Suggested languages for you:

Americas

Europe

Problem 503

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.

Expert verified

In order to develop the Fourier Transform from the Fourier Series, we start by considering the limit as the period \(T\) goes to infinity, which leads to a continuous spectrum with angular frequency \(\omega\). Then, we replace the trigonometric functions with the exponential function using Euler's identity. This gives us the Fourier Transform definition:
\[F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx\]
And the inverse Fourier Transform:
\[f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega x} d\omega\]

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 17

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

Chapter 17

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

Chapter 17

Use Fourier transform methods to find the time-domain response network having a system function $$ j 2 \omega /(1+2 j \omega) $$ if the unit is $$ \mathrm{V}(\mathrm{t})=\cos \mathrm{t} $$ (For a sinusoidal input cos \(t\), the Fourier transform is $$ \pi[\delta(\omega-)+\delta(\omega-1)]) $$

Chapter 17

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 $$ (\mathrm{f} * \mathrm{~g})=\\{1 / \sqrt{(} 2 \pi)\\}^{\infty} \int_{-\infty} \mathrm{f}(\xi) \mathrm{g}(\mathrm{x}-\xi) \mathrm{d} \xi $$

Chapter 17

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 \(\mathrm{f}(\mathrm{t})\).

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner