From 9fd5c13dadb91892c5d928bcfc308a760ffb0b3c Mon Sep 17 00:00:00 2001 From: Francis Stokes Date: Fri, 10 May 2024 22:44:00 +0200 Subject: [PATCH] Hotfix: some unsupported MatJax --- 2024/5/10/cordic.md | 45 ++++++++++++++++++++++++++++++++++++-------- assets/cordic/3.png | Bin 0 -> 92767 bytes 2 files changed, 37 insertions(+), 8 deletions(-) create mode 100644 assets/cordic/3.png diff --git a/2024/5/10/cordic.md b/2024/5/10/cordic.md index 6ff7552..6ed3d3d 100644 --- a/2024/5/10/cordic.md +++ b/2024/5/10/cordic.md @@ -77,9 +77,16 @@ It's sort of like a binary search: You move towards the target angle by some la If you've worked with these kinds of operations before, you'll know that rotating a vector involves multiplying it with a matrix consisting of sines and cosines of the angle to be rotated to. That seems counter productive, since those are the functions we're trying to compute! $$ -\newcommand\colvec[1]{\begin{bmatrix}#1\end{bmatrix}} - -\colvec{x'\\y'} = \begin{bmatrix}\cos(\theta) & -\sin(\theta)\\\sin(\theta) & \cos(\theta)\end{bmatrix} \colvec{x\\y} +\begin{bmatrix} +x' \\ +y' +\end{bmatrix} = \begin{bmatrix} +\cos(\theta) & -\sin(\theta) \\ +\sin(\theta) & \cos(\theta) +\end{bmatrix} \begin{bmatrix} +x \\ +y +v\end{bmatrix} $$ We'll put that aside for a second, and get a big picture overview before solving this problem. Now, it's fairly obvious that rotating by say `22.75˚` is the same as rotating by `45˚` and then `-22.5˚` - i.e. we can break up a rotation into smaller parts, with both positive and negative components. @@ -101,16 +108,38 @@ With some understanding in hand, we can return to the issue of, well, rotations $$ \cos(\theta) = \frac{1}{\sqrt{1 + tan^2(\theta)}} -\\ \\ +$$ + +$$ \sin(\theta) = \frac{\tan(\theta)}{\sqrt{1 + tan^2(\theta)}} -\\ \\ -\colvec{x'\\y'} = \cos(\theta)\begin{bmatrix}1 & -\tan(\theta)\\\tan(\theta) & 1\end{bmatrix} \colvec{x\\y} +$$ + +$$ +\begin{bmatrix} +x' \\ +y' +\end{bmatrix} = \cos(\theta)\begin{bmatrix} +1 & -\tan(\theta) \\ +\tan(\theta) & 1 +\end{bmatrix} \begin{bmatrix} +x \\ +y +\end{bmatrix} $$ We have a few constants ones now, but we still have the `tan(a)`, plus the `cos(a)` out front. Let's ignore the `cos(a)` and focus on getting rid of `tan(a)`. As you saw when we ran through the algorithm, we're always rotating by a total of `~90˚`: First by `45˚`, then `22.5˚`, then `11.25˚`, and so on. Since we're doing this a fixed number of times, we can just precompute those values, and put them in a table. You might be thinking: *"You said there wouldn't be any tables!"*. Well, no. I said there wouldn't be any *expensive* tables. This table, in our case, will only contain 16 `uint32_t`s - a whopping 64 bytes! Even the most stripped down embedded projects can *usually* afford that. (In contrast, an *unoptimised* table for `sin(x)` that contains 4096 entries covering values from -1 to 1 would need 16KiB - and that's pretty poor precision!) $$ -\colvec{x'\\y'} = \cos(\theta)\begin{bmatrix}1 & -table[i]\\table[i] & 1\end{bmatrix} \colvec{x\\y} +\begin{bmatrix} +x' \\ +y' +\end{bmatrix} = \cos(\theta)\begin{bmatrix} +1 & -table[i] \\ +table[i] & 1 +\end{bmatrix} \begin{bmatrix} +x \\ +y +\end{bmatrix} $$ That means our rotation matrix now only contains constants! We do however still have that `cos(a)` term. In fact, every iteration brings it's own new `cos(a)` term. 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