Delay Operator

The Delay Operator $\mathcal{R}$ is a Linear Unary Operator that maps a Discrete-Time Signal to a copy delayed by one sample:

$$\mathcal{R}\{x[n]\}=x[n-1]$$

It is equivalent to multiplication by $z^{-1}$ in the Z Transform domain, so $\mathcal{R}\leftrightarrow z^{-1}$.

Operator Algebra

First of all, see Operator Notation first.

Systems expressible as difference equations can be written as polynomials in $\mathcal{R}$ and manipulated algebraically. The difference machine $y[n]=x[n]-x[n-1]$ becomes:

$$Y=(1-\mathcal{R})X$$

Cascading two such systems multiplies their polynomials:

$$Y_2=(1-\mathcal{R})(1-\mathcal{R})X=(1-2\mathcal{R}+{\mathcal{R}}^2)X$$

which corresponds to $y_2[n]=x[n]-2x[n-1]+x[n-2]$.

$\mathcal{R}$ distributes over addition. It does not commute with arbitrary operators in general.

Operator Reciprocals

Operator Reciprocals

Two operators are reciprocals if their product is the identity operator. The reciprocal of $(1-H)$ is the geometric series:

$$\frac{1}{1-H}=\sum _{k=0}^{\infty }{H}^k$$

Verified by synthetic division:

$$(1-H)(1+H+H^2+\cdots )=1$$

More generally, the reciprocal of $(1-pH)$ for scalar $p$ is:

$$\frac{1}{1-pH}=\sum _{k=0}^{\infty }{p}^k{H}^k$$

This shows that an implicit system $(1-pH)Y=X$ is equivalent to an infinite explicit sum $Y={\sum }_{k=0}^{\infty }{p}^k{H}^{k}X$.

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