The fast DCT-IV/DST-IV computation via the MDCT

The discrete cosine transform of type IV (DCT-IV) and corresponding discrete sine transform of type IV (DST-IV) have played key role in the efficient implementation of orthogonal lapped transforms and perfect reconstruction cosine-modulated filter banks such as the oddly stacked modified discrete cosine transform (MDCT) or equivalently, the modulated lapped transform (MLT). However, the DCT-IV and DST-IV of double sizes are related to two variants of filter banks defined by Dolby Labs AC-3 digital audio compression algorithm. Since these two variants of filter banks are efficiently computed by recently proposed new fast algorithm for the oddly stacked MDCT (Signal Processing 82 (2002) 433), it is shown that the efficient DCT-IV and DST-IV computation can be realized via the MDCT of double size. The careful analysis of regular structure of the new fast MDCT algorithm allows to extract a new DCT-IV/DST-IV computational structure and to suggest a new sparse matrix factorization of the DCT-IV matrix. Finally, the new DCT-IV/DST-IV computational structure provides an alternative efficient implementation of the forward and inverse MDCT in layer III of MPEG (MP3) audio coding.