Consider a causal system having a transfer function :
subjected to the sinusoidal input sequence :
Theorem on steady state response
The system steady-state output response is :
So the output is a sinusoid at the same digital frequency. The amplitude gain of the system is
and the phase shift through the system is 
.
Discussion
We need to first calculate the z-transform of the response :
From our table of transforms :
Poles of Y(z)
From H(z) the poles are 
say
From X(z) the poles are 
Note that the residues of the X(z) poles are a complex conjugate pair. When H(z) and Y(z) are multiplied out the resulting Y(z) (represented as a sum of partial fractions) will have the form :
where 
and 
are a complex conjugate pair.
Take the inverse z-transform (causal sequence) :
The terms 
are transients, decaying to zero as 
.
Evaluate and in the usual way for partial fractions - divide by z ,
multiply by 
and evaluate the result at 
:
Put and get :
Similarly we find :
Conventionally we write : 