LTI output, h={1,1,1}, x={1,1,1,1}

For an LTI system y(n) = Σ x(k) h(n − k). With no origin marked, take both sequences to start at n = 0: h = {1, 1, 1} (3 samples), x = {1, 1, 1, 1} (4 samples). The output has 3 + 4 − 1 = 6 samples, n = 0 … 5.

Table method (sum along diagonals)

Adding the entries with the same i + j:

• y(0) = 1
• y(1) = 1 + 1 = 2
• y(2) = 1 + 1 + 1 = 3
• y(3) = 1 + 1 + 1 = 3
• y(4) = 1 + 1 = 2
• y(5) = 1

Answer: y(n) = {1, 2, 3, 3, 2, 1}, n = 0 … 5.

Check: Σy = 12 = (Σx)(Σh) = 4 × 3. ✓ Also X(z)H(z) = (1 + z⁻¹ + z⁻² + z⁻³)(1 + z⁻¹ + z⁻²) = 1 + 2z⁻¹ + 3z⁻² + 3z⁻³ + 2z⁻⁴ + z⁻⁵. ✓