Fourier Series

Discussion in 'Physics & Math' started by That One Guy, Jul 10, 2002.

  1. That One Guy Registered Member

    Messages:
    8
    Can anyone recommend any introductory texts(online or offline) on Fourier Series(or Fourier Optics)? Or possibly explain them to me? I don't mind a math intensive explanation, but I cannot stand texts that go with the math first, concept later approach.
    BTW, I look forward to becoming an active member of the forums, see ya around.
     
  2. Google AdSense Guest Advertisement



    to hide all adverts.
  3. wet1 Wanderer Registered Senior Member

    Messages:
    8,616
    Welcome to sciforums, That One Guy.
     
  4. Google AdSense Guest Advertisement



    to hide all adverts.
  5. Merlijn curious cat Registered Senior Member

    Messages:
    1,014
    Welcome,
    I have a course here, but it was written by my professor, and only fotocopied for the students. I have no idea where to find such a thing... maybe it's an idea to try to overcome your dislike of the standard method.
     
  6. Google AdSense Guest Advertisement



    to hide all adverts.
  7. itchy Registered Senior Member

    Messages:
    47
    This is a good math site (perhaps not so conceptual though):

    Fourier Series

    Basically, every function can be represented as an infinite fourier series. It can be practical to translate from the time domain to the frequency domain since frequencies are often easier to analyse. This can be used in audio compression for instance.
     
  8. Joeman Eviiiiiiiil Clown Registered Senior Member

    Messages:
    2,448
    Dude, he is gonna have a hard time following all that.

    fourier series...hmm....okay....I will get you something to get started without being too technical. There are strictions and such....basically every waveform can be constructed by the sum of sine waves of different frequency. fourier series is a series of summations of different weighted sine waves (actually exp(j theta)). Each sine wave is called a harmonic. To obtain the fourier series, you need to figure out the fourier coefficient which is the weight of each exponentials.(frequency component).... Man I think I suck at explaining things. *sigh* I didn't realize I am so worthless.

    I will provide you the mathematic detail tomorrow. I left all my books at work.
     
  9. That One Guy Registered Member

    Messages:
    8
    Thanks to all who contributed.
    Itchy: Thanks for the site, but its kind of hard for me to follow as I have absolutely no experience with the Fourier Series.
    Joeman: Thanks, and great job with the explanation. I'm actually starting to understand it. I look forward to seeing the rest with the math.
    In case you all were wondering understanding the Fourier Series is the key to understanding holograms, which will eventually aid me when I begin my yearlong project.
     
  10. Joeman Eviiiiiiiil Clown Registered Senior Member

    Messages:
    2,448
    Math:

    Okay I'll try. *taking a deep breath*

    Let X(t) be a periodic signal with period T.

    X(t) = sum = x_k exp(j(2*pi*k*t/T)
    ........k = -inf to inf

    where x_k = (1/T)*integral from zero to T(f(t)*exp(-j(2*pi*k*t/T))dt)

    X(t) is a summation of series. It is a weighted sum of sinusoids with frequencies that are integer multiples of 1/T

    That means lets say if you have a square wave with frequency f, this square wave can be broken down into summations of sine waves with frequency f,2f,3f,4f, and etc.....f is called the fundamental frequency, and 2f is called the second harmonics, and etc....

    To find the fourier series, you just need to computer the fourier coefficient which is x_k. you plug in the function into that formula and integrate over one period. where k is integer from - infinity to + infinity.

    Example:
    x(t) = cos(2*pi*f*t)

    now use euler's identity

    x(t) = (1/2)exp(j(2pi*f*t) + (1/2)exp(-j(2pi*f*t)

    therefore x_1 = x_-1 = 1/2 and x_k = 0 for all other k.

    This example shows this consine wave has frequency content of f and -f. The weight for each frequency is 1/2.

    You can try a square wave for practice. You will find a square wave has infinite frequency content so you can never have a perfect square wave in real life. (gibbs phenomenon) However, I believe all you need is 6 harmonics to construct a pretty decent square wave. That is square wave z(t) = x_0*DC + x_1*z(f) + x_2*z(2f) + x_3*z(3f) + x_4*z(4f)....... and the negative frequencies also.

    fourier series has some important properties but I don't think you need them right now.

    You can try this using matlab. Try to find fourier coefficients for square wave. Try to find first 6 harmonics and add them up to see how good of a square wave you get. I did this kind of stuff for home work in college.
     
  11. lotuseatsvipers CloseMindedBob Registered Senior Member

    Messages:
    340
    Do you want to know how to compute a fourier series, or just understand what it is?

    is it for application, or math fun time?

    It basically just shows the frequency spectrum a function. thers a long way to compute it, that has three equations to get the coefficients, and a main equation. then there are the basic transformations which are pretty simple to do, theres a chart of things like, the transform of 1/t = s...and you apply these in pieces. god i hope that makes any sense at all, sorry.
     
  12. Tom2 Registered Senior Member

    Messages:
    726
    I think the universal standard is "Introduction to Fourier Analysis by Lighthill. As for optics, the book "Introduction to Photonics" by Saleh and Teich is pretty good, although I have no idea of why the word "photonics" is used (it's primarily classical optics). And of course, the E+M book by Griffiths is a good place to look.
     
  13. That One Guy Registered Member

    Messages:
    8
    Hey, I haven't replied in a while because I've been gone on a mini vacation. Thanks for all the book recommendations, tutorials, and stuff. At the moment I'm reading the AMAZING book called Who's Fourier which explains everything in a really straightforward manner. Imagine something like Quantum Mechanics Theory for Dummies and you've pretty much got it. After reading this, I think I'll check out the fourier book and photonics book recommended by Tom2.
    Once again, thanks to everyone.

    As a side note: What do Fourier Transforms have to do with Holography?
     
  14. Joeman Eviiiiiiiil Clown Registered Senior Member

    Messages:
    2,448
    I ain't know nothing about no Holography, but you do fourier transform whenever you want to the frequency content of a waveform or signal. That means if you are a function that is a function of time, after you do fourier transform it becomes a function of frequency.
     
  15. lotuseatsvipers CloseMindedBob Registered Senior Member

    Messages:
    340
    Joe I think we think too much like engineers. I'm confused on what this guy is looking for. The statistical views on FFT and what not perhaps?? The stuff I didn't learn and don't understand, thinking about picking up that book who is fourier just for fun.
     
  16. Joeman Eviiiiiiiil Clown Registered Senior Member

    Messages:
    2,448
    Dude, FFT Is soooooo engineering. Physicists don't know anything about FFT. If they do they are dumb. He wants to know hologram. For some reason you need to know fourier transform to understand hologram.
     
  17. lotuseatsvipers CloseMindedBob Registered Senior Member

    Messages:
    340
    a lot of statistics people use FFT on test analysis. I suppose that is the region of engineers too. I am confused though, no denying that. I think the book would do me good? who knows.
     
  18. Crisp Gone 4ever Registered Senior Member

    Messages:
    1,339
    Hi Joeman,

    "Dude, FFT Is soooooo engineering. Physicists don't know anything about FFT."

    Ofcourse we don't, we use the real Fourier transforms and not the algorithm to do Fast Fourier Transforms

    Please Register or Log in to view the hidden image!

    .

    Bye!

    Crisp
     
  19. Joeman Eviiiiiiiil Clown Registered Senior Member

    Messages:
    2,448
    Believe it or not, that is exactly what I was talking about

    Please Register or Log in to view the hidden image!



    Physicists prefer to spend hours do it by hand rather than couple minutes using matlab

    Please Register or Log in to view the hidden image!

     
  20. anthof Registered Member

    Messages:
    3
    Are you guys kidding?

    First off, That One Guy, if you are strictly talking about Fourier Series and not Fourier Transformations, I can give you a complete derivation and explantion on a PDF. I'll just scan some of my notes that I gave to another friend. It's quite small and VERY easy to follow if you have a good understanding of Calculus.

    Second, what type of physicists are you guys talking about? I've never met a physicist who prefers to do Fourier Transforms by hand?!?! I seriously hope you guys are joking! It's practically impossible to do them in a reasonable amount of time unless you have the simplest waveforms imaginable. Personally, as a physicist, I use Matlab and Labview to do all my DFT's and FFT's and so do my colleagues. If you know someone who does these by hand, just ask them what type of waveforms they're analyzing...it's probably and infinite sine

    Please Register or Log in to view the hidden image!

     
  21. That One Guy Registered Member

    Messages:
    8
    Just a clarification: The book "Who is Fourier" is not a biography on Fourier, but a guide starting from the sine wave and ending with an analysis of the Fast Fourier Transform method. It does so in an extremely easy to understand manner.

    I need to know Fourier Series/Transforms in order to understand diffraction which is a very important part of holography. My main goal is to gain a general understanding of holography so I can start playing around with holograms(holographic optical elements, data storage systems and such). Then I could also check out voice recognition programs that deal with Fourier Analysis(Im pretty good at C/C++).

    anthof: I would be forever grateful if you could give me a copy of the notes.

    BTW, I'm just an amateur inventor with a little free time outside of school. Adios.
     
  22. PsychoElk Registered Member

    Messages:
    1
    Fourier

    Does anyone knows how to compare two very similar fourier signals in labview?
    I want to be able to tell the difference between two very similar ffts.
     
  23. anthof Registered Member

    Messages:
    3
    How similar are the waveforms? How many points do you have? What type of signal are you working with. You're going to have to give a lot of details. Are you sampling above the Nyquist frequency? If you give me some of this information, I may be able to help you out. Also, what version of LabView are you using? Developer's suite, educational....?
     

Share This Page