The free agent penned a one-year deal yesterday.
Speaking after completing a deadline day move, Carroll said: "Hopefully, I can carry on where I left off. It's been a long time.
"Every transfer window I was sick of hearing I could be back but it's finally happened .
"For me, it's unfinished business.
"I left on a high and I want to get back to how it was, scoring goals, playing games and enjoying games."
When asked how he felt on the road back to Newcastle, Carroll said: "It's an amazing feeling.
"On my way up, I had butterflies in my stomach.
"The minute I left Newcastle eight and a half years ago, it was always in my head to come back although I thought it would be a lot quicker.
"But I'm here now and cannot wait to get started.
"When I put that shirt back on it will probably be an even better feeling than the first time because it's only when you go away, you realise what you had and what you miss.
"I've missed a lot of games, so there is plenty left in the tank.
"Coming back is what all your dreams are made of and hopefully, I can hit the ground running."