# A New Effective Narrowband Active Noise Control System for Accommodating Frequency Mismatch
## 1 Introduction  
In our daily work and life, many disturbing noises can not only cause psychological discomfort for humans, but could also strengthen the structural load and, hence, give rise to fatigue damage [1; 2; 3; 4; 5]. These noises are generated by rotating machines such as cutting machines, electric fans, and motors. The other characteristic low frequency humming noise is due to the periodic vibration of the iron core with the change of the excitation frequency in the power transformer box [6]. A similar noise to this may be modeled as a sinusoidal signal in additive noise. Usually, the frequencies of the noise signals are arbitrary, and their magnitudes are time-varying. The noise generated by rotating or reciprocating equipment is mainly low frequency harmful noise [7]. Narrowband active noise control (NANC) is commonly used in noise-cancelling headsets [8; 9], modern automobiles [10; 11; 12; 13], and vehicle noise [14; 15; 16]. The conventional NANC system has shown great superiority in low frequency signal processing [17], while the use of sound-absorbing and sound-insulating materials shows a poor performance in dealing with the low frequency part of this noise [18]. In general, there are two ways to obtain reference signals. One is to obtain the noise directly by placing an acoustic signal sensor in the noise region, and the other is to use a piece of electronics hardware to convert the detected velocity or acceleration into the samefrequency signal via a linear conversion relation. However, to avoid the acoustic feedback problem caused by acoustic sensors, the aforementioned non-acoustic sensors can be used instead in most conventional feedforward NANC systems. Compared with the former method, it is a good choice to detect the fundamental frequency information of the primary noise to generate a reference signal in [19]. Typically, the reference signal is generated by a waveform synthesis method to control the narrowband noise, whose amplitude is the same as and the phase is opposite to the primary noise. The secondary signal produced by the filtered reference signal is then fed to the loudspeaker to attenuate the noise. In real life, non-acoustic sensors or cosine wave generators maybe not be very precise, fatigue and aging, which could cause indeterminacy. If the non-acoustic sensor error is nonnegligible or sufficiently large, the frequency of the reference signal fed to each channel will be at odds with the frequency possessed by the primary noise signal, and the NANC system will present worse performance. Therefore, the difference in frequencies between the reference signal and the primary noise is defined as the frequency mismatch (FM). The simulation results have shown that the noise reduction performance of active noise control system will be severely degraded even if it is just 1% FM [20]. Various different adaptive filtering algorithms have been proposed to deal with FM. Based on the theory of acoustic superposition, a typical parallel adaptive NANC system designed by [PERSON] et al. could eliminate the target single noise by means of a second-order filter in [21], but a problem that existed in the system is that the frequency of the reference signal was inconsistent with the target noise signal. Considering the flaws in the system, a new auto-regressive (AR) ANC system was proposed by [PERSON] and other scholars in [22], whose reference signal is preprocessed by the AR model in advance, thus correcting the reference signal. In this way, the system was found to compensate the FM by 10%. The statistical analysis of the NANC system is presented in [23; 24], where [PERSON] and [PERSON] elaborated on the condition in which the system reaches a steady state with the FM, such as the optimum step size and excess mean square error for the NANC system. However, this construction makes it difficult to handle large misalignments.