Fourier transforms have a reputation for being difficult beasts, and you will likely need to cross reference many other articles and play around in MATLAB to deeply understand them — this article is meant to tie together some more complex topics in the context of implementing our reverb.

Thus far, we have been discussing audio signals in the time domain: each float in the collection represents a discrete amplitude value recorded at a consistent rate sample rate. However, a signal is more than just a series of floats sampled at a constant rate; it also represents a sum of frequency components known as its frequency spectrum.

## Algorithms for Discrete Fourier Transform and Convolution

In some cases, it is more useful to consider a signal in terms of its frequency components. Specifically for our scenario we are taking advantage of duality: multiplication of two frequency spectra is equivalent to convolution of their corresponding time based signals.

Edward83 Edward83 3, 13 13 gold badges 63 63 silver badges 93 93 bronze badges. The transforms look fine, but there's nothing in the program that is doing convolution. Sign up or log in Sign up using Google.

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Featured on Meta. Unicorn Meta Zoo 9: How do we handle problem users? Related In addition, new results are obtained for the multiplicative complexity of multidimensional cyclic convolution, the one-dimensional discrete Fourier transform DFT , and convolutions and DFTs with input constraints or output restrictions. The multiplicative complexity of multidimensional cyclic convolution is determined for any possible combination of lengths in any number of dimensions, extending a result of Winograd for one- and two-dimensional cyclic convolution.

This result is shown to be applicable in determining the multiplicative complexity of the one-dimensional discrete Fourier transform DFT.

## Fast Fourier Transform and Convolution Algorithms - Dimensions

The multiplicative complexity of the DFT for all possible lengths is determined starting with Winograd's result for odd prime lengths and then extending it to power-of-prime lengths, power-of-two lengths, and finally to arbitrary lengths. The multiplicative complexity of systems of polynomial multiplication with constrained inputs is considered.

An input constraint must imply a nontrivial factorization of one input polynomial for which one factor has coefficients only in the ground field if the multiplicative complexity is to be reduced over unconstrained polynomial multiplication. This result is applied to symmetric polynomial multiplication. The multiplicative complexity of polynomial products for which only selected outputs are needed is analyzed.

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