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.
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
Factoring out the common terms and gives
the 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.)
Z Transform of Convolution