The 30-year-old Denmark international is a vastly experienced midfielder, having featured in the German Bundesliga, the Spanish Primera Division and Serie A.
"With this move here I have a lot to prove," said Poulsen, who has penned a three-year contract.
"I didn't play the last matches in the Europa League because of the move to this club.
"I am feeling fit but of course it's a decision for the manager and the staff."
Poulsen has already played under Reds boss Roy Hodgson, and the pair achieved success with Copenhagen in the Danish Superliga.
"I had two good years with him so I hope to carry that on, and hopefully he'll be satisfied with my qualities," he remarked.
"He's a manager who's been all over the world and has a lot of experience, so of course I have a lot of respect for him.