Let’s summarise what we’ve done so far, as well as pointing out some properties of our general mixed-radix FFT algorithm that will be important going forward. We’ll also think a little bit about just what we are going to do next. Here’s what we’ve done:

Written down the basic expression for the DFT and implemented it directly in Haskell. POST

Built some “toy algebra” tools to help us understand the Fourier matrix decomposition at the heart of the Cooley-Tukey FFT algorithm. POST

Implemented a basic powers-of-two FFT. POST

Extended the “toy algebra” code to the case of general input vector lengths. POST

Implemented a full mixed-radix FFT in Haskell for general input vector lengths. POST

We say “**a** full mixed-radix FFT” because there are lots of choices available for implementing this calculation, and we’ve chosen just one representative approach. There are two properties of our implementation that are important for what we’re going to do next:

We’re using the basic naïve DFT for the final prime-length transformations at the “bottom” of our input vector decomposition. One thing we can do to improve the efficiency of our overall FFT algorithm is to explore better ways of dealing with these prime-length vectors that rely on the special algebraic properties of the Fourier matrix ${F}_{p}$ for prime $p$.

Our implementation of the mixed-radix FFT algorithm uses a “sorted prime factor” decomposition of the input vector length to decide how to break down the input vector for transformation. However, nowhere in any part of the algorithm is it required that the factors be sorted, or be prime – none of the elements of the algorithm (permutation calculations, sub-factor Fourier matrices, and generalised Danielson-Lanczos step) are dependent on factor ordering or primality.

This second point is the one that’s really critical going forward: for a given length of input vector, we can choose any order of factors of the length to drive the Fourier matrix decomposition. For example, if we have a 1024-element input vector, we could use the “standard” powers-of-two decomposition, splitting the input hierarchically ten times to get to single-element subvectors; or, we could split the input into 32 vectors of length 32, perform direct $O({N}^{2})$ DFTs on those 32-element vectors, then combine the results to give a final answer in one “32-way” step; or we could do the decomposition based on any other combination of factors whose overall product is 1024.

For any particular length of input vector, there’s no *a priori* way to know what the most efficient decomposition will be. It’s very dependent on machine architecture and cache size. However, the FFT is a deterministic algorithm – once you decide on a scheme to factorise the input length, the whole of the rest of the computation always proceeds in exactly the same order: the same input vector elements are always accessed in the same order and are used to build the same intermediate results in the same order. This means that once you know the best factorisation for a given input vector length, it doesn’t change depending on the input data values.

It might be an interesting exercise to work out the best factorisation for a given input vector length, machine architecture and cache size by hand, but it’s really a rather futile task. As well as the “theoretical” cache coherency calculations you would need to do, you would also need detailed information about the machine code that your compiler produces for different factorisations. Any future compiler improvements would invalidate your calculations: for example, if a later version of your compiler can make better use of SIMD instructions, this will change the balance between memory access and potential pipeline stalls, and you would have to take account of that.

The approach to this problem taken by FFTW (and by the ATLAS low-level linear algebra libraries) is to use *empirical* information to select the factorisation to use. Instead of trying to determine exactly which factorisation is the best from first principles, we choose a number of likely good factorisations and *measure* which one is best.

In the next article in this series, we’ll introduction the Haskell Criterion benchmarking framework and use it to compare our unoptimised mixed-radix FFT algorithm with some other examples, in order to get some idea of how far we have to go in terms of optimising our code.

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