Problem 1. (Orthogonality in DFT, 10 points)
Consider the set of vectors:
$$D = \{e^{-j\frac{2\pi kn}{M}}; k, n = 0, 1, 2, ..., M - 1\},$$
where $M \ge 2$. Consider an $M \times M$ DFT matrix D whose $(k, n)^{th}$-element is $e^{-j\frac{2\pi kn}{M}}$,
$k, n = 0, 1, 2, ..., M - 1$. Determine the adjoint and the inverse of D.
Note: D constitutes an orthogonal set over $n \in [0, M - 1]$