Re: Fourier Question

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1615] Re: Fourier Question
• From: rknapp (Robert Knapp)
• Date: Thu, 6 Jul 1995 22:32:57 -0400
• Organization: The Santa Fe Institute

```In article <3t2ium\$mie at news0.cybernetics.net> jt12799 at glibm3.cen.uiuc.edu (Jehan Tsai) writes:

>
> I have a question concerning the Mathematica command 'Fourier'
> Recently, I have been experimenting with this command for a program I have
> been working on that requires it.  For the most part, it seems to work as
> expected, except when we multiply the list by Exp[I b x], which is supposed
> to cause a shift in the frequency domain.  However, it does not seem to
> perform as expected shifting the function around strangely in the frequency
> domain.  According to references, Exp[I 2 Pi f0/N x] Should cause a shift
> of f0, but does not.  Below are the commands I used in my trial.
>
> g=UnitStep[x+5]-UnitStep[x-5];
> b= ;
> f1=Table[N[g*Exp[I x b]],{x,-63,64}];
> f2=Fourier[f1];
> ListPlot[Abs[f2]];
>
> A comparison of this plot with the Fourier of sampled function g by itself
> indicates that for whatever b I choose has no correlation on the shift itself.
> I'm not sure what the problem is, whether I am misunderstanding the actual
> code, the way Mathematica does the FFT or whether I am just plotting the
> samples over an incorrect interval.  I am interested in any help or
> suggestions in what the problem is.  Thanks in advance.
>
> J.T. Tsai

Your references are correct.  The values of the transform should shift
by f0 in the negative direction.  I am not quite sure what values you
were trying for b, but I suspect you may not have included the
2Pi/128 factor.  Below, I have produced a table which indicates that
the Fourier function is reflecting this accurately.  Rather than
creating a plot, which is hard to send via nntp, I have simply tracked
the position of the maximum of the modulus of the transform.  I have
also put in more general variables so that the proper scaling is clear.

In[89]:=
Table[
g=UnitStep[x+5]-UnitStep[x-5];
n = 128;
b = j 2*Pi/n;
f1=Table[N[g*Exp[I x b]],{x,1-n/2,n/2}];
f2=Fourier[f1];
f3 = Sort[Transpose[{Abs[f2],Range[n]}]];
Last[f3],
{j,1280,1280+128,16}] // TableForm

Out[89]//TableForm=
0.883883   1

0.883883   113

0.883883   97

0.883883   81

0.883883   65

0.883883   49

0.883883   33

0.883883   17

0.883883   1

Note that if non integer multiples of 2Pi/n are used, the value of the
maximum will change as well as its position.  For example setting b to
integers gives shifts that look somewhat random, but are really quite
predicatble:

In[92]:=
Table[
g=UnitStep[x+5]-UnitStep[x-5];
n = 128;
b = j;
f1=Table[N[g*Exp[I x b]],{x,1-n/2,n/2}];
f2=Fourier[f1];
f3 = Sort[Transpose[{Abs[f2],Range[n]}]];
Join[Last[f3],{N[b*n/(2*Pi)]}],
{j,1,10}] // TableForm

Out[92]//TableForm=
0.882669   109   20.3718

0.883306   88    40.7437

0.883766   68    61.1155

0.881798   48    81.4873

0.883709   27    101.859

0.883415   7     122.231

0.882498   114   142.603

0.883878   94    162.975

0.882829   74    183.346

0.883187   53    203.718

The last column gives the predicted shift.

Rob Knapp
WRI

```

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