Signal Processing 87 (2007) 1355–1362 Canonical transformations of the discrete cosine transform Mohamed F. Mansour Texas Instruments Inc., DSP Solutions R&D Center, Dallas, TX 75243, USA Received 27 March 2006; received in revised form 23 July 2006; accepted 17 November 2006 Available online 12 December 2006 Abstract We provide different transformation formulae between the different discrete cosine transform (DCT) types of the same size. The transformations use only diagonal and special lower/upper triangular matrices that minimize the overhead of transformation. These transformations provide a tool for using any of the DCT types as a core module for computing all other types. r 2006 Elsevier B.V. All rights reserved. Keywords: Discrete cosine transform; DCT; Type II; Type III; Type IV 1. Introduction The discrete cosine transform (DCT) [1] has an important role in many signal processing applica- tions especially in multimedia coding. International standards (e.g., MPEG and ISO standards) for audio and video coding have widely adopted the cosine-modulated filter banks for subband coding. The efficient polyphase implementation of cosine- modulated filter banks requires a fast DCT at both the analysis and synthesis sides. The three common types of the DCT are: Type II : xðnÞ¼ X M1 k¼0 X ðkÞ cos p M nðk þ 0:5Þ   , Type III : xðnÞ¼ X M1 k¼0 X ðkÞ cos p M kðn þ 0:5Þ   , Type IV : xðnÞ¼ X M1 k¼0 X ðkÞ cos p M ðn þ 0:5Þðk þ 0:5Þ   . In audio coding standards there are two broad categories of the subband coding module, namely, the frequency-domain aliasing cancellation (FDAC) filter banks [3], and the time-domain aliasing cancellation (TDAC) filter banks [2]. The FDAC filter banks are adopted in the earlier versions of the audio coding standards, e.g., MPEG-1, whereas the TDAC filter banks are adopted in later standards (e.g., AAC, WMA). The core cosine transform for the FDAC analysis and synthesis filter banks are the DCT-III and the DCT-II (respectively), whereas the core cosine transform for TDAC analysis and synthesis filter banks is the DCT-IV. The computation procedure for the different types of the DCT transforms is quite different. Usually FFT-based approaches (e.g. [4–6]) are used in practice because of the availability of an FFT module in most practical DSP systems. The core transform for the FFT-based implementations of an M-point DCT is an M=2-point complex FFT. However, the preprocessing, postprocessing, and data mapping is different for the different DCT types. The computational requirements of these ARTICLE IN PRESS www.elsevier.com/locate/sigpro 0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2006.11.007 E-mail address: mfmansour@ti.com.