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Roy:
[snip] The last sentence should read: A lossless lowpass filter has zero attenuation only at DC. : : Roy Lewallen, W7EL Ummmm... no that statement is only true for one type of approximation polynomial. A lossless low pass filter has zero attenuation at its' reflection coefficient zeros. If it is a maximally flat low pass. a.k.a. Butterworth. then all of the reflection zeros are located at DC, but for any other type, e.g. Chebychev, Cauer/Darlington, General Parameter, etc, etc... this is not true. Such a filter will have zero loss at the designed reflection zeros which are distributed at various appropriate frequencies across the passband according to the dictates of the approximation polynomials. Aside: Reflection zeros are also known as Return Loss [Echo Loss] poles. These are the pass band frequencies of zero loss for lossless LC filters designed according to modern insertion loss methods. No one really knows where the reflection zeros of an image parameter LC filter are, one has to find them by analysis after the design. Whereas with insertion loss design the frequencies of zero loss [the reflection zeros] are specified by the approximation polynomials, specifically the reflection zero polynomial usually designated by F(s). In fact modern insertion loss design begins with a specification of attenuation ripple between zero loss and the maximum loss in the pass band. The frequencies of zero loss then become the zeros of the reflection zero polynomial F(s). The attenuation in the stop band results in the specification of the loss pole polynomial P(s) whose zeros are the so called loss poles or attenuation poles. The natural mode polynomial of the filter E(s) whose zeros are known as the natural modes or sometimes just "the filter poles" is formed from the loss poles and reflection zeros using Feldtkeller's Equation. E(s)E(-s) = P(s)P(-s) +k^2F(s)F(-s) In the approximation process the stopband attenuation is set first by "placing" the loss poles in the stopband, i.e. determining the polynomial P(s). Then from the desired passband attenuation and type of approximation desired; maximally flat, equiripple, etc... the reflection zeros F(s) are determined and finally from Feldtkeller's Equation and the ripple factor k, the natural modes or E(s) is determined. Then the LC filter is synthesized from either or both of the short circuit or open circuit reactance functions which are formed from even and odd parts of E and F, for example. X = (Eev - Fev)/(Eod + Fod), etc... You can review all of this in the very practical and professionally oriented textbook: Adel S. Sedra and Peter O. Brackett, "Filter Theory and Design: Active and Passive", Matrix Publishers, Champaign, IL 1978. Another good practical and professionally oriented textbook is: Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill, New York, 1962. If you can get a copy of: R. Saal and E. Ulbrich, "On the design of filters by synthesis", IRE Trans. Vol. CT-5, No. 4, pp.284-327, Dec. 1958. Bind it firmly and keep it in your library forever... you will have the whole story in a nutshell. Saal and Ulbrich is "the bible" on LC filter design. -- Peter Freelance Professional Consultant Signal Processing and Analog Electronics Indialantic By-the-Sea, FL |
#2
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You're correct, and I apologize. In fact, the example I gave of a
"quarter wave" filter contradicts the statement about the attenuation. I was thinking of a Butterworth when I wrote it, but as you point out and as my own example shows, there are many other types for which the statement is wrong. I apologize for the error. Thanks for the correction. Roy Lewallen, W7EL Peter O. Brackett wrote: Roy: [snip] The last sentence should read: A lossless lowpass filter has zero attenuation only at DC. : : Roy Lewallen, W7EL Ummmm... no that statement is only true for one type of approximation polynomial. . . . |
#3
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You're correct, and I apologize. In fact, the example I gave of a
"quarter wave" filter contradicts the statement about the attenuation. I was thinking of a Butterworth when I wrote it, but as you point out and as my own example shows, there are many other types for which the statement is wrong. I apologize for the error. Thanks for the correction. Roy Lewallen, W7EL Peter O. Brackett wrote: Roy: [snip] The last sentence should read: A lossless lowpass filter has zero attenuation only at DC. : : Roy Lewallen, W7EL Ummmm... no that statement is only true for one type of approximation polynomial. . . . |
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