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:
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:
The natural response is the solution to the
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:
Given a first-order difference equation
find its homogeneous solution. Your answer should be in terms of a constant .
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 .