"Break forth together into singing, you waste places of Jerusalem, for the LORD has comforted his people; he has redeemed Jerusalem". ESV. My emphasis.
Barnes notes on the Bible has:
"The idea here is, that Yahweh was about to restore his people from their long captivity, and again cause Jerusalem to be rebuilt."
My comment: If the waste places of Jerusalem being restored is what is initially being described here, in what sense did Jerusalem need redeeming? Could there have been restoration without redemption?
What has been redeemed? How was this redemption achieved, and be achieved in the past? ["he has"].
Leviticus 25:25 "gaal" has an element of payment.