- 2071 Chaitra · 6 marks
LTI output for 2δ(n) − δ(n−1) input
Short answer
By linearity and time-invariance, y(n)=2h(n)-h(n-1): y(n)={2,5,1,-4,3,-1} for n=-1 to 4.
A discrete time LTI system has impulse response h(n) = {1, 3, 2, −1, 1} for −1 ≤ n ≤ 3. Determine the system output y(n) if the input x(n) is given by x(n) = 2δ(n) − δ(n−1).
Given: h(n) = {1, 3, 2, −1, 1} for n = −1 … 3, so h(−1) = 1, h(0) = 3, h(1) = 2, h(2) = −1, h(3) = 1.
Input x(n) = 2δ(n) − δ(n − 1).
Because the system is LTI, linearity and time-invariance give
y(n) = 2h(n) − h(n − 1)
(an impulse δ(n − k) produces h(n − k)).
| n | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| 2h(n) | 2 | 6 | 4 | −2 | 2 | 0 |
| −h(n−1) | 0 | −1 | −3 | −2 | 1 | −1 |
| y(n) | 2 | 5 | 1 | −4 | 3 | −1 |
Answer: y(n) = {2, 5, 1, −4, 3, −1} for n = −1 … 4.
Check: Σy = 6; (Σx)(Σh) = (2 − 1)(1 + 3 + 2 − 1 + 1) = 6. ✓
✓ Answer checked by Claude Opus 5.5 (high effort)
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