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    By: Sidney BurrusAs a part of collection: "Notes on the Design of Optimal FIR Filters"

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Extension to Non-lowpass Filters

Module by: John Treichler. E-mail the author

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

δ = min { δ i } , for all pass and stopbands i, and δ=min{ δ i },for all pass and stopbands i, and
(1)
Δ f = min { Δ f k } for all transition bands k Δf=min{Δ f k }for all transition bands k
(2)

If so then equation Equation 1 from the module titled "Filter Sizing" can be used directly and the equation for αα becomes

α = 0 . 22 - l o g e δ π . α = 0 . 22 - l o g e δ π .
(3)

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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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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