Frames for compactly supported functions with irrational density

We find sufficient conditions on a compactly supported function $g$, $\text{supp}g=[a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\alpha ,\beta )=\{e^{2\pi i\beta mx}g(x-\alpha n){\}}_{m,n\in \mathbb{Z}}$$ is a frame for all $\alpha <b-a,\alpha \beta <1,\alpha \beta \mathrm{\notin }\text{Q}$. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions $g$ which give us a frame e.g. $g(x)=\exp (\frac{1}{x^4-1}){\chi }_{(-1,1)}(x)$.