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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:

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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