Pole-Zero Analysis

This chapter discusses pole-zero analysis of digital filters. Every digital filter can be specified by its poles and zeros (plus a gain factor). Poles and zeros give useful insights into a filter's response, and can be used as the basis for digital filter design. The Durbin step-down recursion for checking filter stability by finding the reflection coefficients is presented, including matlab code.

Going back to Eq. (6.5), we can write the general transfer function for the recursive LTI digital filter as

$$H(z) = g\frac{1 + \beta_1 z^{-1}+ \cdots + \beta_M z^{-M}}{1 + a_1 z^{-1}+ \cdots + a_N z^{-N}} \protect$$ (9.1)

which is the same as Eq. (6.5) except that we have factored out the leading coefficient $b_0$ in the numerator (assumed to be nonzero) and called it g. (Here $\beta_i \isdef b_i/b_0$.) In the same way that $z^2 + 3z + 2$ can be factored into $(z + 1)(z + 2)$, we can factor the numerator and denominator to obtain

$$H(z) = g\frac{(1-q_1z^{-1})(1-q_2z^{-1})\cdots(1-q_Mz^{-1})}{(1-p_1z^{-1})(1-p_2z^{-1})\cdots(1-p_Nz^{-1})}. \protect$$ (9.2)

Assume, for simplicity, that none of the factors cancel out. The (possibly complex) numbers $\{q_1,\ldots,q_M\}$ are the roots of the numerator polynomial. When $z$ takes on any of these values, the transfer function evaluates to 0. For this reason, these roots are called the zeros of the filter. Similarly, when $z$ approaches any root of the denominator polynomial, the magnitude of the transfer function becomes larger and larger, approaching infinity. Consequently, the denominator roots $\{p_1, \ldots, p_N\}$ are called the poles of the filter.

The term ``pole'' really makes sense when you plot the magnitude of $H(z)$ as a function of z. Since $z$ is complex, it may be taken to lie in a plane (the $z$ plane). The magnitude of $H(z)$ is real and therefore can be represented by distance above the $z$ plane. The plot appears as an infinitely thin surface spanning in all directions over the $z$ plane. The zeros are the points where the surface dips down to touch the $z$ plane. At high altitude, the poles look like thin, well, ``poles'' that go straight up forever, getting thinner the higher they go.

Notice that the $M+1$ feedforward coefficients from the general difference quation, Eq. (5.1), give rise to $M$ zeros. Similarly, the $N$ feedback coeficients in Eq. (5.1) give rise to $N$ poles. This illustrates the general fact that zeros are caused by adding a finite number of input samples together and poles are caused by feedback. Recall that the filter order is the maximum of $N$ and $M$. If $b_0\neq 0$ in Eq. (6.5), it then follows that the filter order equals the number of poles or zeros, whichever is greater.

Recall that the order of a polynomial is defined as the highest power of the polynomial variable. For example, the order of the polynomial $p(x)=1+2x+3x^2$ is 2. From Eq. (8.1), we see that $M$ is the order of the transfer-function numerator polynomial in $z^{-1}$. Similarly, $N$ is the order of the denominator polynomial in $z^{-1}$. Therefore, the filter order is given by the maximum of the numerator and denominator polynomial orders.