I. Introduction
An adaptive filter is often employed in an environment of unknown statistics for various purposes such as system identification, inverse modelling for channel equalization, adaptive prediction, and interference cancelling. Knowing nothing about the environment, the filter is initially set to an arbitrary condition and updated in a step-by-step manner toward an optimum filter setting. For updating, the least mean-square (LMS) algorithm is often used for its simplicity and robust performance [1]. However, the LMS algorithm exhibits slow convergence when used with an ill-conditioned input such as speech and requires a high computational cost, especially when the system to be identified has a long impulse response [2]. One promising method that improves the performance and reduces the computational cost is subband adaptive filtering (SAF), in which the input is decomposed into a number of subband signals, and the adaptive filtering is performed on each subband [3]–[10]. It has the potential for a faster convergence and a lower computational complexity than a fullband structure. However, a subband structure suffers from two deficiencies. First, the interband aliasing that is introduced by the downsampling process required in reducing the data rate is unavoidable and degrades the performance. Second, the filter bank introduces additional computation and system delay. For these reasons, various SAF structures were proposed. In [3], an SAF using nonoverlapping subbands was proposed to mitigate the interband aliasing; however, it leads to output distortion in the form of spectral gaps that are very large. In [4] and [5], cross-adaptive filters between the subbands were used to compensate for the interband aliasing. Their use requires additional computational cost, and despite the increased computational complexity, its convergence is slow. In [6], an SAF using auxiliary channels was proposed to avoid both interband aliasing and spectral gaps; however, this results in increased complexity. In [7], [8], an oversampled filter bank was used to reduce interband aliasing; however, its computational complexity was increased, and it has been found to have slow convergence [2]. In [9], [10], subband structures that can exactly model an finite-impulse-response (FIR) system were proposed. In [9], a subband structure based on the polyphase decomposition of an FIR system to be adaptively modelled was proposed; however, its computational complexity is similar to that of the fullband structure. In [10], a subband structure proposed in [11] was used to exactly model an arbitrary FIR system. This structure requires additional computation for adaptively filtering the overlappings between adjacent subbands.