Transfer Function

The transfer function is the ratio of the output to the input.

Group the $Y(z)$ terms together on the left hand side and factor out common terms $X(z)$ and $Y(z)$:

$$Y(z)[1 + a_1z^{-1}+\cdots+a_Nz^{-N}] =$$

$$X(z)[b_0 + b_1z^{-1}+\cdots+b_Mz^{-M}]$$

In defining the following polynomials:

$$A(z) \triangleq 1 + a_1z^{-1}+\cdots+z_{N}z^{-N}$$

$$B(z) \triangleq b_0 + b_1z^{-1}+\cdots+b_{M}z^{-M},$$

the $z$ transform of the difference equation becomes

$$A(z)Y(z) = B(z)X(z).$$

In solving for $Y(z)/X(z)$ we obtain the transfer function

$$H(z) = \frac{Y(z)}{X(z)} = \frac{B(z)}{A(z)},$$

where $X(z)$ and $Y(z)$ are the $z$ transforms of the input and output signal, respectively.

$H(z)$ is the $z$ transform of the impulse response describing the filter.