The convolution sum y[n] = sum_k x[k] h[n-k] is used to compute the output of a discrete-time system, provided the system is linear and time-invariant.

The convolution sum y[n] = sum_k x[k] h[n-k] is used to compute the output of a discrete-time system, provided the system is linear and time-invariant.

• A. True
• B. False
• C. Only for stable systems
• D. Only for causal systems

Answer: (A)

For a discrete-time system that is linear and time-invariant, the output is determined by convolving the input with the system’s impulse response. The input x[n] can be written as a sum of shifted impulses: x[n] = sum_k x[k] δ[n − k]. Because the system is linear, the response to a scaled impulse is scaled accordingly, and because it is time-invariant, the impulse δ[n − k] produces a shifted version of the same impulse response h[n − k]. Putting these together, the total output is y[n] = sum_k x[k] h[n − k], which is exactly the discrete-time convolution x[n] * h[n]. The condition of linearity and time invariance is what guarantees this superposition of shifted impulse responses holds. Stability affects whether the sum converges for a given input, and causality is not required for the convolution formula to apply. Therefore, the statement is true.

For a discrete-time system that is linear and time-invariant, the output is determined by convolving the input with the system’s impulse response. The input x[n] can be written as a sum of shifted impulses: x[n] = sum_k x[k] δ[n − k]. Because the system is linear, the response to a scaled impulse is scaled accordingly, and because it is time-invariant, the impulse δ[n − k] produces a shifted version of the same impulse response h[n − k]. Putting these together, the total output is y[n] = sum_k x[k] h[n − k], which is exactly the discrete-time convolution x[n] * h[n]. The condition of linearity and time invariance is what guarantees this superposition of shifted impulse responses holds. Stability affects whether the sum converges for a given input, and causality is not required for the convolution formula to apply. Therefore, the statement is true.