## 1 Introduction  
Wireless headphones have become one of the most prevailing wearable personal devices in recent years. For most high-qualified headphones, an active noise control (ANC) module is generally equipped as a standard configuration, which is designed to reduce the outside noise and offer a better listening environment. The control strategies for ANC headphones can be classified into the feedforward, the feedback and the hybrid ones, among which the feedback control strategy has the longest history of research and application and is discussed in the rest of this paper. In feedback ANC headphones, the noise reduction (NR) within the low frequency band is generally realized at the price of noise enhancements at other frequencies, which is well-known as the waterbed effect [1] in the literature. Thus, the basic principle to design a feedback controller for ANC headphones is to maximize the NR performance within the low frequency band while maintain the robust stability of the system and restricting the noise enhancement at high frequencies. Meanwhile, the NR bandwidth is restricted by the delay of the open-loop plant response [1], which indicates that the feedback microphone should be placed near the speaker and the latency of the controller should also be minimized. The feedback controllers of ANC headphones were first implemented with analog circuits. Different design procedures were proposed by [PERSON] et al. [2], [PERSON] [3] and [PERSON] et al. [4, 5], respectively. However, the main drawback of an analog ANC controller is the uncertainty of its frequency response (FR), which is mainly caused by the uncertainty of the capacitors used in the circuits. In massive production, the actual capacitance would deviate from its nominal value (up to 5%, 10% or even 20%). As a result, the actual NRperformance would not be in accordance and the averaged NR performance would also be limited. Compared with analog circuits, digital controllers have the advantages of high consistency as well as high flexibility. [PERSON] et al. [6] proposed an H2/Hoo design method for feedback controllers, where a convex programming problem using FIR filters is established with the help of internal model control (IMC). [PERSON] et al. [7] proposed an intuitive approach where the feedback controller is first designed in frequency domain and then approximated by an FIR filter. With the IMC structure, the feedback controller could even be made fully adaptive [8]. However, the modeling errors of the secondary path deteriorate the stability bound of the system [9] and the waterbed effect should also be taken care of [10]. A semi-adaptive strategy [11] can also be considered with a digital controller, where the controller coefficients are selected from a set of pre-trained fixed filters according to the current noise environment. The major problem of a digital controller is that an extra delay is introduced, which could be reduced by a higher sampling frequency. However, the computational burden would be greatly enhanced with the above methods where FIR controllers are used. Specially designed very large scale integration circuits [12] might be necessary in this case. This problem could be alleviated with the usages of a warped FIR filter [13] or cascade biquad filters [14], the latter of which is particularly advantageous for commercial products since the controller cost could be greatly lowered and the battery lifetime could also be lengthened. The effectiveness of cascade biquad feedback controllers has been confirmed in [14], in which genetic algorithm and Nelder-Mead simplex methods are combined to search the optimal parameters of the feedback controller. To the best of the authors' knowledge, cascade biquad filters have been widely used as controllers for commercial ANC headphone products in recent years. In the works mentioned above, the NR performance was generally evaluated by the feedback microphone.