2. Complex Numbers: For discrete-time signals $x(n)$ as given below, compute
\begin{equation}
X(k) = \sum_{n=0}^{N-1} x(n) e^{-j2\pi kn/N}
\end{equation}
for $k = 0, \dots, N - 1$. Simplify the resulting expressions as much as possible.
1. $x(n) = \delta(n)$
2. $x(n) = \delta(n - n_0)$
3. $x(n) = a^n$
4. $x(n) = \begin{cases} 1 & \text{if } 0 \le n \le N/2 - 1 \\ 0 & \text{otherwise} \end{cases}$
5. $x(n) = e^{j2\pi k_0 n/N}$
6. $x(n) = \cos(2\pi k_0 n/N)$
Note: Identify geometric series wherever possible. The Dirac delta function is defined as
$\delta(n) = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{otherwise.} \end{cases}$
Assume $a \in \mathbb{R}$ and $n_0, k_0 \in \{0, 1, \dots, N - 1\}$.
Please submit a scan of your detailed calculations with intermediate steps.
