Fourier Transform with Differential Equation

• To: mathgroup at smc.vnet.net
• Subject: [mg48968] Fourier Transform with Differential Equation
• From: Lee Fisher <lfis at helix.nih.gov>
• Date: Fri, 25 Jun 2004 02:58:42 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```I'm attempting to find the frequency response of the output of an RC
circuit (a low-pass filter).  For some reason, the output of the Fourier
transform in mathematica does not seem to correspond to anything I would
expect from this differential equation.  It seems that the placement of
the initial conditions has an enormous effect on the shape of the
frequency response, so much so that the phase portion looks nothing like
i would expect it to. I'm using the following code

w=1
result=DSolve[{10 V'[t]+V[t]==E^(I w t), V[0]==0},V,t}]
time=Table[i,{i,0,100}];
out=Re[Evaluate[V[time]/.result[[1]]]];
fout=Fourier[out];
ListPlot[Abs[fout],PlotJoined->True,PlotRange->All]
ListPlot[Arg[fout],PlotJoined->True,PlotRange->All]

and adjusting w to observe the response at different frequencies. While
the magnitude graph is somewhat reminiscent of what I would expect, the
phase graph looks nothing like I'd expect (it should be at Pi highly
negative frequencies, negative Pi at highly positive frequencies, and at
0 at the zero frequency).  Could someone please explain to me either
where my code is wrong, or where my expectation of the magnitude and
phase output is wrong.

Thanks so much,
Lee Fisher

```

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