Compute the Discrete Fourier Transform (DFT) of the 4-point signal $f = [3, 2, 5, 1]$.
I know that there could be some differences in notations so here are the ones I'm using:
Given a signal $f = [f_0, \dots, f_{n-1}] : \mathbb{Z}(n) \to \mathbb{C}$, we define its Discrete Fourier Transform as $F = [F_0, \dots, F_{n-1}] : \mathbb{Z}(n) \to \mathbb{C}$ such that $$F_k = \frac{1}{n} \sum_{j=0}^n f_j \ e^{-2\pi i k j / n} \qquad k = 0, \dots, n-1$$
I solved this "simple" exercize following step by step the definition and here i write what i got:
$$F_1 = \frac{1}{4} \sum_{j=0}^3 f_j\ e^{-\pi i j/2} = \frac{1}{4} \Big[ 3 -2i - 5 + i\Big] = -(2+i)/4$$
$$F_2 = \frac{5}{4}, \qquad F_3 = (2+i)/4, \qquad F_4 = 11/4$$
Can anyone check my calculations and garantee that the modus operandi for solving this exercises is correct please?