Question

Given a sequence x(n) for 0 < n < 3 where x(0) = 1, x(1) = 1, compute its Discrete Fourier Transform (DFT) x(k).

a. DFT not applicable
b. [1, 1, 0]
c. [1, -1, 0]
d. [2, 0, 0]

Answer 1

The correct Discrete Fourier Transform (DFT) for the given sequence x(1) = 1 and x(2) = 1 is [2, 0, 0]. The computation of the DFT involves summing the product of the sequence elements and complex exponential factors. The provided options did not account for normalization, but option d is the closest to the expected result.

Step-by-step explanation:

The Discrete Fourier Transform (DFT) of a sequence x(n) where 0 < n < 3 with given values x(0) = 1, x(1) = 1, and not defined for n=2, since 0x(1) = 1, x(2) = 1. We can use the formula for the DFT, which is X(k) = ∑_{n=0}^{N-1} x(n) · e^{-j2πnk/N}, where N is the length of the sequence, n is the current sample, k is the current frequency, and j is the imaginary unit. For a 2-point sequence, the DFT is calculated as follows:

• 
  
• X(0) = x(1) + x(2) = 2
• 
  
• X(1) = x(1) · e^{-j2π(1)1/2} + x(2) · e^{-j2π(2)1/2} = 1 · (cos(π) - j · sin(π)) + 1 · (cos(π) - j · sin(π)) = -1 -1 = -2 (which is 0 after normalization for a 2-point DFT)

However, the correct normalized DFT result should be [2, 0]. Therefore, the correct answer to the student's question would be option d. [2, 0, 0].