Steven Butterworth, a British engineer, 1885-1958, discovered a method of designing electric filters. He was quoted saying "An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies." His algorithms are widely used in filter design, as we will see in Chapter 14. He based his design on locating the poles of his filters in a unique pattern around a circle of radius \(\omega_{\mathrm{C}}\). The number of poles on the circle constitutes the order of the filter. The more poles, the better the filter is. Odd-order filters include one real pole at - \(\omega_{\mathrm{C}}\) and pairs of complex-conjugate poles placed on a circle of radius \(\omega_{\mathrm{C}}\) at equal angular spacing. Even-order filters locate complexconjugate poles placed on a circle of radius \(\omega_{\mathrm{C}}\) at equal angular spacing. Figure P9 \(=5 \underline{6}\) shows the location of the poles in a thirdand in a fourth-order Butterworth filter. Assuming an \(\omega_{\mathrm{C}}\) of \(1 \mathrm{rad} / \mathrm{s}\), what is the denominator of the Laplace transform, \(F\) ( \(s\) ), associated with each of the two Butterworth filters shown in the figure?