timeinvariant

Time-invariant refers to a system or process whose output depends on the input only through its shape and timing relative to the current time, not on the absolute time at which the input is applied. A system is time-invariant if, for every input signal x(t) and every time shift τ, the output satisfies F{x(t−τ)} = y(t−τ), where y(t) = F{x(t)}. In discrete time, replace t with n and τ with k, giving F{x[n−k]} = y[n−k].

Time-invariant is often considered together with linearity, yielding linear time-invariant (LTI) systems. For LTI systems, the output to any input is the convolution of the input with a fixed impulse response h(t) (continuous time) or h[n] (discrete time): y(t) = x(t) * h(t), y[n] = x[n] * h[n]. The impulse response completely characterizes an LTI system, and shifting the input results in a corresponding shift of the output.

Not all systems are time-invariant. A time-varying system has coefficients or dynamics that change with time, so F{x(t−τ)} ≠ y(t−τ) in general. Examples include a filter whose coefficients depend on t, a gain that changes over time, or a system that monitors a time index and alters behavior accordingly.

Applications and testing: Time-invariant models simplify analysis and design in signal processing, communications, and control. To test time invariance, apply a test input x(t) and its shifted version x(t−τ) and compare outputs; if the outputs are shifted by τ as well, the system is time-invariant. Related concepts include linear time-invariant systems, convolution, and impulse response.