There are two different groups: Local Bulgarians that converted to Islam and Turks that migrated to Bulgaria. During the exchange of populations during the early 20th century, the Muslims in Bulgaria, whether local or migrants were supposed to have been exchanged for Bulgarian Christians in Turkey but not everybody was actually exchanged. At present the Turkish descent minority in Bulgaria comprises 10% of the population. I am not aware which part of Turkey they came from originally. The locals that converted to Islam are much less numerous.
I am not aware of any autosomal study but I do recall a Y-DNA study:
https://en.wikipedia.org/wiki/Bulgarian_Turks
and go to genetic origins section.