Factored Form

By the fundamental theorem of algebra, every $N$th order polynomial can be factored into a product of $N$ first-order polynomials. Therefore, Eq. (6.5) above can be written in factored form as

$$H(z) = b_0\frac{(1-q_1z^{-1})(1-q_2z^{-1})\cdots(1-q_Mz^{-1})}{(1-p_1z^{-1})(1-p_2z^{-1})\cdots(1-p_Nz^{-1})}. \protect$$ (7.6)

The numerator roots $\{q_1,\ldots,q_M\}$ are called the zeros of the transfer function, and the denominator roots $\{p_1, \ldots, p_N\}$ are called the poles of the filter. Poles and zeros are discussed further in Chapter 8.