** **

An input-output relationship is called a **system **if
there is a unique output for any given input. By a system we usually mean a
complex mechanism which has been built up by interconnecting various simpler
component mechanisms. These components generally have a very simple behavior;
but when they are interconnected, they can influence each other in such a way
the over-all system has an interestingly complex behavior.

A system can be physical mechanism like an automobile automatic transmission (mechanical) system, or a television receiver (electrical, mechanical) system. Or it can be an abstract system, which has been mentally constructed to make the solution of a mathematical problem easier to obtain and appreciate. In this latter case, instead of a system, which operates on physical quantities, such as shaft rotations or electrical voltage, we may have one which operates on non-physical quantities such as probabilities.

Some of these system quantities will be under external control; others wil be observable, but not controllable. We are normally interested in the behavior of system quantities as functions of time when we have some of them under external control.

It will be sufficient in the following to limit
ourselves to two system quantities, one, which we control and the other which
we observe. The first we call the input to the system and designate it as a
function by x(n_{1}, n_{2}).The second we call the output or
response and designate it by y(n_{1}, n_{2}). The system
provides the connecting mechanism between these quantities. A system T that
relates an input x(n_{1}, n_{2}) to an output y(n_{1},
n_{2}) is represented by

y(n_{1}, n_{2})
**= **T[x(n_{1}, n_{2})] (1.11)

We represent the system by a box between two lines
showing that we apply x(n_{1}, n_{2}) at the input point.

The problem of system
analysis is to describe mathematically how the system produces y(n_{1},
n_{2}) from x(n_{1}, n_{2}). Often the output is the unknown
which we desire to find, given the input. However, we may also seek to find the
input which produces a given output or to find the system which will convert a
given input to a given output.

Often we speak of a
system as a **filter** because of its job as being the removal from the
input, x(n_{1}, n_{2}) of unwanted noise. For example x(n_{1},
n_{2}) may be th sum of a desired function _{}and an undesired function _{}. We would then ideally
like to discover that filter system which converts

x(n_{1}, n_{2})=
_{}+_{} to y(n_{1},
n_{2})= _{}