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An explicit formula for the general bilinear transformation of polynomials with its application
An Explicit Formula for the General Bilinear Transformation of Polynomials with its Application by S.ERFAN1.f AT&T
Bell Laboratories,
Holmdel,
New Jersey
07733,
U.S.A
M. AHMADI
Department
of Electrical
Engineering,
University
of Windsor,
Windsor,
Ontario,
Concordia
University,
Montreal,
Quebec
N9B 3P4, Canada andv.
RAMACHANDRAN
Department of Electrical H3G IM8, Canada
Engineering,
An explicit formula for establishing the relationship between the original coeficients of a single-variable polynomial and the coeficients of the transformed polynomial upon application of the general bilinear transformation is derived. The derivation is based on an alternative approach which is simpler than existing schemes. The result is applied to produce eficient algorithms for the digital simulation of analog systems as well as for the realization of variable recursive digitaljlters. ABSTRACT:
I. Introduction In a variety of applications such as the design of digital filters from analog counterparts, stability tests of discrete systems, and analog and digital frequency band transformations, the bilinear transformation of a polynomial must be computed. Application of a general bilinear transformation to a polynomial with real coefficients will result in a rational function whose numerator represents the transformed polynomial. Several different approaches have been reported for computing the coefficients of the transformed single-variable polynomial in terms of the coefficients of the original polynomial (14). In the case of multivariable polynomials, there are also several techniques for computing the coefficients of the transformed polynomial in terms of the coefficients of the original one (57). These techniques, however, are of limited application since they are restricted to special cases of the bilinear transformation. In this paper, the procedure presented in (2) is extended to include the general case of bilinear-transformation of polynomials. t This work was performed
while S. Erfani was at the University
0 The Franklin Institute OOlCCP332/88 S3.OOf0.00
of Michigan-Dearborn.
385
S. Erfani et al. We derive a closed-form relationship between coefficients of the original polynomial and those of the transformed one. A feature of the relationship derived here is that it lends itself to generating variable recursive digital filters.
II. The Algorithm Consider
the following
polynomial
of degree N in the complex
variable
s
:
(1) Let the bilinear
be of the general form :
transformation
crz+fi s=yz+6’ The polynomial rational function
Q(s) is transformed
cd # py.
by the bilinear
(2) transformation
(2) into the
1 I;(z) = Q(s) where the transformed
s=5
= (yz+ 8)” P(z)
P(z) is given as :
polynomial P(z) = ;
ai(az+p)‘(yz+6)N-Z.
(4)
i=O
Note that polynomial
(4) can be expanded P(z) = ;
and alternatively bjzi.
written
as (9
j=O
In order to find the coefficients bj in terms of the original coefficients ai, the derivative approach of (2) can be utilized. By taking the jth derivative from both sides of Eq. (5) and evaluating it at z = 0, we obtain : d’P(z) dz j
Similarly,
386
using Leibniz
formula,
= j!b,. 2=0
(6)
from Eq. (4), we obtain
Journal of the Franklin Institute Pergamon Press plc
General Bilinear Transformation of Polynomials where C,” = (J’), and C,” = 0 for all m < n. Equating following relationship between coefficients bj and a, : bj
5i
=
Eqs (6) and (7), yields the
c:c~~~ClkBi-kyj~ks[(n-i)~(j-k)lai.
i=Ok=O
Notice that the coefficients of the corresponding polynomial P(z) may be represented by the (N+ 1)-column vector b = [b,, b,, . . . , b,JT which is related to the matrix multicoefficient vector a = [ao, a,, . . . , a,]’ of Q(s) by the following plication : (9)
b=Qa where, qii, entries of the (N+ 1) by (N+ 1) matrix 4ij
=
i
Q are :
c:c~_~~akj3i-kyj-kGN-‘j+k
(10)
k=O
This relationship has been given previously using a different approach (8). It should be mentioned that due to some symmetry properties inherent in combinatorial relations the number of calculations of the entries qij may be reduced for some particular cases (8). Also, note that Eq. (8) gives a means to compute a particular coefficient bj with no need to know the values of the preceding bj’s or the entries of a transformation matrix (i.e. matrix Q).
III. Application
of the Algarith
to tfte DigitaI Realization
of Analog Systems
The formulation derived in the previous section is used to realize a digital implementation of an analog system. It is well known that the use of the bilinear transformation s-1
l-z-’ 1 +z-
(11)
will enable the filter designer to obtain a digital filter from an analog counterpart possessing similar amplitude characteristics. It is also desirable to preserve the structure of the prototype filter through this transformation. The results in Section II provide a basis to extend the structural realization of analog transfer functions to the digital domain. In general, consider an analog transfer function H,(s) defined by
where the coefficients Vol. 325, No. 3, pp. 381-391, Printed in Great Britain
a, and bi may be positive or negative.
A direct form realization
1988
387
S. Erfani et al.
I--?-------
iy~*y;o FIG. 1. Block diagram
representation
of the analog transfer
function
given in Eq. (12).
of this general transfer function is shown in Fig. 1. The corresponding digital realization is obtained by replacing each integrator - l/s in this structure with unit delays z-’ and replacing each coefficient with transformed coefficients. Transformed coefficients are obtained from the original coefficients by letting --CI = /I = y = 6 = 1 in Eq. (8). Hence the resulting digital transfer function is
f&(2-
‘) = H,(s)
j=Oz=Ok=O I&-’
=
N
‘=l+z’
N
=
j
j;oi~ok;o(_
l)kC:C~:;b,zPi
(13)
~
j$odjz-”
Using Eq. (13), a digital realization of the transfer function under consideration can be obtained as shown in Fig. 2. It is seen from Eq. (13) that coefficients cj and dj are : cj = 2
i
(-
l)kC~C~T~ai
(144
i=Ok=O
and dj = ;
i
(14b)
(- l)kC;C;:;bi.
i=Ok=O
It should be mentioned at this juncture that the matrix multiplication in the case, can be decomposed into the following form :
of Eq. (9)
c=Qa=ULa where c is the (N+ 1)-column
FIG. 2. Block diagram
388
vector comprised
representation
(15) of the coefficients
of the digital transfer
function
of the numerator
given in Eq. (13).
Journal of the
Franklin Pergamon
Institute Press plc
General Bilinear Transformation
of Polynomials
(denominator) of the digital transfer function, a is the (N+ 1)-column coefficient vector of the numerator (denominator) of the original analog transfer function, and U and L are the upper and lower triangular combinatorial matrices, respectively, and are given by : c:
c:,
c;
...
c:
0
c;
c:
...
cy
(164
and
Equation (15) is a very important result giving the relationship between the coefficients of the original polynomial and those of the transformed polynomial in an explicit matrix form. Note that the matrix formulation in Eq. (15) is simpler and computationally more efficient than the hitherto known Q-matrix technique (1).
IV. Implementation
of Variable Recursive Digital Filters
In a variable lowpass frequency transformation
digital filter the following lowpass-to-lowpass is applied to the recursive prototype filter :
bilinear
(17) in the It has been noted that when zP ’ is replaced by the above transformation transfer function of a recursive digital filter, the resulting direct form structure has delay-free loops and is therefore not realizable without modification (9). In the light of this method, we present a simple algorithm for implementing the variable digital filters, one which incorporates the all-pass substructure inherent in the bilinear transformation (17), and which results in very fast coefficient recomputation. Given the original transfer function :
(18)
the transformed Vol. 325, No. 3, pp. 38M91, Printed in Great Britain
transfer 1988
function
using transformation
(17) is 389
S. Erfani et al.
BN
A~
FIG. 3. Realization of the transformed filter of Eq. (19a).
jjio Aiz-i
f&(z) =
Bo+
5
(194
Bjz-'
j= I
where coefficients
Aj and Bj of (19a) are calculated Aj
=
5
i
CLCy_-ipi+j-
i
C:C~_p$t+j-2kbi,
using relation
2kai
(8) as follows
:
Wb)
i=Ok=O
Bj
=
$
j
#
0
(194
i=Ok=O N
B. =
1 Bibi.
(194
The transfer function (19a) has no delay-free loops, so this form of transfer function is directly realizable, as shown in Fig. 3. When the numerator and denominator of the original filter are of the same degree, the resulting transformed filter has the advantage of having the same filtering complexity as the original filter, however, both numerator and denominator coefficients must be recomputed. On the other hand, when the frequency transformation (17) is applied to an all-pole prototype, the resulting structure will not have the same hardware complexity as the original filter due to a numerator polynomial which will be introduced to the structure. V. Conclusion An explicit formula for the calculation of the coefficients of a transformed polynomial in terms of the coefficients of the original polynomial using a general bilinear transformation is derived. The developed technique is then applied to produce an efficient algorithm for digital realization of analog systems. It has been shown that the matrix relating the coefficients of the original polynomial to those 390
Journal of the Franklin Institute Pergamon Press pLc
General Bilinear Transformation
of Polynomials
of the transformed polynomial can be decomposed to an LU multiplication simple combinatorial matrices. Furthermore, we have extended the result implement a variable cutoff frequency digital filter.
of to
References (1) E. I. Jury, “Remarks on ‘the mechanics of bilinear transformation’“, IEEE Trans. Audio Electroacoust., Vol. AU-21, pp. 380-382, 1973. (2) S. Erfani, M. Ahmadi and V. Ramachandran, “On the algebra of multiple bilinear transformations”, J. Franklin Znst., Vol. 322, No. 3, pp. 137-142, Sept. 1986. (3) B. O’Conner and T. S. Huang, “An efficient algorithm forbilinear transformation of multivariable polynomials”, IEEE Trans. Acoust. Speech, Signal Processing, Vol. ASSP-24, pp. 266-267, 1976. (4) R. Parthasarathy and K. N. Jayasimha, “Bilinear transformation by synthetic division”, IEEE Trans. Aut. Control, Vol. AC-29, pp. 575-576, 1984. (5) N. K. Bose and E. I. Jury, “Positivity and stability tests for multi-dimensional filters (discrete-continuous)“, IEEE Trans. Acoust, Speech, Signal Processing, Vol. ASSP22, pp. 174-180, 1974. (6) N. M. Smart and S. Barnett, “Bilinear transformation of multi-variable polynomials using the Horner method”, Znt. J. Control, Vol. 37, No. 4, pp. 861-865, 1983. (7) V. V. Bapeswara Rao and K. Nageswara Rao, “A note on transformation matrices for bilinear transformation”, IEEE Trans. Circuits, Syst., Vol. CAS-32, p. 857, 1985. (8) N. M. Smart and S. Barnett, “Transformation matrices in the general bilinear transformation of multivariable polynomials”, IEEE Trans. Acoust. Speech, Signal Processing, Vol. ASSP-32, pp. 634-636, 1984. (9) W. H. Schiissler and W. Winkelnkemper, “Variable digital filters”, Arch. Elektr. iibertrag., Vol. 24, pp. 524-525, 1970.
Vol. 325, No. 3, pp. 38S391, Printed in Great Britain
1988
391
Bell Laboratories,
Holmdel,
New Jersey
07733,
U.S.A
M. AHMADI
Department
of Electrical
Engineering,
University
of Windsor,
Windsor,
Ontario,
Concordia
University,
Montreal,
Quebec
N9B 3P4, Canada andv.
RAMACHANDRAN
Department of Electrical H3G IM8, Canada
Engineering,
An explicit formula for establishing the relationship between the original coeficients of a single-variable polynomial and the coeficients of the transformed polynomial upon application of the general bilinear transformation is derived. The derivation is based on an alternative approach which is simpler than existing schemes. The result is applied to produce eficient algorithms for the digital simulation of analog systems as well as for the realization of variable recursive digitaljlters. ABSTRACT:
I. Introduction In a variety of applications such as the design of digital filters from analog counterparts, stability tests of discrete systems, and analog and digital frequency band transformations, the bilinear transformation of a polynomial must be computed. Application of a general bilinear transformation to a polynomial with real coefficients will result in a rational function whose numerator represents the transformed polynomial. Several different approaches have been reported for computing the coefficients of the transformed single-variable polynomial in terms of the coefficients of the original polynomial (14). In the case of multivariable polynomials, there are also several techniques for computing the coefficients of the transformed polynomial in terms of the coefficients of the original one (57). These techniques, however, are of limited application since they are restricted to special cases of the bilinear transformation. In this paper, the procedure presented in (2) is extended to include the general case of bilinear-transformation of polynomials. t This work was performed
while S. Erfani was at the University
0 The Franklin Institute OOlCCP332/88 S3.OOf0.00
of Michigan-Dearborn.
385
S. Erfani et al. We derive a closed-form relationship between coefficients of the original polynomial and those of the transformed one. A feature of the relationship derived here is that it lends itself to generating variable recursive digital filters.
II. The Algorithm Consider
the following
polynomial
of degree N in the complex
variable
s
:
(1) Let the bilinear
be of the general form :
transformation
crz+fi s=yz+6’ The polynomial rational function
Q(s) is transformed
cd # py.
by the bilinear
(2) transformation
(2) into the
1 I;(z) = Q(s) where the transformed
s=5
= (yz+ 8)” P(z)
P(z) is given as :
polynomial P(z) = ;
ai(az+p)‘(yz+6)N-Z.
(4)
i=O
Note that polynomial
(4) can be expanded P(z) = ;
and alternatively bjzi.
written
as (9
j=O
In order to find the coefficients bj in terms of the original coefficients ai, the derivative approach of (2) can be utilized. By taking the jth derivative from both sides of Eq. (5) and evaluating it at z = 0, we obtain : d’P(z) dz j
Similarly,
386
using Leibniz
formula,
= j!b,. 2=0
(6)
from Eq. (4), we obtain
Journal of the Franklin Institute Pergamon Press plc
General Bilinear Transformation of Polynomials where C,” = (J’), and C,” = 0 for all m < n. Equating following relationship between coefficients bj and a, : bj
5i
=
Eqs (6) and (7), yields the
c:c~~~ClkBi-kyj~ks[(n-i)~(j-k)lai.
i=Ok=O
Notice that the coefficients of the corresponding polynomial P(z) may be represented by the (N+ 1)-column vector b = [b,, b,, . . . , b,JT which is related to the matrix multicoefficient vector a = [ao, a,, . . . , a,]’ of Q(s) by the following plication : (9)
b=Qa where, qii, entries of the (N+ 1) by (N+ 1) matrix 4ij
=
i
Q are :
c:c~_~~akj3i-kyj-kGN-‘j+k
(10)
k=O
This relationship has been given previously using a different approach (8). It should be mentioned that due to some symmetry properties inherent in combinatorial relations the number of calculations of the entries qij may be reduced for some particular cases (8). Also, note that Eq. (8) gives a means to compute a particular coefficient bj with no need to know the values of the preceding bj’s or the entries of a transformation matrix (i.e. matrix Q).
III. Application
of the Algarith
to tfte DigitaI Realization
of Analog Systems
The formulation derived in the previous section is used to realize a digital implementation of an analog system. It is well known that the use of the bilinear transformation s-1
l-z-’ 1 +z-
(11)
will enable the filter designer to obtain a digital filter from an analog counterpart possessing similar amplitude characteristics. It is also desirable to preserve the structure of the prototype filter through this transformation. The results in Section II provide a basis to extend the structural realization of analog transfer functions to the digital domain. In general, consider an analog transfer function H,(s) defined by
where the coefficients Vol. 325, No. 3, pp. 381-391, Printed in Great Britain
a, and bi may be positive or negative.
A direct form realization
1988
387
S. Erfani et al.
I--?-------
iy~*y;o FIG. 1. Block diagram
representation
of the analog transfer
function
given in Eq. (12).
of this general transfer function is shown in Fig. 1. The corresponding digital realization is obtained by replacing each integrator - l/s in this structure with unit delays z-’ and replacing each coefficient with transformed coefficients. Transformed coefficients are obtained from the original coefficients by letting --CI = /I = y = 6 = 1 in Eq. (8). Hence the resulting digital transfer function is
f&(2-
‘) = H,(s)
j=Oz=Ok=O I&-’
=
N
‘=l+z’
N
=
j
j;oi~ok;o(_
l)kC:C~:;b,zPi
(13)
~
j$odjz-”
Using Eq. (13), a digital realization of the transfer function under consideration can be obtained as shown in Fig. 2. It is seen from Eq. (13) that coefficients cj and dj are : cj = 2
i
(-
l)kC~C~T~ai
(144
i=Ok=O
and dj = ;
i
(14b)
(- l)kC;C;:;bi.
i=Ok=O
It should be mentioned at this juncture that the matrix multiplication in the case, can be decomposed into the following form :
of Eq. (9)
c=Qa=ULa where c is the (N+ 1)-column
FIG. 2. Block diagram
388
vector comprised
representation
(15) of the coefficients
of the digital transfer
function
of the numerator
given in Eq. (13).
Journal of the
Franklin Pergamon
Institute Press plc
General Bilinear Transformation
of Polynomials
(denominator) of the digital transfer function, a is the (N+ 1)-column coefficient vector of the numerator (denominator) of the original analog transfer function, and U and L are the upper and lower triangular combinatorial matrices, respectively, and are given by : c:
c:,
c;
...
c:
0
c;
c:
...
cy
(164
and
Equation (15) is a very important result giving the relationship between the coefficients of the original polynomial and those of the transformed polynomial in an explicit matrix form. Note that the matrix formulation in Eq. (15) is simpler and computationally more efficient than the hitherto known Q-matrix technique (1).
IV. Implementation
of Variable Recursive Digital Filters
In a variable lowpass frequency transformation
digital filter the following lowpass-to-lowpass is applied to the recursive prototype filter :
bilinear
(17) in the It has been noted that when zP ’ is replaced by the above transformation transfer function of a recursive digital filter, the resulting direct form structure has delay-free loops and is therefore not realizable without modification (9). In the light of this method, we present a simple algorithm for implementing the variable digital filters, one which incorporates the all-pass substructure inherent in the bilinear transformation (17), and which results in very fast coefficient recomputation. Given the original transfer function :
(18)
the transformed Vol. 325, No. 3, pp. 38M91, Printed in Great Britain
transfer 1988
function
using transformation
(17) is 389
S. Erfani et al.
BN
A~
FIG. 3. Realization of the transformed filter of Eq. (19a).
jjio Aiz-i
f&(z) =
Bo+
5
(194
Bjz-'
j= I
where coefficients
Aj and Bj of (19a) are calculated Aj
=
5
i
CLCy_-ipi+j-
i
C:C~_p$t+j-2kbi,
using relation
2kai
(8) as follows
:
Wb)
i=Ok=O
Bj
=
$
j
#
0
(194
i=Ok=O N
B. =
1 Bibi.
(194
The transfer function (19a) has no delay-free loops, so this form of transfer function is directly realizable, as shown in Fig. 3. When the numerator and denominator of the original filter are of the same degree, the resulting transformed filter has the advantage of having the same filtering complexity as the original filter, however, both numerator and denominator coefficients must be recomputed. On the other hand, when the frequency transformation (17) is applied to an all-pole prototype, the resulting structure will not have the same hardware complexity as the original filter due to a numerator polynomial which will be introduced to the structure. V. Conclusion An explicit formula for the calculation of the coefficients of a transformed polynomial in terms of the coefficients of the original polynomial using a general bilinear transformation is derived. The developed technique is then applied to produce an efficient algorithm for digital realization of analog systems. It has been shown that the matrix relating the coefficients of the original polynomial to those 390
Journal of the Franklin Institute Pergamon Press pLc
General Bilinear Transformation
of Polynomials
of the transformed polynomial can be decomposed to an LU multiplication simple combinatorial matrices. Furthermore, we have extended the result implement a variable cutoff frequency digital filter.
of to
References (1) E. I. Jury, “Remarks on ‘the mechanics of bilinear transformation’“, IEEE Trans. Audio Electroacoust., Vol. AU-21, pp. 380-382, 1973. (2) S. Erfani, M. Ahmadi and V. Ramachandran, “On the algebra of multiple bilinear transformations”, J. Franklin Znst., Vol. 322, No. 3, pp. 137-142, Sept. 1986. (3) B. O’Conner and T. S. Huang, “An efficient algorithm forbilinear transformation of multivariable polynomials”, IEEE Trans. Acoust. Speech, Signal Processing, Vol. ASSP-24, pp. 266-267, 1976. (4) R. Parthasarathy and K. N. Jayasimha, “Bilinear transformation by synthetic division”, IEEE Trans. Aut. Control, Vol. AC-29, pp. 575-576, 1984. (5) N. K. Bose and E. I. Jury, “Positivity and stability tests for multi-dimensional filters (discrete-continuous)“, IEEE Trans. Acoust, Speech, Signal Processing, Vol. ASSP22, pp. 174-180, 1974. (6) N. M. Smart and S. Barnett, “Bilinear transformation of multi-variable polynomials using the Horner method”, Znt. J. Control, Vol. 37, No. 4, pp. 861-865, 1983. (7) V. V. Bapeswara Rao and K. Nageswara Rao, “A note on transformation matrices for bilinear transformation”, IEEE Trans. Circuits, Syst., Vol. CAS-32, p. 857, 1985. (8) N. M. Smart and S. Barnett, “Transformation matrices in the general bilinear transformation of multivariable polynomials”, IEEE Trans. Acoust. Speech, Signal Processing, Vol. ASSP-32, pp. 634-636, 1984. (9) W. H. Schiissler and W. Winkelnkemper, “Variable digital filters”, Arch. Elektr. iibertrag., Vol. 24, pp. 524-525, 1970.
Vol. 325, No. 3, pp. 38S391, Printed in Great Britain
1988
391