Problem $40$: $(15$ points$)$

This problem shows how to calculate the Fourier transform of periodic signals.

$(1$ point$)$ Find the Fourier transform of $(t).$

$(2$ point$)$ In Problem Set $4,$ Problem $18$ Part $5$ you showed that $|)a|||.$ Using this result and the symmetry property in Problem $37,$ determine the Fourier transform of $f(t)=1.$

$(2$ points$)$ Using the result from part $2$ and the frequency shift property from Problem $38,$ determine the Fourier transform of $e^{{}_{o}t}.$

$(4$ points$)$ Find the Fourier transform of the periodic signals $sin({}_{o}t)$ and $cos({}_{o}t)$ given the result in part $3$ and the fact that ${}_o$ is a real$-$valued constant.

$(2$ points$)$ Suppose that $f(t)$ is a periodic signal with period $T$ and has the Fourier series representation

$f(t)={\displaystyle \underset{n=-}{\overset{}{}}}{D}_n{e}^{jn{}_{o}t}$

Use the result from part $3$ to show that

$F()=2{\displaystyle \underset{n=-}{\overset{}{}}}{D}_n(-\frac{2}{T}n).$

$(4$ points$)$ Find the Fourier transform of the periodic signal $f(t)$ shown on slide $21$ of Recitation $9.$ You may use the compact trigonometric Fourier series coefficients on slide $30$ to determine $F().$