In this talk I present Wiener Lemma type results on several Banach *-algebras of time-frequency shift operators with absolutely summable (integrable) coefficients, and relationships to two applications. One application is the Heil-Ramanathan-Topiwala conjecture that states that finitely many time-frequency shifts of one L^2 function are linearly independent. This turns to be equivalent to the absence of eigenspectrum for finite linear combinations of time-frequency shifts. I will prove a special case of this conjecture. The second application is related to the channel equalization problem.