Bidirectional conversion between DCT coefficients of blocks and their subblocks

In the context of the discrete Fourier transform (DFT) and the discrete cosine transform (DCT), we present efficient methods for bidirectional conversion between transform coefficients of a signal block (one- or two-dimensional) and transform coefficients of the signal's subblocks. The DFT case is used to exemplify the underlying theoretical issues while the DCT case is considered for its practical importance. For typical DCT block sizes, our algorithms result in a 20% savings in multiplications over the fastest existing methods and have no penalty in reordering operations.