Why does it say "Chatzos Laylah Akum"?
Radak #1: I awaken and rise from my bed to thank You for Your straight and righteous Mishpatim. I compose Mizmorim to thank You, when I rise.
Radak #2: Some say that [I thank You for] Your righteous judgment of Resha'im. Malbim - the verse depicts night to be when Hashem exacts judgment on Resha'im - "v'Hafach Laylah v'Yidaka'u" (Iyov 34:25), "ba'Chatzi ha'Laylah va'Shem Hikah Kol Bechor" (Shemos 12:29).
Malbim: The need to sleep did not stop me [from doing Mitzvos]. I always rise at midnight to thank You for Your righteous Mishpatim!
Below (147), it says "Kidamti va'Neshef va'Ashave'ah" (he rose at the beginning of the night)!
Brachos 3b: In our verse, David says that he was never sleeping when midnight arrived.
