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aaltomani's avatar
aaltomani
Copper Contributor
Feb 21, 2023

PeterBartholomew1 

I needed the FFT for a fast convolution of probability distributions. I had previously used a VBA implementation, and recently migrated to pure Excel formulas. It is at least as fast as the VBA implementation, except for some edge case when copying formulas to a different sheet (relevant thread here: https://techcommunity.microsoft.com/t5/excel/puzzling-performance-issues-with-lambda-array-formulas/m-p/3720783).

 

How do you plan to make a non-recursive version? Implementations in "fast" languages such as C use arrays that are mutated in place. Have you found a way to emulate that in Excel?

PeterBartholomew1's avatar
PeterBartholomew1
Silver Contributor
Feb 21, 2023

aaltomani 

Basically I used REDUCE for applying both the even-odd sort matrix and the butterfly matrix.  Any judgement of how Excel handles that iteration/recursion would be pure guesswork on my part.  At the more detailed level, I transformed the part-processed result as each stage so that the blocks of the right-hand side vector were stacked horizontally and then the multiplication by a block diagonal matrix reduced to a single broadcast product.

 

For me, the DFT wasn't something I needed.  The exercise, rather like the fourth order Runge-Kutta routine I wrote, were tasks conducted to see how far I could get with spreadsheet formulas and the extent to which they could replace VBA or other formal programming languages.  I am interested that you have a real application that has driven your development to such a level.  I had planned to complete the work by implementing the inverse version of the formula and then, maybe applying it to some convolution problems.

Resources

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Excel is now meant to be Turing complete so it appears to be possible to write serious computing functionality.  As an experiment,  I have implemented a Discrete Fourier Transformation algorithm using recursive FFT.  I avoided the built-in text-based functionality to process complex numbers and instead wrote a Lambda function to replace complex multiplication

Multiplyλ\n\n= LET(\n u, TAKE(w, , 1),\n v, TAKE(w, , -1),\n x, TAKE(z, , 1),\n y, TAKE(z, , -1),\n HSTACK(u * x - v * y, u * y + v * x)\n )

I allowed each complex number of the array to spill across two columns, but was it necessary?

 

For anyone who might be interested, the solution I presented uses a double recursion to navigate a tree of calculations (typically this is used to reduce 10 000 000 multiplications to a mere 50 000, but I am not planning to try that with Excel)

DFTλ(z, N, s)\n\n= IF( N = 1, z,\n LET(\n z₀, DFTλ(Splitλ(z, 2, 0), N / 2, 2 * s),\n z₁, DFTλ(Splitλ(z, 2, 1), N / 2, 2 * s),\n W, TAKE(Splitλ(W₀, N₀ / N), N / 2),\n p, z₀,\n q, Multiplyλ(W, z₁),\n VSTACK(p + q, p - q)\n )\n )

The key element of the function DFTλ is that it calls itself twice, first following a branch of the calculation until it goes full depth; it evaluates that and then backs up a level and calculates the other branch.  The final part of the routine combines two results before moving back up a further level.

 

I agree I haven't exactly followed the spreadsheet mantra for formula of 'keep it simple', but someone has to try it!

 

# Lambda functions

# Recursion

# Complex Numbers

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aaltomani 

Basically I used REDUCE for applying both the even-odd sort matrix and the butterfly matrix.  Any judgement of how Excel handles that iteration/recursion would be pure guesswork on my part.  At the more detailed level, I transformed the part-processed result as each stage so that the blocks of the right-hand side vector were stacked horizontally and then the multiplication by a block diagonal matrix reduced to a single broadcast product.

 

For me, the DFT wasn't something I needed.  The exercise, rather like the fourth order Runge-Kutta routine I wrote, were tasks conducted to see how far I could get with spreadsheet formulas and the extent to which they could replace VBA or other formal programming languages.  I am interested that you have a real application that has driven your development to such a level.  I had planned to complete the work by implementing the inverse version of the formula and then, maybe applying it to some convolution problems.

","body@stringLength":"1226","rawBody":"

 

Basically I used REDUCE for applying both the even-odd sort matrix and the butterfly matrix.  Any judgement of how Excel handles that iteration/recursion would be pure guesswork on my part.  At the more detailed level, I transformed the part-processed result as each stage so that the blocks of the right-hand side vector were stacked horizontally and then the multiplication by a block diagonal matrix reduced to a single broadcast product.

 

For me, the DFT wasn't something I needed.  The exercise, rather like the fourth order Runge-Kutta routine I wrote, were tasks conducted to see how far I could get with spreadsheet formulas and the extent to which they could replace VBA or other formal programming languages.  I am interested that you have a real application that has driven your development to such a level.  I had planned to complete the work by implementing the inverse version of the formula and then, maybe applying it to some convolution problems.

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PeterBartholomew1 

I needed the FFT for a fast convolution of probability distributions. I had previously used a VBA implementation, and recently migrated to pure Excel formulas. It is at least as fast as the VBA implementation, except for some edge case when copying formulas to a different sheet (relevant thread here: https://techcommunity.microsoft.com/t5/excel/puzzling-performance-issues-with-lambda-array-formulas/m-p/3720783).

 

How do you plan to make a non-recursive version? Implementations in \"fast\" languages such as C use arrays that are mutated in place. Have you found a way to emulate that in Excel?

","body@stringLength":"992","rawBody":"

 

I needed the FFT for a fast convolution of probability distributions. I had previously used a VBA implementation, and recently migrated to pure Excel formulas. It is at least as fast as the VBA implementation, except for some edge case when copying formulas to a different sheet (relevant thread here: https://techcommunity.microsoft.com/t5/excel/puzzling-performance-issues-with-lambda-array-formulas/m-p/3720783).

 

How do you plan to make a non-recursive version? Implementations in \"fast\" languages such as C use arrays that are mutated in place. Have you found a way to emulate that in Excel?

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