## 1 Introduction  
While driving a passenger car, a driver can hear many types of sounds inside the car [1]. Among them, the engine is one of the dominant noise sources and influences the sound quality of the interior environment [2]. The frequency components of engine noise are correlated to the rotating speed of the crankshaft (i.e., rpm) of an engine and its harmonics [3]. These harmonic components are called 'orders'. This correlation relationship between the engine noise and the rotating speed enables the control of the engine sound by using active control systems. Active noise control (ANC) has received considerable attention in recent decades, owing to its superior advantages in cancelling low frequency noise as compared to passive methods [4; 5]. Adaptive feedforward control systems are very popular in active control systems (such as ANC) for the reduction of engine noise inside a car cabin [4; 5; 6; 7]. This system uses an order signal as a reference. Order signals are sinusoidal signals within a narrow band, and can be synthesized by using the controller area network (CAN) bus (or a tachometer) in a car. The frequency of the order signal is changed in the run-up condition of the engine speed, and is a non-stationary signal. Many adaptive filters can be used in ANC systems [8]. In previous studies, the filtered x-LMS algorithm (FxLMS) based on the adaptive notch filter (ANF) has been widely used for the ANC of engine noise [7]. The filter length of the adaptive filter and the step-size should be carefully selected to ensure that the least mean squares (LMS) algorithm convergesrapidly and is stable. [PERSON] [9] discussed the method to select the appropriate filter length and step size parameter. Although there are some implementations of variable length LMS adaptive filters, their performance is not steady [10]. [PERSON] et al. introduced a variable LMS adaptive filter [11]. They suggested an implementation of multiple adaptive filters with variable lengths. [PERSON] [12] proposed a method to speed up this convergence rate by varying the length of the adaptive filter, utilizing the larger step sizes allowed for short filters. These studies show the possibility of acquiring a rapid convergence when the optimal length of an adaptive filter is applied to the control of engine noise inside the cabin of a car. An adaptive notch filter used for the ANC of engine noise removes the primary spectral components within a narrowband centered at the frequency of reference. The filter length of the adaptive notch filter used in this study is short and two weights have been used. The other inherent feature of the LMS algorithm is the step size, and it requires careful adjustment. A small step size, required for small excess mean square error, results in slow convergence. A large step size, required for fast adaptation, may result in the loss of stability. Therefore, many modifications of the LMS algorithm, where the step size changes during the adaptation process depending on particular characteristics, have been developed [9; 13]. The performance of the adaptive notch filter depends heavily on the accuracy of the frequency of the reference signal because the bandwidth of the adaptive notch filter is proportional to the convergence step size [14; 15]. The frequency change of the reference signal deteriorates the performance of an ANC system [16]. Therefore, in order to improve the convergence rate, the optimization of the step size in the adaptive notch filter is required for the ANC of engine noise [6]. [PERSON] [17] reviewed seventeen of the best-known variable step size LMS (VS-LMS) algorithms to the degree of detail that allows implementing them. The performance of the algorithms is compared in typical applications. According to the conclusion of this review, no VS-LMS algorithm appears to be as versatile, easy to use, and well-suited for real-time applications as the normalized least mean squares filter (NLMS).