4.8 Two DT signals, $s_k(n)$ and $s_l(n)$, are said to be orthogonal over an interval $[N_1, N_2]$ if
\begin{equation}
\sum_{n=N_1}^{N_2} s_k(n)s_l^*(n) = \begin{cases} A_k, & k=l\\0, & k \neq l\end{cases}
\end{equation}
If $A_k = 1$, the signals are called orthonormal.
(a) Prove the relation
\begin{equation}
\sum_{n=0}^{N-1} e^{j2\pi kn/N} = \begin{cases} N, & k = 0, \pm N, \pm 2N, ...\\0, & \text{otherwise}\end{cases}
\end{equation}
(b) Illustrate the validity of the relation in part (a) by plotting for every value of
$k = 1, 2, ..., 6$, the signals $s_k(n) = e^{j(2\pi/6)kn}$, $n = 0, 1, ..., 5$. [Note: For a given
k, n the signal $s_k(n)$ can be represented as a vector in the complex plane.]
(c) Show that the harmonically related signals
$s_k(n) = e^{j(2\pi/N)kn}$
are orthogonal over any interval of length N.
