Extension to Non-lowpass Filters 1.2 2008/05/20 13:54:46 GMT-5 2008/07/27 14:24:12.538 GMT-5 John R Treichler jrt@appsig.com John R Treichler jrt@appsig.com Daniel Collins Williamson dwilliamson1285@gmail.com Richard G. Baraniuk richb@rice.edu C. Sidney Burrus csb@rice.edu Raymond Wagner rwagner@rice.edu All of the discussion to this point has focused on lowpass filters. Practical applications require other types, of course, including highpass, bandpass, and bandstop designs. In fact the analysis presented in the previous sections applies to all of these design criteria and the rules for filter length estimation can be used almost directly. In general Equation 1 and Equation 2 from the module titled "Filter Sizing" apply when one of the equal ripple specifications dominates all others and when one of the transition band specifications dominates all others. As a practical matter this means that δi dominates if it is less than one-tenth of all other rippple specifications and that Δfi dominates if it is simply less than all others. Suppose we define δ and Δf by the equations: δ = m i n { δ i } , for all pass and stopbands i, and Δ f = m i n { Δf k } for all transition bands k If so then equation Equation 1 from the module titled "Filter Sizing" can be used directly and the equation for α becomes α = 0 . 22 - l o g e δ π . A final hint - Watch out for the implicit boundary conditions present in the design of linear phase FIR digital filters in two cases: even order, symmetric response and odd order, antisymmetrical response. In both of these cases the underlying equations for the filter's frequency response constrain it to equal exactly zero at fs2. This is obviously not a problem for lowpass filters, since the desired gain at fs2 is zero already. However, in the design of multiband and highpass filters an inordinate amount of engineering time has been spent trying to design even-order filters when in fact it is impossible to do so. The Parks-McClellan algorithm will gamely try, but will fail. As a rule, use odd values of N for highpass and multiband filters requiring nonzero response at fs2 and use even-order filters for differentiators.