Small-Size Algorithms for Quaternion Discrete Fourier Transform

The quaternion discrete Fourier transform (QDFT) is a powerful tool in modern digital signal processing, even though until recently this transformation seemed exotic. In recent years, quite a lot of publications have appeared devoted to effective ways to calculate this transformation. In particular, in one of our previous publications, we presented an economical algorithm for calculating one-dimensional QDFT and showed that this algorithm has the lowest computational complexity among all known algorithms of this type. This generalized algorithm is suitable for computing the QDFT of any sequence in which the number of elements is a power of two. However, as it turned out, there are additional possibilities that make it possible to further reduce the computational complexity of the developed algorithm for each specific N. In this article, we provide some examples of the synthesis of such algorithms for short-length input sequences (samples of signals). In particular, algorithms for N ∊ {2, 3, 4, 5, 6, 7, 8} are presented. A parallel implementation of the proposed algorithm allows for saving more than half of the number of required multipliers in each case compared with the parallel implementation of the naive methods of calculation.