities, multiplication by the directivities is replaced by convolutions with the directional impulse responses of source \(g_{\rm S}(\theta_{\rm S})\to*g_{\rm S}(t,\theta_{\rm S})\) and receiver \(g_{\rm R}(\theta_{\rm R})\to g_{\rm R}(t,\theta_{\rm R})*\), respectively. _In the Ambisonic domain:_ Equation (1) is transformed into the spherical harmonic domain by integrating either dependency on the variable sending and receiving direction over the spherical harmonics. As a result, the spherical delta functions are replaced by spherical harmonics (SH) evaluated at either DOA (\(\theta_{\rm R,\it i}\)) or DOD (\(\theta_{\rm S,\it i}\)) of the respective propagation path \(i\). We get  
\[