On the splitting of a complex-coefficient polynomial M. de P. Barros. M.Sc. and L.F. Lind. Ph.D.. C.Eng., M.I.E.E. Indexing terms: Mathematical techniques, Polynomials, Filters and filtering Abstract: An iterative technique is presented for the splitting of a complex-coefficient polynomial into two lower-degree polynomial factors, one being the paraconjugate of the other. Unlike more usual procedures, the previous calculation of the roots of the initial polynomial is not required, making the splitting technique a faster process, readily implemented as a computer program. A typical application comprises the calculation of the reflection coefficient of a lossless network from its transmission coefficient. A numerical example is given. 1 Introduction There are many bandpass filter specifications which have different selectivity slopes at the two passband edges. For example, single sideband filters should have extremely sharp selectivity on one side, to remove the unwanted side- band, whereas the other side can have a gentle rolloff, as there is not much unwanted spectrum to contend with. One way to create an asymmetric bandpass filter is to start from a complex coefficient lowpass design, followed by a conventional lowpass to bandpass mapping trans- formation. If the passband is very narrow (such as for crystal filters), the complex coefficients can be realised by frequency invariant immittances. A lumped coil or capac- itor will have approximately this behaviour over the narrow passband. With this approach, all of the desirable features of lowpass to bandpass mapping are retained, the most important being the relative simplicity of the lowpass design. We will now concentrate on the complex coefficient lowpass design. A common procedure is to use the magni- tude squared response of the transfer function as a cri- terion. Given the transmission zeros, there are analytic methods to locate the reflection zeros so that the magni- tude squared response is an equiripple in the passband [1]. By recovering the transfer function, one can then find the group delay response (or the impulse response). If this response is not the one desired, the transmission zeros are moved slightly, and the sequence just described is repeated. The iteration continues until no further improvement is necessary. Thus we see that an important problem with the above method is the determination of the transfer function. As both the numerators of the transmission and reflection coefficients are known (say, H(p) and F(p), respectively), the only unknown is their common denominator, E(p), con- taining the poles of the transfer function. One solution is, by using analytic continuation, to form the equation E(p)E itt (p) = H{p)HJp) + F{p)F^{p), where the lower aster- isk denotes the paraconjugate polynomial (at this point note that, for real coefficient polynomials, P*(p) = P{ — p))- The next usual step is to find all the roots of this product polynomial, and then to form E(p) by multiplying the factors of the roots on the left-hand p-plane. This procedure recovers the transfer function H(p)/E(p), but the recalling of both the root finding and the poly- nomial forming routines during each step of an iterative adjustment method increases the run time considerably. Paper 4449G (KIO), received 20th May 1985 The authors are with the Department of Electronic Systems Engineering, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom Let us look at another situation. During the initial steps of the cascade synthesis of a lossless network, one is often required to calculate the chain matrix of the network, starting from the reflection coefficient R(p) = F(p)/E(p). Supposing that only the desired transfer function, or the transmission coefficient T(p) = H(p)/E(p), is known, the numerator F(p) of the reflection coefficient can be calcu- lated from the product P(p) = F(p)F m (p) = E(p)E m {p) - H(p)H^p). Again the first usual step is to find the roots of this product, and then to allocate half of these roots to the polynomial G(p) by multiplying the convenient factors. Many years ago, Tuttle [2] showed how a real coef- ficient polynomial could be split directly into two parts, without requiring a root finding procedure. The purpose of this paper is to show how Tuttle's method can be extended to complex coefficient polynomials, therefore making pos- sible its integration into asymmetric filter design pro- cedures. As has been found for the real coefficient case, there is a considerable time saving with the splitting tech- nique. The formulas are straightforward to derive and program, as will be explained later on. Like in the real coefficient case, and concerning the splitting of the polynomial P{p) = G(p)G^.(p), care must be taken, however, if the splitting process is to converge towards a desired G(p). In this case, as shown by Tuttle, good initial guesses for the coefficients of G(p) must be known in advance, as these will be used as the starting points for the method. This is particularly important when G(p) is to be assigned only the left-half plane roots of P(p), i.e. the poles of a transfer function. When the polynomial splitting procedure is part of an iterative routine that determines a transfer function and checks its group delay response, like the one previously described, the polynomial coefficients found in one loop can be stored and used as the starting points for the next loop, and so on, because the transmission zeros are just slightly altered during each iteration. No such problem exists, however, when the poly- nomial G(p) being calculated represents the numerator of a reflection coefficient, because the reflection zeros can be positioned anywhere on the p-plane. 2 Theory Not any polynomial P(p) can be split as the product of two polynomials, one being the paraconjugate of the other. The sole condition is that the zeros of P(p) shall present symmetry about the imaginary axis of the p-plane. This imposes some conditions on the P(p) coefficients, as we will see later. But, having assured that, our problem consists in finding a polynomial G(p) such that its zeros are the mirror IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986 95
Transcript
On the splitting of a complex-coefficientpolynomial
M. de P. Barros. M.Sc. and L.F. Lind. Ph.D.. C.Eng., M.I.E.E.
Indexing terms: Mathematical techniques, Polynomials, Filters and filtering
Abstract: An iterative technique is presented for the splitting of a complex-coefficient polynomial into twolower-degree polynomial factors, one being the paraconjugate of the other. Unlike more usual procedures, theprevious calculation of the roots of the initial polynomial is not required, making the splitting technique a fasterprocess, readily implemented as a computer program. A typical application comprises the calculation of thereflection coefficient of a lossless network from its transmission coefficient. A numerical example is given.
1 Introduction
There are many bandpass filter specifications which havedifferent selectivity slopes at the two passband edges. Forexample, single sideband filters should have extremelysharp selectivity on one side, to remove the unwanted side-band, whereas the other side can have a gentle rolloff, asthere is not much unwanted spectrum to contend with.
One way to create an asymmetric bandpass filter is tostart from a complex coefficient lowpass design, followedby a conventional lowpass to bandpass mapping trans-formation. If the passband is very narrow (such as forcrystal filters), the complex coefficients can be realised byfrequency invariant immittances. A lumped coil or capac-itor will have approximately this behaviour over thenarrow passband. With this approach, all of the desirablefeatures of lowpass to bandpass mapping are retained, themost important being the relative simplicity of the lowpassdesign.
We will now concentrate on the complex coefficientlowpass design. A common procedure is to use the magni-tude squared response of the transfer function as a cri-terion. Given the transmission zeros, there are analyticmethods to locate the reflection zeros so that the magni-tude squared response is an equiripple in the passband [1].By recovering the transfer function, one can then find thegroup delay response (or the impulse response). If thisresponse is not the one desired, the transmission zeros aremoved slightly, and the sequence just described is repeated.The iteration continues until no further improvement isnecessary.
Thus we see that an important problem with the abovemethod is the determination of the transfer function. Asboth the numerators of the transmission and reflectioncoefficients are known (say, H(p) and F(p), respectively), theonly unknown is their common denominator, E(p), con-taining the poles of the transfer function. One solution is,by using analytic continuation, to form the equationE(p)Eitt(p) = H{p)HJp) + F{p)F^{p), where the lower aster-isk denotes the paraconjugate polynomial (at this pointnote that, for real coefficient polynomials, P*(p) = P{ — p))-The next usual step is to find all the roots of this productpolynomial, and then to form E(p) by multiplying thefactors of the roots on the left-hand p-plane.
This procedure recovers the transfer function H(p)/E(p),but the recalling of both the root finding and the poly-nomial forming routines during each step of an iterativeadjustment method increases the run time considerably.
Paper 4449G (KIO), received 20th May 1985
The authors are with the Department of Electronic Systems Engineering, Universityof Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom
Let us look at another situation. During the initial stepsof the cascade synthesis of a lossless network, one is oftenrequired to calculate the chain matrix of the network,starting from the reflection coefficient R(p) = F(p)/E(p).Supposing that only the desired transfer function, or thetransmission coefficient T(p) = H(p)/E(p), is known, thenumerator F(p) of the reflection coefficient can be calcu-lated from the product
P(p) = F(p)Fm(p) = E(p)Em{p) - H(p)H^p).
Again the first usual step is to find the roots of thisproduct, and then to allocate half of these roots to thepolynomial G(p) by multiplying the convenient factors.
Many years ago, Tuttle [2] showed how a real coef-ficient polynomial could be split directly into two parts,without requiring a root finding procedure. The purpose ofthis paper is to show how Tuttle's method can be extendedto complex coefficient polynomials, therefore making pos-sible its integration into asymmetric filter design pro-cedures. As has been found for the real coefficient case,there is a considerable time saving with the splitting tech-nique. The formulas are straightforward to derive andprogram, as will be explained later on.
Like in the real coefficient case, and concerning thesplitting of the polynomial P{p) = G(p)G^.(p), care must betaken, however, if the splitting process is to convergetowards a desired G(p). In this case, as shown by Tuttle,good initial guesses for the coefficients of G(p) must beknown in advance, as these will be used as the startingpoints for the method. This is particularly important whenG(p) is to be assigned only the left-half plane roots of P(p),i.e. the poles of a transfer function. When the polynomialsplitting procedure is part of an iterative routine thatdetermines a transfer function and checks its group delayresponse, like the one previously described, the polynomialcoefficients found in one loop can be stored and used asthe starting points for the next loop, and so on, becausethe transmission zeros are just slightly altered during eachiteration. No such problem exists, however, when the poly-nomial G(p) being calculated represents the numerator of areflection coefficient, because the reflection zeros can bepositioned anywhere on the p-plane.
2 Theory
Not any polynomial P(p) can be split as the product of twopolynomials, one being the paraconjugate of the other.The sole condition is that the zeros of P(p) shall presentsymmetry about the imaginary axis of the p-plane. Thisimposes some conditions on the P(p) coefficients, as we willsee later. But, having assured that, our problem consists infinding a polynomial G(p) such that its zeros are the mirror
IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986 95
images of those of G#(p) with respect to the jcu-axis,without actually knowing these zeros.
We start by defining the initial polynomial P(p) as
P(P) = W
+ • • • + [ar(N) + jai(N)-]pN (1)
where N is even. The unknown polynomials are
G(p) = z(0) + z(\)p + z(2)p2 + ••• + z(N/2)pN'2
G*(p) = z*(Q) - z*(l)p + z*(2)p™
+ ••• + {-l)N/2z*(N/2)pN'2 (2)
where the complex coefficients z(k) are given bybr(k) + jbi(k).
By inspecting the product G(p)Gi).(p), we see that thecoefficients of this product can be calculated from
ar(k) - \)mz(k - m)z*(m) (3)m - 0
for k = 0 to N, with the condition z{i) — 0 for i > N/2.It is easier to deduce the final appearance of these coef-
ficients if we consider the even- and the odd-power coeffi-cients separately. Thus, from eqn. 3, we find that:
(a) For even-power coefficients only, k — 2, 4, 6, . . . ,N-2:
ar(k) +jai(k) = ( - \)k'2z{k/2)z*{k/2)
+ E (-l)'"[z(k - m)z*{m) + z*(k - m)z{m)~] (4)
(b) F o r odd -power coefficients only, k = 1, 3, 5, . . . ,N - 1:
(ft- l ) / 2
ar(k)+jai(k)= £ (-1)"1
x [z(fc - m)z*(m) - z*(/c - m)z{m)~] (5)
where, for both cases, the inferior limit of the summation is
q = max [0, k - (N/2)]
The independent and the highest-power coefficients aregiven, respectively, by
ar(0) + jailO) = z(0)z*(0)(6)
ar(N) +jai{N) = ( - l)N/2z(N/2)z*(N/2)
Making z(n) = br(n) + jbi{n) in eqns. 4-6 leads to(a) Even-power coefficients, k = 2, 4, 6, . . . , N — 2:
ar{k) = ( - l)fc/2[6r(/c/2)2 + fa(/c/2)2]
m — q
+ bi(m)bi(k - m)]
fli(fc) = 0 (7)
(b) Odd-power coefficients, k — 1, 3, 5, . . . , N — 1:
From these equations we see that both the independentand the highest-power coefficients are real, ar(0) beingpositive and the sign of ar(N) given by (— 1)N/2. The divi-sion of the polynomial by a convenient constant factormust be effected in advance, to assure those conditions.Also, we note that the inner coefficients present an alter-nate real x purely imaginary pattern, i.e. P{p) has the form
P{p) = ar(0) ar(2)p ar(N)pN (10)
Before going any further, we note that the characteristicsjust concluded are necessary but not sufficient to makepossible the splitting of a polynomial. For example, thecoefficients of the polynomial
satisfy those conditions but, despite that, P(p) cannot besplit into a product G(p)G^.{p), because it contains onlysingle zeros on the imaginary axis. If P(p) contains zeroson thejco-axis, these zeros must be of even order, such thathalf of them can be allocated to G(p), and half to Gs|c(p).
The formulas just derived will be effectively used to cal-culate the coefficients of G(p), implicitly immersed intothem. Eqns. 7-9 form a system of N + 1 nonlinear equa-tions against N + 2 unknowns, the coefficients of G(p),br(n) + jbi(n), n = 0 to N/2. However, dividing P(p) by itsindependent coefficient ar(0) gives
Now the polynomial G(p) can take the form (amongothers)
G(p) = +jbi(2)-]p
(12)
with br(0) = 1 and 6/(0) = 0. That reduces the number ofequations by one, whereas the number of unknowns isreduced by two, turning the set of equations into a(N x N) system. The scaling factor ar(0) can be taken intoaccount at the end of the process.
Following a procedure similar to the one employed byTuttle, the nonlinear equation system is then linearised.First, from eqns. 7-9, we define the error functions f(k)that, ideally, should vanish, were the brs and bis correctlyadjusted:
(a) For the real coefficients of P(p), k = 2, 4, 6, .. . ,N-2:
Eqns. 16-18 form a linear set of N equations with Nunknowns, the Abrs and Aftis, which represent the correc-tions to be made on the G{p) coefficients, thus updatingthem to more correct figures.
At this stage, the complete numerical procedure can bedecomposed into the following steps:
(i) Step 1: Assuming that P(p) has already been scaledsuch that its independent coefficient is unity, guess theinitial values for the coefficients of G{p), br(n) and bi(n),i = l to N/2. Note that, from eqns. 8, an exact guess forbi(\) is c(l)/2, because br{0) = 1 and bi{0) = 0
(ii) Step 2: Using eqns. 13-15, calculate the error func-t ions/^) , for k = 1 to N
(iii) Step 3: Make Af{k) = -f(k) for k = 1 to N, thenecessary increments to make the error functions null
(iv) Step 4: Solve the linear equation represented byeqns. 16-18, therefore calculating Abr(n) and Abi{n) forn = 1 to N/2
(v) Step 5: Update the old G(p) coefficients:
br{n)i + , = br(n)i + Abr{n) and
bi(n)i+i = bi(n)i + Abi(n) for n = 1 to N/2
(vi) Step 6: Calculate and check the total error againstthe required tolerance. The error can be calculated as:
N/2
E= (19)
(vii) Step 7: Repeat the process, starting from step 2 ifthe error E is greater than the required tolerance; other-wise proceed to the next step
(viii) Step 8: Finally, multiply the (scaled) coefficients ofG{p) by the square root of ar(0), leading to the final (butnot unique G{p).
3 Numerical example and general comments
The procedure just described is easily implemented as acomputer program, and the speed of the method can becompared with that of more conventional techniques. Ingeneral, for polynomial degrees of up to 30 (degree of theinitial P{p)), the polynomial splitting process proved to befrom 10% to 30% faster than the conjunct of a rootfinding procedure plus a polynomial forming routine,starting from the desired roots.
As an example, suppose we want to split the polynomial
P(p) = 20 + j\6p - 9p2 - j 4 p 3 + p4
IEE PROCEEDINGS, Vol. 133, Pt. G, No. 2, APRIL 1986
which contains the roots ± 2 and ± 1 + j2. First, this poly-nomial is divided by the independent coefficient ar{0) — 20,leading to
already in the form given by eqn. 11.As no special G(p) is desired, i.e. the process is allowed
to converge towards any G(p), we choose unitary startingcoefficients for Gs{p), but for bi{\)0, which is already knownas equal to c(l)/2 = 0.4 (see step 1).
The required tolerance is chosen to be 10 \ and theerror calculated according to eqn. 19. The convergence ofthe unknowns towards the coefficients of Gs(p) is sum-marised in Table 1.
Table 1: Convergence of the unknowns towards the coef-ficients of Gs(p)
where the precision of the coefficients is better than 10"3.Multiplying this polynomial by its paraconjugate leads toa product close within six decimal places to Ps{p).
Finally, G(p) is obtained by multiplying Gs{p) by thesquare root of c(0), or 4.4721:
The roots of this polynomial are —2 and —1 +j2, caus-ally all on the lefthand side of the p-plane.
As a higher order example, we (exactly) form a 10th-degree P(p) from the following set of prechosen roots: +j\(double), ± 1, ±0.4 - j 0 .5 , ±0.5 +7O.6 and -/0.9 (double).This polynomial was then split into G(p) and G^(p) by thesplitting technique, and also by calculating its roots andmultiplying the desired factors from half of the roots, thusforming G(p). As P{p) and G(p) are exactly known, the rela-tive errors introduced by both techniques can be com-pared, as well as their relative speed.
The polynomial splitting technique found the coeffi-cients of G{p) in 29 iterations (the required error, as calcu-lated by eqn. 19, was chosen to be 10~4). When comparedto the exact G(p), the total relative error on the coefficientsof the calculated G{p) was 0.0018%.
The root finder routine chosen was the one presented inReference 3, which employs a steepest descent method thatmakes use of the so-called Siljak polynomials. It is a veryfast routine and, independently of the starting values, italways converges to the roots. For this example, the rootsof the lOth-degree P(p) were calculated within a toleranceof 10~5, taking an average of 12 iterations per root. Afterforming G(p), the error introduced on its coefficients wasseen to be 0.0063%, i.e. more than three times higher thanthe relative error introduced by the polynomial splittingprocess. Also, the splitting technique in this case was about10% faster than the conjunct of the root finding and poly-nomial forming routines.
97
At this point it is worthwhile to mention that the coef-ficient matrix of the equation system given by eqns. 16 to18, i.e. formed by the coefficients of the unknowns Abr andAbi, can be kept constant during most of the iterations,once the process is converging. As an example, a goodapproach is to update this matrix during the first threeiterations, and then every fifth iteration. The increase inspeed doing this is more noticeable during the splitting ofhigher degree polynomials.
The slowest part of the process is during step 4, thesolving of a high-order linear equation system. Severalapproaches were considered, and the one that proved to bethe fastest consists in dividing the N x N matrix of thesystem into precisely four smaller N/2 x N/2 matrices. Theequation set can then be written as
(20)
where the independent vectors AfrQ and Afi() correspondto the even and odd differentiated error functions of k,respectively, given by eqns. 16-18, i.e. to the error func-tions related to the even- and odd-power coefficients ofP(P).
From eqn. 20, the unknown vectors Abr() and AbiQ canbe calculated as
AM = {X- WU i V)-\Afi - WU~ lAfr)
Abr = U~\Afr- VAbi)
(21)
(22)
It was seen to be faster to invert two N/2 x N/2 matricesand to perform some matrix multiplications than to solvean N x N equation system.
The process can be further improved to check whetherthe initial P(p) is a real polynomial. In this case, eqns. 21and 22 become simply
UAbr = Afr (23)
because all the factors related to imaginary coefficients aredropped. The speed is greatly increased, because the calcu-lation of the brs involves the solving of a N/2 x N/2 equa-tion system only.
4 Conclusion
An iterative numerical process for the splitting of acomplex-coefficient polynomial P(p) into two equal-degree
factors G{p) and G^(p) has been presented. Compared tomore usual processes that involve root finding routines,the polynomial splitting technique is faster and easilyimplemented as a computer program.
Once the process is completely general, coping withcomplex polynomials, it finds immediate application in theapproximation and synthesis of filtering networks thatshall realise asymmetric transmission zeros along the rejec-tion band.
5 Acknowledgment
Mr. M. de P. Barros would like to acknowledge the finan-cial support received from the Conselho Nacional deDesenvolvimento Cientifico e Tecnologico (CNPq),Brasilia, Brazil.
6 References
1 CAMERON, R.J.: 'Fast generation of Chebyshev filter prototypes withasymmetrically-prescribed transmission zeros', ESA J., 1982, 6, pp.83-95
3 MOORE, J.B.: 'A convergent algorithm for sowing polynomial equa-tions', J. Assoc. Comput. Mach., 1967, 14, (2), pp. 311-315
Larry Lind was born in Minnesota, USA, in 1941. He came tothe UK in 1965, and obtained a Ph.D. degree from Leeds Uni-versity in 1968. Since then, he has been at Essex University,where he is now a Reader in the Department of ElectronicSystems Engineering. He has served on and was Vice Chairmanof IEE Professional Group E10 (Circuit Theory and Design). Hisresearch interests include circuit theory, communications theoryand logic design.
Marcos de P. Barros was born in Rio de Janeiro, Brazil, in 1951.He received the degree of Engenheiro Eletronico from the Uni-versidade Catolica do Rio de Janeiro, in 1974. Since then he hasbeen with Radio Cristais do Brasil SA, Rio de Janeiro, where heworks on the design and manufacture of quartz crystal oscillatingunits and crystal filters. In 1983 he received the M.Sc. degreefrom the University of Essex, Colchester, UK, where he is cur-rently working towards the Ph.D. degree. His research interestsinclude passive network theory, crystal oscillators and filters, andcomputer-aided circuit design.
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