The sequence x[n] = {1, 2, 3} (n = 0, 1, 2) is applied to an LTI system with impulse response h[n] = {1, −1} (n = 0, 1). Which of the following are the outputs y[0] to y[3]?

Answer: C. 1, 1, 1, −3

For h[n] = {1, −1}, the convolution y[n] = Σx[k]h[n − k] becomes y[n] = x[n] − x[n − 1], the difference from the previous input. y[0] = 1 − 0 = 1, y[1] = 2 − 1 = 1, y[2] = 3 − 2 = 1 and y[3] = 0 − 3 = −3. The output length is 3 + 2 − 1 = 4, from the input length 3 and the impulse-response length 2.

• A. This is computed without time-reversing h (a cross-correlation); its signs are opposite to the correct output.
• B. This is the convolution with {1, 1}, dropping the sign of h[1].
• C. Correct. It computes the difference from the previous input, x[n] − x[n − 1].