This is the interactwave project, a Vite-based application for interactive visualization of waves.

Make sure you have Node.js installed.

To start the development server, run:

npm run dev

This will start Vite's development server.

To build the project for production, run:

npm run build

This will create a dist directory with the production build.

To preview the production build, run:

npm run preview

This will start a local server to preview the production build.

• regl
• sass
• vite-plugin-glsl

• TypeScript
• Vite

This is based on my understanding of Cooley–Tukey FFT algorithm, I am also learning it.

$$ \begin{bmatrix} X_0 \\ \vdots\\ X_i \\ \vdots \\ X_{N-1} \end{bmatrix} =\begin{bmatrix} W_N^0 & \ldots & W_N^0 & \ldots & W_N^0 \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ W_N^0 & \ldots & W_N^{ij} & \ldots & W_N^{-i} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ W_N^0 & \ldots & W_N^{-j} & \ldots & W_N^1 \\ \end{bmatrix} \begin{bmatrix} x_0 \\ \vdots\\ x_j \\ \vdots \\ x_{N-1} \end{bmatrix} $$

$$ W_N = \cos{2\pi\over N} + \textit{i} \sin{2\pi\over N} $$

the $\textit{i}$ before $\sin$ represents imaginary number, in other places, the $i$ is used for indexing for convenience.

$$ X_i = \sum_{j=0}^{N-1}W_N^{ij}x_j $$

If we isolate columns with $j=2n$ and $j=2n+1$

$n=0, 1, \ldots, N/2-1$

$$ X_i = \sum W_N^{i2n}x_{2n} + \sum W_N^{i(2n+1)}x_{2n+1} $$

And isolate rows with $i=m$ and $i=m+N/2$

$m=0, 1, \ldots, N/2-1$

case 1

$$ X_m = \sum W_N^{2mn}x_{2n} + \sum W_N^{m(2n+1)}x_{2n+1} $$

case 2

$$ X_{m+N/2} = \sum W_N^{(m+N/2)2n}x_{2n} + \sum W_N^{(m+N/2)(2n+1)}x_{2n+1} \ = \sum W_N^{2mn+nN}x_{2n} + \sum W_N^{m(2n+1)+nN+N/2}x_{2n+1} $$

The first term of both cases:

$$ \sum W_N^{2mn+nN}x_{2n} = \sum W_N^{2mn}x_{2n} $$

The second term of both cases:

$$ \sum W_N^{m(2n+1)+nN+N/2}x_{2n+1} = - \sum W_N^{m(2n+1)}x_{2n+1} $$

First, we compute:

$$ A_m = \sum_{n=0}^{N/2-1} W_N^{2mn}x_{2n} $$

$$ B_m = W_N^m \sum_{n=0}^{N/2-1} W_N^{2mn}x_{2n+1} $$

Then we have:

$$ X_{m} = A_m + B_m $$

$$ X_{m+N/2} = A_m - B_m $$

$A_m$ is a smaller FFT, with only half the size, so does $B_m$ but need to multiply by a factor $W_{N}^m$:

let $i'=m,j'=n, N'=N/2$

$$ A_{i'}= \sum_{j'=0}^{N'-1} W_{N'}^{i'j'}x_{2i'} $$

$$ B_{i'}= W_N^{i'}\sum_{j'=0}^{N'-1} W_{N'}^{i'j'}x_{2i'+1} $$

Which is A, which is B in each recursive steps:

• A: 0, 2, 4, 6
  • A: 0, 4
  • B: 2, 6
• B: 1, 3, 5, 7
  • A: 1, 5
  • B: 3, 7

Look up index in each step, perform A+B or A-B, and the W to multiply:

B is always (A+(N/2))%N, A depends on:

• Size of FFT (N')
• How many matrices? (N/N')
• Each matrix advances index by (N'/2)

for entry i:

• which matrix? : floor(i/N')
• index within matrix? : mod(i, N')
• index within half of the matrix? : mod(i, N'/2)
• index of corresponding A? mod(i, N'/2) + floor(i/N') * (N'/2)

For N=8 example:

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

You are free to:

• Share — copy and redistribute the material in any medium or format
• Adapt — remix, transform, and build upon the material
  for any purpose, even commercially.

Under the following terms:

• Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.

See the full license at https://creativecommons.org/licenses/by-sa/4.0/.