Block Diagrams and System Functions

The operator for a system with one delay can be written as follows: $Y = X + \mathcal{R}X ,$

which is often rewritten in the form

$Y = \frac{1}{1 - \mathcal{R}}X .$

To make the operator a system function, we collect the output $Y$ and the input $X$ on the same side as a quotient.

$\frac{Y}{X} = \frac{1}{1 - \mathcal{R}} .$

In this example, there is an added gain of $c$ along with the delay. This is a gain, so all that needs to be done is a multiplication with the delay, $R$ .

$\frac{Y}{X} = \frac{1}{1 - \mathcal{cR}} .$