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