Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

lalaly

Fourier

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    @mr.math

    • 2 years ago
  2. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    it s ok mr math, thanks anyways:)

    • 2 years ago
  3. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Oh I know the solution -.- My page went down twice!! :(

    • 2 years ago
  4. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    I will write again using another browser. I love my solutions to be complete, so sorry for taking so long :)

    • 2 years ago
  5. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    oh :-D take your time lol, i am sorry for bothering you xD

    • 2 years ago
  6. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    I'm starting to hate Google Chrome. Lets see how Firefox works out for me :D

    • 2 years ago
  7. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    So you know the definition of Fourier transform: The Fourier Transform of an integrable function \(f(x)\) is define as \[\large F(\omega)=\int_{-\infty}^{\infty} f(x)e^{-2\pi i x\omega}.\] For our function \(f(x)=e^{-x^2}\), we have \[F(\omega)=\int_{-\infty}^{\infty}e^{-x^2}e^{-2\pi i x \omega}dx.\] The question becomes now how to evaluate this integral?! You probably know the famous Gaussian integral \(\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi}\). We will manipulate our integrand to make it of a similar form. \[F(\omega)=\int_{-\infty}^{\infty}e^{-x^2}e^{-2\pi i x \omega}dx=\int_{-\infty}^{\infty}e^{-x^2-2\pi i x \omega}dx=\int_{-\infty}^{\infty}e^{-(x^2+2\pi i x \omega)}dx\] \[=\int_{-\infty}^{\infty}e^{-(x+\pi i \omega)^2-\pi^2\omega^2}dx=e^{-\pi^2\omega^2}\int_{-\infty}^{\infty}e^{-(x+\pi i w)^2}dx.\]

    • 2 years ago
  8. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Now substitute \(u=x+\pi i \omega\) and use Gaussian integral to evaluate the integral above.

    • 2 years ago
  9. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    :-D i will do that,, Thankyou soo much Mr math :D youre the best

    • 2 years ago
  10. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    You can see this for the integration of exp(-x^2) https://www.youtube.com/watch?v=fWOGfzC3IeY

    • 2 years ago
  11. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    if you want! :)

    • 2 years ago
  12. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    i know how to find it by normal distribution xD

    • 2 years ago
  13. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    I have alwyas known that you're smart!

    • 2 years ago
  14. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    haha not as smart as mr math xD

    • 2 years ago
  15. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    If I remember well, you study Communication Engineering, right?

    • 2 years ago
  16. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    yepp :-D lol you have a good memory

    • 2 years ago
  17. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Only the things about "important" people! ;)

    • 2 years ago
  18. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    I feel special now :-$

    • 2 years ago
  19. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Fourier transform has many applications in your field, right?

    • 2 years ago
  20. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    I have taken a course in Signals and Systems two semesters ago and I liked it.

    • 2 years ago
  21. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah but i am taking a course where fourier transform is solved in mathematics way... and its different from what ive taken in signals and systems and communication -.- i solved this question in a way ,, the professor said he wants it solved in mathematicians way not engineers :S

    • 2 years ago
  22. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Lol, Mathematicians are always the best. Engineers come second so you don't get upset :P

    • 2 years ago
  23. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    lol no doubt:P I dont mind coming second after you xD

    • 2 years ago
  24. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Well, I think as long as the solution provides all logical steps needed, it should be enough. I did take Fourier transform in two different courses one of which was this Engineering course I just told you about. I remember in that course we were allowed to use tables, but in a Math course we would have to derive them ourselves in one way or another.

    • 2 years ago
  25. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    It is amazing,,, Thanks again :-D ... i wont drive u crazy after this dont worry hehe

    • 2 years ago
  26. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    You look more "Arab" in this picture for some reason. I like all you pictures anyways :)

    • 2 years ago
  27. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    You're welcome, and good luck!

    • 2 years ago
  28. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    lol thats sweet:)

    • 2 years ago
  29. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Important note: I should have used \(f\) instead of \(\omega\) there, because the transform I did was in terms of frequency not angular frequence. You can use \(\omega=2\pi f\) to write it in angular frequency, as you know.

    • 2 years ago
  30. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    i appreciate ur help ^_^

    • 2 years ago
  31. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    found the easy way :P thought id show it to u let f(x)=e^(-x^2) and let f^ (f-hat) = F \[f'(x)=-2xe^{-x^2}\] take fourier transform of both sides\[F[f'(x)]=-2F[xf(x)]\]we know that \[F[f']=iwF\]and \[\large{F[xf]=iF '}\]so now we have\[iwF=-2iF'\]the i cancels \[wF=-2F'\]now we seperate\[wdw=\frac{-2}{f}dF\]integrate both sides\[\frac{w^2}{2}=-2lnF+C\]take e of both sides\[\large{e^{\frac{w^2}{2}}=e^{lnF^{-2}+C}}\]so simplifying\[\huge{F=C_2e^{-\frac{w^2}{4}}}\]now we find the constant observe F(0)=C_2 so \[C_2=\frac{1}{\sqrt{2 pi}} \int\limits_{-\infty}^{\infty}e^{-x^2}dx\]we know that theat integral =sqrt(pi) so \[C_2=\frac{1}{\sqrt2}\] now \[\huge{F(w)=\frac{1}{\sqrt 2}e^{-\frac{w^2}{4}}}\]hoooooooooooooof lol

    • 2 years ago
  32. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    @mr.math

    • 2 years ago
  33. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    That's smarter but not easier :-)

    • 2 years ago
  34. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    lol maybe :P ... i dont like having so many integrals ... so thats why i tried to find it in another way

    • 2 years ago
  35. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    Mr math do u know bivariate random variables?

    • 2 years ago
  36. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    I don't think the solution I gave has that many integrals. And I don't know bivariate random variables.

    • 2 years ago
  37. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    lol its ok,,, i just wanted to ask a question xD

    • 2 years ago
  38. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    If I was doing it with myself, I would just do it like this: \[F(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-x^2}e^{-i\omega x}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-(x+\frac{i\omega}{2})^2-{\omega^2\over 4}}dx\] \[=\frac{e^\frac{-\omega^2}{4}}{\sqrt{2\pi}}\sqrt{\pi}=\large \frac{1}{\sqrt{2}} e^{\frac{-\omega^2}{4}}.\]

    • 2 years ago
  39. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    But I like the tricks you used.

    • 2 years ago
  40. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah thats shorter hehe,,, ill write down both,, what u did was awesome, i just wanted to share with u what i thought about

    • 2 years ago
  41. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    Thanks! What you did is awesome too. Is this a homewrok asignment or what?

    • 2 years ago
  42. Mr.Math
    Best Response
    You've already chosen the best response.
    Medals 2

    assignment*

    • 2 years ago
  43. lalaly
    Best Response
    You've already chosen the best response.
    Medals 1

    lol yeah something like that :P

    • 2 years ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.