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\title{VictorBook of the Fourier Transform}
\subtitle{A Musician's Approach}
\author{Juan Reyes}
\email{juanig@CCRMA.Stanford.EDU}
\institution{CCRMA\\Center for Computer Research in Music and Acoustics}
\newcommand{\abs}[1]{\ensuremath{\left#1\right}}
\newcommand{\ejoT}{\ensuremath{e^{j\omega T}}}
\newcommand{\defined}{\ensuremath{\stackrel{{\scriptscriptstyle\Delta}}{=}}}
\begin{document}
\maketitle
\begin{slide}{Fourier Series}
Fourier series are defined as:
\[ f_{per}(\omega_0t) = a_0 + \sum_{m=1}^{\infty} b_m\cos(2\pi
m\omega_0t) + c_m \sin(2\pi m \omega_0t) \]
\begin{itemize}
\item where $a_0,$ is the D.C. component and
\item where the coefficients $b_m,$ and $c_m,$ represent
magnitudes of sine and cosine components for each frequency
$\omega_0t,$ respectively.
\end{itemize}
\end{slide}
\begin{slide}[Dissolve]{Fourier Transforms}
\begin{itemize}
\item By using Euler's identity and in terms of sines
and cosines the Fourier Transform is defined as:
%\vspace{0.125in}
\[S(\omega_0) = \int_{\infty}^{\infty} s(t)
[\cos(2\pi\omega_0t) + j\sin(2\pi\omega_0t)] \ ,dt., \]
\item The Inverse Fourier Transform is defined as:
%\vspace{0.125in}
\[s(t) = \int_{\infty}^{\infty} S(\omega_0)
[\cos(2\pi\omega_0t) + j\sin(2\pi\omega_0t)] \ ,dt., \]
\end{itemize}
\end{slide}
\begin{slide}{IDFT}
The \emph{ IDFT} in terms of its angular frequency $\omega,$
is defined as:
\[ y(n) \defined \frac{1}{N} \sum_{n=0}^{N1} Y(k) e^{j\omega_k n},
k=0,1,2,..., N1 \]
\vspace{0.125in}
where $y(n) \leftrightarrow Y(k),$ conform a pair of transforms.
The term $k,$ is the frequency index and $n,$ is the time index.
\end{slide}
\end{document}
