Which statement describes a shift-invariant discrete-time system?

Which statement describes a shift-invariant discrete-time system?

• A. The impulse response depends on absolute time
• B. The impulse response is independent of time shifts
• C. The system is always causal
• D. The system is nonlinear

Answer: (B)

Time invariance means the system’s behavior doesn’t depend on when the input is applied: shifting the input in time shifts the output by the same amount, but otherwise leaves it unchanged. In a discrete-time setting this leads to the impulse response being the same regardless of the absolute time index; it depends only on the relative time difference. If the impulse response already depended on the absolute time, the system would respond differently to the same input at different times, which is the hallmark of a time-varying system. Hence, stating that the impulse response is independent of time shifts captures the idea of a shift-invariant (time-invariant) discrete-time system. Note that causality or nonlinearity are separate properties and are not what defines shift invariance.