**Lesson 9**

Difference-Equation Models

LTI discrete-time systems are usually modeled by linear difference
equations with constant coefficients. For example, a digital filter
is modeled by a difference equation.

An example of a difference equation is:

A general th order linear difference equation with constant
coefficients (LCCDE) is:

which we can write as:

where and are real constants.

An important case to be familiar with is the first-order system

in which the output is a function of a delay of only one time unit.

The Classical method for the solution is to express the output as
the sum of *complementary* or *natural*
() and *particular* or *forced* () solutions:

Natural response

The natural response is the solution to the
homogeneous equation:

where .

We assume solutions of the form .

We can see that:

and substituting in the homogeneous equation yields:

Clearly, values of satisfy this equation.

The solution is of the form:

assuming there are no repeated roots (which is all we will cover).

Given a first-order difference equation

find its homogeneous solution. Your answer should be in terms of a constant .

Forced response

The forced response solves the equation

The form of the solution is determined by the input . For an exponential input , the solution would be where and are constants.

For the previous example, given an input , find the particular solution .

Now, assuming that the system is initially at rest, i.e., initial conditions of 0 (), solve for the constant in your overall solution .

.

Given

with and , find .