The [[transfer function]] of a [[discrete-time LTI system]] can be found by applying the [[z-transform]] to the general form of the [[difference equation]]: $\sum_{k=0}^{N} a_k y(n-k) = \sum_{k=0}^{M} b_k x(n-k)$ using the [[time shifting property of the z-transform]]: $Y(z) \sum_{k=0}^{N} a_k z^{-k} = X(z)\sum_{k=0}^{M} b_k z^{-k}$ Using the [[convolution property of the z-transform]] it is easy to conclude that the transfer function of an LTI system is the z-transform of the output signal divided by the z-transform of the input signal: $ H(z) = \frac{Y(z)}{X(z)} $ that is $ H(z) = \frac{\sum_{k=0}^{M}b_{k}z^{-k}}{\sum_{k=0}^{N}a_{k}z^{-k}} = \frac{P(z)}{Q(z)} $ where the numerator and denominator are polynomials in $z$. The [[transfer function]] of [[discrete-time LTI system]] defined by a [[difference equation]] is rational: $ H(z)=\frac{P(z)}{Q(z)} $ where $P(z)$ and $Q(z)$ are polynomials in $z$. The roots of $P(z)$ are called **zeros** of the transfer function $H(z)$. The roots of $Q(z)$ are called **poles** of the transfer function $H(z)$.