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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

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

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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.

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

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