- 2070 Chaitra · 6 marks
Output y(n) for h={5,4,3,2}, x={1,0,3,2}
Short answer
Linear convolution of h(n)={5,4,3,2} and x(n)={1,0,3,2} starting at n=0 gives y(n)={5,4,18,24,17,12,4} for n=0 to 6.
Find output y(n) when h(n) = {5,4,3,2} and x(n) = {1,0,3,2}.
y(n) = Σ x(k) h(n − k), with both sequences starting at n = 0: h = {5, 4, 3, 2}, x = {1, 0, 3, 2}. The output length is 4 + 4 − 1 = 7.
Table method
Each entry is x(j)·h(i), summed along diagonals i + j = n.
| x0=1 | x1=0 | x2=3 | x3=2 | |
|---|---|---|---|---|
| h0=5 | 5 | 0 | 15 | 10 |
| h1=4 | 4 | 0 | 12 | 8 |
| h2=3 | 3 | 0 | 9 | 6 |
| h3=2 | 2 | 0 | 6 | 4 |
- y(0) = 5
- y(1) = 0 + 4 = 4
- y(2) = 15 + 0 + 3 = 18
- y(3) = 10 + 12 + 0 + 2 = 24
- y(4) = 8 + 9 + 0 = 17
- y(5) = 6 + 6 = 12
- y(6) = 4
Answer: y(n) = {5, 4, 18, 24, 17, 12, 4}, n = 0 … 6.
Check: Σy = 84 = (5 + 4 + 3 + 2)(1 + 0 + 3 + 2) = 14 × 6. ✓
✓ Answer checked by Claude Opus 5.5 (high effort)
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