Extension to Non-lowpass Filters

Module by: John Treichler

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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 δiδi dominates if it is less than one-tenth of all other rippple specifications and that ΔfiΔfi dominates if it is simply less than all others. Suppose we define δδ and ΔfΔf 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 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 αα becomes

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 fs2fs2. This is obviously not a problem for lowpass filters, since the desired gain at fs2fs2 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 NN for highpass and multiband filters requiring nonzero response at fs2fs2 and use even-order filters for differentiators.

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This work is licensed by John Treichler under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Daniel Williamson on Oct 15, 2008 11:06 am GMT-5.