When are the chirp signals orthogonal?

Question

Assume that we have two bounded-time chirp signals, \begin{align} x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\ y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\Big),\quad 0\leq t\leq T. \end{align} When are these signals orthogonal? I understand that if $\alpha=\alpha'$ and $\beta-\beta'=2n/T$ for any integer $n$ these two signals are orthogonal. Is there any other case for orthogonality? Another question is whether there is any closed form for the distance between these two signals, i.e., $$ d(x,y)=\int_0^T |x(t)-y(t)|^2 \, dt. $$ I know these are simple questions, but I could not find any resources for comparison to reassure me that I am on the correct path.

Answer 1

just for the record, the distance function has a complicated expression in terms of the imaginary error function erfi:

_{$$d(x,y)=2T+\operatorname{Re}\,\left\{(\alpha+\alpha')^{-1/2}(-1)^{3/4} \left[\text{erfi}\left(\frac{(-1)^{1/4} \sqrt{\pi } (2 T (\alpha+\alpha')+\beta+\beta')}{2 \sqrt{\alpha+\alpha'}}\right)-\text{erfi}\left(\frac{(-1)^{1/4} \sqrt{\pi } (\beta+\beta')}{2 \sqrt{\alpha+\alpha'}}\right)\right]\right.$$ $$\left.\qquad\times \exp \left(\frac{i \pi \left(4 (\alpha+\alpha') (\gamma+\gamma')-(\beta+\beta')^2\right)}{4 (\alpha+\alpha')}\right)\right\}.$$}