# When are the chirp signals orthogonal?

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.

• this is indeed the only orthogonality condition; the expression for the distance integral is too lengthy for a comment, see below. Dec 3, 2023 at 11:29

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