timeinvariance

Time invariance, a fundamental concept in signal processing and systems theory, refers to a property of a system whereby shifting the input signal in time results in an equivalent shift of the output signal, without altering the signal's shape or features. If a system maps input x(t) to output y(t) = T{x(t)}, the system is time-invariant if for any real t0, T{x(t - t0)} = y(t - t0). Equivalently, if y(t) is the response to x(t), then delaying the input by t0 delays the output by the same amount.

In linear time-invariant (LTI) systems, time invariance implies that the response to any input can be obtained by convolution with a fixed impulse response h(t): y(t) = ∫ h(τ) x(t - τ) dτ. In discrete time, y[n] = ∑ h[k] x[n - k]. The Fourier or Laplace transforms of LTI systems are multiplicative: Y(ω) = H(ω) X(ω).

Not all systems are time-invariant. Examples:

- Time-invariant: a pure time delay y(t) = x(t - t0) is time-invariant; applying the delay to the input yields the same delay in the output.

- Time-varying: a system that multiplies by a time-varying factor, such as y(t) = t x(t), or a filter whose coefficients depend on time, or scaling of time y(t) = x(a t). These do not satisfy T{x(t - t0)} = y(t - t0) in general.

Time invariance is a foundational assumption that simplifies analysis, design, and interpretation of systems, particularly in signal processing and control theory.