FFTs of Arbitrary Dimensions on GPUs

We present the fast Fourier transform (FFT), of arbitrary dimensions, on the graphics processing unit (GPU). The FFT on GPUs exploits the architecture in its image processing capability, as well as its particular graphics/image rendering capacity. It also couples the processing and rendering furthermore. We view the GPU as a special architecture that supports fine-granularity, two-dimensional (2D) memory accesses at the level of application programming interface (API). The unique architectural features are utilized by mathematical and algorithmic means richly associated with the FFT, which has an important role in signal and image processing and in scientific computing in general. At the kernel of the FFT on GPUs, i.e., at the level innermost to the the native architecture, are the primitive array operations for the 2D FFT, instead of the 1D FFT. Basically, the 2D array operations have natural mappings to the architecture by their joint potential in performance. A lower or higher dimensional FFT is described in terms of the kernel operations, in order to exploit the architecture at the application programming level. This algorithmic abstraction of the operation primitives and their compositions enables, especially, the 2D twiddle scaling, which uses less memory space, and the 2D bit-reversal permutation, which manifests the unique GPU feature in memory access. The 2D FFT on GPUs is detailed in [3], where mixed-radix factorizations are also used to further utilize the memory resource. In this paper we turn the focus onto FFTs of other dimensions on GPUs. We describe the FFT reformulation and data mappings. We provide experimental results to demonstrate that the 2D FFT performance is conveyed to the other FFTs as well.