CIRCUIT THEORY AND APPLICATIONS. VOL. 2, 341-351 (1974)
THE EQUIDISTANT LINEAR PHASE POLYNOMIAL FOR
DISTRIBUTED A N D DIGITAL NETWORKS
M. F. FAHMY AND J . D. RHODES
Department of Elecrrical and Electronic Engineering, University of Lee& England
SUMMARY
A closed form expression for the recurrence formula for the generation of polynomials whose phase interpolates to linear
characteristicsat equidistant frequenciesis presented.This polynomial may be used directly in the design of both distributed
and digital networks and, due to the recurrence formula, readily generated. Since the complete proof of this formula is
extensive and not required for normal use of this polynomial, it has been outlined in an Appendix.
INTRODUCTION
In a number of applications a distributed parameter filter as well as recursive digital filter are required to
satisfy certain amplitude requirements, and at the same time approximate constant group delay in the
passband.
Several approaches have been used to design such filters. The classical approach, is t o design the filter,
whether distributed or digital, to satisfy solely the amplitude requirements, and then connect a phase correcting network, in tandem, to compensate for the deviation of the group delay from constancy in the passband.
However, this method suffersfrom requiring a high degree correcting network, if a reasonable delay characteristic is to be maintained over the passband.
Another different approach is to design the filter to satisfy certain constraints upon amplitude and delay in
passband. A closed form solution for the filter transfer function is known, for the maximally flat behaviour of
amplitude and delay in the passband, only for cases where the ratio of the constraints upon amplitude to that
upon delay are 1 : 2,2 : 1 and 1 : 1 r e ~ p e c t i v e l y . ~
However,
* ~ * ~ in these cases, the amplitude response is generally
unsatisfactory for medium and narrow bandwidth filters, and the only means to increase the selectivity is by
increasing the order of the filter. On the other hand, the delay response is much better than required.
R h o d e ~ ,through
~ . ~ the use of what he named as 'equidistant phase polynomial', managed to design a low
degree lumped constant low pass, with superior amplitude response, compared with filters designed by
previous methods.
In this paper, the concept of 'equidistant phase polynomial' is extended to the distributed plane, and thereby
to the z plane.
From the detailed analysis given in subsequent sections, one can summarize these polynomials as follows :
The real polynomial A , ( t t o f degree n in t , satisfying the condition
Arg A,( + j tan r e ) = are
{r
= 1 -+
n)
is defined by the recurrence relation
A,(t) = A,-,(t)+B,-,(t2+tan2 ( n - l ) O ) A n - 2 ( t )
with the initial conditions
Received 19 October 1973
Revised 12 February 1974
8 1974 by John Wiley & Sans, Ltd.
34 1
342
M. F. FAHMY AND J. D. RHODES
where
p = complex variable in the lumped constant domain = a + j o
t = complex variable in the distributed domain =
+jR
= tanh ap
a = one way delay of the transmission line
8 = am
R, = tanre
{ r = 1,2, ...,n}
= set of points of zero phase error from linearity
B,, B,(n > 1) are constants, given in the following Section, by equations (4) and (15) respectively.
The application of such new polynomials in filter designs will be discussed in comparison to the use of the
maximally flat linear phase polynomials.6
1
(1 +B,t)
+
~ ~ ( ttan2@
,
1
-1
0
-1
I
I
I
I
I
I
I
I
I
I
I
0
0
I
B,(t2 + tan238)
1
I
I
I
o--------------o
o--------------o
0
B2(t2+ tan228)
1I
I
I
I
o-----
0
tan a0
tan 8
For the 2nd order polynomial, either (1) or (2) yields
=-
+ + Bl(t2+tan2@
A 2 ( t )= 1 B,t
* The complete proof may be obtained directly from the authors.
I
I
I
+
(1)
I
1 En- l(t2 tAn2(n- l)8)
-1
1
(4)
(5)
THE EQUIDISTANT LINEAR PHASE POLYNOMIAL FOR DISTRIBUTED AND DIGITAL NETWORKS
343
B, can be determined such that phase of A$) at t = j tan 28 should be 2a8, i.e.
Bo tan 28
1 B,( - tan220 tan2@
tan 2u8 =
+
or
B,
=
+
( t a n W - tan%)
1
( t a n 9 8 - tan2@(1- t a n 9 )
For the 3rd order polynomial, equation (1) yields
A,(t) = (1 +B,t)(l +B,(tan228+t2))+B,(r2+tan28)
(8)
B, can be determined from the condition that phase of A&) at t = j tan 38 should be 3u8. This gives the
following
B, tan 38
tan 3uB =
(9)
B,(tan28- tan23B)
I t
1 + B,(tan228 - tan238)
.
,
Substitution for B , , B, from (4) and (7) will yield
1 (tan2u8- tan%) tan 8(tan 38- tan e)
2 (tan%- tan28)(tan238- tan228)
Bz = -
For the 4th order polynomial, equation (1) yields
A 4 ( t ) = (1 +B,t)(t2+tan220)B2+(1 +B,(tZ+tan238))(1+ B , t + B , ( t 2 +tan%))
For the phase A4(t) at t
=j
tan 48, be 4~x8,one should have
Bo tan 48
B,(tan28tan248)
..
+
~ , ( t a n 2 2 8- tan248)
+ 1 + ~,(tan230- tan2401
tan 4ue
Using (4), (7) and (10)for B,, B , and B , respectively in (1l), will yield
B3
=
1 (tan2u8- tan238)tan 8(tan 48 -tan 28)
(tan230 - t a n 2 2 0 ) ( t a n ~ tan230)
5
(13)
From this analysis, it is not difficult to see that, for nth polynomial, B n p 1can be determined from the
requirement that
Bo tan n8
tan nu8 =
B (tan28- tan%@
I t
~ , ( t a n ~2 tan2n8)
8
l+ I
,
..
I
B,- ,(tan2(n -2)O- tan’n8)
1 Bn- ,(tan2(n- l)8- tan%@
+
Also, consideration of the B,’s expression suggests that they imply a general law given by
+
1 (tan2u8- tan%@ tan O(tan (n i)e- tan (n- i)e)
B =2 (tan2ne-tanZ(n- l)e)(tan2(n+1)8-tan2ne)
-
+
(n 2 1)
cos (n- i)e . cos (n i)e . sin (u+ n)e . sin (a - n)e . c o s W
sin (2n - i)e . sin (2n i)e . cos2ae
* Equation (15b) was suggested by one of the reviewers.
+
344
M. F. FAHMY AND J. D. RHODES
It should be noticed that the phase condition of the nth degree polynomial, only imposes n conditions on the
(n + I ) coefficients, whereas the recurrence relation (2), yield implicitly the remaining coefficient. Also it is
worthwhile to state that the maximally flat case (symmetrical Jacobi polynomial^^-^) may be recovered by
letting 6 tend to zero.
To determine the conditions under which the set of polynomials A,,(?)are Hurwitz, we have, after replacing
(n 1) by n in (21,
+
A,+
--
l(t)
- 1
+
B,,(tan2nB t 2 )
+
A,(t)
I+
B,- ,(t2+tan2(n- 1)O)
I+
B n - 2 ( f 2 +tan2(n-2)8)
\
1
-+1
BOt
Multiplying both sides of (15) by P,,(t),where
P,(t) =
B , B , . .. E n - , (t2+tan26)(t2+tan238)...(t2+tan2(n-1)O)
t ( t 2+ tan228)(t2+ tan240). . . ( r 2 + tan2&)
BOB2. . . B,
- BOB2 . . . B , - , ~t(t2+tan22~)(t2+tan240)...(t2+tan2(n-1)O)
( t 2+ tan20)(t2+ tan238). . . ( t 2+ tan2nO)
B , B , . . . B,,
n even
n odd
(17)
we get
Thus, the L.H.S. of (17) represents the input impedance of a lossless resistance terminated network so
long as B, ( I = 1 + n) is positive-and therefore it is p.r.f. with the result that A,(t) and A, + , ( t ) are Hurwitz.
Hence for the set of polynomials A&) to be Hurwitz, one should have B,'s to be positive. However,
consideration of (4,7, 14) reveals that, for B,, to be positive, one should have
a
>= n,
(n+1)8 < n/2,
a6 < 4 2
(19)
To end this section, it is worth while to tabulate some of these polynomials as well as their amplitude and
phase response.
Table I gives the coefficients of these polynomials up to the 8th degree for different values of a and 8,
whereas Figures 1, 2 and 3 show their amplitude and delay response.
EQUIDISTANT POLYNOMIALS IN Z-PLANE
The analogy between the distributed and Z-planes was discussed in a previous paper.4 In short, the
transformation
z-1
t=z + 1,
where
z = complex variable in the Z-plane
= epT
T = sampling period = 2a
345
THE EQUIDISTANT LINEAR PHASE POLYNOMIAL FOR DISTRIBUTED AND DIGITAL NETWORKS
Table l(a). Coefficients of the polynomial A&) = a o + a , t + a 2 t 2 + . . . +t”, with normalized highest power,
a = 20,O = 0.052229
0.030365
0.005514
0.011848
0.000398
0~0001682
8.9541 x lo-’
7 5.7478 x lo-’
8 4.38661 x lo-’
1
2
3
4
5
6
1
0.091594 1
0.025128 0.184711
0.007807 0.71056
0.003385 0.030016
0@01787 0.016378
0.001502 0.0106025
0~0008772 0.008161
1
0.311308
0.160015
0.088179
0.059395
0.04648
1
0.473618 1
0.314555 0.674632 1
0.219612 0.5649898 0.918553
0.179943 0.489249 0.952263
1
1.20952
1
Table I(b). Coefficients of the polynomial for K = 10, O = 0.104458
No.
a.
a,
a2
a3
a4
a5
a6
a7
1
2
3
4
5
6
7
8
0.060896
0.022364
0.010018
0007245
0.006997
OQ09165
0.016151
0.039564
1
0.186756
0.10575
0.071298
0.070267
0.091555
0.161539
0.395626
1
0.386467
0.316265
0.305424
0407306
0.721883
1,775654
1
0.67552
0.771 119
1W061 I
1.891878
4.699492
1
1.079644
1.691359
3.1751
8.096387
1
1.642294
3.532206
9401912
1
2441581
2.496133
1
3.635277
1
will map the L.H.S. of the t-plane into the interior of un.it tide in the z-plane. Due to this analogy, the
z-plane frequency distribution corresponding to that discussed in the previous section will be located on the
circumference of IzI = 1 at equal angular spacing of 28.
To obtain the recurrence relation of such polynomials in the z-plane, A,(t) is written as
where P,(z) is the equidistant polynomial in the z-plane, obtained after substituting 19) in the A$) poly
nomial.
U.sing equations (2) and (20) will yield
P,(Z) = (1 + z ~ ~ , _ , ( z ) + ( z ~ + 2 ~ c o s 2 e + i ) ~ , _ , ~ , _ , ( z )
(22)
where P,(z) will have all its zeros inside ( z ( = 1, as a result of A,(t) being Hurwitz.
CONCLUSION
A design procedure has been presented for the construction of equidistant linear phase polynomials in both
distributed and z-planes. For this particular distribution a closed form solution has been obtained for the
degree varying recurrence formula. Using this polynomial, the bandwidth of the linear phase approximation
is increased over the maximally flat solution with the introduction of a ripple component. The use of this
polynomial in selective linear phase response applications is anticipated to yield a considerable improvement
over those using the Jacobi Polynomials.’*2q3
346
M. F. FAHMY AND J. D. RHODES
N:
1
V
n
E
2
c
C
.Q
.32
.... .
$ZI 0
-0
.I
.' .
.
0;e
.' ...:
U
W
w)
W
E
i=
8
-1
t
o = 10 $6 0.104458
N =6,8
0.2
0
-
c
0.2
0.4
0.6
0.8
UT
Figure I . The equidistant phase polynomial, a = 10, 0 = 0.104458, n = 6, 8
THE EQUIDISTANT LINEAR PHASE POLYNOMIAL FOR DISTRIBUTED AND DIGITAL NETWORKS
2
V
n
E
2
r
1
c
.Q
t
0
..
..
..
i
:
.. .
i
4
0
i:
0
..
4
. :.
.. ..
I
0
-
U
al
a)
E
.
w
0.6
ur
.. ..
.. ..
iIN.8
-1
I=
!:
$?
-2
10
* = 20,8= 0.05222
0.8
N=6,8
0-2
0
0.2
0.4
0.6
*
0.8 &T
Figure 2. The equidistant phase polynomial, a = 20, 0 = 0.05222, n = 6, 8
347
M. F. FAHMY AND J. D. RHODES
348
u
0
1
E
e
c
s o
c
j
%
5
-1
'0
t!
._
I-
s
-2
-3
1.0
t
0.8
Q
= 2 0 , e=0.05969
N=8
0.2
0 I
c
I
0.2
0.4
0.6
0.8
ur
Figure 3. The equidistant phase polynomial, a = 20 and 0 = 0.05969, n = 8
349
THE EQUIDISTANT LINEAR PHASE POLYNOMIAL FOR DISTRIBUTED AND DIGITAL NETWORKS
APPENDIX
The method outlined here is based on mathematical induction, in which through the assumption of the
validity ofequation (15) up to B,- I ,one can verify that it is still valid for B,,and thereby valid for all values of n.
As a start, consider the function
Expanding ( A l ) in a power series of (l/f), taking into consideration the phase shift of A,(t) at the prescribed
set of frequencies discussed in the Section 'Equidistant phase polynomials in the distributed plane', will
yield
'I
=
tanare 1 1
K : l -tanr8
.-+,- t t
tanaretan r0 tan%
t3
-___
t4
+ tanar8tan3r0
t5
+-.. I
tan4r0
(A2)
t6
where K: represents the residue of F,,(t) at its rth pole.
But due to equation (Al), F,(t) contains an nth order zero at infinity. This requires the following matrix
equation
tan 3a8
tan 38
1
...
tan nu8
tan no
1
tan 3a8 tan 38
...
tan nu0 tan n0
tan28
tan 2a8 tan 20
tan228
tan230
...
tan2n0
tan"-30
tan" - 328
tan"- 338
...
tan"- 3n8
tan a8
tan 8
tan 2a0
tan 28
1
1
tan a8 tan 8
I tan ue tan"- zo
tan 2u0 tan"-'28
...
(n
tan 3a8 tarY338 . . . tan na0 tan"-3n8
where
1N.B.: In the case of n
=
= (-
1)1'"-1)/21x BOB, . . . B,-
(n
- (-
1)1("-2)/21x B B 3 . . . B,-
(n = even)
=
odd)
even, the last row of the coefficient matrix of (A3) will be
tan"-'e,
tann-'2e,. . . ,tan"-2n81
It is thus evident that, from (A3) one has
where
A" is the determinant of the coefficient matrix in equation (A3)
A;." is the principal minor obtained from A" by cancelling the nth row and column.
=
odd)
(A31
3 50
M. F. FAHMY AND J. D. MODES
Similarly, for the (n+ 1)th polynomial, one should have
where
A"" is the determinant of the coefficient matrix in case of (n+ 1)th polynomial
Ant
,+ 1 is the minor obtained from A"+ ' by cancelling the (n+ I)th row and nth column.
Thus, using equations (A5) and (A6) in conjunction with equation (2), one can prove that
K:+' =
K:
(tan2(n+ i)e - tan2ne)
Substituting for K: " and K: in equation (A7) would result in
B, =
-
An An+]
1
BOB,.. . En- ,
n.n
BIB,. . .
tan2(n+ 1)8-tan2ne A" An+'
n+l,n
All All+]
1
n.n
tan2(n+ l)e - tan2n8 A" A"+
n+l.n
B , B , . . . EnBOB2. . . En-
'
n = odd
n
=
even
Therefore, under the assumption that B, (0 < r < n - l), as expressed by equation (15), are correct, we have
B,
=
1
-tanaO
2
n
( n - 1)/2
(tan2aB-tan22r8)
tan(2r+ 1)e-tan (2r- i)e
(tan2ae- tan2(2r- i)e) tan (2r + 218 -tan 2re
(l-tan28)(tan(n+l)e-tan(n- 1)O)
An+'
' (tan2(n+1)e-tan2ne)(tan2ne- tan2(n - i)e) ' AMA::
1
--
tan ae
("-
')I2
iVfl (n = odd)
(tan2ae- tan2(2r- i)e) tan (2r + 2)e - tan 2re
(tan*ae- tan22re)
tan (2r+ i)e- tan (2r- i)e
+
tan (n 1)e- tan (n- l)e
A" ' A:."
(1 -tan28)(tan2n8-tan2(n- l)e)(tan2(n+1)8-tan2nB) A" A",!.,
(n = even)
(A9)
Now, the geometry of the coefficient determinant, as given by (A3), shows that
n (tan2a8- tan2je)I("-j+
n- 1
(tan a6)1n/21
where F"(B) is independent of a.
Expressing equation (A10) and cancelling common terms, will result in
1 (tan2aB- tan2ne)(tan(n+ i)e- tan ( n - i)e)tan e
B =" 2 (tan2nB-tan2(n- l)e)(tan2(n+1)8-tan2ne)
--_ 1 (tan2ae-tan2ne)(tan(n+ l)O-tan(n- i)e)tan 8
2 (tan2& - tan2(n- l)e)(tan2(n+ i)e- tan2&)
(All)
THE EQUIDISTANT LINEAR PHASE POLYNOMIAL FOR DISTRIBUTED AND DIGITAL NETWORKS
351
where F"", F:,, and F ; : ; . , are functions of 8 only, that result from expressing A"'', A;,, and A;:;.,
respectively, in a way similar to that in equation (A10).
Since the presence of the main terms in the B, expression has been indicated, it is sufficient to calculate B,
for a specified a to show that, the in-bracket terms of R.H.S. of equation ( A l l ) are in fact equal to unity.
This can be achieved by the following choice.
With
tana8
=
1
tan 0
~
we have
tanjae
1
tan j 8
j = odd
tan j e
j = even J
= __
=
Expressing equation (A12) in the form of A", A:,,,A:ti,, and A"" respectively and making use of
Vandermonde theorem,' of matrix expansion, one can prove the validity of equation (15).
REFERENCES
I . J. D. Rhodes, 'The design and synthesis of a class of microwave bandpass linear phase filters', IEEE Trans. Microwave Theory and
Techniques, MTT-17, 189-204 (1 969).
2. J . D. Rhodes, 'Generalized interdigital linear phase networks with optimum maximally flat amplitude characteristics', IEEE
Trans. Circuir Theory, CT-17, 391-399 (1970).
3. J. D. Rhodes and M. F. Fahmy, 'Digital filters with maximally flat amplitude and delay characteristics', Inf. J . Cir. Theor. Appl.,
2, 3-11 (1974).
4. J. D. Rhodes, ' A low-pass prototype network for microwave linear phase filters', IEEE Trans. Microwave Theory and Techniques,
MTT-18. 290-301 (1970).
5. J . D. Rhodes, 'Filters with periodic phase delay and insertion loss ripple', Proc. IEE, 119, 28-32 (1972).
6. T. A. Abele, 'Transmission line filters approximating a constant delay in maximally flat sense', IEEE Trans. Circuir Theory, CT-14,
298-306 (1967).
7. L. Mirsky, An inrroducrion ro linear algebra, Oxford University Press, 1955, p. 17.