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LTI output with sparse samples, verify

Short answer

Superposing scaled shifted impulse responses gives y[n]={0.5,0,3,4.5,4,12,9} for n=-2 to 4.

Find the output of LTI system having impulse response h[n] with h[−2] = 1, h[0] = 2, h[1] = 3 and input signal x[n] with x[0] = 1/2, x[2] = 2, x[3] = 3. Also check the answer.

Reading the samples (zero where not listed):

  • h[n]: h[−2] = 1, h[−1] = 0, h[0] = 2, h[1] = 3
  • x[n]: x[0] = ½, x[1] = 0, x[2] = 2, x[3] = 3

y[n] = Σ x[k] h[n − k]. Since x has three non-zero samples, y is the sum of three shifted, scaled copies of h:

  • x[0] = ½ → ½·h[n] : value ½ at n = −2, 1 at n = 0, 1.5 at n = 1
  • x[2] = 2 → 2·h[n − 2]: 2 at n = 0, 4 at n = 2, 6 at n = 3
  • x[3] = 3 → 3·h[n − 3]: 3 at n = 1, 6 at n = 3, 9 at n = 4

Where each copy lands: x[2] shifts h by 2 (h[−2], h[0], h[1] land at n = 0, 2, 3); x[3] shifts h by 3 (n = 1, 3, 4). Adding by position:

n−2−101234
½·h[n]0.5011.5000
2·h[n−2]0020460
3·h[n−3]0003069
y[n]0.5034.54129

Answer: y[n] = {0.5, 0, 3, 4.5, 4, 12, 9} for n = −2 … 4.

Check

Σy = 0.5 + 3 + 4.5 + 4 + 12 + 9 = 33. (Σx)(Σh) = (0.5 + 2 + 3)(1 + 2 + 3) = 5.5 × 6 = 33. ✓

✓ Answer checked by Claude Opus 5.5 (high effort)

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