I would  like to calculate the Fourier transform of  Sinc[ b (ω1 - ω2)], but there are some problems as follows:

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**My target is**

$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\text{Sinc}(b (\text{$\omega_1$}-\text{$\omega_2$})) e^{-i \text{$ \omega_1 t_1$}  } e^{-i  \text{$ \omega_2 t_2 $} } d \text{$t_1$} d\text{$t_2$}   =\frac{2 \pi ^2}{b} \Pi \left(\frac{\text{$t_1$}}{2 b}\right) \delta (\text{$t_1$}+\text{$t_2$})$

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**The problem is**

There is no output after runing for a long time with the following codes:
       
        Assuming[{b ∈ Reals}, FourierTransform[  Sinc[b (ω1 - ω2)], {ω1, ω2}, {t1, t2}, \!\(TraditionalForm\`FourierParameters -> {1, \(-1\)}\)]]
 
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***Clue 1***


        FourierTransform[ Sinc[ω1 - ω2], {ω1, ω2}, {t1,   t2}, \!\(TraditionalForm\`FourierParameters -> {1, \(-1\)}\)]

The result is $ π^2 DiracDelta[t1 + t2] (Sign[1 - t1] + Sign[1 + t1]) $

But in Mathematica, `(Sign[1 - t1] + Sign[1 + t1])`  does not equal to   `2 UnitBox[t1/2]`,  because   `Sign[1 - t1] + Sign[1 + t1] - 2 UnitBox[t1/2] != 0 `

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***Clue 2***

        

        FourierTransform[ Sinc[b ω], {ω}, {t}, \!\(TraditionalForm\`FourierParameters -> {1, \(-1\)}\)]


The result is  $ \frac{\pi}{2 b} (Sign[b - t] + Sign[b + t])$

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Any comment or suggestion would be highly appreciated.