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

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$})$

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, \(-1\)}\)]]

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[t/2] != 0

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])$

Any comment or suggestion are highly appreciated.