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In recent years, tree-structured analysis/reconstruction systems have been extensively studied for use in subband coders for speech. In such systems, it is imperative that the individual channel signals be decimated in such a way that the number of samples coded and transmitted do not exceed the number of samples in the original speech signal. Under this constraint, the systems presented in the past have sought to remove the aliasing distortion while minimizing the overall analysis/reconstruction distortion. In this paper, it is shown that it is possible to design tree-structured analysis/reconstruction systems which meet the sampling rate condition and which result in exact reconstruction of the input signal. The conditions for exact reconstruction are developed and presented. Furthermore, it is shown that these conditions are not overly restrictive and high-quality frequency division may be performed in the analysis section. A filter design procedure is presented which allows high-quality filters to be easily designed.
Smith et al. (Sun,) studied this question.