# Directional Suppression of Monotone Noises with A Parametric Array Loudspeaker
## 2 Theory and Methods
### PAL as the Directional Secondary Source  
PAL can generate highly directional audible sound fields based on the nonlinear acoustic wave attenuation in air. In simple terms, for an acoustic beam containing multiple frequency components, as the wave propagates, the high-frequency contents attenuate faster to leave over the lowest frequency component as the subtraction result of the primary frequencies. Hence, it is possible to reproduce audible sound waves from high-frequency carrier waves through self-demodulation during propagation. The classic nonlinear acoustic model was derived by [PERSON], [PERSON], and [PERSON], known as the KZK equation, by considering the absorption of finite amplitude acoustic waves. In the far field, the derived quasilinear solution can predict the overall pressure distribution [27]. The resulting pressure field \(p\) is considered a combination of the solution to the linear ultrasound field \(p_{1}\) and the nonlinear demodulated field result \(p_{2}\) as \(p=p_{1}+p_{2}\). For a known ultrasound field, \(p_{1}\) can be used as the driving source to solve for \(p_{2}\), and the KZK equation is converted into [28; 29]  
\[\frac{\partial^{2}p_{2}}{\partial z\partial\tau}=\frac{c_{0}}{2}\
abla_{r}^{2} p_{2}+\frac{\delta}{2c_{0}^{3}}\frac{\partial^{3}p_{2}}{\partial\tau^{3}}+ \frac{\beta}{2\rho_{0}c_{0}^{3}}\frac{\partial^{2}p_{1}^{2}}{\partial\tau^{2}}. \tag{7}\]  
This equation describes \(p_{1}\) and \(p_{2}\)'s relation in the \(z\)-propagating axial direction, where \(\tau=t-z/c_{0}\) is the retarded time, and \(\delta\) and \(\beta\) are the diffusivity and coefficient of nonlinearity, respectively. \(\
abla_{r}^{2}=\partial^{2}/\partial r^{2}+r^{-1}(\partial/\partial r)\), where \(r\) is the transverse radial coordinate in the \(xy\) plane. \(c_{0}\) and \(\rho_{0}\) are the sound velocity and density of the air medium, respectively. [PERSON]'s solution of Equation (7) gives a simplified time-domain expression of the reproduced audio wave in the far field \(p_{2}(r,z,\tau)\), which is proportional to the second derivative of the squared envelope through amplitude modulation (AM) of the primary wave \(p_{1}(r,z,\tau)\)[30]. This is considered the foundation for implementing AM techniques for PALs [31; 32]. For a primary ultrasound carrier wave of the angular frequency \(\omega_{c}\) and initial amplitude \(P_{0}\), the AM-processed ultrasound pressure is [29]  
\[p_{1}(r,z,\tau)=P_{0}E(\tau)\sin(\omega_{c}\tau)e^{-az}H(a-r), \tag{8}\]  
where \(E(\tau)\) is the AM envelope of the primary wave. \(a\) is the radius of the source aperture and \(\alpha\) is the absorption coefficient. \(H(\cdot)\) is the unit step function. Hence, the self-demodulated audible wave can be expressed as [28; 29; 33]  
\[p_{2}(r,z,\tau)\approx\frac{\beta P_{0}^{2}a^{2}}{16\rho_{0}c_{0}^{4}a^{2} \alpha}\frac{\partial^{2}}{\partial\tau^{2}}E^{2}(\tau). \tag{9}\]  
For a proof-of-principle study to explore PAL as the secondary ANC source, in this paper we choose a simple AM method using the double-sideband amplitude modulation (DSBAM) technique [24]. For the target audio under modulation \(g(\tau)\), the DSB modulated signal for PAL actuation is expressed as  
\[g_{DSB}(\tau)=[1+mg(\tau)]\sin{(\omega_{c}\tau)}, \tag{10}\]  
where \(0<m\leq 1\) is the modulation index.