Solve Exercise 1.22 using the concept of the transfer function.

Exercise 1.22

Compute the inverse Fourier transform of

\[X\left(\mathrm{e}^{\mathrm{j} \omega}\right)=\frac{1}{1-\mathrm{e}^{-\mathrm{j} \omega}}\]

in Example 2.19.

Hint: Replace \(\mathrm{e}^{\mathrm{j} \omega}\) by \(z\), compute the inverse \(z\) transform using as closed contour \(C\) the one defined by \(z=ho \mathrm{e}^{\mathrm{j} \omega}, ho>1\), and take the limit as \(ho \rightarrow 1\).

Example 2.19.

Compute the Fourier transform of the sequence x(n) = u(n)