As someone once said:
"If I be goin' there I be-n't start by here"
For what the OP describes you don't need any processing at all, just A/D -> memory -> D/A with a simple clock and counter to step through the memory. The read needs to be 1 tick ahead of the write.
So its:
step, read, write, step, read, write ... etc.
Your delay is the memory size (in words) / clock rate.