Direct methods for computing discrete sinusoidal transforms

Hou (1987) has developed an algorithm for DCT-II which is similar to a decimation-in-frequency (DIF) algorithm and is numerically stable. However, an index mapping is needed to transform the DCT to a phase-modulated DFT, which may not be performed in-place. A variant of Hou´s algorithm is presented which is both in-place and numerically stable. The method is generalised to compute the entire class of discrete sinusoidal transforms. By making use of the decimation-in-time (DIT) and DIF concepts and the orthogonal properties of the DCTs, it is shown that simple algebraic formulations of these algorithms can readily be obtained. The resulting algorithms are regular in structure and are more stable. By means of the Kronecker matrix product representation, the 1-D algorithms introduced can readily be generalised to compute transforms of higher dimensions. These algorithms, which can be viewed as the vector-radix generalisation of the present algorithms, share the in-place and regular structure of their 1-D counterparts