# Transfer function of a nonhomogeneous difference equation

Consider the following difference equation:

$$y_k=\alpha y_{k-1}+\beta x_k$$

The transfer function for this is given by:

$$\displaystyle\frac{Y(z)}{X(z)}=\frac{\beta}{1-\alpha z^{-1}} = \frac{\beta}{z-\alpha}$$

How does one calculate the transfer function for

$$y_k=\alpha y_{k-1}+\beta x_k + \gamma$$?

The last difference equation is not a linear system due to the addition of the constant $$\gamma$$, therefore it does not have a transfer function.

• However there are ways to transform it into a LTI system. – fibonatic Nov 22 '18 at 0:33