\section{Question 1}

Find the Fourier transform of the following signal. (Hint: make good use of the function)

\begin{figure}[h]

\centering

\caption{Time-domain representation of the signal in Question 1.}

\end{figure}

\textbf{Answer: }

We commence our investigation by deriving the Fourier transform of the scaled function .

(1)

Leveraging the shift relation in Fourier transform,

we can extend the derivation to a generalized form for :

Thus, employing this generalized form, we deduce that (\Cref{fig:fourier_transform_2D})

\begin{figure}[h]

\centering

\hbox{

\hspace{1cm}

}

\caption{Fourier Transform of : Real and Imaginary Parts}

\label{fig:fourier_transform_2D}

\end{figure}