Add FFT latex notes (incomplete)

fft-notes.bib

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+@misc{gstrang,
+	title = {Linear Algebra and Its Applications, by Gilbert Strang (chapter 3.5)},
+	howpublished = "\url{https://archive.org/details/linearalgebrait00stra}",
+}
+@misc{tpornin,
+	title = {{Thomas Pornin} mathoverflow answer},
+	howpublished = "\url{https://crypto.stackexchange.com/a/63616}",
+}
+@misc{rfateman,
+	title = {notes by {Prof. R. Fateman}},
+	howpublished = "\url{https://www.csee.umbc.edu/~phatak/691a/fft-lnotes/fftnotes.pdf}",
+}
+@misc{fftrs,
+	title = {fft-rs},
+	howpublished = "\url{https://github.com/arnaucube/fft-rs}",
+}
+@misc{fftsage,
+	title = {fft-sage},
+	howpublished = "\url{https://github.com/arnaucube/math/blob/master/fft.sage}",
+}

fft-notes.tex

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+\documentclass{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{enumerate}
+\usepackage{hyperref}
+\hypersetup{
+    colorlinks,
+    citecolor=black,
+    filecolor=black,
+    linkcolor=black,
+    urlcolor=blue
+}
+
+\theoremstyle{definition}
+\newtheorem{definition}{Def}[section]
+\newtheorem{theorem}[definition]{Thm}
+\newtheorem{innersolution}{}
+\newenvironment{solution}[1]
+{\renewcommand\theinnersolution{#1}\innersolution}
+{\endinnersolution}
+
+
+\title{FFT: Fast Fourier Transform}
+\author{arnaucube}
+\date{August 2022}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+  Usually while reading papers and books I take handwritten notes, this document contains some of them re-written to $LaTeX$.
+
+  The notes are not complete, don't include all the steps neither all the proofs. I use these notes to revisit the concepts after some time of reading the topic.
+
+  This document are notes done while reading about the topic from \cite{gstrang}, \cite{tpornin}, \cite{rfateman}.
+\end{abstract}
+
+\tableofcontents
+
+\section{Discrete \& Fast Fourier Transform}
+
+\subsection{Discrete Fourier Transform (DFT)}
+
+Continuous:
+
+$$
+x(f) = \int_{-\infty}^{\infty} x(t) e^{-2 \pi f t} dt
+$$
+
+Discrete:
+The $k^{th}$ frequency, evaluating at $n$ of $N$ samples.
+$$
+\hat{f_k} = \sum_{n=0}^{n-1} f_n e^{\frac{-j \pi kn}{N}}
+$$
+
+where we can group under $b_n = \frac{\pi kn}{N}$. The previous expression can be expanded into:
+$$
+x_k = x_0 e^{-b_0 j} + x_1 e^{-b_1 j} + ... + x_n e^{-b_n j}
+$$
+
+By the \emph{Euler's formula} we have $e^{jx} = cos(x) + j\cdot sin(x)$, and using it in the previous $x_k$, we obtain
+
+$$
+x_k = x_0 [cos(-b_0) + j \cdot sin(-b_0)] + \ldots
+$$
+
+Using $\hat{f_k}$ we obtained
+$$
+\{f_0, f_1, \ldots, f_N\} \xrightarrow{DFT} \{ \hat{f_0}, \hat{f_1}, \ldots, \hat{f_N} \}
+$$
+
+To reverse the $\hat{f_k}$ back to $f_k$:
+$$
+f_k = \left( \sum_{n=0}^{n-1} \hat{f_n} e^{\frac{-j \pi kn}{N}} \right) \cdot \frac{1}{N}
+$$
+
+
+$$
+DFT =
+\begin{pmatrix}
+\hat{f_0}\\
+\hat{f_1}\\
+\hat{f_2}\\
+\vdots\\
+\hat{f_n}\\
+\end{pmatrix}=
+\begin{pmatrix}
+1 & 1 & 1 & \ldots & 1 \\
+1 & w_n & w_n^2 & \ldots & w_n^{N-1} \\
+1 & w_n^2 & w_n^4 & \ldots & w_n^{2(N-1)} \\
+\vdots & \vdots & \vdots & & \vdots\\
+1 & w_n^{n-1} & w_n^{2(n-1)} & \ldots & w_n^{(N-1)^2} \\
+\end{pmatrix}
+\begin{pmatrix}
+f_0 \\ f_1 \\ f_2 \\ \vdots \\ f_n
+\end{pmatrix}
+$$
+
+\subsection{Fast Fourier Transform (FFT)}
+While DFT is $O(n)$, FFT is $O(n \space log(n))$
+
+Here you can find a simple implementation of the these concepts in Rust: \href{https://github.com/arnaucube/fft-rs}{arnaucube/fft-rs} \cite{fftrs}
+
+
+\section{FFT over finite fields, roots of unity, and polynomial multiplication}
+
+FFT is very useful when working with polynomials. [TODO poly multiplication]
+
+An implementation of the FFT over finite fields using the Vandermonde matrix approach can be found at \cite{fftsage}.
+
+\subsection{Intro}
+Let $A(x)$ be a polynomial of degree $n-1$,
+
+$$
+A(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + \cdots + a_{n-1} \cdot x^{n-1} = \sum_{i=0}^{n-1} a_i \cdot x^i
+$$
+
+We can represent $A(x)$ in its evaluation form,
+
+$$
+(x_0, A(x_0)), (x_1, A(x_1)), \cdots, (x_{n-1}, A(x_{n-1})) = (x_i, A(x_i))
+$$
+
+
+We can evaluate A(x) at n given points $(x_0, x_1, ..., x_{n-1}$):
+
+$$
+\begin{pmatrix}
+A(x_0)\\ A(x_1)\\ A(x_2)\\ \vdots\\ A(x_{n-1})
+\end{pmatrix}=
+\begin{pmatrix}
+x_0^0 & x_0^1 & x_0^2 & \ldots & x_0^{n-1} \\
+x_1^0 & x_1^1 & x_1^2 & \ldots & x_1^{n-1} \\
+x_2^0 & x_2^1 & x_2^2 & \ldots & x_2^{n-1} \\
+\vdots & \vdots & \vdots & & \vdots\\
+x_{n-1}^0 & x_{n-1}^1 & x_{n-1}^2 & \ldots & x_{n-1}^{n-1} \\
+\end{pmatrix}
+\begin{pmatrix}
+a_0 \\ a_1 \\ a_2 \\ \vdots \\ a_{n-1}
+\end{pmatrix}
+$$
+
+This is known by the Vandermonde matrix.
+
+But this will not be too efficient. Instead of random $x_i$ values, we use \emph{roots of unity}, where $\omega_n^n = 1$. We denote $\omega$ as a primitive $n^{th}$ root of unity:
+
+$$
+\begin{pmatrix}
+A(1)\\ A(\omega)\\ A(\omega^2)\\ \vdots\\ A(\omega^{n-1})
+\end{pmatrix}=
+\begin{pmatrix}
+1 & 1 & 1 & \ldots & 1 \\
+1 & \omega & \omega^2 & \ldots & \omega^{n-1} \\
+1 & \omega^2 & \omega^4 & \ldots & \omega^{2(n-1)} \\
+\vdots & \vdots & \vdots & & \vdots\\
+1 & \omega^{n-1} & \omega^{2(n-1)} & \ldots & \omega^{(n-1)^2} \\
+\end{pmatrix}
+\begin{pmatrix}
+a_0 \\ a_1 \\ a_2 \\ \vdots \\ a_{n-1}
+\end{pmatrix}
+$$
+
+Which we can see as
+
+$$
+\hat{A} = F_n \cdot A
+$$
+
+This matches our system of equations:
+
+\begin{itemize}
+  \item at $x=0$, $a_0 + a_1 + \cdots + a_{n-1} = A_0 = A(1)$
+  \item at $x=1$, $a_0 \cdot 1 + a_1 \cdot \omega + a_2 \cdot \omega^2 + \cdots + a_{n-1} \cdot \omega^{n-1} = A_1 = A(\omega)$
+  \item at $x=2$, $a_0 \cdot 1 + a_1 \cdot \omega^2 + a_2 \cdot \omega^4 + \cdots + a_{n-1} \cdot \omega^{2(n-1)} = A_2 = A(\omega^2)$
+  \item $\cdots$
+  \item at $x=n-1$, $a_0 \cdot 1 + a_1 \cdot \omega^{n-1} + a_2 \cdot \omega^{2(n-1)} + \cdots + a_{n-1} \cdot \omega^{(n-1)(n-1)} = A_2 = A(\omega^{n-1})$
+\end{itemize}
+
+We denote the $F_n$ as the Fourier matrix, with $j$ rows and $k$ columns, where each entry can be expressed as $F_{jk} = \omega^{jk}$.
+
+To find the $a_i$ values, we use the inverted $F_n = F_n^{-1}$
+
+\subsection{Roots of unity}
+todo
+
+\subsection{FFT over finite fields}
+todo
+
+\subsection{Polynomial multiplication with FFT}
+todo
+
+
+\bibliography{fft-notes.bib}
+\bibliographystyle{unsrt}
+
+\end{document}