The 24-year-old, who has previously played for Leeds United in the Championship, moved back to the second tier on Friday after being allowed to leave by Hammers boss Manuel Pellegrini.
Speaking to the Nottingham Post, Byram also admitted that there was a lot of speculation surrounding his move and expressed his delight in finally agreeing the switch.
"It feels really good. It has been rumoured for a while now but I am glad to be here and I can't wait to get started," the defender said. "The manager played a big part in me coming here and you can see the players he has brought in.
"There is a real push for promotion and I just want to come here, do my best and help the team. I had a bit of bad luck with injuries last year but now I am ready to go and I can't wait to get back out on the pitch. There are lots of games, I have been used to it with Leeds.
"It is tough, there is no guaranteed wins or easy teams, so I am looking forward to getting back into it. I watched the [Forest v West Bromwich Albion] match the other night and I thought the team played really well."
Byram appeared 130 times in the Championship with Leeds before joining West Ham.