Topic: Digital Signal Processing
What is 4 point Discrete Fourier Transform of signal $x(n)=\{1,2,3,4\}$?
- A. $[10, 2j, 2, -2 ]$
- B. $[10, -2, -2, -2]$
- C. $[10, -2+2j, -2, -2-2j]$
- D. $[10, -2, -2, -10 ]$
Answer: C. $[10, -2+2j, -2, -2-2j]$
Explanation: $X(k) = \sum_{n=0}^{3} x(n) e^{-j\frac{2\pi}{4}kn} = \sum_{n=0}^{3} x(n) (-j)^{kn}$.
$X(0) = 1 + 2 + 3 + 4 = 10$.
$X(1) = 1(1) + 2(-j) + 3(-1) + 4(j) = -2 + 2j$.
$X(2) = 1(1) + 2(-1) + 3(1) + 4(-1) = 1 - 2 + 3 - 4 = -2$.
$X(3) = 1(1) + 2(j) + 3(-1) + 4(-j) = -2 - 2j$.
Therefore, the 4-point DFT is $[10, -2+2j, -2, -2-2j]$.