Let $\mathrm{F}(\mathrm{k})=\Phi\\{\mathrm{f}(\mathrm{x})\\}, \mathrm{G}(\mathrm{k})=\Phi\\{\mathrm{g}(\mathrm{x})\\}$ and suppose $\mathrm{F}(\mathrm{k}) \mathrm{G}(\mathrm{k})=\Phi\\{\mathrm{h}(\mathrm{x})\\}$. Prove the convolution theorem for Fourier transforms: If \(g(x)\) and \(F(k)\) are absolutely integrable on $(-\infty, \infty)\( and if the Fourier inversion integral for \)\mathrm{f}(\mathrm{x})$ is valid for all \(\mathrm{x}\) except possibly a countably infinite number of points, then $$ \mathrm{h}(\mathrm{x})=(\mathrm{f} * \mathrm{~g}) $$ where ( \(f * g\) ) is the convolution of \(f\) and \(g\) defined by $$ (\mathrm{f} * \mathrm{~g})=\\{1 / \sqrt{(} 2 \pi)\\}^{\infty} \int_{-\infty} \mathrm{f}(\xi) \mathrm{g}(\mathrm{x}-\xi) \mathrm{d} \xi $$

Prove the following properties of Fourier transforms: a) The Fourier transform of \(\mathrm{f}(\mathrm{x})\) exists if \(\mathrm{f}\) is absolutely integrable over \((-\infty,+\infty)\), b) If \(\mathrm{f}(\mathrm{x})\) is real valued then $$ F(-k)=F^{*}(k) $$ where \(F^{*}(k)\) is the complex conjugate of \(F(k)\).

a) If \(\mathrm{a}>0\) show that the Fourier transform of the function defined by $$ \begin{array}{cc} \mathrm{f}(\mathrm{t})=\mathrm{e}^{-a t} \text { coswdt } & \mathrm{t} \geq 0 \\\ & =0 & \mathrm{t}<0 \end{array} $$ is \((a+j w)^{2} /\left[(a+j w)^{2}+\omega^{2} d\right]\) Then find the total \(1 \Omega\) energy associated with the function $$ \begin{array}{rlrl} \quad \mathrm{f} & =\mathrm{e}^{-\mathrm{t}} \text { cost } & & \mathrm{t} \geq 0 \\ \text { and } & =0 & \mathrm{t} & <0 \end{array} $$ b) time domain integration. That is, find the total energy by integrating $$ \left.\mathrm{W}={ }^{\infty} \int_{0}[\mathrm{f}(\mathrm{t})\\}\right]^{2} \mathrm{dt} $$ c) frequency domain integration. That is, find the total energy by integrating $$ \mathrm{W}=(1 / 2 \pi)^{\infty} \int_{-\infty}|\mathrm{F}(\mathrm{w})|^{2} \mathrm{~d} \mathrm{w} $$ where \(F(\omega)\) is the Fourier transform of the function \(\mathrm{f}(\mathrm{t})\).

a) Prove the attentuation property of Fourier transforms: $$ \Phi\left\\{\mathrm{f}(\mathrm{x}) \mathrm{e}^{\mathrm{ax}}\right\\}=\mathrm{F}(\mathrm{k}-\mathrm{ai}) $$ where $$ F(k)=\Phi\\{f(x)\\} $$ b) Prove the shifting property of Fourier transforms: $$ \Phi\\{\mathrm{f}(\mathrm{x}-\mathrm{a})\\}=\mathrm{e}^{\mathrm{ika}} \mathrm{F}(\mathrm{k}) $$ c) Prove the derivative properties of Fourier transforms: $$ \begin{aligned} &\Phi\left\\{f^{\prime}(x)\right\\}=-i k \Phi\\{f(x)\\} \\ &\Phi\left\\{f^{\prime}(x)\right\\}=-k^{2} \Phi\\{f(x)\\} \end{aligned} $$

Find the Fourier transform, \(\mathrm{F}(\mathrm{k})=\Phi\\{\mathrm{f}(\mathrm{x})\\}\) of the Gaussian probability function $\mathrm{f}(\mathrm{x})=\mathrm{Ne}^{(-\alpha \mathrm{x}) 2} \quad(\mathrm{~N}, \alpha=\( constant \))$ Show directly that \(\mathrm{f}(\mathrm{x})\) is retrievable from the inverse transform. I.e., show that $\mathrm{f}(\mathrm{x})=\left\\{1 / \sqrt{(2 \pi)\\}}^{\infty} \int_{-\infty} \mathrm{F}(\mathrm{k}) \mathrm{e}^{-\mathrm{ikx}} \mathrm{dx}=\Phi^{-1}[\Phi\\{\mathrm{f}(\mathrm{x})\\}]\right.$

Given the current pulse, $\mathrm{i}(\mathrm{t})=\mathrm{te}^{-\mathrm{bt} \text { . }}$ (a) find the total \(1 \Omega\) energy associated with this waveform; (b) what fraction of this energy is present in the frequency band from \(-\mathrm{b}\) to \(\mathrm{b} \mathrm{rad} / \mathrm{s} ?\)