Question

For a 4-point sequence {0, 1, 2, 3}, FFT is ________.

• A. 6, -2 + j2, -2, -2 - j2
• B. 6, -2+j2, 2, -2-j2
• C. 6, -2+j2, -2, 2-j2
• D. 6, -2+j2, 2, 2-j2

Hint:

For small sequences, compute DFT directly using the formula. For longer sequences, FFT algorithms reduce computation time significantly.

Answer 1

The Correct Option is B

Let the given sequence be:
\[ x[n] = \{0, 1, 2, 3\} \]
We compute the 4-point DFT (Discrete Fourier Transform) using:
\[ X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j2\pi nk/N},\quad k = 0, 1, 2, 3 \]
Here, \( N = 4 \)

Step 1: Compute X[0]:
\[ X[0] = x[0] + x[1] + x[2] + x[3] = 0 + 1 + 2 + 3 = 6 \]
Step 2: Compute X[1]:
\[ \begin{aligned} X[1] &= x[0] + x[1]e^{-j2\pi(1)/4} + x[2]e^{-j2\pi(2)/4} + x[3]e^{-j2\pi(3)/4} \\ &= 0 + 1(-j) + 2(-1) + 3(j) = -j -2 + 3j = -2 + 2j \end{aligned} \]
Step 3: Compute X[2]:
\[ \begin{aligned} X[2] &= x[0] + x[1]e^{-j2\pi(2)/4} + x[2]e^{-j2\pi(4)/4} + x[3]e^{-j2\pi(6)/4} \\ &= 0 + 1(-1) + 2(1) + 3(-1) = -1 + 2 -3 = -2 \end{aligned} \]
Step 4: Compute X[3]:
\[ \begin{aligned} X[3] &= x[0] + x[1]e^{-j2\pi(3)/4} + x[2]e^{-j2\pi(6)/4} + x[3]e^{-j2\pi(9)/4} \\ &= 0 + 1(j) + 2(-1) + 3(-j) = j -2 -3j = -2 -2j \end{aligned} \]
So the FFT result is:
\[ \{6,\ -2 + 2j,\ -2,\ -2 - 2j\} \]