# Block diagrams

We have been using block diagrams informally, but now we can define their meaning more formally. A block in a block diagram represents a system, a function on signal spaces. In the picture*x* and *y* are variables representing signals in some function
space, and *f* is a system that relates these variables by

*y*=

*f*(

*x*)

The picture

represents the function composition *g* ·
*f*, so that *z* = *g*(*f* (*x*)). The following
picture represents a more complicated form of function composition:

Here, *z* = *g*(*w*, *f* (*x*)). A still
more complicated block diagram is:

Here, *z* is defined implicitly as the solution to the equation

*z*=

*g*(

*z*,

*f*(

*x*)).

Depending on the function spaces and functions involved, this equation
may or may not have a solution, and if it has a solution the solution may
or may not be unique. Such **feedback** systems are often powerful solutions
to complicated problems.