       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.
>

```

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