Example 9.20 (b).  Find the general solution to  .   

Solution 9.20 (b).

    The transfer function is seen to be   .  Using the z-transform in example 8 and the fact that   we can write the z-transform using convolution   

          

Residue calculus can again be used to find the solution, but the details are tedious.  Let us announce that the following computations hold true  

    

which is part of the homogeneous solution.  And the steady state or particular part of the solution is  

    

Therefore, the general solution to part (b) is

        .

    For applications, it is useful to determine the limiting amplitude of  .  We need to simplify in a form that displays it's amplitude and to do this we apply to the trigonometric identity (also known as the harmonic addition theorem)  

        .  

Therefore, the steady state solution is

        
        
and the output signal tends to this limit as ,  i.e.  .  Loosely speaking, for large values of    the values of the input signal    are amplified by the factor   to produce the values of the output signal  .    An illustration using is shown in figure 3 below.

Typo.  Please note that there is a typo in the book which mistakenly stated that the homogeneous solution involves .  We apologize for this inconvenience.

(c) 2006 John H. Mathews, Russell W. Howell