Yes, your equation is right. If you want a (short) proof and some information about the Poisson Summation formula (PSF), you can look at the Wikipedia page. Another form this equation takes is when you set $\omega=0$; then you get the very symmetric $\sum_{k\in\mathbb Z}\hat{f}(2\pi k)=\sum_{k\in\mathbb Z}b(k).$ One interesting interpretation of this result is through the Selberg trace formula. Indeed, this trace formula says that, on the circle $S^1$, the trace of the Laplacian $-\Delta=-\frac{d^2}{dx^2}$ can be computed in two ways: the right-hand side of the PSF is just the sum of the eigenvalues, whereas the left-hand side caan be interpreted as the sum over the periodic orbits of the geodesic flow on the circle. In short, in this context, the PSF implies that the two ways to compute the trace of the Laplacian will give you the same result. Which is what you would expect.
Another application of the PSF is in proving the analytic continuation of the Riemann zeta function, and more generally, of Dirichlet L-functions.