Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 5

Find a Fourier series representation of the Dirac delta function, \(\delta(x)\), on \([-L, L]\)

Short Answer

Expert verified
The Fourier series representation of the Dirac Delta function on the interval \([-L, L]\) is \(\delta(x) \approx 1/2L\).

Step by step solution

01

Understand the Dirac Delta Function

The Dirac Delta function, denoted as \(\delta(x)\), is a function that is zero everywhere except for the point at 0, where it is undefined, and its integral over the entire real line is equal to 1. It is often considered as 'sampling' function.
02

Establish Fourier Transform Pair for Dirac Delta Function

We have the general representation for a Fourier transform pair as follows: Fourier transform of \(\delta(x)\) is given by \(F(w) = \int_{-\infty}^{+\infty} \delta(x) e^{-iwx} dx\). We know that integral of \(\delta(x)\) over \(-∞\) to \(+∞\) equals 1. So, Fourier transform of \(\delta(x)\) equals 1 for all w, that is \(F(w) = 1\). The inverse Fourier transform of 1 is given by \(ƒ(x) = \int_{-\infty}^{+\infty} 1 * e^{iwx} dw\), which yields the dirac delta function. Thus the Fourier transform pair is established, \(\delta(x) \longleftrightarrow 1\).
03

Find Fourier Series Representation

Having established the Fourier transform pair, we can next move to finding the Fourier Series representation. The representation for any function in Fourier series within an interval \([-L, L]\) is given by \(ƒ(x) ≈ (1/2L) \int_{-L}^{L} dw e^{iwx}\), substituting \(\delta(x)\) in this we have \(\delta(x) \approx (1/2L) \int_{-L}^{L} dw e^{iwx}\).
04

Simplify the Fourier Series Representation

The integral of \(e^{iwx}\) runs from \(-L\) to \(L\), this implies that this is a periodic function and its average value over one period is constant as 1. Therefore, Fourier series of \(\delta(x)\) over \([-L, L]\) simplifies to \(\delta(x) \approx 1/2L\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the inverse Laplace transform of the following functions using the properties of Laplace transforms and the table of Laplace transform pairs. a. \(F(s)=\frac{18}{s^{3}}+\frac{7}{s} .\) b. \(F(s)=\frac{1}{s-5}-\frac{2}{s^{2}+4}\). c. \(F(s)=\frac{s+1}{s^{2}+1} .\) d. \(F(s)=\frac{3}{s^{2}+2 s+2}\) e. \(F(s)=\frac{1}{(s-1)^{2}}\) f. \(F(s)=\frac{e^{-3 s}}{s^{2}-1} .\) g. \(F(s)=\frac{1}{s^{2}+4 s-5}\) h. \(F(s)=\frac{s+3}{s^{2}+8 s+17} .\)

Find the Laplace transform of the following functions: a. \(f(t)=9 t^{2}-7\) b. \(f(t)=e^{5 t-3}\). c. \(f(t)=\cos 7 t\). d. \(f(t)=e^{4 l} \sin 2 t\). e. \(f(t)=e^{2 t}(t+\cosh t)\). f. \(f(t)=t^{2} H(t-1)\). g. \(f(t)=\left\\{\begin{array}{cl}\sin t, & t<4 \pi \\ \sin t+\cos t, & t>4 \pi\end{array}\right.\) h. \(f(t)=\int_{0}^{t}(t-u)^{2} \sin u d u\). i. \(f(t)=(t+5)^{2}+t e^{2 t} \cos 3 t\) and write the answer in the simplest form.

Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).

Show that the convolution operation is associative: \((f *(g * h))(t)=\) \(((f * g) * h)(t)\)

In this problem, you will directly compute the convolution of two Gaussian functions in two steps. a. Use completing the square to evaluate $$ \int_{-\infty}^{\infty} e^{-a t^{2}+\beta t} d t $$ b. Use the result from part a to directly compute the convolution in Example 8.16: $$ (f * g)(x)=e^{-b x^{2}} \int_{-\infty}^{\infty} e^{-(a+b) t^{2}+2 b x t} d t $$
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free