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\title{Frequency Domain Processing of Audio Signals}
\author{Dan Ports, Andrew Clough, and Albert Chiou}

\maketitle


\section*{Conversion Module}
\large{
\begin{itemize}
\item Three functions:
\begin{enumerate}
\item Perform A/D and D/A conversion
\item Compute the FFT and FFT$^{-1}$
\item Transmit and receive data from other modules via parallel bus
\end{enumerate}
\item All three functions will be performed in parallel, in realtime
\end{itemize}
}

\section*{FFT Theory}
\large
{
\begin{itemize}
\item Discrete Fourier Transform converts discrete time signal to sampled frequency spectrum
\[X[k] =\sum_{n=1}^{N-1}x[n]W_N^{kn}, \; \; \; W_N e^{-j(2\pi/n)}\]
\item FFT uses decimation in time to perform DFT with $\Theta(n \log n)$ operations and in-place (only one storage array required)
\[X[k] = \mathrm{DFT}\set{x[k\: \mathrm{even}]}  + W_N^k \mathrm{DFT}\set{x[k\: \mathrm{odd}]}\]
\item Duality allows inverse FFT to be computed the same way
\[ \mathrm{DFT}\set{x[n]} = X[k] \;\; \Leftrightarrow\;\; \mathrm{DFT}\set{X[k]} = Nx[-n]\]
\end{itemize}
}

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