shiftinvariance

Shift invariance is a property of a function, operator, or system with respect to translations in its input. A system is shift-invariant if translating the input by a fixed amount results in a corresponding translation of the output by the same amount. Formally, let S_a denote the shift operator that maps an input x(t) to x(t − a). A system F is shift-invariant if for all inputs x and all a, F(S_a x) = S_a(F x). This means the system’s behavior does not depend on the absolute position of the input.

In signal processing, shift invariance is closely related to the concept of time invariance. A time-invariant system responds to a time-shifted input with a time-shifted output. Linear time-invariant (LTI) systems are a primary example: their output can be expressed as a convolution with a fixed impulse response h, y(t) = x(t) * h(t). If the input is shifted, the output is shifted by the same amount: y_a(t) = y(t − a). In discrete time this is written as y[n] = sum_k h[k] x[n − k].

Shift invariance is not universal; some systems are shift-variant, meaning their response depends on the input’s absolute position. In applications, identifying shift invariance helps characterize system behavior, guide design (as with LTI filters), and explain why certain architectures, such as convolutional neural networks, are translation-equivariant and can be made approximately invariant through pooling and other mechanisms.