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Sinusoids in Fourier Series

  1. Aug 23, 2014 #1
    Fourier said that any periodic signal can be represented as sum of harmonics i.e., containing frequencies which are integral multiples of fundamental frequncies. Why did he chose the basis functions i.e., the functions which are added to make the original signal to be sinusoidal? I know sinusoids are orthogonal functions but could we use some other basis functions for this representation?
     
  2. jcsd
  3. Aug 23, 2014 #2
    Well, I think Fourier saw the value in being able to express any periodic function as a sum of very simple terms. This is, for instance, extremely practical if you're solving linear differential equations (like Fourier did with the heat equation). Due to the property of superposition, you can solve your differential equation "one simple term at a time" and them recombine all these to form a solution to your original problem. This is a very powerful tool to have at your disposal, especially since you don't have to include all the terms of the Fourier series, i.e. you could find an approximate solution using a partial sum instead.

    There are other sets of functions that you can use to form an analog to the Fourier series, but the Fourier series is, arguably, "special" in its mathematical simplicity. Sines and cosines have a tendency to pop up naturally in a lot of places and they're very easy functions to work with.
     
    Last edited: Aug 23, 2014
  4. Aug 24, 2014 #3
  5. Aug 25, 2014 #4

    analogdesign

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    In practice decaying exponentials are used instead of raw sinusoids (although this is just a book-keeping convention as either formulation can be recast as the other).

    It isn't enough for the functions to be orthogonal. They also have to span the space. Any good book for a second course on signals and systems will talk about this.
     
  6. Aug 25, 2014 #5

    jim hardy

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    I remember reading a Hewlett Packard instruction manual for a function generator that described in great detail how it synthesized sine waves from square waves.
    Ever since i retired i've sought a reference to that instrument.

    But:
    As i get older, the number things i can remember grows, and the number of things i can imagine grows also,
    but the distinction between them grows less clear .

    So I too await an answer to your query .
     
  7. Aug 25, 2014 #6

    analogdesign

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    That one's easy. A narrowband filter to pick out on of the harmonics (or dominant) tone in the square wave will do the trick. ;)

    I suspect you meant synthesize square waves from sine waves. Since to get a square wave you add as many odd harmonics of the final wave as needed (keeping in mind Gibb's phenomenon will limit the purity of the square wave) the speed you need determines the method. At slow speeds, instruments keep the square wave in ROM (or RAM) and play it back. At faster speeds they can use sinewave ROMS and then offset them with frequency doublers and add them back in. Fascinating stuff.
     
  8. Aug 26, 2014 #7

    jim hardy

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    No, that's why the gizmo was so fascinating - it went the other way.

    Found a patent that describes a technique
    but they have to low - pass to remove last 3% of harmonics

    https://www.google.com/patents/US3512092


    block diagram here but it's too wide to post
    https://patentimages.storage.googleapis.com/pages/US3512092-0.png

    might work well with a switched capacitor filter?

    Here's a clever one that works as you suggested, remove harmonics from a square wave with a sharp filter:
    http://electronicdesign.com/analog/standalone-circuit-converts-square-waves-sine-waves
    figure_01.gif

    looks like a pure digital synthesis is not possible, just an approximation that can be refined by old fashioned filtering.

    This is more than i was able to find a few years ago.

    Thanks, guys !

    old jim
     
  9. Aug 26, 2014 #8

    f95toli

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    Sine is a nice easy, function to use; but there are many others can that can -and ARE- used.
    Any book on the more "general" theory of Fourier analysis will introduce you to a whole families of functions are can be used to e.g. solve differential equations and/or in signal analysis.
    You also have things like Wavelet's that can used.
     
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