"He's a left-back with a terrific engine and he'll be part of a squad that will hopefully do well."
The Scot has played twice in the Champions League and five times in the Uefa Cup for Rangers.
Smith fought back from a groin problem that sidelined him for 16 months in 2006 to feature regularly in the Scottish Premier League title-winning side of 2009.
However, he only made 12 starts and six substitute appearances in the season just passed under Walter Smith.
"When the new manager came in I wasn't fit and he was really successful and it's hard to get into a successful side," Smith added.
"I've played in the Champions League and big Old Firm games, so I may not have played a lot of games in my career but the games I have played in stand me in good stead for this league.
"People always put question marks over you because of injuries but it's not something I'm worried about."
Smith becomes former Celtic midfielder Lambert's third close season signing after midfielder Andrew Crofts and defender Elliott Ward.
It is the Scots' first since signing an improved contract with the Canaries on Tuesday.
Norwich have won nearly 68% of their games under Lambert, who joined from Colchester United in August, turning a side close to the relegation zone around to such an extent that they won the League One title by nine points.
The news of Lambert's new deal coincided with a Football Disciplinary Commission ruling that Norwich should pay his former club £425,000 in compensation for the departure of the Scot and his backroom team.
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