Set of design curves eliminates work involved in designing linear phase filters by being normalized in such a way as to apply to low, band, and high-pass filters of any bandwidth. Similar curves for any number of poles are plotted by solving a system of simultaneous equations

Phares, R. L.

English

NASA Technical Reports Server

December 1968
	 Brie! 68-10572 
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Design of Dissipative Linear Phase Filters 
The use of linear phase filters has become rather 
common over the past few years. Some of their appli-
cations include pulse shaping networks, amplifiers 
with good waveform response and low phase distor-
tion, and amplifiers with Gaussian frequency re-
sponse. 
On the other hand, the design of linear phase filters 
is a tedious operation, especially when designing to 
optimize some particular characteristic of the filter, 
such as insertion loss. 
A set of design curves has been constructed for the 
purpose of eliminating most of the work involved in 
designing linear phase filters. These curves are nor-
malized in such a way as to apply to low-pass, band-
pass, and high-pass filters of any bandwidth. While the 
curves are for filters with a maximum of four poles, it 
is possible to plot similar curves for any number "n" 
of poles by solving a system of "n" simultaneous 
equations. 
The design is maximally flat delay only for low-pass 
filters. When the frequency variable is transformed 
from low-pass to band-pass or high-pass some delay 
distortion is introduced by the transformation. The 
amount of distortion may be determined from the 
curves. For narrow band-pass filters the distortion is 
negligible. For high-pass filters, it is considerable. 
One disadvantage of this particular set of design 
curves is that they do not provide a minimum loss de-
sign for lowpass and high-pass filters. This comes 
about because all internal Q's are assumed to be equal. 
Therefore, branches which contain only a capacitor 
must be shunted by a resistance which introduces 
some loss. This loss, however, in most practical cases 
is insignificant. 
The design of maximally flat and Chebishev type 
dissipative ladder filter networks by equating the
coefficients of the Butterworth and Chebishev poly-
nomials respectively to the coefficients of the transfer 
impedance polynomial of the network has been des-
cribed by Dishal. This is accomplished by expressing 
the coefficients of the transfer impedance of the net-
work in terms of the (n—I) coefficients of coupling 
between poles and the "n" decrements of the poles. 
In the usual case, the internal decrements and the 
bandwidth are known; therefore, the net result is a 
system of "n" equations with (n+ I) unknowns; the 
(n—I) coefficients of coupling and the first and last 
decrements. One of these unknowns may be specified 
arbitrarily, within certain limits, and the others de-
termined from the "n" simultaneous equations that 
remain. For instance, it may be desirable to choose 
one of the elements to give a minimum (or any desig-
nated amount) of insertion loss. Taub and Bogner 
have published a similar set of curves for the design 
of 3-pole Butterworth filters for the condition of 
minimum insertion loss. 
This paper deals primarily with a similar design for 
the case of 2-, 3-, and 4-pole maximally flat delay 
ladder networks. The Dishal network coefficients are 
equated to those of the n order Bessel polynomial 
for maximally linear phase. This polynomial has an 
amplitude frequency characteristic that approaches 
Gaussian shape and a delay-frequency characteristic 
that approaches zero as "n" increases indefinitely.. 
Of the (n+l) coefficients required to specify the nth 
degree polynomial, one is used to adjust the gain and 
one is used to specify bandwidth, leaving (n— 1)coef-
ficients to adjust flatjiess of delay. 
After equating the Dishal coefficients to those of the 
Bessel polynomial, the resulting equations are nor-
malized to the fractional 3 db bandwidth. Then the 
normalized end decrements and normalized coefficients 
(continued overleaf) 
Thi document was prepared under the sponsorship of the National
	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States
	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States
	 will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19680000528 2020-03-12T06:33:26+00:00Z
of coupling are plotted using the normalized internal 
decrements, all equal, as a parameter. The insertion 
loss curves plotted with the normalized unloaded dec-
rement as a parameter assume, in addition, that all 
unloaded decrements, including the first and last are 
equal.. The amplitude-frequency and delay-frequency 
curves aye plotted as a function of normalized frac-
tional bandwidth. This set of curves gives a complete 
solution to any valid design problem within the scope 
of this type of network. 
Notes.: 
1. While the idea presented is not new, the set of 
curves employed is believed to be the only one of 
its kind in existence.
2. Documentation for the innovation is available 
from:
Clearinghouse for Federal Scientific 
and Technical Information 
Springfield, Virginia 22151 
Price $3.00 
Reference: B68-10572 
Patent status: 
No patent action is contemplated by NASA. 
Source: Richard L. Phares 
ofSPACO, Inc. 
under contract to 
Marshall Space Flight Center 
(MFS-14698) 
Brief 68-10572	 .	 .	 Category 01


Design of dissipative linear phase filters

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