By the way, when people speak slowly, /ы/ can actually be of a different quality, compared to the normal spoken language. The main difference a Russian native speaker hears between /ти/ and /ты/ is in the quality of the consonant: soft and hard. The quality of the vowel may vary.
I study Turkish and face the same trouble.Turkish vowel system is absolutely symmetrical, it has pairs e - a, i - ı (this pair is very close to и - ы), ö - o, ü - u. Surprisingly, distinguishing between /i/ and /ı/ sometimes can be as hard for me as distinguishing between /ö/ and /o/ or /ü/ and /u/. This is because the quality of these vowels play a more important role in Turkish, than in Russian.