4. A. As shown in Figure $1, \odot O$'s diameter $A B$ intersects with chord $C D$ at point $P$, forming an angle of $45^{\circ}$. If $P C^{2}+P D^{2}=8$, then the radius of $\odot O$ is ( ).
(A) $\sqrt{2}$
(B) 2
(C) $2 \sqrt{2}$
(D) 4