Hello everyone: First of all, thanks to all those who have posted before and for saving the archives. As you know this is my first post here. I searched the archives by terms and by circular convolution but did not find anything, hope I didn't miss a post. You see, I'm trying to implement a parallel cyclic convolution algorithm and was wondering if someone can point out the fastest cyclic convolution algorithm available for a composite length, that is N=R*S where both R and S can also be composite integers. I know we can use the FFT as: x cconv h =invFFT{ FFT(x) .* FFT(h) } but is there anything else. I'm planning to use typical lengths but will increase it as much as I can. Any help will be appreciated, thanks Ivan

# fastest cyclic convolution algorithm

Started by ●February 9, 2006

Reply by ●February 9, 20062006-02-09

Ivan72 wrote:> I know we can use the FFT as: > > x cconv h =invFFT{ FFT(x) .* FFT(h) } > > but is there anything else.No, that is basically it. If you find anything else, please report!

Reply by ●February 9, 20062006-02-09

> >Ivan72 wrote: >> I know we can use the FFT as: >> >> x cconv h =invFFT{ FFT(x) .* FFT(h) } >> >> but is there anything else. > >No, that is basically it. If you find anything else, please report! > >Trust me I will. One more question, Besides the obvious N^2 multiplications, what options would you consider if you don't want to go to the Frequency domain?. Thanks, Ivan

Reply by ●February 9, 20062006-02-09

Ivan72 wrote:> > > >Ivan72 wrote: > >> I know we can use the FFT as: > >> > >> x cconv h =invFFT{ FFT(x) .* FFT(h) } > >> > >> but is there anything else. > > > >No, that is basically it. If you find anything else, please report! > > > > > Trust me I will. > > One more question, > Besides the obvious N^2 multiplications, what options would you consider > if you don't want to go to the Frequency domain?.There don't seem to be any other algorithmic tricks that work in general. Perhaps you can reduce the size of the filter h by using a sparse approximation that can be computed more efficiently. That depends on what you want to do, and how accurate the result has to be.

Reply by ●February 9, 20062006-02-09

"Ivan72" <utility72@hotmail.com> wrote in message news:eoWdnecePvIoyXbenZ2dnUVZ_sednZ2d@giganews.com...> > >>Ivan72 wrote: >>> I know we can use the FFT as: >>> >>> x cconv h =invFFT{ FFT(x) .* FFT(h) } >>> >>> but is there anything else. >> >>No, that is basically it. If you find anything else, please report! >> >> > Trust me I will. > > One more question, > Besides the obvious N^2 multiplications, what options would you consider > if you don't want to go to the Frequency domain?. Thanks, > > IvanThere are a few tricks: - don't continually FFT the filter. Just do it once and for all. The formulation above implies that you're doing FFTs all the time. Better like this: HF=FFT(h) [new x loop] XF=FFT(x) ZF=XF.*HF z=ifft(ZF) - if you're thinking of zero-stuffing to increase the sample rate and planning to FFT thereafter, don't. Simply replicate the FFT after it's computed. The result is exactly the same and replication can be done with indexing. Computational efficiency results vary according to processor and algorithm. - if you're thinking of filtering by multiplication as above and if the filter has large stop bands and if the data has natural locations where the data is zero or nearly so, then consider using a "gate" filter instead. That is, replace everything in the stop band with zeros. This saves multiplies. It works best if the edges of the gate (perfect 1,0 filter) are located where the data is also zero or very small. It causes temporal aliasing if the edges are on top of or near nonzero data - so use prudently. (all of the tricks above can be used together for interpolation) Fred

Reply by ●February 9, 20062006-02-09

There are numerous methods for fast convolutions besides the usual FFT idea. For example, one simple variation is that you can perform a cyclic convolution exactly by zero-padding it to a cyclic convolution of a larger size. This is useful if your original cyclic convolution size has large prime factors. If you are convolving a long sequence with a short one, you should investigate overlap-add/save methods (to do a sequence of short convolutions instead of one long one). There are other transforms that have convolution theorems similar to the DFT. e.g. the Discrete Hartley Transform, etc., although it's not clear that these have any convincing advantages. Number-theoretic transforms are somewhat attractive for integer convolutions, especially if you don't have floating-point hardware. Depending on the transform size, there are convolution algorithms that are not O(N log N) but are better than O(N^2), such as Karatsuba's algorithm and Toom's algorithm. These can have better performance for small-to-moderate N due to the constant factors, and are especially useful for linear convolutions. As JVB pointed out in another thread recently, you can also apply an FFT algorithm "partway" so that you reduce the problem to a set of smaller convolutions, which you may then evaluate by another method and/or hard-coded convolutions of small sizes. (The usual FFT convolution method is the special case where you reduce to a set of convolutions of length 1, i.e. multiplications.) More generally, FFTs and many other fast cyclic convolution algorithms (see e.g. the book by Nussbaumer) can be expressed via polynomial products that are recursively reduced modulo factorizations of z^N - 1 and related polynomials. The Cooley-Tukey FFT corresponds to one factorization, but there are others due to Bruun, Winograd, etcetera. Cordially, Steven G. Johnson