A 4×4 DFT matrix is given by \(\dfrac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1…
2017
A 4×4 DFT matrix is given by
\(\dfrac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & x & -1 & y \\ 1 & -1 & 1 & -1 \\ 1 & -j & -1 & j \end{bmatrix} \\ j^2 = -1\)
Where the values of 𝑥 and 𝑦 are ______, _____ respectively.
- A.
1,−1
- B.
−1,1
- C.
−𝑗,𝑗
- D.
𝑗,−𝑗
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Correct answer: D
Solution:
For a 4-point DFT the twiddle factor is W = e^{j2π/4} = e^{jπ/2} = j. Compute its powers:
W^0 = 1
W^1 = j
W^2 = -1
W^3 = -j
The second row of the matrix corresponds to 1, W^1, W^2, W^3, so the entries are 1, j, -1, -j. Therefore x = j and y = -j.
This is consistent with the fourth row 1, -j, -1, j, which comes from the same twiddle-factor powers for the third index.