Shift operators

The shift operators are:

• $z^{-1}$ backward shift operator
• $z^1$ forward shift operator

Given a stochastic process $y(t,s)$ then $z^-1y(t,s)=y(t-1,s)$ which has the same realization shifted one time instant backwards.

Properties

The shift operators:

1. are linear: $$z^{-1}(av(t)+by(t))=az^{-1}v(t)+bz^{-1}y(t)$$

2. can be recursively combined: $$z^{-1}(z^{-1}(z^{-1}(y(t))))=z^{-3}y(t)$$