By J. Arndt

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5 Half cyclic convolution for half the price? 8a (without computing h(1) ) is called half cyclic convolution. Apparently, one asks for less information than one gets from the acyclic convolution. One might hope to find an algorithm that computes h(0) and uses only half the memory compared to the linear convolution or that needs half the work, possibly both. It may be a surprise that no such algorithm seems to be known. Here is a clumsy attempt to find h(0) alone: Use the weighted transform with the weight sequence vx = V x where V n is very small.

Memory access is without skips) 6. Multiply each matrix element (index r, c) by exp(−σ 2 π i r c/n). 7. Apply a (length R) FFT on each column (of the transposed matrix). (memory access with C-skips) Note that steps 3, 4 and 5 constitute a length-C convolution on each row. cc] The algorithm can be used for mass storage convolutions if care is taken to minimize the number of disk seeks. Minimization of the number of disk seeks The number of disk seeks has to be kept minimal because these are slow operations which, if occur too often, degrade performance unacceptably.

The process, however, is less than delightful and error-prone. It would be much better to have another program that takes the existing FFT code as input and emit the code for the generator. Let us call this a meta-generator. Implementing such a meta-generator of course is highly nontrivial. It actually is equivalent to writing an interpreter for the language used plus the necessary data flow analysis. A practical compromise is to write a program that, while theoretically not even close to a meta-generator, creates output that, after a little hand editing, is a usable generator code.