me22b149.tex

\section{me22b149}
Name: Madabathula Revanth\\
GitHub user-ID : ME22b149Revanth
\subsection*{Description\footnote{Taken from Quora}}
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on the temporal frequency or spatial frequency respectively. That process is also called analysis.\\
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The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
\subsection*{The analysis formula\footnote{Taken from Wikipedia}}
The Fourier transform is an extension of the Fourier series, which in its most general form introduces the use of complex exponential functions. For example, for a function \textit{f(x)}, the amplitude and phase of a frequency component at frequency \textit{n/P} , \textit{n} $\in$ $\mathbb{Z}$ is given by this complex number: \\

\begin{equation}
\label{a1}
    c_n = \frac{1}{P}\int_P f(x) e^{-i2\pi \frac{n}{P} x }dx \
\end{equation}

The extension provides a frequency continuum of components ,  ( $\xi$ $\in$ $\mathbb{R}$ ), using an infinite integral of integration:\\
\begin{equation}
\label{a2}
    \hat{f} (\xi)= \int_{-\infty}^\infty f(x) e^{-i2 \pi \xi x} dx
\end{equation}
Here, the transform of the function $\textit{f(x)}$ at frequency $\xi$ is denoted by the complex number $\hat{f}$($\xi$), which is just one of several common conventions. Evaluating \ref{a2} for all values of $\xi$ produces the frequency-domain function. When the independent variable ($\textit x$) represents time (often denoted by $\textit{t}$), the transform variable ($\xi$ ) represents frequency (often denoted by $\textit{f}$).