Results 1 to 9 of 9

Math Help - Fourier Series : analysis or synthesis equations ?

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    5

    Fourier Series : analysis or synthesis equations ?

    Apologies for the textual format (no idea how to write these equations properly in a forum) but I hope those who are familiar with Fourier Series will recognise these equations.

    Q: In Fourier Series, our teacher has told us that it should be obvious to us whether to use the synthesis :

    x(t) = inf.sum of ak*e^(jkw0t)

    or the analysis equation :

    ak = 1/T0 * integral of x(t)*e^(-jkw0t) dt

    But I'm afraid I haven't got the foggiest myself when to use one and not the other and it is not explained anywhere in the notes we have been given or in numerous textbooks I have checked.

    Wiki suggests something about one is to break up terms and the other is the restructure but that is not the context my teacher in the engineering course is putting them in and I suspect it is wiki-garbage.

    Please can someone help before I start banging my head against the wall with intent of permanent coma.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jan 2009
    Posts
    56
    Well, if you are given x(t) and you are asked to find ak, then you use the first one, otherwise use the second equation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Sling View Post
    Apologies for the textual format (no idea how to write these equations properly in a forum) but I hope those who are familiar with Fourier Series will recognise these equations.

    Q: In Fourier Series, our teacher has told us that it should be obvious to us whether to use the synthesis :

    x(t) = inf.sum of ak*e^(jkw0t)

    or the analysis equation :

    ak = 1/T0 * integral of x(t)*e^(-jkw0t) dt

    But I'm afraid I haven't got the foggiest myself when to use one and not the other and it is not explained anywhere in the notes we have been given or in numerous textbooks I have checked.

    Wiki suggests something about one is to break up terms and the other is the restructure but that is not the context my teacher in the engineering course is putting them in and I suspect it is wiki-garbage.

    Please can someone help before I start banging my head against the wall with intent of permanent coma.
    When given a function of time, the analysis step decomposes the function to give the coeficients of the frequency components that make up the signal.

    The synthesis process reconstructs the time function from the coefficients of the frequency decomposition.

    The situation with functions of position is analogous, a function of a spacial variable is decomposed by the analysis process to give the coeficients of the spacial frequencies, ...

    CB
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2009
    Posts
    5
    This is not quite the approach that my teacher is using. We are given a signal and are expected to know ourselves how to calculate the Fourier Series.

    Would it be correct to say that if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?

    I've attached one example to illustrate what we're expected to be able to do.
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Sling View Post
    This is not quite the approach that my teacher is using. We are given a signal and are expected to know ourselves how to calculate the Fourier Series.

    Would it be correct to say that if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?

    I've attached one example to illustrate what we're expected to be able to do.
    What is the problem with that, you are asked to find the Fourier series of a given waveform. That is to write it in the form:

    x(t)=\sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0t}

    where you have evaluated the a_k 's using:

    a_k=\frac{1}{T_0}\int_{T_0} x(t)e^{-jk\omega_0t}\;dt

    where here T_0=6 so the angular frequency of the fundamental is \omega_0=2\pi/6, and so the integral becomes the integral over an interval of length 6, lets say:

    a_k=\frac{1}{6}\int_{0}^6 x(t)e^{-jk\pi t/3}\;dt

    CB
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Sep 2009
    Posts
    5
    Quote Originally Posted by Sling View Post
    if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?
    So, is this correct?

    If so then the two equations seem to be two methods of doing the same thing, rather than converting the equation from a focus on time to frequency or vice versa.

    (I didn't do the above example btw, it was provided... And the teacher started with giving a definition of x(t) but that's not what he seemed to end up calculating although that could be just me being fussy - if I give a definition to aim for, I then calculate that definition and present it at the end of the calculations.)
    Last edited by Sling; October 1st 2009 at 01:23 AM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Sling View Post
    Would it be correct to say that if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?
    Sorry I did not address this comment. No it would not be correct.

    You are trying to represent a time domain signal as the sum of sinusoids (in this case hidden withing the complex exponentials). The analysis bit tells you what the coeficients of each frequency component are and the synthesis expression tell you how to represent your signal as a sum of the frequency components.

    CB
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Sep 2009
    Posts
    5
    Quote Originally Posted by CaptainBlack View Post
    You are trying to represent a time domain signal as the sum of sinusoids (in this case hidden withing the complex exponentials). The analysis bit tells you what the coeficients of each frequency component are and the synthesis expression tell you how to represent your signal as a sum of the frequency components.

    That makes a lot more sense, in fact that's what I was suspecting but the way my teacher was talking it sounded like we had to choose between one or the other equation which just got me confused (the way it's worded in wiki doesn't help much either tbh).

    In fact both equations would hold true for any given signal.

    I'm guessing that isolating the components will make the Fourier Transform possible which seems to be the really useful part. (We haven't got to that yet though).

    The joys of being a student , kk thanks for taking the time to clear that up (if I've got it right now anyway).
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Sling View Post
    That makes a lot more sense, in fact that's what I was suspecting but the way my teacher was talking it sounded like we had to choose between one or the other equation which just got me confused (the way it's worded in wiki doesn't help much either tbh).

    In fact both equations would hold true for any given signal.

    I'm guessing that isolating the components will make the Fourier Transform possible which seems to be the really useful part. (We haven't got to that yet though).

    The joys of being a student , kk thanks for taking the time to clear that up (if I've got it right now anyway).
    Yes, you have it now.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fourier Series and Linear Differential Equations
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: September 15th 2010, 06:20 AM
  2. Frequency modulation synthesis in series
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: February 22nd 2010, 04:20 AM
  3. Complex Fourier Series & Full Fourier Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 9th 2009, 06:39 AM
  4. Fourier Synthesis or Analysis
    Posted in the Calculus Forum
    Replies: 0
    Last Post: September 29th 2009, 12:49 PM
  5. Replies: 1
    Last Post: March 12th 2009, 05:41 AM

Search Tags


/mathhelpforum @mathhelpforum