Reassembling Fourier Transforms

• To: mathgroup at smc.vnet.net
• Subject: [mg48860] Reassembling Fourier Transforms
• From: Lee Fisher <lfis at helix.nih.gov>
• Date: Sat, 19 Jun 2004 04:31:18 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```I am trying to turn a list of points into a function, and one of the
ways I thought to do this was to use Fourier to find the frequency
components of the list, and then to Create a sum of exponentials as
follows:

f[x_] := Random[NormalDistribution[0, 1]]
rawnoise = Table[f[x], {x, 1, 1000}];
frawnoise = Fourier[rawnoise];
magrawnoise = Abs[rawnoise];
phaserawnoise = Arg[frawnoise];
c[t] := \[Sum]\+\(n = 1\)\%1000\(( .005\ magrawnoise[\([n]\)]\ E^\((I\
n\ t\  + phaserawnoise[\([n]\)])\))\);\)\)
c[t] = Re[c[t]];

I have two questions concerning this.  First, is there an easier way to
turn a set of random values into a function so that NDSolve can move
through the function and always find the same value at a given point
(i.e. c[6] will always equal .6002)?  Second, if not, what sort of
factors are necessary in adjusting the phase and magnitude so that they
match the original numbers.

I've tried doing this with much simpler inputs, such as two sine waves
(Sin[t]+Sin[2 t]) and still the output does not properly match the
input.

Thanks for any help,
Lee

```

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