In Mathematica after version 9. We had ListFourierSequenceTransform to do discrete-time Fourier Transform, but we do not have the inverse function. If we peform it according to the formula, e.g.

Integrate[(1/4 + 7/8*Exp[-I*\[Omega]] - 1/2*Exp[-2*I*\[Omega]])/(
1 + 1/6*Exp[-I*\[Omega]] - 1/6*Exp[-2*I*\[Omega]])*
Exp[I*\[Omega]*n], \[Omega], {\[Omega], 0, 2*Pi}]

It will take long time and give complex result. May some kind one give me some hint how to implement the inverse discrete-time Fourier transform?

In Mathematica after version 9. We had ListFourierSequenceTransform to do discrete-time Fourier Transform, but we do not have the inverse function. If we peform it according to the formula, $$\frac 1{2\pi} \int_0^{2\pi} H(e^{j\omega}) \, e^{j \omega n}\,d\omega$$ e.g.

Integrate[(1/4 + 7/8*Exp[-I*\[Omega]] - 1/2*Exp[-2*I*\[Omega]])/(
1 + 1/6*Exp[-I*\[Omega]] - 1/6*Exp[-2*I*\[Omega]])*
Exp[I*\[Omega]*n], \[Omega], {\[Omega], 0, 2*Pi}]

It will take long time and give complex result. May some kind one give me some hint how to implement the inverse discrete-time Fourier transform?