## 1 Introduction  
Active noise control (ANC) is intended to suppress an external noise in an active way that an anti-noise with opposite phase and same magnitude cancels the external noise [1; 2; 3]. Feedforward ANC, which is the typical ANC scheme, captures a reference signal with a microphone at noise source to provide the input of the ANC system, which drives the anti-noise. Unlike the feedforward ANC, feedback active noise control (FBANC) does not need a microphone or sensor to install at noise source for a reference signal but utilizes a feedback mechanism to internally generate a reference signal. The structure of the feedback ANC is considered as one of the typical internal model control (IMC) in the sense that the estimated model of the secondary path is used for the generation of a reference signal [2]. For this structural nature, in applications where noise sources are many and propagation paths of noises are unspecified, e.g., ANC headphones, the feedback ANC is preferred to be implemented rather than the feedforward scheme, in which independent reference sensors need to be installed [4; 5; 6]. In practical feedback ANC applications, the stability problem caused by the feedback mechanism is an important issue in the design and implementation. In the feedback mechanism, there are an electro-acoustic coupling path and its estimated model, which is in a form of an FIR filter. The electro-acoustic coupling path called the secondary path includes a loud speaker, a microphone, an acoustic propagation path, and AD/DA converters. The feedback ANC structure is reduced to the feedforward structure provided the secondary path model is perfect. Theoretically, if the secondary path model is perfectly estimated, the feedback ANC scheme has no stability problem [2; 3; 4]. However, in practice, it is not available to exactly estimate them as a linear filter model, since the electro-acoustic devices and the acoustic path are a non-linear and time-varying system [3; 7; 8; 9]. If the estimated secondary path model is not perfect, its adverse effects considering both the stability aspect of the feedback loop and the performance issues, such as convergence and stability, of the FXLMS-based adaptive algorithm must be analyzed. The previous research [7] gives examples of combined analysis to show the convergence stability of adaptive active noise control systems. There are, in the literature, some analyses showing how to guarantee convergence of the feedfowards FXLMS without secondary path modeling by applying varying-delay LMS approach, or fuzzy approach [10; 11]. However, it is true that the comprehensive analysis of the FXLMS-based feedback ANC considering all the feedback structural aspects is generally a hard problem. The difficulty is caused by its inherent structural complexity and, as well, by the difficulty of reflecting the wide range of environmental variations of ANC application. This paper exclusively discusses the stability aspect of the feedback loop to derive a closed-form stability bound equation of the feedback ANC system with a delay mismatch in the secondary path model. The availability of the stability bound in a form of closed equation, relating with various physical domain design factors such as noise characteristics, or system parameters, would be very helpful for practical feedback ANC applications. A typical method to determine the stability of feedback systems is to examine the pole locations of a system's characteristic equation. In many applications of feedback ANC system, it is not possible to directly obtain the roots of the characteristic equation, because the equation is generally of a very high order [2]. Particularly, the delay of the acoustic path, which is relatively long compare to that of the electric path, makes the order of the characteristic equation even higher. Therefore, many literatures on the stability analysis of feedback ANC systems have been analyzed using the frequency domain analysis approach [3; 7; 8].