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\subsection*{Abstract}

The discrete Fourier transform (DFT) maps a finite number of discrete time-domain samples to the same number of discrete Fourier-domain samples.
Being practical to compute, it is the primary transform applied to real-world sampled data in digital signal processing.
The DFT has special relationships with the discrete-time Fourier transform and the continuous-time Fourier transform that let it be used as a practical approximation of them through truncation and windowing of an infinite-length signal.

\section{The Discrete-Time Fourier Transform}

The Discrete-Time Fourier Transform (DTFT)
is the primary theoretical tool for understanding the frequency content
of a discrete-time (sampled) signal.
The DTFT is defined as
\be
X( \omega ) = \sum_{n=-\infty}^{\infty} x [ n ] e^{-j \omega n}
\ee
where $x[n]$ is the sampled signal, $n$ is the integer sample index,
and $\omega$ is the continuous-valued frequency.
The inverse DTFT (IDTFT) is defined by an integral formula, because it operates on a continuous-frequency DTFT spectrum:
\be
x[n] = \frac{1}{2 \pi} \int_{- \pi}^{\pi} X( \omega ) e^{j \omega n} d \omega
\ee
The DTFT is very useful for theory and analysis, but is not practical for
      numerically computing a spectrum digitally, because
\begin{enumerate}
\item
Infinite time samples means infinite computation and infinite delay.
\item
Frequency is a continuous variable.
\end{enumerate}

We thus need a different definition of frequency for computer-based implementation.

\section{The Discrete Fourier Transform}

The DFT transforms $N$ samples of a discrete-time
signal to the same number of discrete frequency samples, and is defined asAn example signal is shown in

\begin{figure}[htb]\centerline{\includegraphics[scale=0.44]{examplesig.eps}}
\caption{An example signal}
\label{fig:examplesig}
\end{figure}


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