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:

and we get the characteristic equation:

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 .