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\title{Notes on the Discrete Fourier Transform}
\author{A. Researcher}
\date{\today}

\begin{document}
\maketitle

\section{Definition}
For a length-$N$ signal $x_0,\dots,x_{N-1}$, the discrete Fourier transform is
the sequence $X_0,\dots,X_{N-1}$ defined by
\begin{equation}
  X_k = \sum_{n=0}^{N-1} x_n \, e^{-\mathrm{i} 2\pi k n / N},
  \qquad k = 0, 1, \dots, N-1 .
\end{equation}
Each coefficient $X_k$ is a complex number whose magnitude $\lvert X_k \rvert$
gives the amplitude at frequency bin $k$.

\section{Inverse and Parseval}
The inverse transform recovers $x_n$ from the coefficients:
\begin{equation}
  x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \, e^{\mathrm{i} 2\pi k n / N}.
\end{equation}
Parseval's theorem relates the energy in the two domains,
\begin{align}
  \sum_{n=0}^{N-1} \lvert x_n \rvert^2
    &= \frac{1}{N} \sum_{k=0}^{N-1} \lvert X_k \rvert^2 ,
\end{align}
so an $\ell^2$-bounded signal maps to an $\ell^2$-bounded spectrum.

\end{document}
