Baun, 25, has played five NHL games between 2014-2016, all of them with the Blackhawks.
He had spent the last two seasons with the AHL's Rockford IceHogs. In 117 games, he scored 15 goals, 14 of them in 2016-17, and collected 28 assists for 43 points.
The 6-foot-2, 209-pound Toronto native was undrafted.
Martinsen was acquired at the trade deadline last season, in return for Sven Andrighetto. He had been sent down to the Rocket earlier in the week.
