##
*Z* Transform of Difference Equations

Since *z*transforming the convolution representation for digital filters was so fruitful, let's apply it now to the general difference equation, Eq.(5.1). To do this requires two properties of the

*z*transform,

*linearity*(easy to show) and the

*shift theorem*(derived in §6.3 above). Using these two properties, we can write down the

*z*transform of any difference equation by inspection, as we now show. In §6.8.2, we'll show how to

*invert*by inspection as well.

Repeating the general difference equation for LTI filters, we have (from Eq.(5.1)) Let's take the

*z*transform of both sides, denoting the transform by . Because is a linear operator, it may be distributed through the terms on the right-hand side as follows:

^{7.3}where we used the superposition and scaling properties of linearity given on page , followed by use of the shift theorem, in that order. The terms in may be grouped together on the left-hand side to get

*z*transform of the difference equation yields

Thus, taking the

*z*transform of the general difference equation led to a new formula for the transfer function in terms of the difference equation coefficients. (Now the minus signs for the feedback coefficients in the difference equation Eq.(5.1) are explained.)

**Next Section:**

Factored Form

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Z Transform of Convolution