Rectangular Transforms for Digital Convolution on the Research Signal Processor

The Rectangular Transform (RT) method for computing convolutions belongs to a family of Reduced Computational Complexity (RCC) algorithms. Convolution calculations by the RT method were programmed for the Research Signal Processor (RSP) and run on the RSP simulator, giving tabulations of numbers of RSP machine cycles. One of the original objectives was to see how well the original RSP architecture was suited to the RCC algorithms and to be able to make suggestions for possible changes. The results are also intended to demonstrate the efficiency of the RT convolution algorithms on a microprocessor with a limited instruction set and to show how to construct efficient RT programs for digital convolution. All results are given for the original RSP, as it was before the modijcation which resulted in the Real-time Signal Processor described in another paper in this issue [ l ]. Introduction and background Considerable theoretical advances have been made in Reduced Computational Complexity (RCC) algorithms for some basic calculations such as the computation of multi-linear forms, convolutions, and the Discrete Fourier Transform (DFT). (See [2-61). From these have come what are now known as the Rectangular Transform (RT) algorithms and the Winograd Fourier Transform (WFT) algorithms. In these algorithms, the theory deals with the reduction in the number of multiplications which, unfortunately, is usually achieved at the cost of an increase in program complexity and, sometimes, in the number of additions. For these reasons, and due to the fact that multiplication is almost as fast as addition on many general-purpose computers, there has been little use made of the powerful new computational complexity theory and the algorithms resulting from it. It is quite natural, therefore, to look about to see where RCC algorithms can be used to advantage. One is led to consider microprocessors in which multiplications are relatively slow and where the machine typically is used in dedicated applications where it is worth the effort to reduce computing time.