Generalized fast mixed-radix algorithm for the computation of forward and inverse MDCTs

Abstract

This paper presents a generalized mixed-radix decimation-in-time (DIT) fast algorithm for computing the modified discrete cosine transform (MDCT) of the composite lengths N=2× q m , m≥2, where q is an odd positive integer. The proposed algorithm not only has the merits of parallelism and numerical stability, but also needs less multiplications than that of type-IV discrete cosine transform (DCT-IV) and type-II discrete cosine transform (DCT-II) based MDCT algorithms due to the optimized efficient length-( N/ q) modules. The computation of MDCT for composite lengths N= q m ×2 n , m≥2, n≥2, can then be realized by combining the proposed algorithm with fast radix-2 MDCT algorithm developed for N=2 n . The combined algorithm can be used for the computation of length-12/36 MDCT used in MPEG-1/-2 layer III audio coding as well as the recently established wideband speech and audio coding standards such as G.729.1, where length-640 MDCT is used. The realization of the inverse MDCT (IMDCT) can be obtained by transposing the signal flow graph of the MDCT.