Ideal PLAN Frigate

As a generalization, you can be flexible with ASW ship's size, but not with AAW ships.

ASW ships can range from small sub chasers, to large ASW destroyers. The Russian Medvedka ASROC system can be installed on any ships displacing 350+ tons. An ASW helicopter can drop sonar buoys to far distances. As an ASW ship, you can run off to chase subs.

But small ships cannot carry large SAMs. Any ship can carry light SAMs for point defense, but you need larger ships to carry heavy SAM's, and in sufficient numbers. Unlike ASW ships that can make do with 8 ASROC's and few dozen ASW rockets, an AAW ship needs a lot of missiles.

I agree that a frigate - in a navy where there are destroyer, cruisers, carriers - should perform the scout and ASW roles because then you don't have to put to many system on one, not so big vessel.
Probably the frigates in the outer ring coould carry few long rang SAMs, that are remote-controlled by AEGIS-like vessels in the center of the formation, to push the AA capability even further.