Re: FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]
- To: mathgroup at smc.vnet.net
- Subject: [mg21381] Re: [mg21368] FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]
- From: BobHanlon at aol.com
- Date: Fri, 31 Dec 1999 21:30:14 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Jordan,
FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]
2*(-Pi*DiracDelta[w] - I/w) - 2*Pi*DiracDelta[w]
Expand[%]
-4*Pi*DiracDelta[w] - (2*I)/w
InverseFourierTransform[%, w, t, FourierParameters -> {1, -1}]
-Sign[t] - 2
The results of the Fourier Transform has the sign wrong in the term 2*((-I)/w
- Pi*DiracDelta[w]). Further, the error seems to arise in using a modified
scaling {a, b} where b is negative since
FourierTransform[Sign[t], t, w, FourierParameters -> {a, b}]
2*((2^((a - 1)/2)*Pi^((a + 1)/2)*Sqrt[Abs[b]]*DiracDelta[w])/b +
(I*(2*Pi)^((a - 1)/2)*Sqrt[Abs[b]])/(b*w)) - (Sqrt[(2*Pi)^(a +
1)]*DiracDelta[w])/
Sqrt[Abs[b]]
Simplify[Expand[%], Element[a, Reals]]
((2*Pi)^((a + 1)/2)*Sqrt[Abs[b]]*DiracDelta[w])/b - ((2*Pi)^((a +
1)/2)*DiracDelta[w])/
Sqrt[Abs[b]] + (I*2^((a + 1)/2)*Pi^((a - 1)/2)*Sqrt[Abs[b]])/(b*w)
InverseFourierTransform[%, w, t, FourierParameters -> {a, b}]
((2*Pi)^(a/2)*(Abs[b]*(Sign[t] + Sqrt[(2*Pi)^(-a)]*Sqrt[(2*Pi)^a]) -
b*Sqrt[(2*Pi)^(-a)]*Sqrt[(2*Pi)^a]))/(b*Sqrt[(2*Pi)^a])
Simplify[%, Element[a, Reals]]
(Abs[b]*(Sign[t] + 1) - b)/b
which will only be correct for b > 0.
The default scaling or any specified scaling that does not have a
non-positive b (e.g., {1, 1}, {-1, 1}) appears to work correctly.
To eliminate the formatting prior to pasting into an e-mail, convert the
output cells to input format prior to copying them as plain text.
Bob Hanlon
In a message dated 12/30/1999 12:31:20 AM, jr at ece.gatech.edu writes:
>I have only been using Mathematica for under a month and have a question.
> I
>was trying to reproduce the results of some well known Fourier transforms
>(allowing distributions) in Mathematica. For instance,
>
> sgn(t) <----------> 2/(j*w)
>
>where j = sqrt(-1). (See, for example "The Fourier Integral and its
>Applications", Papoulis). So I ran the following code
>
> FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]
>
>and got the following result
>
> \!\(\(-2\)\ \[Pi]\ DiracDelta[w] +
> 2\ \((\(-\(\[ImaginaryI]\/w\)\) - \[Pi]\ DiracDelta[w])\)\)
>
>Some questions:
>
>1) Why does the answer not match what I expect? Am I missing an assumption
>somewhere or using something wrong?
>
>2) In the result given by Mathematica, two of the terms obviously combine
>so
>I tried to use / /Simplify after the input, but it didn't simplify this
>rather "simple" term. Why? (I know I can use / /Expand to get it to
>simplify, but was just surprised that / /Simplify didn't work).
>
>Minor question: Is there a better way to copy an output into a posting?
>All those backslashes seem to be a bit confusing.
>