11.1 Definition, Solutions, and Applications

Difference equations are used in numerous applications such as engineering, mathematics, physics, and other sciences. A difference equation defines the relationship between the values y _k of a function, and the discrete set of the independent variables x _k. For example, the relation

is a linear difference equation with constant coefficients, and describes the relationship of a discrete input x( n) and the corresponding discrete output y( n) in a linear and time invariant ^[*] system. with constant coefficients and a _i and b _i.

In (11.1), the difference order k was chosen to be the same on both sides. However, in most cases certain coefficients a _i and b _i are zero and thus, the order k for the left and right sides will not always be the same.

The general form of a linear, constant coefficient difference equation has the form

where a _k represents a constant coefficient and E is an operator similar to D the operator in ordinary differential equations. The E operator increases the argument of a function by one interval h, and r is a positive integer that denotes the...