Shift-Invariance

Shift-Invariance means that a delay can be applied either before or after a system, and produce the same result; shifting in the input signal shifts everywhere in the output signal by the same amount.

TLDR Definition

$$T(x[n-n_0])=y[n-n_0]$$

Formal Definition

Let $\Delta$ be a $n_0$-step delay Filter for some fixed by delay $n_0\in \mathbb{N}$, where $\vec{0}\in {\mathbb{R}}^{1\times n_0}$

$$\Delta =[\vec{0},1]$$

so that Convolution with signal $x$ yields

$$(\Delta \ast x)[n]=x[n-n_0]$$

A filter $g$ is shift-invariant if

$$\forall n_0\forall xg(\Delta \ast x)=\Delta \ast g(x)$$