# Solving LCCDEs by Unilateral Z-Transform

Due to its time delay property, the unilateral z-transform is a powerful tool for solving LCCDEs with arbitrary initial conditions.

Example: A system is described by this LCCDE

Taking unilateral z-transform of the DE, we get

• The particular (zero-state) solution

If the system is initially at rest, i.e., , the above equation can be solved for the output to get

where is the system's transfer function. In time domain this is the particular (or zero-state) solution (caused by the input with zero initial condition):

• The homogeneous (zero-input) solution

When the initial condition is nonzero

but the input is zero , the z-transform of the difference equation becomes

which can be solved for

In time domain, this is the homogeneous (or zero-input) solution (caused by the initial condition with zero input):

When neither nor is zero, we have

Solving this algebraic equation in z-domain for we get

The first term is the particular solution caused by the input alone and the second term is the homogeneous solution caused by the initial condition alone. The can be further written as

and in time domain, we have the general solution

which is the sum of both the homogeneous and particular solutions.

Note that bilateral z-transform can also be used to solve LCCDEs. However, as bilateral z-transform does not take initial condition into account, it is always implicitly assumed that the system is initially at rest. If this is not the case, unilateral z-transform has to be used.