4-point DFT of {2, 1+j, 1−j}

The sequence has only 3 samples, so append one zero for a 4-point DFT: $x[n] = \{2,\; 1+j,\; 1-j,\; 0\}$.

Formula: $X[k] = \sum_{n=0}^{3}x[n]W_4^{nk}$, with $W_4 = e^{-j\pi/2} = -j$, so $W_4^0=1$, $W_4^1=-j$, $W_4^2=-1$, $W_4^3=j$.

$$\begin{bmatrix} X[0] \\ X[1] \\ X[2] \\ X[3] \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -j & -1 & j \\ 1 & -1 & 1 & -1 \\ 1 & j & -1 & -j \end{bmatrix} \begin{bmatrix} 2 \\ 1+j \\ 1-j \\ 0 \end{bmatrix}$$

Working:

$$\begin{aligned} X[0] &= 2 + (1+j) + (1-j) + 0 = 4 \\ X[1] &= 2 + (1+j)(-j) + (1-j)(-1) + 0 = 2 + (1 - j) + (-1 + j) = 2 \\ X[2] &= 2 - (1+j) + (1-j) - 0 = 2 - 2j \\ X[3] &= 2 + (1+j)(j) + (1-j)(-1) + 0 = 2 + (-1 + j) + (-1 + j) = 2j \end{aligned}$$

Answer: $X[k] = \{4,\; 2,\; 2-2j,\; 2j\}$

Spectra (stem plots):

 |X[k]|
 4 |  |
 3 |  |       |
 2 |  |   |   |   |
 1 |  |   |   |   |
 0 +--+---+---+---+--> k
      0   1   2   3

 phase (degrees)
  90 |              |
  45 |              |
   0 +--o---o---+---+--> k
 -45 |          |
        0   1   2   3

(Here $x[n]$ is complex, so $X[3] \ne X^*[1]$; the spectrum has no conjugate symmetry.)

Check (Parseval): $\sum|x[n]|^2 = 4 + 2 + 2 + 0 = 8$ and $\frac{1}{4}\sum|X[k]|^2 = \frac{1}{4}(16 + 4 + 8 + 4) = 8$.