If the system and sound propagation were ideal, this fractional delay would extend infinitely to the past.
I believe this is where your problem lies.
You go from talking about a real system (D to A, speaker, microphone, A to D), and then you say that it can't be an ideal system.
This train of thought has three problems:
- Ideal systems are nice, but we only get to play with real ones.
- Whose ideal is it, anyway?
- Fractional delays -- who cares?
If you're talking about a typical audio system, then if you measured the time from the moment that a signal is applied to the D to A to the moment that you detect signal at the A to D output, you would find many many samples worth of delay. I would assume that you'd see hundreds of samples, if not thousands, in a typical pro or prosumer audio setup.
Even in control systems applications (where I cut my teeth on DSP), a normal system has a bandwidth that's 1/10th or less than the sampling rate, meaning that from the moment that you advance a command to the loop until the time that you can really see a result is several samples, and the motion isn't really done for tens or hundreds of samples.