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Problem with FourierParameters

I get a wrong result using FourierParameters:

FourierTransform[f, x, w]
InverseFourierTransform[%, w, x]


1/2 Sqrt[\[Pi]/2] (Sign[1 - w] + Sign[1 + w])

which is what I expected. But with the setting of FourierParameters:

FourierTransform[f, x, w, FourierParameters -> {1, -1}]
InverseFourierTransform[%, w, x, FourierParameters -> {1, -1}]


1/2 \[Pi] Sign[1 - w] + 1/2 \[Pi] Sign[1 + w]     <--- This is OK
0                                                 <--- Bug?

Other functions e.g. f=1/(1+x^2) transform correctly in both cases.

Gruss Peter
Peter Breitfeld, Bad Saulgau, Germany --

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