FIR Transfer Function

The transfer function of an FIR filter is given by the z transform of its impulse response. This is true for any LTI filter, as discussed in Chapter 6. For FIR filters in particular, we have, from Eq. (5.6),

$$H(z) \isdef \sum_{n=-\infty}^{\infty} h_n z^{-n} = \sum_{n=0}^M b_n z^{-n} \protect$$ (6.8)

Thus, the transfer function of every length $N=M+1$ FIR filter is an $M$th-order polynomial in $z$.