Tahajjud (Arabic for: optional late night Prayer) is most probably performed after sleeping for some time; the best time to perform it is the last third portion of the night. Yet, Tahajjud may be performed in any part of the night after performing `Isha’ Prayer before going to sleep.
According to Dr. Muzammil H. Siddiqi, stated: It is not required to sleep before praying Tahajjud. If you are awake during the night, you can pray Tahajjud also.
However, the beauty of this Prayer is that one leaves one’s bed and stands up to worship Allah during the night while many others prefer to sleep at that time.