are one's brain waves in LDs different than normal dreams?

I was wondering this recently, while listening to some binaural beats/reading about brain waves.

also: are one’s brain waves in a false awakening different from waking life? I noticed that even though I go to bed with certain LD goals in mind, it’s hard to remember them when I become lucid. BUT, the other day, I had a false awakening after a LD, and in the FA I thought “damn, I had the perfect chance to do [dream goal] but missed it!” so I was able to remember it when I THOUGHTI was awake…I thought maybe it had something to do with brainwaves.

like could there be a difference in consciousness between waking life:dreaming vs false awakening after a dream. in the latter, you’re at least aware that you were previously dreaming (even though you still are dreaming) and in the first you have no idea of dreaming…