Mulumbu, 31, spent last season on loan at Scottish side Kilmarnock, scoring his first goal against Celtic in February.
He has also played Premier League and Championship football for West Brom and Norwich City.
Speaking to the club's website, Mulumbu said: "I walked onto the pitch when I signed and it felt amazing, these are the reasons we play football.
"My immediate focus is to stay fit and ready and be 100 per cent ready. I've played at Celtic Park as the opposition and it's a special place to play, so I'm looking forward to being part of the team and playing there.
"I cannot describe how amazing this day is for me, but it's one step to the other. I have signed but now I must work hard to play.
"The first goal for me is to show to the manager that I'm ready. Anytime he needs me I will respond on the pitch. I know the game and what is expected of me so I'm going to be ready when the gaffer needs me.
"This is what you work for as a footballer. To play in a big team like Celtic is just amazing. I always work very hard and I want to help the young lads here."