Visualization of Demodulated Sound Based on Sequential Acoustic Ray Tracing with Self-Demodulation in Parametric Array Loudspeakers

Abstract

With the development of acoustic simulation methods in recent decades, it has become feasible to simulate the sound pressure distribution of loudspeakers before actually setting physical speakers and measuring the sound field. The parametric array loudspeaker (PAL) has attracted attention due to its sharp directivity and unique applications. However, the sound reproduced by PALs is generated by the nonlinear interactions of ultrasound in the air, which makes it difficult to simulate the reproduced sound of a PAL with low computational load. Focusing on the sharp directivity of ultrasound, we extended conventional acoustic ray-tracing methods to consider the self-demodulation phenomenon of PALs. In this study, we developed a visualization method for the demodulated sound of a PAL. Specifically, the demodulated sound pressure distribution can be simulated to estimate and visualize the area covered by the reproduced sound of PAL before setting a real PAL. In the proposed method, acoustic rays were generated sequentially to express the generation of demodulated sound. Therefore, the proposed method is expected to simulate the demodulated sound of a PAL with acceptable accuracy and low calculation complexity. Quantitative evaluation between simulation results and practical measurement has been carried out, and the results demonstrate the effectiveness of the proposed method.

1. Introduction

With the advancement of acoustic simulation and visualization technologies, it has become feasible to simulate and visualize a sound field with given conditions for the sound sources and environment in recent decades [1]. Therefore, loudspeaker placement, including the number, direction, and position of loudspeakers, can be optimized with simulation and visualization for various purposes [2,3,4], which is superior to typical trial-and-error methods. Popular acoustic simulation approaches can be generally categorized into wave-based approaches [5,6,7] and geometry-based approaches [8,9,10]. Representative wave-based acoustic simulation approaches are mainly based on the finite element method [11,12], boundary element method [13,14], and finite-difference time-domain (FDTD) method [15,16,17]. Wave-based methods divide the target space or boundary into a mesh of small elements or cells and take the wave features into consideration. Therefore, high simulation accuracy can be obtained by these methods but at a huge computational cost [18,19]. Geometry-based approaches, which are mainly based on the image source method [20,21] and the ray-tracing method [22,23], ignore several wave features but can achieve tolerable approximation accuracy for sounds in large spaces without high algorithmic complexity or heavy computational load [1].

Electro-dynamic loudspeakers (EDLs) are widely used in audio applications. A typical EDL holds wide directivity as it emits audible sound directly. In addition, parametric array loudspeakers (PALs) [24,25] use ultrasound to achieve much sharper directivity than can conventional EDLs [26,27]. In spite of the poorer sound quality compared with conventional EDLs [28], PALs have been attracting attention because of their sharp directivity and unique audio applications [29,30,31,32]. In the practical use of PALs, the target audible signal is converted into ultrasound with amplitude modulation [25] or frequency modulation [33]. The modulated signal holds multiple frequency components and is emitted into the air intensely. Due to the nonlinearity of air, the target audible signal is “demodulated" as one of the difference frequencies among these components. Because this phenomenon occurs due to the nature of the medium [34] rather than by artificial processing, the demodulation process of a PAL can be seen as self-demodulation [25,27]. To simulate the demodulated sound of a PAL, it is essential to consider both the propagation of emitted ultrasound and the self-demodulation phenomenon. When a wave-based simulation such as FDTD is used, the grid (or element) size should be set as very small because the ultrasound wavelength is short [35]. This will lead to heavy computational cost and a long simulation time to simulate the demodulated sound of a PAL in a wide space with a scale of thousands of wavelengths.

In the practical applications of PALs, we are mainly concerned with the sound pressure distribution of the demodulated sound. For example, in [31], an ideal situation is that the whole space can be covered by multiple PALs to obtain a sufficient sound pressure above a certain level at any position inside the space. A quick and feasible simulation of demodulated sound with low computational cost would be helpful in this case. Therefore, we focused on the ray-tracing method [22,23], which can be used to simulate the propagation of ultrasound emitted from PALs [36]. However, it is difficult to use the traditional ray-tracing method to express self-demodulation.

To solve this problem, we introduced self-demodulation to traditional acoustic ray-tracing methods based on Westervelt’s theory [24], enabling the simulation of the demodulated PAL sound with low computational cost and short simulation time. In this paper, we propose a simulation method for the demodulated sound of a PAL based on the sequential acoustic ray-tracing method. Discrete acoustic rays are generated sequentially along the propagation paths of the ultrasound, thereby generating the demodulated sound through the interactions among the ultrasound rays. Compared with traditional ray-tracing methods, the proposed method considers the self-demodulation phenomenon of PALs to achieve higher accuracy. Compared with FDTD-based methods, the proposed method can reduce the calculation complexity drastically because only the discrete acoustic rays are involved in the calculation rather than the whole simulation space. As mentioned, in practical applications of PAL such as [31], what we are most concerned about is the area covered by the reproduced sound rather than the detailed sound field. The proposed method is more feasible in such scenes as it has a much lower calculation complexity.

The rest of this paper is organized as follows. Section 2 gives a brief introduction of the basic principles of PALs that act as the theoretical support of the proposed method. Section 3 describes the proposed method in detail. Section 4 demonstrates the effectiveness of the proposed method through a comparison between numerical simulation and real-world measurements. Finally, Section 5 provides conclusions.

2. Principles of Parametric Array Loudspeakers

As shown in Figure 1, in a typical scenario for reproducing an audible sound at frequency $f_{\mathrm{d}}$ with a PAL, the emitted signal can be expressed as

$$p(0,t)=\sum _{n=1,2}{P}_{n}cos\left(2\pi f_{n}t\right),$$

(1)

where $n=1,2$ denotes the index of the frequency components and $P_n$ and $f_n$ denote the initial sound pressure and the frequency of the corresponding component, respectively. In addition, it should be noted that $\left|f_1-f_2\right|=f_{\mathrm{d}}$.

Because the attenuation is exponential, the ultrasound propagating along the propagation axis can be given as

$$p(r,t)=\sum _{n=1,2}{P}_n{e}^{-{\alpha }_{n}r}cos\left(2\pi f_{n}t-k_{n}r\right),$$

(2)

where r denotes the distance to the PAL and ${\alpha }_n$ and $k_n$ denote the attenuation rate and wavenumber of the frequency component indexed with n, respectively.

Based on Westervelt’s theory [24], a virtual source at frequency is generated as the product of nonlinear interactions [34], whose intensity can be given as

$$q(r,t)=\frac{\beta }{{\rho }^2{c}^4}\frac{\partial }{\partial t}{p}^2(r,t),$$

(3)

where $\beta$ and $\rho$ denote the nonlinear coefficient and intensity of the air, respectively, and c denotes the speed of sound in air. By substituting Equation (2) into Equation (3), we find that the virtual source holds multiple frequency components, including $f_1+f_2$ and $\left|f_1-f_2\right|$. Here, we focus on the difference frequency $f_{\mathrm{d}}=\left|f_1-f_2\right|$ as it will become the sound reproduced by a PAL with self-demodulation. The intensity of the virtual source at the difference frequency can be given as

$$q_{\mathrm{d}}(r,t)=\frac{2\pi f_{\mathrm{d}}\beta P_1{P}_2}{{\rho }^2{c}^4{e}^{\left({\alpha }_1+{\alpha }_2\right)r}}sin(2\pi f_{\mathrm{d}}t-k_{\mathrm{d}}r).$$

(4)

Under Westervelt’s theory [24], the demodulated sound can be regarded as a volume integral of virtual sources at the difference frequency $f_{\mathrm{d}}$. Considering a far-field observation point at distance $R_0$, we can approximate the volume distribution of the virtual source further as a line distribution along the propagation axis. As shown in Figure 1, the collimated ultrasound can be approximated as a cylinder with a base area S and height L. In this case, the demodulated sound can be given as

$$\begin{array}{ccc}\hfill p_{\mathrm{d}}(R_0,t)& =& \frac{\rho }{8\pi }\int \int \int \frac{e^{-j{k}_{\mathrm{d}}R(x,y,z)}}{R(x,y,z)}\frac{\partial q_{\mathrm{d}}(r,t)}{\partial t}dxdydz+\mathrm{c}.\mathrm{c}.\hfill \\ & \approx & \frac{\rho S}{8\pi }{\int }_0^L\frac{e^{-j{k}_{\mathrm{d}}R\left(r\right)}}{R\left(r\right)}\frac{\partial q_{\mathrm{d}}(r,t)}{\partial t}dr+\mathrm{c}.\mathrm{c}.,\hfill \end{array}$$

(5)

where j denotes imaginary unit $\sqrt{-1}$ and $\mathrm{c}.\mathrm{c}.$ denotes a complex conjugate. Equation (5) shows that the observed demodulated sound can be regarded as an accumulation of the sounds generated from a huge number of continuously distributed virtual sources. Therefore, in the proposed method, we extend this consideration and approximate the observed demodulated sound to an accumulation of sounds generated from finite discrete virtual sources.

3. Proposed Self-Demodulation-Based Visualization Method

3.1. Overview of the Proposed Method

In this paper, we propose the visualization of the demodulated sound of a PAL with the sequential ray-tracing method. Figure 2 shows the basic concept of sequential acoustic ray tracing in the proposed method, and Figure 3 gives an overview of the proposed method. A PAL, which is defined as the sound source, emits multiple acoustic rays from its propagation surface. The directions of the rays show the propagation direction and the directivity of the PAL. As the initial rays travel, new rays are generated sequentially along the directions of the initial rays. Each ray holds two indices: the ray index m ($m=0,1,2,\cdots ,M-1$, where M is the total number of initial rays) to identify the original position and direction, and the frame index i $\left(i=0,1,2,\cdots \right)$ to mark the update time.

The target sound reproduced by the PAL is set as an audible sound with frequency $f_{\mathrm{d}}$. Considering a case that uses a single sideband modulation [25], we can express the ultrasound emitted from the PAL with Equation (1). Here, the carrier wave at frequency $f_{\mathrm{c}}$ corresponds to the component indexed with $n=1$ in Equation (1), and the sideband wave at frequency $f_{\mathrm{s}}$ corresponds to the component indexed with $n=2$ in Equation (1).

The following variables are defined for each acoustic ray, and they are initialized at the beginning and updated in each frame.

• The ray index m $\left(m=0,1,2,\cdots ,M-1\right)$ is defined to identify the original positions.
• A position vector $\mathit{u}={\left[u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}}\right]}^{\mathrm{T}}$ is defined as the starting position for each ray.
• The direction vector $\mathit{v}={\left[v_{\mathrm{x}},v_{\mathrm{y}},v_{\mathrm{z}}\right]}^{\mathrm{T}}$ is defined as the direction of each ray, where $v_{\mathrm{x}}^2+v_{\mathrm{y}}^2+v_{\mathrm{z}}^2=1$.
• The sound pressure $P_{\mathrm{c}}$ is defined as the sound pressure of the carrier wave at frequency $f_{\mathrm{c}}$.
• The sound pressure $P_{\mathrm{s}}$ is defined for each sideband wave at frequency $f_{\mathrm{s}}$.
• The sound pressure $P_{\mathrm{d}}$ is defined as the demodulated sound pressure at frequency $f_{\mathrm{d}}$.
• The virtual source intensity $Q_{\mathrm{d}}$ is defined as the intensity of the virtual source at frequency $f_{\mathrm{d}}$.

In the simulation, each time frame has a duration of $\Delta t$. The frame index i $\left(i=0,1,2,\cdots \right)$ is defined for the update time of ray tracing; therefore, the position vector of the m-th ray in the i-th frame is denoted as $\mathit{u}(m,i)$ in the following description.

3.2. Simulation Initialization

The simulation conditions are set at the beginning of the initialization. As shown in Figure 3, we can divide the simulation initialization process into three steps: environment, sound source, and initial acoustic rays.

• Step 1: The initialization of the environment

The initialization of the environment is almost same as other acoustic simulations. In this step, the simulation space and acoustic parameters are defined. The acoustic parameters include the speed of sound c, the frequency-dependent attenuation rate $\alpha$, the nonlinear factor of air $\beta$, and the density of the air $\rho$. In the case that the reflector exists in the simulation space, the reflection attenuation $\eta$ should also be defined. In this method, we also define several parameters to keep the possibility of utilizing the practically measured results to define acoustic parameters, and their definitions will be given in the following description.

• Step 2: The initialization of the sound source

In the $xyz$-coordinate system, the center of the PAL is set at

$${\mathit{u}}_{\mathrm{o}}={\left[u_{\mathrm{ox}},u_{\mathrm{oy}},u_{\mathrm{oz}}\right]}^T.$$

(6)

The propagation axis of the PAL is set as

$$\begin{array}{c}\hfill {\mathit{v}}_{\mathrm{o}}={\left[v_{\mathrm{ox}},v_{\mathrm{oy}},v_{\mathrm{oz}}\right]}^T,\end{array}$$

(7)

$$\begin{array}{c}\hfill \mathrm{s}.\mathrm{t}.v_{\mathrm{ox}}^2+v_{\mathrm{oy}}^2+v_{\mathrm{oz}}^2=1.\end{array}$$

(8)

Other aspects of the sound source are also defined at the beginning, such as the shape, size, and directional angle of the target PAL. The carrier frequency $f_{\mathrm{c}}$ and sideband frequency $f_{\mathrm{s}}$ and their corresponding sound pressure are also defined. The above definitions will be used in the next step to set the initial acoustic rays.

• Step 3: The initialization of the initial acoustic rays

The position vectors and direction vectors of acoustic rays for $i=0$ are initialized based on the shape, size, and directional angle of the target PAL. When the target PAL has a rectangular shape, the position vectors are evenly distributed on the PAL surface. When the target PAL is circular (e.g., MEE or PS-60E), the position vectors are distributed as concentric circles on the PAL surface.

In Figure 4, we give an example of initialization for a circular PAL. The position vectors are distributed on the $yz$-plane and then extended to three-dimensional space with affine transformation and Rodrigues’ rotation formula. As shown in Figure 4, when the z-axis is the propagation axis (${\mathit{v}}_{\mathrm{o}}={\left[0,0,1\right]}^T$), the position vectors on the $yz$-plane can be given as

$${\mathit{u}}^{\prime }\left(\mu \right)={\left[u_y^{\prime }\left(\mu \right),u_z^{\prime }\left(\mu \right)\right]}^{\mathrm{T}},$$

(9)

where $\mu =-M^{\prime },\cdots ,-1,0,1,\cdots ,M^{\prime }$ denotes an index of initial positions. In this example, the initial positions are equally distributed along y-axis; therefore, we have

$$\begin{array}{ccc}\hfill u_y^{\prime }\left(\mu \right)& =& y_0+\mu W/(2M^{\prime }+1),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill u_z^{\prime }\left(\mu \right)& =& z_0,\hfill \end{array}$$

(11)

where W denotes the diameter of PAL. The corresponding direction vectors can be given as

$$\begin{array}{c}\hfill {\mathit{v}}^{\prime }\left(\mu \right)={\left[sin\left(\phi \left(\mu \right)\right),cos\left(\phi \left(\mu \right)\right)\right]}^{\mathrm{T}},\end{array}$$

(12)

$$\begin{array}{c}\hfill \phi \left(\mu \right)=\mu \mathsf{\Phi }/M^{\prime },\end{array}$$

(13)

where $\mathsf{\Phi }$ is the PAL directional angle. By rotating the $yz$-plane around the z-axis, we can obtain all the initial position vectors and direction vectors for acoustic rays, and we re-index them with index $m=0,1,2,\cdots ,M-1$, as shown in Figure 4c. Here, the acoustic ray indexed with $m=0$ is located at the center of the target PAL, and the direction of the propagation axis remains constant. The initialization defines the initial position vector $\mathit{u}(m,0)$ and the initial direction vector $\mathit{v}(m,0)$ for each acoustic ray.

The sound pressures $P_{\mathrm{c}}$ and $P_{\mathrm{s}}$ are initialized based on the directivity of the PAL, which can be given as

$$\begin{array}{c}\hfill P_{\mathrm{c}}(m,0)=P_{\mathrm{c}}(0,0)D_{\mathrm{c}}\left[\theta (m,0)\right],\end{array}$$

(14)

$$\begin{array}{c}\hfill P_{\mathrm{s}}(m,0)=P_{\mathrm{s}}(0,0)D_{\mathrm{s}}\left[\theta (m,0)\right],\end{array}$$

(15)

where $D_{\mathrm{c}}\left(\theta \right)$ and $D_{\mathrm{s}}\left(\theta \right)$ denote the directivities of the carrier wave and sideband wave at the off-axis angle $\theta$ ($0\le \theta \le \pi$), respectively. The off-axis angle for each acoustic ray $\theta (m,i)$ can be calculated as the angle between the propagation axis ${\mathit{v}}_{\mathrm{o}}$ and the direction vector $\mathit{v}(m,i)$, which can be given as

$$\theta (m,i)={cos}^{-1}\frac{\mathit{v}(m,i)·{\mathit{v}}_{\mathrm{o}}}{\left|\mathit{v}(m,i)\right|\left|{\mathit{v}}_{\mathrm{o}}\right|}={cos}^{-1}\left[v_{\mathrm{x}}(m,i)v_{\mathrm{ox}}+v_{\mathrm{y}}(m,i)v_{\mathrm{oy}}+v_{\mathrm{z}}(m,i)v_{\mathrm{oz}}\right],$$

(16)

where · denotes the operator of an inner product.

The initial sound pressure of demodulated sound $P_{\mathrm{d}}(m,0)$ is set as 0 due to the principle of self-demodulation in the PAL. The virtual source intensity $Q_{\mathrm{d}}(m,0)$ is initialized as

$$Q_{\mathrm{d}}(m,0)=\zeta P_{\mathrm{c}}(m,0)P_{\mathrm{s}}(m,0),$$

(17)

where $\zeta$ is defined as the generation efficiency of the virtual source. When the optimal value can be determined for every parameter, we have $\zeta =2\pi f_{\mathrm{d}}\beta {\rho }^{-2}{c}^{-4}$ based on Equation (4). However, in this method, we retain the possibility to calibrate the value of $\zeta$ based on practical measurement rather than substituting a theoretical value.

3.3. Sequential Update of Acoustic Rays

In the sequential update of acoustic rays, new acoustic rays are generated in each time frame as the successors of the rays generated in the previous time frame.

As with traditional ray-tracing methods, the acoustic rays are projected to go approximately straight. Therefore, the position vector of a new acoustic ray $\mathit{u}(m,i+1)$ can be obtained as its position after traveling from $\mathit{u}(m,i)$ in the direction $\mathit{v}(m,i)$ within time duration $\Delta t$, which can be given as

$$\mathit{u}(m,i+1)=\mathit{u}(m,i)+c\Delta t\mathit{v}(m,i).$$

(18)

Then, collision detection is carried out to check whether the acoustic ray collides with walls or another boundary. When the new position vector is beyond the defined boundary, the position vector is corrected based on the reflection off the boundary. As shown in Figure 5a, if no collision occurs, the new acoustic ray inherits the direction from the previous ray and continues in this direction. Therefore, in this case, the updated direction vector can be given as

$$\mathit{v}(m,i+1)=\mathit{v}(m,i).$$

(19)

However, if a collision occurs, the position vector and the direction vector need to be corrected based on reflection off a boundary surface, such as a wall. As shown in Figure 5b, the position of the collision point on the boundary surface is noted as ${\mathit{u}}_{\mathrm{b}}$ and the normal vector of boundary surface is noted as ${\mathit{v}}_{\mathrm{b}}$ ($\left|{\mathit{v}}_{\mathrm{b}}\right|=1$). The position vector updated with Equation (18) is rewritten as $\tilde{\mathit{u}}(m,i+1)$, and the direction vector updated with Equation (19) is rewritten as $\tilde{\mathit{v}}(m,i+1)$. The corrected position vector $\mathit{u}(m,i+1)$ is the mirror image of $\tilde{\mathit{u}}(m,i+1)$ reflected off the boundary surface; therefore, it can be given as

$$\begin{array}{ccc}\hfill \mathit{u}(m,i+1)& =& \tilde{\mathit{u}}(m,i+1)+\left(\left|{\mathit{u}}_{\mathrm{b}}-\tilde{\mathit{u}}(m,i+1)\right|cos\psi \right){\mathit{v}}_{\mathrm{b}}\hfill \\ & =& \mathit{u}(m,i)+c\Delta t\mathit{v}(m,i)+\left(\left|{\mathit{u}}_{\mathrm{b}}-\mathit{u}(m,i)-c\Delta t\mathit{v}(m,i)\right|cos\psi \right){\mathit{v}}_{\mathrm{b}},\hfill \end{array}$$

(20)

$$cos\psi =\frac{\left[\mathit{u}(m,i)-{\mathit{u}}_{\mathrm{b}}\right]·{\mathit{v}}_{\mathrm{b}}}{\left|\mathit{u}(m,i)-{\mathit{u}}_{\mathrm{b}}\right|},$$

(21)

where $\psi$ denotes the angle of incidence. The corrected direction vector $\mathit{v}(m,i+1)$ is the mirror image of $\tilde{\mathit{u}}(m,i+1)$ reflected across the normal vector of boundary surface ${\mathit{v}}_{\mathrm{b}}$; therefore, it can be given as

$$\begin{array}{ccc}\hfill \mathit{v}(m,i+1)& =& \tilde{\mathit{v}}(m,i+1)-2\frac{\mathit{v}(m,i)·{\mathit{v}}_{\mathrm{b}}}{\left|\mathit{v}(m,i)\right|\left|{\mathit{v}}_{\mathrm{b}}\right|}{\mathit{v}}_{\mathrm{b}},\hfill \\ & =& \mathit{v}(m,i)-2\left[\mathit{v}(m,i)·{\mathit{v}}_{\mathrm{b}}\right]{\mathit{v}}_{\mathrm{b}}.\hfill \end{array}$$

(22)

The sound pressures of ultrasound $P_{\mathrm{c}}$ and $P_{\mathrm{s}}$ are updated according to the attenuation due to distance and reflection, which can be given as

$$\begin{array}{c}\hfill P_{\mathrm{c}}(m,i+1)={\eta }_{\mathrm{c}}(m,i)P_{\mathrm{c}}(m,i)e^{-{\lambda }_{\mathrm{c}}c\Delta t},\end{array}$$

(23)

$$\begin{array}{c}\hfill P_{\mathrm{s}}(m,i+1)={\eta }_{\mathrm{s}}(m,i)P_{\mathrm{s}}(m,i)e^{-{\lambda }_{\mathrm{s}}c\Delta t},\end{array}$$

(24)

where ${\lambda }_{\mathrm{c}}$ and ${\eta }_{\mathrm{c}}$ denote the coefficients of distance attenuation and reflection attenuation for the carrier wave, respectively, and ${\lambda }_{\mathrm{s}}$ and ${\eta }_{\mathrm{s}}$ denote the coefficients of distance attenuation and reflection attenuation for the sideband wave, respectively. The coefficient of distance attenuation can be either calculated theoretically (e.g., ${\lambda }_{\mathrm{c}}={\alpha }_{\mathrm{c}}$, ${\lambda }_{\mathrm{s}}={\alpha }_{\mathrm{s}}$) or measured practically. The coefficient of reflection attenuation can be calculated from the absorption ratio of reflection, which is either determined theoretically or measured practically. If no collision occurs, there is no attenuation due to reflection, so we have ${\eta }_{\mathrm{c}}(m,i)={\eta }_{\mathrm{s}}(m,i)=1$ in this case.

The virtual source intensity $Q_{\mathrm{d}}(m,i)$ is updated with the current ultrasound pressure, which is similar to Equation (17) and can be given as

$$Q_{\mathrm{d}}(m,i)=\zeta P_{\mathrm{c}}(m,i)P_{\mathrm{s}}(m,i).$$

(25)

The demodulated sound pressure $P_{\mathrm{d}}(m,i)$ can be approximated as an accumulation of the virtual source intensity along the propagation path. Therefore, it is updated as a sum of attenuated virtual source intensity among acoustic rays with the same ray index m based on Equation (5) and can be given as

$$P_{\mathrm{d}}(m,i+1)=\frac{1}{2}{\widehat{P}}_{\mathrm{d}}(m,i+1)+c.c.,$$

(26)

$${\widehat{P}}_{\mathrm{d}}(m,i+1)=\sum _{\tau =0}^i\left[2\pi f_{\mathrm{d}}{Q}_{\mathrm{d}}(m,\tau )\chi (m,\tau ,i+1)\mu (m,\tau ,i+1)\right],$$

(27)

where $\chi (m,\tau ,i+1)$ and $\mu (m,\tau ,i+1)$ denote the total distance attenuation ratio and total reflection attenuation ratio from the virtual source with index $(m,\tau )$ to the acoustic ray to be updated with index $(m,i+1)$ and ${\widehat{P}}_{\mathrm{d}}$ is the complex-valued format of $P_{\mathrm{d}}$. The total reflection attenuation ratio $\mu$ is calculated by counting all reflection attenuation at the frequency of demodulated sound along the rays with the same index m, which can be given as

$$\mu (m,\tau ,i+1)=\prod _{{\tau }^{\prime }=\tau +1}^i{\eta }_{\mathrm{d}}(m,{\tau }^{\prime }),$$

(28)

where ${\eta }_{\mathrm{d}}$ denotes the coefficient of reflection attenuation at frequency $f_{\mathrm{d}}$. The total distance attenuation ratio $\chi$ can be measured practically or calculated theoretically based on Equation (5):

$$\chi (m,\tau ,i+1)=\frac{\rho e^{-j{k}_{\mathrm{d}}d(\tau ,i+1)}}{4\pi d(\tau ,i+1)},$$

(29)

where $d(\tau ,i+1)$ denotes the distance between the virtual source with index $(m,\tau )$ and the acoustic ray with index $(m,i+1)$, which can be given as

$$d(\tau ,i+1)=(i+1-\tau )c\Delta t.$$

(30)

As shown in Equation (27), to update the demodulated sound pressure, all acoustic rays with the same ray index m should be included in the calculation; however, the pressure can also be approximated in an iterative way if the distance to the PAL is far enough, which can be given as

$${\widehat{P}}_{\mathrm{d}}(m,i+1)\approx \left[{\widehat{P}}_{\mathrm{d}}(m,i)+2\pi f_{\mathrm{d}}{Q}_{\mathrm{d}}(m,i)\right]\chi (m,I,i+1){\eta }_{\mathrm{d}}(m,i,i+1).$$

(31)

This can be explained by the demodulated sound of acoustic ray $(m,i)$ traveling to the position of acoustic ray $(m,i+1)$, and the virtual source generated at the position of acoustic ray $(m,i)$ also traveling to the position of acoustic ray $(m,i+1)$. This approximation can reduce the calculation complexity significantly when the acoustic rays travel far from the PAL.

3.4. Computational Complexity of the Proposed Method

From the above description, it is indicated that the simulation of the proposed method can be regarded as calculating Equation (31) or Equation (27) for $MI$ times, where M denotes the number of initial acoustic rays and I denotes the total number of time frames until the end of simulation.

In the case that the above approximation is not utilized, where Equation (27) is calculated instead of Equation (31), the computational complexity of the proposed method can be expressed as

$$G_{\mathrm{prop}.\mathrm{w}/\mathrm{o}\mathrm{appr}.}\propto M{I}^2.$$

(32)

In the case that the approximation as Equation (31) is utilized, the computational complexity of the proposed method can be expressed as

$$G_{\mathrm{prop}.}\propto MI.$$

(33)

As mentioned, this approximation can significantly reduce the computational complexity of the proposed method.

Considering an FDTD-based method in three-dimensional space as a comparison, the whole field needs to be updated for I times if the same destination condition is applied. In this case, the computational complexity of the FDTD-based method can be given as

$$G_{\mathrm{3dFDTD}}\propto N_x{N}_y{N}_{z}I,$$

(34)

where $N_x$, $N_y$, and $N_z$ denote the number of meshes in the x, y, and z-axis, respectively. It can be obviously noticed that the proposed method with approximation can be regarded as an $O\left(N\right)$-algorithm, while FDTD-based methods are generally $O\left(N^3\right)$-algorithms. Moreover, to simulate ultrasound in a large space, the number of meshes in the FDTD-based methods should be set large enough, while the number of rays in the proposed method can be set based on the desired resolution rather than physical constraints.

4. Evaluation Experiments

4.1. Experimental Conditions

To demonstrate the effectiveness of the proposed method, we carried out evaluation experiments in which we compared the results of numerical simulations and practical measurements. The experimental environment and conditions are summarized in

Table 1. Experimental environment and conditions.

Environment	Soundproof room (Width: 2.3 m/Depth: 3.2 m/Height: 2.2 m)
Ambient noise level	$L_{\mathrm{A}}=22.8$ dB
Reverberation time	$T_{60}=150$ ms
Frame interval	$\Delta t=0.15$ ms
Carrier frequency	$f_{\mathrm{c}}=40$ kHz
Sideband frequency	$f_{\mathrm{s}}=33,34,\cdots ,39$ kHz
Demodulated frequency	$f_{\mathrm{d}}=1,2,\cdots ,7$ kHz
Applied voltage	15 V

, and the equipment used in the experiments is listed in

Table 2. Experimental equipment.

Ultrasonic sound pressure meter	RION, UN-14
Microphone	Sennheiser, MKH 8020
A/D, D/A converter	RME, Fireface UFX
Power amplifier	JVC, PS-A2002
Ultrasonic transducer	SPL Limited, UT1007-Z325R

. The PAL used in the experiments is shown in Figure 6; it consists of 8 columns of 10 ultrasonic transducers.

As shown in Figure 7, we prepared two experimental setups for the evaluation: one without reflection and the other with reflection. In Figure 7a, the PAL is placed at ${\left[0,0,0\right]}^{\mathrm{T}}$ in the $xyz$-coordinate system, and the propagation axis of the PAL overlaps the z-axis. In Figure 7b, the PAL is rotated around the y-axis by 45 deg, and a reflector parallel to the z-axis is set to reflect the sound emitted from the PAL. The propagation axis collides with the reflector at ${\left[0.5,0,0.5\right]}^{\mathrm{T}}$. The reflector is made of acrylic plastic, and the reflection attenuation coefficient is measured in advance with the same procedure as [37]. Beyond the area shown in Figure 7, the environment is regarded as an open space, and all of the acoustic rays beyond those areas are absorbed. Therefore, the simulations for evaluation are terminated after all acoustic rays reach the boundary of the area where $-0.5\le x\le 0.5$, $-0.5\le y\le 0.5$, and $0\le z\le 2.0$.

We set multiple points for the determination of the parameters to approximate the real environment, as described in Section 3. As shown in Figure 7a, we used the measurement results on the propagation axis to calibrate the parameters, including the generation efficiency of virtual source $\zeta$ and the distance attenuation ratio $\chi$. A simple recursion based on Equation (31) is utilized to calculated these parameters in a backward manner.

A practical PAL generally has a frequency-dependent response [28]; therefore, we set calibration coefficients in the simulation according to the sideband frequency $f_{\mathrm{s}}$ ($f_{\mathrm{s}}=f_{\mathrm{c}}-f_{\mathrm{d}}$ for lower sideband modulation). The coefficients were calculated based on the measurement results at the points for parameter determination and are given in

Table 3. Calibration coefficients on sound pressure of sideband wave $P_{\mathrm{s}}$.

Demodulated frequency $f_{\mathrm{d}}$ (kHz)	1	2	3	4	5	6	7
Sideband frequency $f_{\mathrm{s}}$ (kHz)	39	38	37	36	35	34	33
Ratio of sound pressure $P_{\mathrm{s}}$	1.0 (reference)	0.76	0.28	8.2 $\times {10}^{-2}$	2.8 $\times {10}^{-2}$	1.5 $\times {10}^{-2}$	8.8 $\times {10}^{-3}$

. A practical PAL also has directivity in the ultrasonic range [26,27]; therefore, we measured the frequency-dependent directivity in advance. The PAL was set on a pan-rotation unit to measure the directivity characteristics, which is same as the experiments in [27].

We also defined multiple evaluation points to evaluate whether the simulation results matched those of the practical measurement in Figure 7a. Here, some points were set for both parameter determination and evaluation, while other points were set for evaluation only. In Figure 7b, the setup with reflection, no point is set for parameter determination, and some points located on the path of the demodulated sound are set for evaluation.

The root mean square error (RMSE) was used to evaluate the error between the simulation and measured results, which can be given as

$$E_{\mathrm{rms}}=\sqrt{\frac{1}{G}\sum _{g=0}^{G-1}{\left({\mathrm{SPL}}_{\mathrm{d}}^{\mathrm{sim}}-{\mathrm{SPL}}_{\mathrm{d}}^{\mathrm{meas}}\right)}^2},$$

(35)

where g ($g=0,1,\cdots ,G-1$) denotes the index of each evaluation point, G denotes the total number of evaluation points, ${\mathrm{SPL}}_{\mathrm{d}}^{\mathrm{sim}}$ denotes the simulated sound pressure level (SPL) in decibels of demodulated sound, and ${\mathrm{SPL}}_{\mathrm{d}}^{\mathrm{meas}}$ denotes the measured SPL of demodulated sound. The SPL can be calculated from the sound pressure in Pascals. For example, the SPL of demodulated sound can be given as

$${\mathrm{SPL}}_{\mathrm{d}}=20{log}_{10}\frac{P_{\mathrm{d}}}{P_0},$$

(36)

where $P_0$ denotes the reference sound pressure ($P_0=2\times {10}^{-5}$ Pa). Moreover, the average absolute error is also used to evaluate the error between the simulation and measured results, which can be given as

$$E_{\mathrm{abs}}=\frac{1}{G}\sum _{g=0}^{G-1}\left|{\mathrm{SPL}}_{\mathrm{d}}^{\mathrm{sim}}-{\mathrm{SPL}}_{\mathrm{d}}^{\mathrm{meas}}\right|.$$

(37)

4.2. Experimental Results

Figure 8 shows the SPL measurement results for sideband waves (${\mathrm{SPL}}_{\mathrm{s}}$) along the z-axis. Because a PAL produces a frequency-dependent response, the results are shown in separate panels according to sideband frequency $f_{\mathrm{s}}$. These measurement results were used for parameter determination, as mentioned in Section 4.1.

Figure 9 shows the simulation and measurement results for the SPL of demodulated sound (${\mathrm{SPL}}_{\mathrm{d}}$) along the z-axis. Similarly to Figure 8, the results are shown in separate panels according to demodulated frequency $f_{\mathrm{d}}$. The measurement results were also used for parameter determination, as mentioned in Section 4.1. The simulation results of the proposed method generally match the measurement results, especially for $f_{\mathrm{d}}=2,3,\cdots ,7$ kHz. However, it is difficult for the proposed method to express the SPL fluctuation as the distance from the PAL increases. When $f_{\mathrm{d}}=1$ kHz, the error between simulation and measurement is larger than that at other frequencies. This can be explained by the low demodulation efficiency when the demodulated frequency $f_{\mathrm{d}}$ is low, which can be demonstrated theoretically from Equations (4) and (5).

The simulation results of demodulated sound pressure on the $xz$-plane in the experimental setup without reflection (as shown in Figure 7a) are shown in Figure 10 and Figure 11, and the results for the experimental setup with reflection (as shown in Figure 7b) are shown in Figure 12 and Figure 13. The simulation and measurement results are given in pairs for comparison. As a quantitative evaluation, we used Equations (35) and (37) (Section 4.1) to calculate the RMSE and average absolute error, respectively, of the demodulated sound pressures between the simulation and measurement results. The results are given in

Table 4. Root mean square error $E_{\mathrm{rms}}$ for the sound pressure level of demodulated sound between the simulation and measurement results (dB).

Evaluation Area	Demodulated Frequency ${\mathit{f}}_{\mathbf{d}}$ (kHz)
	1	2	3	4	5	6	7
Points for parameter determination	1.94	0.87	0.83	0.81	0.93	0.65	0.52
Points for evaluation (without reflection)	2.15	1.14	1.39	1.13	1.16	0.98	1.26
Points for evaluation (with reflection)	2.48	1.89	2.28	2.94	4.23	4.25	4.95

and

Table 5. Average absolute error $E_{\mathrm{abs}}$ for the sound pressure level of demodulated sound between the simulation and measurement results (dB).

Evaluation Area	Demodulated Frequency ${\mathit{f}}_{\mathbf{d}}$ (kHz)
	1	2	3	4	5	6	7
Points for parameter determination	1.67	0.64	0.62	0.59	0.73	0.46	0.42
Points for evaluation (without reflection)	1.84	0.81	0.90	0.75	0.90	0.63	0.71
Points for evaluation (with reflection)	1.93	1.26	1.90	2.22	3.13	3.22	3.67

, respectively.

The above results show that the proposed method can generally approximate the propagation of the demodulated sound of a PAL. The frequency-dependent demodulation efficiency [25,28] can also be reflected in the simulation results of the proposed method because our method can include self-demodulation in the calculation of demodulated sound pressure. In the cases without reflection, the error between simulation and measurement is very small, especially when the demodulated frequency $f_{\mathrm{d}}$ is in the range of 2 kHz $\le f_{\mathrm{d}}\le 6$ kHz. When $f_{\mathrm{d}}=1$ kHz, the demodulation efficiency is much lower so it is considered that the simulation results contain more errors. In the cases with reflection, the error between simulation and measurement is much larger than that in the cases without reflection, and the error increases obviously as the demodulated frequency $f_{\mathrm{d}}$ increases, which is different from the error in the cases without reflection. Two major reasons can be identified: one is that the reflection coefficients were not set properly, and the other is that the measurement results also lack accuracy. In summary, the experimental results demonstrated the effectiveness of the proposed method in approximating the demodulated sound of a PAL. To further improve the accuracy in various simulation conditions (e.g., reflection), determining the parameters properly plays an important role, as in other acoustic simulations.

5. Conclusions

In this paper, we proposed a simulation method for demodulated PAL sound. This method is based on the sequential ray-tracing method and is extended to express the self-demodulation phenomenon in the sound reproduction; therefore, it can simulate the distribution of the demodulated sound of a PAL with higher accuracy than can other geometry-based simulation methods and has much lower calculation complexity than FDTD-based methods. The experimental results demonstrated the effectiveness of the proposed method with a quantitative evaluation.

In the future, we plan to further improve the accuracy of the proposed method by investigating a feasible approach to estimate the simulation parameters with higher accuracy. The simulation of the proposed method is in three-dimensional space; however, it is difficult to show the results in three-dimensional space. Therefore, we also plan to develop a user-friendly interface to show the simulation results.