Orbital Algorithms and Unified Array Processor for Computing 2D Separable Transforms

Abstract

The two-dimensional (2D) forward/inverse discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST), discrete Hartley transform (DHT), discrete Walsh-Hadamard transform (DWHT), play a fundamental role in many practical applications. Due to the separability property, all these transforms can be uniquely defined as a triple matrix product with one matrix transposition. Based on a systematic approach to represent and schedule different forms of the $n\times n$ matrix-matrix multiply-add (MMA) operation in 3D index space, we design new orbital highly-parallel/scalable algorithms and present an efficient $n\times n$ unified array processor for computing {\it any} $n\times n$ forward/inverse discrete separable transform in the minimal $2n$ time-steps. Unlike traditional 2D systolic array processing, all $n^2$ register-stored elements of initial/intermediate matrices are processed simultaneously by all $n^2$ processing elements of the unified array processor at each time-step. Hence the proposed array processor is appropriate for applications with naturally arranged multidimensional data such as still images, video frames, 2D data from a matrix sensor, etc. Ultimately, we introduce a novel formulation and a highly-parallel implementation of the frequently required matrix data alignment and manipulation by using MMA operations on the same array processor so that no additional circuitry is needed.