Why Does α Depend on the Cutoff Frequency? 1.3 2008/05/20 13:21:16 GMT-5 2008/10/15 11:08:20.529 GMT-5 John R Treichler jrt@appsig.com John R Treichler jrt@appsig.com Daniel Collins Williamson dcwill@cnx.org Richard G. Baraniuk richb@rice.edu C. Sidney Burrus csb@rice.edu Raymond Wagner rwagner@rice.edu The formulas presented in Equation 1 and Equation 2 from the module titled "Filter Sizing" imply that α and hence the required filter order N are independent of the cutoff frequency fc. The supporting analysis showed that this is only true in the limit of high order filters, i.e. when N is large. The dependence for shorter filters is shown in Figure 2 from the module titled "Filter Sizing". Why should this occur? Consider the filter design problem shown in . Again the goal is a simple lowpass filter with cutoff frequency fc. The frequency sampling points at frequency multiples of fsN are also shown as solid dots. Instead of fixing the gains we presume that the filter gains h^n, or, equivalently, the graphic equalizer levers, are optimized, by whatever means, to yield the best stopband ripple performance. (a) shows the combination of gains h^n needed to constrain the peak stopband ripple to a given level, say δ2¯. The frequency at which this equal ripple band starts is of course fst and the difference between fst and fc is Δf. Now suppose that fc is increased slightly, as shown in (b). Now a different set of the h^n are needed to make the peak ripple equal δ2¯ and these result in different values of fst and Δf. Pursuing this graphical analysis we find that: Cutoff frequencies near multiples of fsN result in smaller transition bands, and hence smaller values of α, than those near the center of two bins. This occurs, to first order, since two or more stopband basis filters are needed to cancel the first sidelobe of the last basis passband filter when the passband stops between two bins, while one is needed if the passband stops near a bin. Because these “hard" and “easy" frequency ranges occur for every bin, the number of the ranges, counting both positive and negative frequencies, is about the same as the filter orderVarious boundary conditions can make the actual number one less or one more than the filter order.N. The variation in the transition band Δf is more pronounced as N decreases since there are fewer basis filters to use in optimizing the response.

Visualizing the Effects of Cutoff Frequency on Design Difficulty

As an aside one might observe from Figure 1 from the module titled Performance Comparison with other FIR Design Methods that all three methods perform about equally for high levels of stopband ripple. Intuitively the reason for this should now be clear. Window-based methods need not use much shaping if high levels of ripple are tolerable. Similarly, frequency sampling need not use many adjustable coefficients. Since this is true the equal-ripple techniques will not perform much better since their only advantage is that of adjusting all of the filter gains. The underlying point is that, for high-ripple designs, all of the methods produce designs closely resembling the sum of simple, shifted sinNqsinq functions and produce a transition band Δf of about the order of fsN, hence an α of about unity. Only as the stopband ripple specification grows tighter does the method and accuracy of adjusting the coefficients and the number of them available for adjustment begin to affect the transition band performance.