Alarm Clock Range

On different days I can wake up at the same time and feel much different.  Even if I go to sleep at the same time, the results in the morning vary.  I suspect that this is based on sleep cycles.  I feel especially groggy in the morning when I am woken from a dream or period of deep sleep.  On the opposite end of the spectrum, sometimes I wake up naturally (around the same time as my alarm) and I feel well-rested.

I am no expert on sleep, but I imagine the cycles go something like this:

How difficult it is to wake up over the span of the night.

Based on this graph, there should be an optimal time to wake up.  Unfortunately schedules do not automatically adjust themselves based on how much we feel like sleeping.  Instead, we can try to find the most optimal time to wake up within a certain period.

Say, for example, you want to wake up at 8:00.  A normal alarm would ring at 8:00, no matter where you are in your current sleep cycle.  My idea is to develop an alarm that has a range of when you want to wake up, and through analyzing your sleep cycle, wake you up at the most optimal time.  Using my alarm, you would set the range to 7:00-8:00.  Then, if the device senses that you are at an optimal time to wake up at 7:40, the alarm will ring then.  Even though you had twenty less minutes of sleep, you will feel better than if you slept longer.