Doom Classic Effect - Realtime Pixelated Shadows

I'm not sure what you're exactly looking for, but as other have said, Doom doesn't have shadows at all. It just has a primitive lighting model. I'm not even sure if they do a normal calculating. In Doom all environment is actually lit by baked lightmaps (or already baked into the textures). There are just a few actual dynamical lights. One light moves with the player. Also each shot emits a short light-flash. Besides that nothing of the lighting changes dynamically. Specular highlights are baked and don't change when you move.

The reason why it looks "pixelated" is because it's a quite low resolution compared to nowadays hardware.

If you want such a bad-resolution-effect, the easiest way is using a rendertexture like @screenname-taken already mentioned.