- 2073 Chaitra · 6 marks
Convolution of given sample values, verify
Short answer
Summing copies of h(n)={1,0.75,0.5} shifted by non-zero input samples yields y[n]={0.5,0.375,1.25,1.5,1.0625,0.875,0.375,0.25} for n=-2 to 5.
Calculate y[n], if x[n] is x[−2] = 0.5, x[0] = 1, x[1] = 0.75, x[3] = 0.5 and h[n] is h[0] = 1, h[1] = 0.75 and h[2] = 0.5, and verify your result.
Samples (others are zero):
- x[−2] = 0.5, x[0] = 1, x[1] = 0.75, x[3] = 0.5
- h[0] = 1, h[1] = 0.75, h[2] = 0.5
y[n] = Σ x[k] h[n − k] is the sum of copies of h, each shifted to the position of a sample of x and scaled by that sample. Here h = {1, 0.75, 0.5} starts at n = 0.
| n | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|
| 0.5·h[n+2] | 0.5 | 0.375 | 0.25 | 0 | 0 | 0 | 0 | 0 |
| 1·h[n] | 0 | 0 | 1 | 0.75 | 0.5 | 0 | 0 | 0 |
| 0.75·h[n−1] | 0 | 0 | 0 | 0.75 | 0.5625 | 0.375 | 0 | 0 |
| 0.5·h[n−3] | 0 | 0 | 0 | 0 | 0 | 0.5 | 0.375 | 0.25 |
| y[n] | 0.5 | 0.375 | 1.25 | 1.5 | 1.0625 | 0.875 | 0.375 | 0.25 |
Answer: y[n] = {0.5, 0.375, 1.25, 1.5, 1.0625, 0.875, 0.375, 0.25} for n = −2 … 5.
Verification
- Σx = 2.75, Σh = 2.25, product = 6.1875. Σy = 0.5 + 0.375 + 1.25 + 1.5 + 1.0625 + 0.875 + 0.375 + 0.25 = 6.1875. ✓
- Direct check of one point: y[2] = x[0]h[2] + x[1]h[1] = 0.5 + 0.5625 = 1.0625. ✓
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Verify convolution theorem, f = g = e^{-x²} · Applied Mathematics, chapter 4
- Complex Fourier series; verify convolution · Applied Mathematics, chapter 4
- CLT: sample mean between 75 and 78 · Probability and Statistics, chapter 4 · asked 2×
- Verify commutative law of convolution · Signals and Systems, chapter 5
All Digital Signal Analysis and Processing chapter 1 questions →