Multidimensional vector radix FHT algorithms
In this paper, efficient multidimensional (M-D) vector radix (VR) decimation-in-frequency and decimation-in-time fast Hartley transform (FHT) algorithms are derived for computing the discrete Hartley transform (DHT) of any dimension using an appropriate index mapping and the Kronecker product. The proposed algorithms are more effective and highly suitable for hardware and software implementations compared to all existing M-D FHT algorithms that are derived for the computation of the DHT of any dimension. The butterflies of the proposed algorithms are based on simple closed-form expressions that allow easy implementations of these algorithms for any dimension. In addition, the proposed algorithms possess properties such as high regularity, simplicity and in-place computation that are highly desirable for software and hardware implementations, especially for the M-D applications. A close relationship between the M-D VR complex-valued fast Fourier transform algorithms and the proposed M-D VR FHT algorithms is established. This type of relationship is of great significance for software and hardware implementations of the algorithms, since it is shown that because of this relationship and the fact that the DHT is an alternative to the discrete Fourier transform (DFT) for real data, a single module with a little or no modification can be used to carry out the forward and inverse M-D DFTs for real- or complex-valued data and M-D DHTs. Thus, the same module (with a little or no modification) can be used to cover all domains of applications that involve the DFTs or DHTs.