For given sequence i.e.,
x₁(n) = δ(n) + δ(n−1) − δ(n−2) − δ(n−3)
x₂(n) = δ(n) − δ(n−2) + δ(n−4)
Given, N = 5
The circular convolution x₃(n) of the two sequences shall be:
- A. {3, 0, 2, 0, 2}
- B. {1, 0, −4, 1, 2}
- C. {2, 0, −3, 0, 1}
- D. {3, 0, −3, −2, 2}
Solution
x₁(n) = [1, 1, −1, −1, 0]
x₂(n) = [1, 0, −1, 0, 1]
Circular Convolution Formula:
y(n) = Σ x₁(k) · x₂((n − k) mod 5)
y(0)
= 1·1 + 1·1 + (−1)·0 + (−1)·(−1) + 0·0 = 3
y(1)
= 1·0 + 1·1 + (−1)·1 + (−1)·0 + 0·(−1) = 0
y(2)
= 1·(−1) + 1·0 + (−1)·1 + (−1)·1 + 0·0 = −3
y(3)
= 1·0 + 1·(−1) + (−1)·0 + (−1)·1 + 0·1 = −2
y(4)
= 1·1 + 1·0 + (−1)·(−1) + (−1)·0 + 0·1 = 2
Final Answer: x₃(n) = [3, 0, −3, −2, 2]
Correct Option: D