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 .