Plots

A pole is a place in the complex plane where a filter B(Z_p) becomes infinity. This occurs where a denominator vanishes. For example, in equation (8) we see that there is one pole and it is located at $Z_p=1/\rho$.In plots like Figure 1, a pole location is denoted by a ``p'' and a zero location by a ``z." I chose to display the pole and zero locations in the $\omega_0$-plane instead of in the Z₀-plane. Thus real frequencies run along the horizontal axis instead of around the circle of |Z|=1. I further chose to superpose the complex $\omega_0$-plane on the graph of $\vert F(\omega)\vert$ versus $\omega$.This enables us to correlate the pole and zero locations to the spectrum. I plotted $(\Re\omega_0, -\Im\omega_0)$in order that the $\omega$ and $\Re\omega_0$ axes would coincide. As we will see later, some poles give stable filters and some poles give unstable filters. At the risk of some confusion, I introduced the minus sign to put the stable poles atop the positive spectrum. Since we will never see a negative spectrum and we will rarely see an unstable pole, this economizes on paper (or maximizes resolution for a fixed amount of paper).

In Figure 1, moving the ``p'' down toward the horizontal axis would cause a slower time decay and a sharper frequency function.