Write the convolution relations for continuous-time and discrete-time LTI systems.

• A. CT: y(t) = ∫ x(τ) h(t−τ) dτ; DT: y[n] = ∑ x[k] h[n−k].
• B. CT: y(t) = ∑ x(k) h(t−k).
• C. DT: y[n] = ∫ x(τ) h(n−τ) dτ.
• D. CT: y(t) = x(t) h(t).

Answer: (A)

The output of an LTI system is built by convolving the input with the system’s impulse response. In continuous time, every input value x(τ) contributes a shifted version of the impulse response h(t−τ), and you add all those contributions via an integral: y(t) = ∫{-∞}^{∞} x(τ) h(t−τ) dτ. In discrete time, the same idea becomes a sum over all samples: y[n] = ∑{k=-∞}^{∞} x[k] h[n−k]. The impulse response fully characterizes the system, and convolution describes how the system blends past and present input information to produce the output. The other forms mix time domains improperly or use a simple pointwise product, which does not represent convolution.