The impulse response of a discrete-time LTI system completely characterizes the system.

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

Answer: (C)

In a discrete-time LTI system, the impulse response is the complete description of how the system transforms any input. The impulse response h[n] is what you get as the output when the input is a unit impulse δ[n]. Because the system is linear and time-invariant, any input x[n] produces an output y[n] that is the convolution of x with h: y[n] = sum over k of x[k] h[n−k]. This convolution formula shows that every feature of the input is processed by the system according to h[n], and nothing else about the system is needed to predict the output.

Therefore, knowing h[n] lets you determine the output for any input, and two systems with different impulse responses will yield different outputs for at least some inputs. Conversely, if two systems share the same impulse response, they behave identically for all inputs. This is why the impulse response fully characterizes a discrete-time LTI system.

The stability of the system is a separate property. A system can be stable or unstable regardless of whether its impulse response exists or is finite; nonetheless, the impulse response still determines the input-output relationship. Hence the statement is true.