## 1 Introduction  
The performance of an acoustic or audio system is highly dependent on the performance of the loudspeaker located at the final stage of the system. Generally, the performance of a loudspeaker is evaluated by its frequency response, especially its amplitude response, but it is not sufficient to evaluate the loudspeaker assuming it is a linear system. This is because nonlinearities in loudspeakers become more pronounced as the diaphragm is driven at higher amplitudes to increase the sound volume. The details of loudspeaker nonlinearity are described in the literature [1]. The two types of nonlinearity are: the nonlinearity caused by the magnetic flux density changing with amplitude as the diaphragm is driven at large amplitudes and the nonlinearity caused by the stiffness of the edges and dampers that support the diaphragm changing with amplitude as the diaphragm is driven at large amplitudes. Loudspeaker nonlinearities generate nonlinear distortions that degrade sound quality. Here, nonlinear distortion is classified into harmonic distortion, in which frequency components of integer multiples of the input frequency are generated, and intermodulation distortion, in which frequency components of the sum and difference of multiple input frequencies are generated. In particular, second- and third-order nonlinear distortions are dominant in loudspeakers, and reducing these nonlinear distortions is essential for improving sound quality. To compensate for nonlinear distortions, it is important to accurately model the nonlinearities of the loudspeaker. A mirror filter [2; 3] that relies on the mechanism that generates nonlinearity based on the nonlinear motion equation of the loudspeaker has been proposed as a method of modeling the nonlinearity of the loudspeaker. The mirror filter also reduces the nonlinear distortion generated in the loudspeaker, and can be implemented using a nonlinear IIR filter [4; 5; 6; 7], which requires a low amount of computation. However, nonlinear compensation using a mirror filter or nonlinear IIR filter is based on the nonlinear motion equation of the loudspeaker and can only compensate for nonlinear distortion caused by the nonlinearity expressed in the motion equation. As a result, the compensation performance is known to be quite limited and the intermodulation distortion cannot be compensated [8]. On the other hand, a method has been proposed by assuming the loudspeaker as a general nonlinear system and approximating its nonlinearity by the Volterra series expansion [9], which is an extension of the Theler expansion to systems with memory and is composed of Volterra kernels that represent the impulse response of a linear system [10]. It is composed of Volterra kernels that represent second-, third-, and higher-order nonlinearities, corresponding to the impulse response of a linear system. The Volterra filters [11; 12], which realize these Voltera kernels as digital filters, can be used to model the nonlinearities of loudspeakers. Here, the coefficients of the Volterra filter are identified using the adaptive Volterra filter [13; 14; 15] or frequency response method [16]. The Volterra filter coefficients obtained by modeling can then be used to construct a linearization system [17; 18] to compensate for nonlinear distortion and placed in the front stage of the loudspeaker to improve sound quality. The Volterra filter can be used to compensate for nonlinear distortion not only in loudspeakers, but also in parametric array loudspeakers [19; 20; 21] using ultrasonic waves because the nonlinear distortion compensation is independent of the cause of the nonlinear distortion [22]. However, Volterra filters are very computationally expensive because of their multi-dimensional filter structure. In particular, the third-order Volterra filter used for compensation of third-order nonlinear distortions has a three-dimensional filter structure, which requires a very large amount of computation.