## 2 Materials and Methods
### Parametric Audio Modulation Method for the Hearing Aid  
In the parametric audio system, the audio is generated because of the interference of the carrier frequency and sideband signals frequency due to the nonlinearity of the air when the strong ultrasonic wave modulated by the audio signal is radiated from the transducer. To generate a focused beam in the hearing aid receiver, it is necessary to select an ultrasonic modulation method suitable for this. Parametric audio is a secondary sound generated by air nonlinearity when a strong ultrasonic beam travels, with the characteristics of a sharp audio beam. So far, several parametric audio-modulation schemes have been developed for directional loudspeakers [13; 14]. Since the DSBAM (double-sided band-amplitude modulation) method was first implemented by [PERSON] et al. [15], SRAM (square root amplitude modulation), SSBAM (single-sided band-amplitude modulation), pulse width modulation (PWM), etc., have been proposed to lower THD and implement an effective system. PWM is inexpensive and easy to implement, but it is not suitable for hearing aids because of its high THD. In this study, the authors decided to use the SRAM method, which is easy to implement and has low THD. In the SRAM approach, the audio signal's envelope is pre-processed by taking its square root before modulating the ultrasonic carrier. This square root pre-modulation, based on [PERSON]'s far-field solution, effectively linearizes the demodulation process and thereby reduces undesirable harmonic distortion in the audible output. That is, the audible signal spectrum, \(\omega_{m}\), among the various signals in which the nonlinear density modulation of air occurs, corresponds to the difference frequency component between the carrier, \(\omega_{s}\), and the sideband's signal spectrum, \(\omega_{s}\pm\omega_{m}\). Parametric audio has a characteristic in which the audio power gradually increases with distance from the radiator and then attenuates, and is transmitted with high directivity. At this time, when the sound pressure of the SRAM wave is \(p_{0}\), the sound pressure, \(p(t)\), of the resulting  
Figure 1: (**a**) Schematic of a PU (pulsed ultrasonic) receiver using multilayer PZT; (**b**) components of the PU receiver; and (**c**) a photo of an ultrasonic PU receiver and BA (balanced armature) receiver in the same-sized silicon cap. output audio signal can be described by the following Equation (2), which is well known as far-field solution, as described in [16]:  
\[p(t)=\frac{\beta p_{0}^{2}\varsigma_{0}}{16\pi\rho_{0}\varsigma_{0}^{4}r\alpha_{0} }\frac{d^{2}}{dt^{2}}E^{2}(\tau) \tag{2}\]  
where \(\beta\) is the nonlinearity coefficient of the air; \(p_{0}\) is the amplitude of the ultrasonic carrier wave (primary pressure); \(S\) is the cross-sectional area of the ultrasonic beam; \(\alpha\) is the acoustic absorption coefficient of the air; \(\rho_{0}\) is the air density; \(c\) is the sound velocity in the air; \(r\) is the propagation distance from the source; \(m(t)\) is the modulation envelope of the audio signal. Here, when the distance, \(r\), from the sound source is very small, \(\tau=t\) can be substituted. If the constant term of Equation (2) is summarized and expressed simply as \(K\), it can be seen that the output sound pressure, \(p(t)\), is proportional to the second-order time-derivative value of the squared envelope, \(E(\tau)^{2}\), as shown in the following equation:  
\[p(t)\propto\frac{d^{2}}{dt^{2}}E^{2}(\tau) \tag{3}\]  
In general, hearing aid receivers are designed to have the ability to provide an audio sound of up to 100 dB SPL to the hearing-impaired people using them. At this time, it is necessary to know how much primary sound pressure, \(p_{0}\), is required to generate a parametric sound pressure of 100 dB SPL, as an ultrasonic radiator of a size that can fit inside the receiver. For this, assuming 80% modulation at 1 kHz, we must calculate the \(p_{0}\) required in Equation (3).