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The Eat at Joe's Kawai K5000 Message Board Digest
Fourier Transforms


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Re: FM revisited
 Tuesday, 28-Jul-98 10:44:26 

      192.28.2.16 writes:

      >On a technical note, I understand the best way of doing FM to additive is to
       use relatively simple formulas to calculate a FM waveform and do a Fourier
       transform to get the harmonics. Unfortunately, I can't seem to grasp the
       equations to do this (sorry Jens), so if any of you can explain it as simply
       as possible, I would be greatful. 

      Can anybody cite a reference with a good explanation of how to do a Fourier
      transform? Or point out some software? (I stopped at 3D integrals--what more
      do you need for chem?) 


      leiter 

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Re: Re: FM revisited
 Tuesday, 28-Jul-98 11:20:37 

      157.187.24.186 writes:

      If you're not afraid of math, the following give an introduction to the Fourier
      Transform:

      http://bass.gmu.edu/~mazel/sci_mag.html

      http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier2.html 

      Geof 

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Re: Re: Re: FM revisited
 Tuesday, 28-Jul-98 13:07:04 

      209.160.126.117 writes:

      Thanks for the sources. I'll see if I can't figure them out. I've checked out a
      library book written way back in 1947 called Frequency Analysis, Modulation and
      Noise by Stanford Goldman. At the beginning of the book, the author takes you
      step by step in analysing a square wave. The only problem is that the he assumes
      to much (at least for me) from the reader and does not explain what all the
      terms mean. If any of you can find the book and make sense out of it, let me
      know. Or if you would like to e-mail me, maybe I can find a way to send copies
      of the relevant pages to you. 

      Leslie 

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Re: Re: Re: FM revisited
 Tuesday, 28-Jul-98 17:37:34 

      192.28.2.16 writes:

      Thanks. So it's the Fourier Series that we've been talking about. What about
      the Fourier Transform--is it applied in audio synthesis? For transients?

      If I understand the Fourier Series, any repeating wave can be reproduced using
      only the overtones of a harmonic series, so theoretically there are no
      "non-series" harmonics. The problem of "non-series" harmonics is that the
      least common frequency of every "series" and "non-series" harmonic may be very
      low, so the K5k can't reproduce it. E.g., a combination of a 999 Hz sine
      and a 1000 Hz sine could be reproduced with one ADD, but you would have to get
      the 999th and 1000th harmonics of the series that starts at 1 Hz.

      I've taken a stab at a setup to analyze a drawn waveform and come up with a
      harmonic series, but it needs some work.



      leiter 

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 Re: Re: Re: FM revisited
 Tuesday, 28-Jul-98 18:06:17 

      209.160.126.137 writes:

      Ok, let's see if any of you can help me out here. I believe these are the
      two formulas for claculating the harmonics of a waveform:

      An=1/Pi from the values -Pi to Pi x(t)*cos(n*t)
      Bn=1/Pi form the values -Pi to Pi x(t)*sin(n*t)

      Is this right so far?

      For the square wave, x(t) would have the values 1 from -Pi to 0 and -1 from
      0 to -Pi, right?

      According to one of the web sites, all the cosine values for the square wave
      equal zero. Does this mean the cosine equation is for the even harmonics?

      I'm wondering how the 1/Pi is used in the equations. Can anyone show me this?

      At some point do you have to plug the numbers back into the original formula:

      inf.
      x(t)=A0+SUM [An cos(n*t)+Bn sin(n*t)]
      n=1

      Finally, if any of you can show me some examples of doing this step by step
      with waveforms like the square, triangle, and the saw, I would aprreciate it.
      Sorry for so many questions, but I think I'm getting closer to understanding
      all this. 

      Leslie 

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Re: Re: Re: Re: FM revisited
 Tuesday, 28-Jul-98 19:46:29 

      205.227.120.254 writes:

      Saying that the cosine term is zero means that the square wave is an odd function.
      An odd function behaves such that f(-t)=-f(t) and an even function behaves such
      that f(-t)=f(t). Note that cos(-t)=cos(t) and sin(-t) = -sin(t). Most engineering
      communications text books has explanation of generating Fourier Series. The
      subject matter first comes up in solving partial differential equations I think
      a heat transfer problem if I remember correctly. So this is pretty mathematically
      intense stuff. I hope I have helped a little bit.  I hope this helps a little bit.

      Andrew 

      Andrew Stovall 

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Re: Re: Re: Re: Re: FM revisited
 Tuesday, 28-Jul-98 23:18:14 

      209.160.126.72 writes:

      Thanks Andrew! I'm already making some progress with this stuff (a mathematical
      term for very complex formulas). I've managed to do the calculations for a
      square wave. I know I'm not doing everything right, though, so I'll just have
      to keep plugging away at it. Thanks for the info. 

      Leslie 

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Fourier Transform
 Wednesday, 29-Jul-98 04:42:23 

      194.172.230.108 writes:

      Follow this link to learn about (discrete) Fourier transform and to find free
      FT software.


      Jens Groh 

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 Re: Fourier Transform
 Wednesday, 29-Jul-98 14:21:02 

      209.160.126.97 writes:


[[STUFF CLIPPED AND PUT IN THE CONVERTING FM TO ADD DIGEST]]


      BTW, so far I've been able to do an Fourier analysis on a square wave and a
      saw wave. So I am making progress. I've go a long way to go before I get this
      stuff down, though.



      Leslie 

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Re: Re: Fourier Transform
 Thursday, 30-Jul-98 04:37:59 

      194.172.230.108 writes:

      Let me repeat here what I wrote in an email to Leslie in order to let everyone
      understand what algorithm he's about to try. I descibed a relatively simple, yet
      exact algorithm to emulate FM by additive synthesis:


      "It works for periodic signals, so the carrier/modulator ratio must be
      whole-numbered. Then they have a common period. Divide this period into 256
      sampling intervals and compute the FM signal's time function for each point:
      x[i] = sin(2*pi*d*(i/256)*(1+m*sin(2*pi*n*(i/256)))) ; i = 0...255 ;
      n/d = carrier/modulator ratio ; m = modulator amplitude.  (I hope this is a
      correct FM formula.) Then compute a 256-point real-valued FFT (Fast Fourier
      Transform) to get the 128 spectral values. (Look for FFT code in any DSP FAQ on
      the 'net or get info here:  http://nr.harvard.edu/nr/bookcpdf.html ) These values
      are complex numbers, thus real/imaginary number pairs. To get the amplitude,
      compute squareroot(real^2+imag^2). Throw away the first value, because its
      frequency is zero. (You may need to throw away the last harmonic as well, because
      it has no phase information.) You're done! For 64 harmonics, you need 128 rather
      than 256 time steps."


      (Of course, any DFT (Discrete Fourier Transform) will do, not only the special
      version called FFT.) 

      Jens Groh 

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 Re: Re: Re: Fourier Transform
 Thursday, 30-Jul-98 18:55:55 

      209.160.126.55 writes:



[[STUFF CLIPPED AND PUT IN THE CONVERTING FM TO ADD DIGEST]]


      One of the Fourier web pages I visited mentioned that if the waveform is
      symetrical you can get away with using only the SIN function to get the
      harmonics. When I graph the waveform using the above formula, I always get
      symetrical looking waves.  Maybe this is because I only use whole number ratios.
      I'm not sure. At the risk of further revealing my ignorance, I'd like to show
      you the formula I've been using to analysis waveforms:

      2Pi
      Bn=|SUM f(t)*SIN(n*t)|
      t=0

      I'm probably butchering the Fourier method, but it seems to work. All my results
      match what the Bessel functions tell me and other waveforms such as saw and
      square waves are correct as well. 

      Leslie 

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Re: Re: Re: Re: Fourier Transform
 Thursday, 30-Jul-98 19:06:21 

      209.160.126.55 writes:

      Oops! For the Hewlett Packard formula that needs to be:

      . 2Pi
      Bn=|SUM f(t)*COS(n*t)|
      . t=0

      f(t) is the amplitude of the waveform at (t) point in time. 

      Leslie 

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Re: Re: Re: Re: Fourier Transform
 Friday, 31-Jul-98 04:03:50 

      194.172.230.108 writes:

      Your formula is a Fourier analysis stripped down to sine components only. This
      works only for waveforms with 'odd symmetry':

      f(-x) = -f(x) 

      The Fourier transform considers both sine and cosine components of a waveform,
      represented by so-called complex numbers:

      z = real + imag * sqrt(-1) 

      The 'real' part of the complex number is the amplitude of the cosine component,
      the 'imaginary' part is the amplitude of the sine component.

      The overall amplitude is:

      | z | = sqrt(real2 + imag2) 



      Jens Groh 

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Re: Re: Re: Re: Re: Fourier Transform
 Friday, 31-Jul-98 13:13:24 

      209.160.126.66 writes:

      This is what I have so far:

      ...2Pi
      An=SUM f(t)*COS(n*t)
      ...t=0

      ...2Pi
      Bn=SUM f(t)*SIN(n*t)
      ...t=0

      |Zn|=sqrt(An^2+Bn^2)

      (Ignore the "..." leading up to the 2Pi and the t=0. It was the only way I
      could think of to keep them over and under the word SUM. Before, it always
      ignored the spaces and placed them at the beginnig of the line).

      When I programmed this into my computer, it worked great. Am I getting closer? 

      Leslie 

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Re: Re: Re: Re: Re: Re: Fourier Transform
 Saturday, 01-Aug-98 09:34:44 

      195.232.49.233 writes:

      "When I programmed this into my computer, it worked great. Am I getting closer?"

      Yep!



      Jens Groh 


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