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 \(F(k)\).

Develop the definition of the Fourier transform of a function \(\mathrm{f}(\mathrm{x})\) be extending the definition of the Fourier series of \(\mathrm{f}\) to the case where the discrete spectrum of Fourier coefficients becomes a continuous spectrum.

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 $\mathrm{f}(\mathrm{x})=\left\\{1 / \sqrt{(2 \pi)\\}}^{\infty} \int_{-\infty} \mathrm{F}(\mathrm{k}) \mathrm{e}^{-\mathrm{ikx}} \mathrm{dx}=\Phi^{-1}[\Phi\\{\mathrm{f}(\mathrm{x})\\}]\right.$

a) Prove that if the functions \(\mathrm{g}(\mathrm{x})\) and \(\mathrm{F}(\mathrm{k})\) are absolutely integrable on \((-\infty,+\infty)\) and that the Fourier inversion integral for \(\mathrm{f}(\mathrm{x})\) is valid for all \(\mathrm{x}\) except possibly at a countably infinite number of points, then $$ \infty_{-\infty} F(k) g(-k) d k={ }^{\infty} \int_{-\infty} f(x) g(k) d x $$ where $$ \mathrm{F}(\mathrm{k})=\Phi\\{\mathrm{f}(\mathrm{x})\\}, \mathrm{G}(\mathrm{k})=\Phi\\{\mathrm{g}(\mathrm{x})\\} $$ This is known as the second Parseval theorem of Fourier transform theory. b) From the above equation (1), prove the first Parseval theorem of Fourier transform theory, $$ \left.{ }^{\infty}\right|_{-\infty}|F(\mathrm{k})|^{2} \mathrm{dk}={ }^{\infty} \int_{-\infty}|\mathrm{f}(\mathrm{x})|^{2} \mathrm{~d} \mathrm{x} \text { . } $$

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 \(\mathrm{b} \mathrm{rad} / \mathrm{s} ?\)

Define the Dirac delta function, \(\delta(\mathrm{x})\), and prove the sifting property of \(\delta(\mathrm{x})\) for all functions \(\mathrm{f}(\mathrm{x})\) which are continuous at \(\mathrm{x}=0\) $$ { }^{\infty} \int_{-\infty} \delta(\mathrm{x}) \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{f}(0) \text { . } $$