Prime factor decomposition of the discrete cosine transform and its hardware realization

The discrete cosine transform (DCT) is an useful tool in solving many digital signal processing problems. One of its more interesting applications is in the realization of a bandpass filter-bank that is necessary to perform conversion between time-division and frequency-division voice multiplexing systems. In this particular example the transform size is dictated by the sampling frequencies and the computationally-efficient power-of-2 algorithms may not applicable. To handle such situations, a new DCT algorithm based on a prime factor decomposition of the transform-length is presented in this paper. To begin with, the definition of the DCT is introduced with a brief discussion of the application mentioned. The existence of the prime factor decomposition technique is derived based on a similar decomposition for the discrete Fourier transform (DFT). The prime factor algorithm for the DCT (PFA-DCT) is then described step by step and followed by an example of a 12-point transform which delineates the method and brings out the essence of the algorithm. A hardware implementation of a 72-point DCT is outlined to illustrate the usefulness of this algorithm in practical systems.