A time-invariant system has the property that:

• A. The input-output relation changes with time.
• B. The impulse response depends on time.
• C. The output is scaled by time.
• D. A time shift in the input results in the same time shift in the output.

Answer: (D)

Time invariance means that delaying the input in time yields an identical delay in the output, with the same waveform. For a linear time-invariant system, the output is y(t) = ∫ x(ξ) h(t−ξ) dξ. If you feed a time-shifted input x_d(t) = x(t−τ), the resulting output becomes y_d(t) = ∫ x(t−τ−ξ) h(t−ξ) dξ. With a change of variables, this simplifies to y_d(t) = y(t−τ). So delaying the input by τ delays the output by the same amount, preserving the shape. This is the defining behavior, whereas changing with time, having an impulse response that depends on absolute time, or output scaling with time would break this property.