The former Germany international was available on a free transfer, having left Bayer Leverkusen at the end of last season.
Hummels is set to be out until January with torn heel ligaments, while calf problems have also sidelined left-back Marcel Schmelzer for three weeks, with both injuries picked up during Germany's 1-0 victory over England on Tuesday.
They join Subotic - expected to be out for six months with cruciate ligament damage in his knee - in the Dortmund medical room.
And Friedrich, who has been training with Dortmund in recent weeks, is looking forward to playing first-team football once again.
"I am very excited to play at Signal Iduna Park as a player of Borussia Dortmund, thus far I only played for visiting teams," he told the club's official website.
"To play for the home crowd, and not against them, must be a great feeling.
"I am very much looking forward to an incredible time in black and yellow."
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