May 1989
LIDS-P-1872
ON A GENERALIZED FACTORIZATION PROBLEM FOR
STRUCTURALLY PASSIVE SYNTHESIS OF DIGITAL FILTERS
Sankar Basu
This research was performed while author was visiting the Laboratory for
Information and Decision Systems, M.I.T., Cambridge, MA 02139.
ABSTRACT
The
problem
filters
of
the
elementary
via
the
of
structurally
passive
quarter-plane
building
synthesis
causal
type
as
of multidimensional digital
an
interconnection
of more
blocks directly in the discrete domain has been addressed
factorization of the chain matrix, the hybrid matrix and the transfer
function
matrix
associated
with
a
prescribed
multidimensional
lossless
two-port. By exploiting recent results on the discrete domain representation of
such
matrices
present
all
sufficient
these
a
generalized
three
lossless two-port matrix has been introduced to
factorizations
conditions
in
an
unified
setting.
Necessary
and
for factorability as well as an algorithm for computing
factors when they exist are obtained. In particular, it is shown that in
one-dimension
discrete
passive
domain
factorizations
of
can
always
be
performed.
Thus,
in 1-D,
algorithms for synthesizing previously unpublished internally
structures
conventional
byproduct
the
as
well
structures
such
as
alternative
methods
of
synthesis for more
as the cascade structure are also obtained as a
our discussion. Since most multidimensional applications dictate
that the filter be either symmetric or (quasi) antimetric, special attention is
paid
to
the
problem
lossless two-ports.
of
synthesis
of
these subclasses of multidimensional
1.INTROIUCTICN:
Various
synthesis
lossless
synthesizing
elementary
etc.
in
lossless
discrete
transfer
building
problem
lossless
in
as
the
Darlington
functions
blocks
resolved
via
the
bounded
interconnection
passive
such
a
as
synthesis
cascade
scheme
for
interconnection of
such as inductors, capacitors, gyrators
the continuous domain are well known in classical network theory. The
corresponding
and
schemes
discrete domain, namely that of synthesizing a
(or
of
positive) transfer function as a structurally
elementary
lossless
building
blocks
was first
transformation from prototype problems in the continuous domain,
the resulting class of filter structures are now known as the wave digital
filters
[1],
[17]. Recently, however, successful attempts to derive these and
similar
other
discrete domain results without making explicit use of tools of
classical network theory have been made. Notable among these are the orthogonal
filters
[2],[4],[14],[15] and the class of filters referred to as the lossless
bounded
real
(LBR)
described
filters
in
[3],
[13]
and
in related other
publications.
In
view
the problem of synthesis of k-D lossless two-port scattering transfer
filters,
via the bisection of a prescribed two-port into a cascade connection of
matrix
two
in the synthesis of multidimensional (k-D) wave digital
interest
of
lossless
continuous
sections of smaller "degree" has been addressed in the
two-port
in
domain
[5].
Factorability
of
continuous
domain
two-port
scattering matrices has also been studied recently [11] in the multidimensional
context.
An
attempt
to develop a complete and self-consistent theory for the
synthesis of k-D structurally passive quarter-plane causal type digital filters
independent
[8],
class
of
the
continuous
domain methods have already been initiated in
and [10] by discussing the discrete domain stability properties of a
[9]
of
multidimensional
synthesizing
a
k-D
polynomials.
In
the present paper the problem of
discrete quarter-plane causal type lossless two-port as a
structurally passive interconnection of more elementary digital building blocks
directly
chain
the discrete domain is approached by the methods of factoring the
in
matrix,
with
a
that
each
the
lossless
of
hybrid
matrix and the transfer function matrix associated
two-port. By following recent results in [9] it can be shown
these
matrices
can be uniquely expressed by means of a set of
2
three
polynomials
matrix
referred
in a form analogous to the Belevitch canonical form [16] of
classical circuit theory. For the purpose of unified presentation of results, a
matrix,
to
as
the
(multidimensional) generalized lossless two-port
can be viewed as a generalization of the multidimensional chain
which
matrix, the hybrid matrix and the transfer function matrix has been introduced.
Interestingly, in 1-D this matrix can be categorized under the class of sigma
lossless rational matrices considered, for example, in [12], [26]. The problem
of
two
factorizing
this
generalized lossless two-port matrix into the product of
matrices of identical type can then be viewed as a problem of structurally
passive
synthesis
factorization
of
of
multidimensional
chain
two-ports.
It
must
be
noted
that
matrix,
when feasible, yield networks having cascade
structures as shown in Figure 1, whereas the factorization of hybrid matrix and
transfer function matrix, when feasible, yield networks having the topological
structure as shown in Figures 2 and 3.
Necessary
and sufficient conditions for this generalized factorization problem
so introduced to be solvable are obtained in the present paper via constructive
techniques.
As
expected
from our previous study of analogous problems in the
continuous case [5],[11] it turns out that the factorization may not be
feasible
in
a
generic multidimensional (k>2) situation. However, the
impossibility of factorization of the chain matrix does not by any means rule
out
the
feasibilty
of factorization of the transfer function matrix. Exactly
the same comment also applies if the role of three types of matrices (i.e.,
chain, hybrid and transfer function) are permuted in any posssible manner (cf.
Section 7). Furthermore, in order for the factorizations under consideration to
yield
computable
digital
filters
the
structures resulting from the
factorization
further
constraint
continuous
Section
may
domain
5, this
not
have
on
the
problems
constrained
any
delay free loop. Apparently, this imposes a
factorization not present in the corresponding
discussed
in
[5] and [11]. However, as shown in
can always be solved if and only if a
problem
solution to the unconstrained problem exists.
In
the
special case of 1-D, the criterion for factorability is always seen to
be satisfied, thus guaranteeing the feasibility of factorization. Additionally,
the factorization is seen to be nonunique. Our algorithm for computing these
factors, however, enjoys two remarkable properties. First, it encompasses the
3
entire family of possible solutions. From this point of view it may be remarked
that
although
synthesis
in
cascade
type
structures
has
previously
been
considered, for example, in [2],[14],[15] and [3], [13] our method, namely that
of factoring the corresponding chain matrix, is somewhat more general. A second
important property is that the factors can be computed essentially by solving a
highly
structured
set
potentially
computed
topological
structures
digital
filters
previous
which
work
in
linear
a
have
discrete
indicated
context,
thus,
synthesis
of
yields
1-D
been
mentioned
in
a
analog
domain
as
a
lossless
and
thus can be
in
[1]
in the context of wave
prototypes,
we
are unaware of any
schemes for internally passive synthesis,
result
Figures
set
equations,
manner. On the other hand, although similar
from
structures
decomposition
simultaneous
fast
obtainable
on
yield
of
of
of
repetitive
application
of
the
2 and 3. Our discussion, even in the 1-D
new
digital
algorithms
filters
for
structurally passive
previously not discussed in the
literature.
Most
multidimensional
symmetries
demands
in
that
two-port.
present
their
the
paper
to
special
tasks
frequency
two-port
Motivated
filters,
filtering
require
be
response
either
the
filter
characteristics
a
symmetric
to have certain
[7]. However, this
or a (quasi) antimetric
by potential applications of the results developed in the
the design of multidimensional structurally passive digital
attention
to
the
synthesis
of
symmetric
and
(quasi)
antimetric lossless two-ports have also been paid.
In
section 2 a precise formulation of the problem along with some notation and
terminology
are
problem
introduced
so
introduced.
In
section 3 it is shown that the factorization
is essentially algebraic in nature. An elementary step
towards the general factorization problem is also taken here. In section 4 some
properties
the
the fundamental equation which, in fact, is a linear version of
algebraic problem and is central to our study, are examined. Necessary and
sufficient
factors,
the
of
conditions
when
results
digital
they
so
filter
considerations
demonstrate
for
structures.
the
and
an algorithm for obtaining the
exist, are obtained in section 5. Section 6 discusses how
obtained
of
factorability
the
need
yield
In
new
section
algorithm
for
as well as known internally passive l-D
for
7
remarks are made on computational
synthesis,
examples are worked out to
factorability of three different kinds of matrices
4
associated with discrete lossless two-ports, and the special cases of symmetric
and
(quasi) antimetric discrete two-ports are dealt with. Finally, conclusions
are drawn in section 8.
5
2. NOTATIONS, TERI
English
capital
GY A
FRUIATICI:
letters are used to denote polynomials and rational functions
(z1 , z2 ..
''., Zk). The notation z n is used to denote the
in k-variables: zn1 n 2
(Zl z2
monomial
POLE
...
nk
Zk ) n
being
the
k-tuple of nonnegative intergers (n1,
n2','' nk). Also,
A -A
(z
, 2.
Zz
) where * denotes complex
1
k
conjugation. The notation degiA will be taken to mean the partial degree of the
polynomial
A in the variable zi. Occasionally we shall also use the notation A
to denote AzThe
with n-(nl,n2,...nk), where ni-degiA.
notation
< 1 denotes Izil
lzl
Similarly for B and C etc.
< 1 for i-1 to k. Similar notations with <
replaced by <, >, >, - etc. are also used.
Definitions
and
properties
polynomials
and
multidimensional
have
arisen
included
the
in
in
studies
Appendix
discussions
that
on
of
various
classes
rational
passive
of
multidimensional
Schur
discrete positive functions which
discrete
multidimensional
systems are
A. These properties are crucial in the understanding of
will follow. Proofs of these properties can be found in
[9].
The
transfer
two-port
can
function
be
matrix Z (cf. Figure 3) associated with a k-D lossless
represented
[9]
as
in
(2.1). Note that (2.1) is slightly
different although an equivalent version of the corresponding representation in
[9].
Consequently,
the
chain matrix e and the hybrid matrix r (cf. Figures 1
and 2 respectively) can also be represented as in (2.2) and (2.3) respectively.
Also,
that
note that a given Z can be uniquely represented as in (2.1) by requiring
A(0)-1.
The
representations
(2.1),
(2.2) and (2.3) can be regarded as
canonic in this sense.
Cz-
Z =(l/A)
:
i
yB
A
(1/C)
I
-ACz-
r-(l/B)
Bi
yAz2.
-yA,
(2.1, 2.2, 2.3)
6
where (i) A is a scattering Schur polynomial (cf. Definition A.1.3)
(2.4a)
(ii) y is a unimodular constant, i.e., JyI - 1
(2.4b)
(iii) AA = BB + CC
(2.4c)
(iv) degi B < ni, deg i C < ni for all i=1 to k
(2.4d)
Note that as a consequence of (2.4c) and (2.4d) we also have:
(v) deg i A < ni for all i=1 to k.
For
(2.4e)
the purpose of a unified presentation of the discussions that will follow,
e, r
matrices
*
matrix
I associated with a lossless two-port will be viewed as a
and
associated
with
the
lossless two-port as expressed in (2.5) while
properties (P1) through (P4) hold true.
X(/W)
(2.5)
't ' (1/W)
(2.5)
Y
PpaXzl
Property 1 (P1): X, Y and W are polynomials; a
Property 2 (P2): XX - pYY -
=
constant,
a-1,
p-+l.
WW.
Property 3 (P3): degi W < ni, deg i X < ni and deg
Y i< ni for all i-1 to k.
Property 4 (P4): X is scattering Schur when p - 1,
whereas W is scattering Schur when p - -1.
A
(2
X
lossless
2)
Since
matrix
it
two-port
diag(1,
can
-p)
be
matrix if the above properties (P1) through (P4) hold true.
shown
i.[diag(l,
-
that
I
satisfy diag(l, -p) -= b.diag(l, -p)].
-p)].*
and
> 0 (i.e.,non-negative definite) in lzl<1,
denotes the Hermitian transpose, the one-dimensional counterpart of f
where
*
thus,
falls
into
[12],
[26].
Since
discussed
such as the one in (2.5) will be said to be a generalized
in
the
multidimensional
the
class
of sigma-lossless transfer functions studied in
factorability
present
of
paper,
generalization
of
transfer function elucidated in [12],
the
our
results
the
[26].
7
t-matrices
can
factorability
form
also
the
be
major topic
viewed
as
a
of 1-D sigma-lossless
We note the following identification of the parameters of the matrix 4 in terms
of
the
of the chain matrix 0, the hybrid matrix r or the transfer
parameters
function matrix E.
(i)
If f = 0 then W = C, X = A, Y = B, a = y, p = 1
(ii)
If f = r then W = B, X = A, Y = C, a = -y,
4 = E then W = A, X = B, Y = C, a = y, p
(iii) If
of
advantage
An
chain
product
or
matrix,
of
formulated
p = 1
two
in
the
unified
or the transfer function matrix into a
matrix
of
matrices
fashion
as
a
identical kind can be conveniently
problem
single
of
factoring
the
two-port matrix $ as 4 = I'f", where 4' and 4" are valid
lossless
generalized
hybrid
non-trivial
a
formulation is that the problem of factoring the
above
the
= -1
generalized lossless two-port matrices represented as in (2.6a) and (2.6b) with
conditions analogous to (P1) through (P4) satisfied for 4' as well as 4".
a'Y'Z n-
X'
4' = (1/W')
_'
n
~
~
n'
,
pa'Xz-
n
"Y"z
X"
;
"
(1,W)
pa"n"
t
(2.6a, b)
,
p"X"zZn
Consider first two generalized lossless two-port matrices 4', V" expressible
respectively in terms of X', Y', W', a' and X", Y", W" , a" which satisfy
properties
analogous to those satisfied by X, Y, W, a in 4 as in (P1) to (P4).
Specifically, one obtains:
(i)
X', Y', W' and
X", Y", W" are polynomials;
(2.7)
a' and a" are unimodular constants.
(ii)
X'X' - pY'Y' = W'W', X"X" - pY"Y" =
(iii)
deg i W' < nji deg i w" < n'' for all i=l to k and
deg i Y' < n', deg i X' < n,
deg
(2.8a, b)
W"W"
n,
deg
<i
(2.9a)
X"
(2.9b)
for all i=l to k
(iv)
n"
X' and X" are scattering Schur when p=1,
whereas W' and W" are scattering Schur when p=-l.
Then the following fact holds true.
8
(2.10)
Fact 2.1: If
i'and V" are generalized lossless two-port matrices then 4
=
t"
is also a generalized lossless two-port matrix.
Proof: Given 4' and 4" define X, Y, W and a and n=(nln
...nk) as follows.
2
W = W'W"
(2.11)
n
X'
X = X'X" + a'Y'Y"z- , Y
Y'X" +
5
pa'X'Y"zn
'
(2.12a, b)
a = pa a"
(2.13)
(2.14)
n. = n! + n' for each i=l to k
1i
1
1
It then follows from
in
of X, Y, W and a as in (2.5). Clearly, (2.12), (2.7) and (2.9b) show
terms
4
that
satisfy
(2.12),
(2.8),
(P1)
shows
that
(P3)
Schur
scattering
and
(2.11)
restriction
degree
=-'=" in a straightforward manner that 4 can be expressed
straightforward
algebraic manipulations involving
and
p = +1 show that 4 satisfy (P2). Considering the
imposed
by (2.9) and (2.14) on (2.11) and (2.12) likewise
is
satisfied.
polynomials
Note
that
when
p = -1 i.e., W' and W" are
then W = W'W" is clearly scattering Schur (cf.
Property A.2.1), and (P4) is thus obviously satisfied. This completes the proof
that
when
a
indeed
i.e.,
p--l,
4=4-'""
generalized
X'
as
expressed
in (2.5) via (2.11) through (2.14) is
lossless two-port matrix. On the other hand, when p = 1
and X" are scattering Schur, X as in (2.12a) need not necessarily be
scattering
Schur,
but can now be shown to be immittance Schur (cf. Definition
A.1.5).
to prove that 4 is a generalized lossless two-port matrix we consider
However,
the rational function F = X/(X'X"). In view of (2.12a) we also have:
(2.15)
F = X/(X'X") = 1 + a'(Y'/X')(Y"/X")zIt follows from (2.8) with p=l that on lzj = 1 we have IY'/X'
•
I
< 1, whenever X'
0 and IY"/X"l < 1, whenever x" • 0. Thus, Re F > 0 for Jzl = 1, whenever X's
0 and
x"
Properties
discrete
•
0. Next,
A.2.1
since
X'
and
X"
are scattering Schur, by invoking
and A.3.1, it follows that Re F > 0 for jzl < 1 i.e., F is a
positive function (cf. Definition A.1.6). Consequently, the numerator
polynomial of F, in irreducible rational form, is a immittance Schur polynomial
9
A.3.2). Note that any possible factor common to X'X" and X must
Property
(cf.
be scattering Schur, because X' and X" are so. Thus, X is immittance Schur (cf.
Definition
A.1.5)
scattering
Schur
therefore,
can,
factor
and
X1
a
be
expressed
as
the
product of a
reactance schur factor D (cf. Definition
D1 be any irreducible (thus, reactance schur (cf. Property A.2.2)
Let
A.1.4).
and
of D and note that there exists a sequentially almost complete set 9 of
factor
such that D1 = X
=
Y
that
W
=
A.1.7 and A.1.8) of unimodular complex numbers
Definitions
(cf.
(k-1)
order
0 for any z0 c Q . Consequently, in view of (P2) we conclude
2. Since D1, X, Y, W have a sequentially almost
0 for all z r
=
(and thus sequentially infinite (cf. Definition A.1.9)) set of common
complete
zeros of order (k-l) and D1 is assumed irreducible, D1 must be a factor of X, Y
and
Property A.4.2). Since D1 is any irreducible factor of D, we then
(cf.
W
that X
have
X 1 D, Y - Y 1D, W
=
=
W 1 D, where X1 , Y 1 , W1 are polynomials. Since D
is reactance Schur D - aDD for some unimodular constant aD. Clearly, then Xzn
(DaD)(Xlz-) and
Yz-
after
Thus,
),
(DaD) (Y
=
where m=(mlm
2 ,...mk)
with mi-ni - degiD for all i=1 to k.
cancelling the common factor D from the numerator and denominator
of each entry of (2.5), 4 can be written as in (2.16), where a1 = aa.D
(2.16)
4 = (l/wl)
I
X 1 is scattering Schur, Property (P4) is satisfied by the representation
Since
for 4. It can be further shown via trivial algebraic manipulations that
(2.16)
X1 ,
PlX_1 Z
Y1 ,
W 1,
al
in
(2.16)
satisfy properties corresponding to (P1) to (P3)
because X, Y, W, a has been shown to satisfy the same properties.
Note
that in the case p=l1 if D is a nonconstant polynomial involving, say, zi ,
then
the
two-port associated with 4 is degenerate in the sense that in (2.16)
mi < ni=n! + n". The main problem addressed in the present paper, however, is
1'
1
1i
the converse problem of finding a non-degenerate factorization 4 = 4'0" of a
prescribed
generalized
lossless two-port matrix 4 into two factors of its own
kind. More specifically, we have the following problem.
MAIN
PROBLEM:
Given a generalized lossless two-port matrix 4 as in (2.5), two
10
constants
a"
c',
such that
Ia1' = lI"l
= 1 and a
pa'a", and the polynomial
=
factorization W = W'W" along with two k-tuples of nonnegative intergers n'=(ni,
n,....nl) and n"=(n 1, ni,...n") such that deg i w' < nj deg i W" < n" and n. =
-
ni
1
+
1
find
k1 -
1'2
k
-
1
1
for all i=1 to k, we seek a factorization · = I'Q", or equivalently,
n"
polynomials x', Y', X" and Y" such that (2.12) along with (2.8) and (2.9)
hold. Furthermore, if p=1 (or p=-l) then we require X' and X" (or W' and W") to
be scattering Schur.
It proves to be convenient to introduce the following two definitions:
Definition 2.1: The pair of polynomial two-tuples {X', Y'} and {X", Y"} is said
to
be
a
solution
to
the
algebraic equation if (2.12) along with (2.8) and
(2.9b) are satisfied.
Note
that
the
restrictions
that
the
polynomials X' and X" or W' and W" be
scattering Schur polynomials are not imposed at all in the above definition.
Definition
2.2:
A
polynomial
triplet
{X',
Y',
Y"} is said to satisfy the
fundamental equation if (2.17) along with (2.18) holds true.
YX' - XY' = pc'Y"W'W'z-
(2.17)
degi X' < n', degi Y' < n', degi Y" < n' for all i=l to k
Note
that
product
(2.17)
is
(2.18a,b,c)
obtained by adding the product of Y' and (2.12a) to the
of (2.12b) and (-x') and subsequently by using (2.8a). Obviously, then
any solution of the algebraic equation also satisfies the fundamental equation.
However, the converse statement is false, consider e.g., X' = 0, Y' - 0, Y"
0. Note further that the algebraic equations (2.12) along with (2.8) and (2.9b)
constitute
a
highly
constrained
nonlinear problem. It is shown in Section 4
that due to the inherent structures underlying the problem under consideration,
solutions
of
to
solutions
linear.
Thus,
this nonlinear equations can be obtained from a certain subclass
to
the
fundamental
solutions
to
the
equation,
which,
algebraic
equation
in contrast, is clearly
can
be
conveniently
characterized in terms of the solutions of the fundamental equation.
11
3.
Clearly,
any
solution
to
solution
the problem of factorization of $ = *'$" is also a
to
the algebraic equation. The converse statement is obvious if f is
that
such
TE ALBRAC ETICN:
l
SOLTICN
p=-l
i.e.,
W
is
scattering
when
p=l
i.e.,
nontrivial
but
can
be proved as follows in Theorem 3.1. Thus, the problem of
4
x
associated
with 4 is scattering Schur is
statement
factoring
when
Schur. The validity of the converse
reduces to that of solving a purely algebraic problem namely that
of finding a solution to the algebaic equations.
Theorem
Let
3.1:
the
solution
to
factoring 4 =
two tuples {X', Y'} and {X", Y"}
algebraic
the
equations
is
solution to the problem of
a
'd".
The case when p = -1 trivially follows from scattering Schur property
Proof:
=
W
polynomial
(correspondingly, W' and W") are scattering Schur. Thus,
polynomials X' and X"
of
of
a solution to the algebraic equation. If p = 1 (or p = -1) then the
constitute
any
pair
When p = 1 i.e., X is scattering Schur, consider the rational
W'W".
function defined as:
(3.1)
F = (X'X")/X
by
Furthermore,
and
(2.12b)
X"
=
adding
(-pY')
-
(XX'
product
the
of
(2.12a) and
subsequently
and
Substituting
pYY')/(W'W').
by
the
using
last
(X') to the product of
(2.8a),
one
expression
obtains
in
(3.1)
straightforward manipulation yields the following:
F = (X'X'/W'W')[1 - (pYY'/XX')]
(3.2)
It follows respectively that from (P2) and (2.8a) that on Izj=1 we have IY/XI <
1,
whenever
yields
X
that
•
ReF
0 and IY'/X'I
>
0
Izl
for
< 1, whenever X' # 0. An examination of (3.2)
=
1, wherever F is well defined. Thus, from
Property A.3.1 it follows that F is a discrete positive function. Consequently,
the
numerator
polynomial
(cf.
polynomial
of
F,
in irreducible form, is an immittance Schur
Property A.3.2). Note that any possible factor common to X'X"
and X must be scattering Schur, because X is so (cf. Property A.2.1), and thus,
12
X'X"
is
widest
have
(2.12a)
Schur
(cf.
Definition
A.1.1)
(more
specifically,
Schur). Next, if for some F0 on the distinguished boundary jzJ - 1,
immittance
we
sense
0, then from (2.8b) it follows that Y"= 0, which in turn due to
X"=
imply
that
X
= 0. Consequently, if X"= 0 for all -dO c 9, where 2 is
sequentially infinite set [6] of order (k-1) of unimodular complex numbers then
0o
c Q. However, this is impossible if X is scattering Schur (cf.
X = 0 for all
Property A.2.3). Therefore, X" cannot have a sequentially infinite set of zeros
of
order (k-1) on the distinguished boundary. The scattering Schur property of
X"
is
established in view of Property A.2.3. Similar arguments hold for
thus
X'.
We
shall need the following fact as a preparation to the factorization problem
addressed in the present paper.
Proposition
as
4 =
matrices
3.2:
14o4
such
Any generalized lossless two-port matrix t can be decomposed
2' where
that
t1 #I o'
1', 12
are
2
are valid generalized lossless two-port
diagonal
and the polynomials XXz n and WWzn
associated with b are coprime.
The proof of Proposition 3.2 is shown in Appendix B.
13
4. PRDPERTIES OF -TE FU NDMENTIL EUATIOCN.
In
this
to
the
section certain properties of the fundamental equation (2.17) crucial
development
generalized
are
lossless
studied
two-port
under
the
assumption
that the prescribed
matrix t be such that the polynomials XXz!n and
WWzn are relatively prime. As shown in Proposition 3.2 no loss of generality is
incurred due to this assumption.
Lemma
4.1:
If
the
polynomial
triplet
{X',
Y',
Y")
is a solution to the
fundamental
equation then there exist a polynomial X" given by (4.1) such that
n'
~
X'n'
polynomial triplet {pY'z
X'z-,
z
pa, X"
is also a solution to the
the
fundamental equation.
X" = P/(Xzn) = -Q/(W'W'zn )
(4.1a)
where
P = (X'W"W"z n +
_
n )
a'YWz
' Yz-Y),
'
n'
Q = Y(pY'z- ) - X(X'z- )
(4.lb, c)
Proof: One obtains (4.2) by adding the product of tilde of (2.17) and pY to the
product
of
(P2)
and
X' and subsequently by using (2.11), (2.14) and trivial
manipulations.
(Xz-)Q = -(W'W'z- )P
Due
to
the
deg i
W"<
that
X"
n",
bounds
on the degrees of X', Y', Y" imposed by (2.18) and
it follows from (4.lb,c) that P and Q are polynomials. The fact
in (4.1a) is a polynomial then follows from (4.2) in view of relative
Xzn
with
n
(-pa' X")W'W'Z
I, the
primeness
pa'
upper
(4.2)
of
W'W'zn.
triplet
Since
{pY'z- ,
Q
=
(pY'z
)
X'z- ,
Y(pY'z-' )
-a'
X"}
X(Xnzn' )
x(x'z- ) =
satisfies the
-
fundamental equation.
The fact that degi(pY'z- ) and degi(X'z- ), for each i=l to k, is upper bounded
by
n i is obvious. In order to prove that degiX" < n' we first note that it
I
1
- 1
follows from (4.lb, c) and the upper bounds on the degrees of X, Y, X', Y', X"
and Y" that for all i=l to k we have:
degiP
<
ni+n', degiQ
14
<
n.+ni
(4.3a, b)
It is then necessary to distinguish between the following two cases.
(i) If
It
i.e., X is scattering Schur then degi(Xz n) = ni for all i=1 to k.
follows from the first equality in (4.1a) and (4.3a) that degiX" < n'
p=l
then
for all i=1 to k.
(ii) If p=-l i.e., W, and thus W', is scattering Schur we consider two sets of
indices I1, I2 such that i E I1 if degiW' = nj, whereas i ¢ 12 if degiW' < n!.
If
i
¢ I1 then due to scattering Schur property of W' we have degi(W'W'z- ) =
2ni . The desired result then follows from (4.3b) and second equality in (4.1a).
On the other hand, if i ¢ 12 then from (2.9a), (2.11) and (2.14) it follows
that Wzn must have a factor z i, and thus, X does not have a factor zi because
n n
n
Wz- and X are assumed to be coprime. Consequently, degiXz- =ni. The result then
follows from (4.3a) and first equality in (4.1a).
Lemma
4.2:
satisfying
If
{Xi, Yi, Yj}
and
{X2,
Y', Y2} are two polynomial triplets
the fundamental equation then the identity (4.4) holds and is equal
to a constant.
Proof:
One
N = p(YX
- XiY2)/X = (XiY2 - X)/
obtains
an
(4.4)
equivalent form of (4.4) by adding the product of the
fundamental equation for {Xi, Yj, Y31} and Xi to the product of the fundamental
equation for {X ,Y ,Y,} and (-Xi). Since X is assumed coprime with W'W'zn ' and
(XiYi
Thus,
N
XiYi) is a polynomial, it follows that X must divide (Y'Xi - XiY,).
= Pa'(YlX2-XiY2)/X in (4.4) is a polynomial. To prove that N is a
constant, note that the following inequalities hold true for all i=l to k.
degi(YX,
-
XiY,) < ni, degi(XiYI - XYi) < 2ni
(4.5a, b)
Consider two sets of indices I1, 12 such that i g I1 if degiX = ni, whereas i c
12
if
degiX
follows
that
< n i.
N
does
If i ¢ I1 then from (4.5a) and first equality (4.4) it
not involve zi. If i C I2 then Xzn has the a factor zi.
Consequently, due to the assumed relative primeness of XXzn and WWzn it follows
from
(2.9a),
(2.11)
and (2.14) that neither W' nor W'z15
may have the factor
zi,
which
Therefore,
in
due
turn
respectively
imply
that
degi (W'z- )=nl
and degiW'=n
. .
to (4.5b) and the second equality (4.3a) N may not involve z i.
Thus, N = constant..
Lemma
4.3:
If {X', Y', Y"} is a polynomial triplet satisfying the fundamental
equation then the expression given in (4.6) is equal to a real constant.
,
K = (X'X' - pY'Y')/(W'W') = (a'Y'z- Y
+ X'X")/X
(4.6)
Proof: Consider in view of Lemma 4.1 two solutions Xi = Xt, Y1 = Y', Y" = y
and
X2 = pY'z- , Y = X'z n ' , Y- = -pa'
immediately
follows
from
Lemma
X" to the fundamental equation. It then
4.2 that K in (4.6) is a constant. Since for
2
2
2
Izl=l we have X'X'=X' 2, Y'Y'=JY' 2 and W'W'=IW' 2, K is a real constant.
Lemma
4.4:
fundamental
{=X'
+
If
the
equation
polynomial
then
fpY'Zn ,z 'Y
+
triplet
Follows
clearly
Y',
Y"}
is a solution to the
there exists an X" as given by Lemma 4.1 such that
, ' ,
azY"
fundamental equation, where a and 0
Proof:
{X',
from
Lemma
-
Spa' X"}
is also a solution to the
are arbitrary complex numbers.
4.1
equation is linear.
16
and
the
fact that the fundamental
5.
FACTORIZATION OF 4:
A
solution {X', , Y
if X'X' f
Y"} to the fundamental equation will be called nonsingular
pY'Y'.
Theorem 5.1: The problem of factorization of 4 admits a solution if and only if
there exists a nonsingular solution {X', Y', Y") to the fundamental equation.
Proof:
1
Y }
Necessity
is
a
4.4, X
obviously follows from (2.8a) and that W=W'W"0O.
nonsingular
=
Xi +
If {Xi, Yi,
solution to the fundamental equation then due to Lemma
pYiz- ,
Y'
=
+
Yi
xi
' z-
pPaji*X 1' is a solution to
Y" = aYl -
the fundamental equation. Straightforward manipulation then yields:
(X'X' - pY'Y')/(W'W') = (1al[
-2
-
p]I6
2
)K1
(5.1)
1 = (X{X
i - PYiYi)/(W'W')
Since
(5.2)
due to Lemma 4.3 and the nonsingularity of {Xi, Yj, Y1}, K1 is a nonzero
if
constant,
(X'X'-pY'Y')
=
0
and
a
W'W'.
chosen
are
Furthermore,
satisfy (lal
to
2
- pJl
2
)
=
-1
K, we have
there exists X" such that (pY'z-
,
X 'zn
-paj*X"), by the virtue of Lemma 4.1, satisfies the fundamental equation.
We
next
show
algebraic
(2.17)
that X', X", Y' and Y" so obtained constitute a solution to the
Equation
equation.
and
X'z
n'
(2.12b)
is
obtained
by adding the product of
the product of second equality of (4.1a) via (4.1c) and
to
(-Y') and subsequently by using (2.8a). Likewise, (2.12a) is obtained by adding
the
product
via
(4.1c)
(2.8b)
of
and
(2.17) and pY'z-n
and
(-X')
to the product of second equality of (4.1a)
subsequently
by using (2.8a). Finally, we obtain
by substitituting (2.12a) and (2.12b) in (P2) and then using (2.11) and
(2.8a).
Thus,
algebraic
the
equation
pair
and
of
two-tuples
via
Theorem
{X',
3.1
Y'} and {X", Y"} satisfies the
is a solution to the problems of
factorization of 4.
Two
polynomial
fundamental
triplets
equation
will
{Xi,
be
Yi,
said
Y'}
and {Xi, Y1, Y2} each satisfying the
to be linearly dependent if there exists
17
X2 E aYi + SYu- aY
constants a and S not simultaneously zero such that aXi +
+ oY~-
0.
Theorem
5.2:
two linearly independent polynomial triplets {Xi, YI, Y"}, i=1,2
exists
there
problem of factorization 4 admits a solution if and only if
The
each of which satisfies the fundamental equation.
to
Let the polynomials X', Y', X'" and Y" constitute a solution
Necessity:
Proof:
the
problem.
factorization
Clearly,
{X',
Y"} is a solution to the
Y',
~
n'
X' n'
*X"
z
-pc' X"} is
X',
equation. Due to Lemma 4.1, therefore, {pY'za solution to the fundamental equation. We claim that these two solutions
fundamental
also
S2
linearly independent, because otherwise there would exist constants ,S1
- O.
02X'zThus,
n2pY'zn
=_ 1Y' +
simultaneously zero, such that 61X' +
are
not
0, which in view of (2.8a), would imply that W' - 0, i.e., due
-
(X'X'-pY'Y')
to (2.11) that W - 0, which is impossible.
If
Sufficiency:
follows
sufficiency
triplet
Y',
{X',
qY', where
p
of the solutions {Xi, Y, Y1}, i=1,2 is nonsingular then
one
and
from Theorem 5.1. If both solutions are singular then the
Y"} obtained as: X' = pXi + qX2, Y' = pYi + qY, Y" = pYj +
q
are
complex numbers, satisfy the fundamental equation.
Straightforward algebraic manipulation via the singularity of the triplets {Xi,
Y!, Y" , i=1,2 then yields:
(X'X'-pY'Y')/(W'W') = pq L + p qL, L = (XiX~ - pYiYj)/(W'W')
(5.3a,b)
X) is also a solution, by invoking
due to Lemma 4.1 {pYz- ,-pa
X n' , -pa XtI it follows
4.2 on the triplets {Xi, Yi, Y1} and
n npY'
I{pY'z
, nXz *
L in (5.3b) is a constant i.e., L=L=L . Thus, the right hand side of
Since
Lemma
that
is 2Re(pq*L), which, if LAO, can be made equal to 1 by proper choice of
p and q. With p, q so chosen {X', Y', Y"} would thus be a nonsingular solution
to the fundamental equation, and by invoking Theorem 5.1, it then follows that
(5.3a)
a solution to the problem of factorization of 4 exists.
The
proof
of the present theorem is completed by showing that LO0. This proof
is shown in Appendix B.
18
The
above
can,
result
strong
corresponding
in fact, be further sharpened as follows. Note that a
result
the continuous case, although true, was not
for
given in [5].
Theorem 5.3: The problem of factorization of t admits a solution if and only if
there exists exactly two linearly independent polynomial triplets {X;, YI, Yi},
i=1,2 each of which satisfy the fundamental equation.
The proof of Theorem 5.3 is shown in Appendix B.
fundamental
The
equation
(2.17),
when
considered
as
a
of
set
linear
simultaneous equations involving the coefficients of the polynomials X', Y', Y"
along with the upper bounds on their degrees, turns out to be overdetermined in
(except when
general
X',
polynomials
total
the
k=l).
we
explicitly,
More
note
that
the
unknown
Y' and Y" contain a total of u unknown coefficients, whereas
of
number
linear simultaneous equations can easily be found to be
equal to e, u and e being as given in (5.4a,b) below.
k
k
(5.4a,b)
k>l we have e>u in a generic situation a solution to the problem of
for
Since
k
n (n'.'+1) + 2 n (n!+l), e = n (n"' + 2n! + 1)
i=1 1
i=1
i
1
1
u =
factoring ~ into two matrices of identical kind may not exist.
free
Delay
In
loop:
order
the
for
digital
network
synthesized
via the
it may not contain delay free loops
arising from interconnection of two sections. It is known [1] that this problem
factorization
can
always
incorporating
from
of
be
4 to
be
'computable'
circumvented,
digital
factorizations
at
equivalents
least
in
the
one-dimensional
case,
by
of unit elements. The structures resulting
e = e'e", r = r'r", ? = s'i"
are shown in Figures 1, 2 and
respectively. An examination of directions of signal flows in Figure 3 shows
that the topological structure arising from the factorization of . as Z = E'Z"
3
cannot
On
contain any delay free loop at the junction of the two-ports E' and E".
the otherhand, Figures 1 and 2 clearly show that the topological structures
arising
from
the factorization of ® as 0 = O'e" and r as r =
r'T"
may contain
delay free loops unless special attention is paid to this issue (note that both
of
these
cases
correspond
to
the
choice
19
p=1).
However,
as shown in the
following, delay free loops at the junction of the two-ports associated with 4'
and
4"
may
(generalized)
with
4'
always
lossless
(the
avoided
by
two-port matrix
extracting
an
appropriate
constant
from 4" and subsequently combining it
obvious alternative of extracting a constant matrix from 4' and
combining it with
Fact
be
"'also apply).
5.4: Any generalized lossless two-port matrix 4" with p=l can be factored
the product of two matrices O' and "' of the same type i.e.,
fr"=V"t"
into
where
4" = constant as defined in (5.5), and 9" is such that the Y-polynomial
f
r
associated with it assumes a zero value for z-0O.
I
= 1//(1 - IKI2 )
where
4"'.
K = Y"(0)/X"(O),
(5.5)
X" and Y" are corresponding polynomials associated with
The proof of Fact 5.4 is shown in Appendix B.
Next, if 4 is factorable as 4 = 4'~" then due to Fact 5.4 we may also write 4 =
r"', where ic
cr
crf- V"', due to Fact 2.1, is a generalized lossless two-port
matrix. Further, it is trivially verified that if 4' and 4" satisfies the
requirements
satisfies
imposed
the
in
same
the
'Main problem' of Section 2 then 4c
" and
requirements.
also
Thus, 4 - %'.V' is a valid solution to the
factorization problem.
If
4"
is
viewed
as
a chain matrix 9", then (assuming that the operation of
shifting the factor o' from 4" into 4' has been carried out) we have that Y"(O)
-=
21(O) = 0 i.e., the corresponding transfer function matrix would in view of
(2.1)
satisfy EZ"i(O)
to
'b'
r"
then
thus
either
Note
= 0. Consequently, there would be no direct path from 'a'
via 9" = O" in Figure 2. Similarly, if 4" is viewed as a hybrid matrix
the
corresponding trasnfer function matrix would satisfy E21 (O) = 0,
guaranteeing
case,
further
no
no
direct path from 'a' to 'b' via 4" =
r"in
Figure 3. In
delay free loop exists at the junction of the two two-ports.
that 9" as in (5.5) correspond to the chain matrix or the hybrid
20
matrix of the well known Gray-Markel section.
when
Furthermore,
from
follows
guarantees
p=l,
X"
is
scattering
Schur and thus x"(O) • 0. It then
(2.12b) that if Y(0) = 0 and Y"(0) = 0 then Y'(0) = 0. This fact
prescribed generalized lossless two-port matrix 4 can be
the
that
successively factored into product of generalized lossless two-port matrices of
progressively lower complexity in such a way that the Y-polynomial associated
each of the factors of 4 except possibly the one at the extreme left when
with
4
is
considerations
at
as
interpreted
each
fragmented
matrix
e
is equal
to
zero
for z_=0. Similar
when 4 is a hybrid matrix r. Absence of delay free loops
apply
junction
chain
of
the
constituent
two-ports,
when
a given two-port is
into an interconnection of more elementary two-ports via the method
of factoring 4, is thus guaranteed.
The algorithm for factoring 4 can then be summarized as follows:
Step
If the prescribed 4 be such that associated XXzn and WWzn are coprime
1:
then proceed to Step 2. Otherwise, factor 4=4-1o042 as described in the proof of
Proposition 3.2. Replace 4 by o'.
Step 2: Find, if possible, two linearly independent solution {X,
2
to
the
YI, YV}, i=1,
fundamental equation (2.17). In the 1-D case such a solution always
exist. Factorization of 4 is impossible if such solutions are nonexistent.
Step 3: If at least one of the two linearly independent solution is nonsingular
i.e., X'X' X pY'Y' for any i then proceed to Step 4. Otherwise, proceed to Step
Step 5.
Step
4: Assuming that {Xi, Yj, Y"} is a nonsingular solution, find XI' from the
second
equality
respectively.
Finally,
form
(4.1a)
and
Also, find K1
X'=
cXx +
(4.1c) where Y' and X' are replaced by YI and X I
1/K1
from (5.2) and a, 0 such that 1a2 - P12
$pYizn 'Xi, Y
=
Yi
+
Y"= aY11 -
pa' X1 and
proceed to Step 6.
Step 5: Find the constant L as in (5.3b) and p, q such that 2Re(pq L) = 1. Form
xI = pXi + qgx,
Y' = PYi + qY1, Y" = PY" ++ qY .
21
Step
6:
Find X" from (2.12a). Thus, {X', Y'j and {X", Y"} i.e., I' and I" are
obtained.
f as
Step 7: If p=l then from X", Y" associated with 4", find K=Y"(O)/X"(O),
in (5.5), V' = (f)-14,, and let 4"' = "'i. Thus, 4 = "r~"without delay free
r
'
c
r
loop at the junction.
Remark:
Since
in
K1
(5.2)
is a real constant it is possible to choose real
values of a and 0 such that the right hand side of (5.1) is equal to 1. If 4 is
real
(i.e.,
solution
X, Y, W
have real coefficients and oa=+1), then Xi, Yi, Yj as a
to the fundamental equation, and thus, X', Y', Y" must also have real
coefficients
if
a
and
S are chosen to be real. Since this implies that the
coefficients of X" are real, the factors 4' and 4" would then also be real.
22
6. CNE-DIMENSIONAL SYNESIS AS A SPECIAL CASE:
In
the
one-dimensional
reveals
that
u
coefficients
than
simultaneous
equation.
-
e
case i.e., if k = 1, a closer examination of (5.4a,b)
=
the
2,
number
equations
which
and,
of
therefore,
linear
determine
there
equations
the
are
in
solution
the
to
two more unknown
set
the
of linear
fundamental
Thus, there are (at least) two linearly independent solutions of the
fundamental
equation, and in view of Theorem 5.2, the problem of factorization
f always admits of a solution. Consequently, structurally passive synthesis
of
for
4 is achieved by performing a sequence of further factorizations of 9' and
4"
into the same kind of matrices of progressively lower complexity i.e., ni <
n1 ,
<
no
(since ni + nl= n1)e until a stage is reached when each of the
n1
resulting
matrices
corresponds
to
decomposition
However,
and
W
that
each
of
any further. This latter situation
the
two-ports
resulting
from the
n1 = 1, i.e., deg1 C < 1 and deg1 B < 1 and deg1 A < 1.
complex roots then it is necessary to allow two-port sections with
deglW=2
order
to
if
avoid
require
that
B
when
above
case
the
factorized
if the prescribed 4 is such that X, Y, W have real coefficients, o=+1
has
O
be
satisfy
nl=2,
=
cannot
realization
delay
involving only real multipliers are sought. In
free loops at the junction of the two-ports we further
two-port
sections
satisfy
= OC and C = O when
types
t
Y(O)= 0 when p = 1 or equivalently,
= zr 1 for0.
=
Two-port sections of the
will be called elementary sections and can in turn be realized in
structures
possibly
techniques
as
other
than those considered here by exploiting synthesis
discussed, for example, in [4].
Thus, the following elementary
sections are obtained.
An
arbitrary
by
using
whereas
lossless chain matrix 0 with nl = 1, B(O) = 0 can be synthesized
the
an
procedure
abitrary
described
lossless
chain
in
[4] in a structure given in Figure 4,
matrix
e with n1 = 2, B(O) = 0 can, by
following the same procedure, be synthesized in the structure of Figure 5.
A
lossless hybrid matrix with n1 =1 and nl = 2 (assuming
respectively
Figures
4
be
realized
Y(O) = C(O) = 0) can
by the same elementary sections as described in the
and 5, but after a clockwise rotation of the corresponding diagrams
by an angle of 90 degrees.
23
On
the
hand, an arbitrary lossless transfer function matrix Z (in this
we may not assume Y(O) to be zero) with n1 = 1 or n 1 = 2 can be realized
case,
by
other
using
section
the
has
elementary
been
sections
extracted
from
described above only after a Gray -Markel
the
corresponding chain matrix (or hybrid
matrix) so as to effect a zero value for B(O).
Thus,
an
arbitrary
interconnection
lossless
two-port
can
indeed
be
synthesized
as
an
of Gary-Markel sections and the sections depicted in Figures 4
and 5 only. Note that sections of Figure 4 and 5 were introduced by Dewilde and
Deprettere
in
the
context
of
cascade
synthesis [19], and can be viewed as
scaled versions of interconnections of wave digital filter adapters [18].
24
7. DISC[USSIONS AND ILLUSTRATIVE EXAMPLES:
The
purpose
the
fundamental
solution
Section
of
both
5
this section is many fold. First, we examine the structure of
equation
for
in somewhat more detail to facilitate the method of
the 1-D and the k-D case. Although it has been remarked in
that the S-matrix is generically not factorable in multidimensions,
the possibility of synthesis for special classes of S-matrices may not be ruled
out. Furthermore, in k-D, nonfactorability of any one of the three matrices, r,
r associated with a lossless two-port does not rule out the factorability
0, or
of
other
two
matrices.
This
fact
is next substantiated via examples, thus
justifying the need to study factorization of all three types of matrices (in a
unified manner). Finally, in practice, all multidimensional frequency filtering
problems require some form of symmetry in the k-D frequency response, and it is
known
that
such
symmetries
dictate that the two-port be either symmetric or
(quasi) antimetric in the sense of classical network theory (to be made precise
later
2-D
in this section). This is indeed the case, for example, in the design of
fan
[20]
and k-D circularly symmetric [21] wave digital filters based on
transformations
E-matrix
from
associated
analog
with
prototypes. Therefore,
these
the factorability of the
subclasses of discrete lossless two-ports is
also undertaken in the present section.
A. Computational considerations:
For
the
purpose
of
the
present discussion, the following notations will be
adopted.
X(z)=Z Pi(z')z,
Y(z)=Z Qi(z' )z,
poW'W'z- =Z R i (z')z
X'(z)=E Pi(z')zl
Y'(z)=Z Q!(z')zl,
Y"(z)=Z RP(z')zl
where
z'
R.=R.(z')
1
1
is
the k-tuple of integers (z2 , Z3'Zk)
...
and
1
(7.1 a,b,c)
(7.2 a,b,c)
and Pi=Pi(z'), Qi--Qi(z'),
P!=P!(z'), Q-Qi(z' ), RI=RI(z') are polynomials in z'.
1
11
11
1 --
Then the
fundamental equation (2.17) can be written in the form of (7.3):
V(z')T(z)=0,
TTz=[T
25
T
T2
T=T(z)=[T
T](7.3
a,b)
1 ....
V(zW) = [Q
Q I -P'
0
1
where
the
superscript
(ni+l)x(n1 +ni+l)
polynomial
lower
matrix:
obtained
by
shift
[Pn
shifting
(ni+l)x(n1 +ni+l)
't'
,...,Po,
he
matrix
T2
0,...0] respectively. For a
a
and
matrix
whose
0,...0]
previous
solution
to
I R", ..., R"]
transposition,
first
and
rows
(7.3 c)
0
T1
is
a
row is the (k-1) variable
subsequent
rows
of which are
by one step towards the right. The
the (n"+l)x(nl+ni+l) matrix T 3 are similarly
from the polynomial row vectors: [Qn ...
obtained
to
denotes
matrix
-Pr
R
nJ
Q0
0,...0] and [R2n,,...Ro,
given T(z') a solItion V(z') to (7.3a) corresponds
the fundamental equation if in (7.3c) the following degree
restrictions for all v are satisfied for each i=2 to k.
degiP
As
< nj1
degiQ•< ni
V -
1
1
degiR
< n"'
V
v~~-
(7.4)
1
remarked earlier, if k>2 for a given T(z) a solution V(z) satisfying (7.3a)
and
(7.4)
adopted
H(z')
may
in
not, in general, exist. However, the following approach may be
attempting a viable solution. First, find the Hermite reduced form
of T(z') via the pseudo-division algorithm as described, for example, in
[22], [23] i.e., find a unimodular matrix U(z') such that U(z')T(z')
in
Hermite
(nl+ni+3)
form.
rows,
Since
the
T(z'),
last
K
and
rows
thus,
(K>2)
= H(z') is
H(z') has (nl+ni+l) columns, but
of
H(z')
are
identically
zero.
Consequently, each of the last K rows of U(z' ) belong to the left null space of
T(z' ).
to
a
However,
solution
generically,
(proof
n2=n2
ommitted
a
two
4
of
fail.
fundamental equation, (7.4) must be satisfied, which,
To
elaborate further on this it may be remarked that
for brevity) in the special case of 2-D i.e., when k=2 and if
necessary
is
and
the
may
independent
of
in order for any vector belonging to this space to correspond
and sufficient condition for the existence of two linearly
solutions
to the fundamental equation i.e., that of factorability
that the dimensionality of left null space of T(z')=T(z 2 ) be exactly
the
T(z')=T(z2 )
two
left
Kronecker
indices
[23]
of
the
polynomial
matrix
be each equal to n'=n". We have so far been unable to establish an
analogous characterization of factorability when n2Hn".
In the 1-D case (k=l), however, both V=V(z' ) and T=T(z') in equation (7.3a) are
constant matrices. Furthermore, since it is known that the lower shift matrices
26
T1 ,
T2
and
T3
matrices, the
exploiting
in
it
can
a
Toeplitz as well as resultant-like
be shown by pursuing the proof technique for Theorem 5.3
two
linearly
independent
solutions
to
the fundamental
choice is made, the solution to the fundamental equation becomes
essentially
unique
polynomials
{X',
except for a constant scale factor multiplying each of the
Y',
polynomials
determinantal
this
to
one of the zeros of the polynomial Y"(z 1 ) may be chosen arbitrarily.
such
three
related
linear simultaneous equation (7.3a) can be potentially solved by
obtaining
equation,
Once
closely
recently developed fast algorithms for solving such equations [24].
Furthermore,
that
are
Y") in the solution. It can be shown that each of these
in
this
solution can in turn be expressed via closed form
as
discussed in [25]. From a computational standpoint
formulas
latter method, as opposed to the Toeplitz-like method mentioned above may
not, however, be the most inexpensive when the integers ni, n', and thus, n are
large.
B. Examples on factorability of I, 0 and r:
We
next
+-matrix
illustrate
as
three
by
three examples that in multidimensions by viewing the
different
types
of
matrices associated with a two-port,
namely the transfer function matrix E, the chain matrix e and the hybrid matrix
r, the
attempted
synthesis
than
by
of
a
larger
class
of discrete lossless two-ports can be
considering the factorization of a matrices of only one of
the above kinds.
(I)
Consider a discrete lossless two-port described by A = PQ, B, and C = 2RS,
n = (3,3) and r=cr--l as in (2.1), where
2
2
P=-1 z 2 -2z2 1 z l+Z+4, Q=z1 Z2 -2z z2 +Z2 +4,
33
23
32
22
2
3
B=4z1 Z2 -4l22 -5Z1 Z2 -6Z2
1 2 +7Z1 2-2Z1
2
2
+4Z1 2 -6Z1
2
3
2
-4z2 -z1 -22 1 -5Z 1+4,
222
2
R=zlz2-1, S=z1 Z2 -Z2 +Z1 z 2 +4Z 1 z2 -Z 2 -Zl-z 1 -2
In attempting a nondegenerate factorization of the corresponding 0-matrix (p=l)
into nonconstant 0' and 0" we encounter the following distinct possibilities:
(i) n'=(2,2), n"=(1,1), W'= C'= S, w"= C"= 2R (ii) n'=(1,1), n"=(2,2), W'= C'
27
=
R,
W"-
C"
= 2S. Neither in case (i) nor in case (ii) we have two independent
solutions to the fundamental equation (2.17). Via Theorem 5.3, we thus conclude
that 0 cannot be factored as 0'0". However, the corresponding £ can factored as
£ = Z'," where E' and £" are described as (clearly, p=p'=p"=-1 in this case):
W'= A' = P, X'- B'= 2z 1 z2-2zlz2-2z2 +2, Y'= C'= z2 +2z2 +1, a'= r'-1, n'= (1,2)
=2
Q
2
,
W"= A" = Q, X"= B"= 2z1 z2 -2zlz2 -2zl+2, Y"= C"= z +2z1 +1, a"= y=-l,
Since
the
above
0
can
also
be
viewed
n"= (2,1)
a hybrid matrix r (with slight
as
modification in the sign of y), the example also demonstrates that there exists
discrete
lossless
two-ports for which the associated transfer function matrix
can be factored but the associated hybrid matrix may not be factorable.
(II)
Consider
the
discrete
lossless
two-port
given by A, B = 2PQ, C
=
RS,
n=(2,2), y=o=l, where
2 2 22'
A=3-zl-z2 -z1 z2 , P=1-z 1Z 2 , Q=l+zlz2 , R=Z 1 Z2 +Z2 +Z1 +1, S=ZlZ2-Z 2 -Z1 +1
An
attempt
rise
to
cases:
the
r into nondegenerate nonconstant factors r' and r" gives
following
two
distinct
possibilities with n'=n"=(1,1) in both
(i) W'= B'= 2P, W"= B"= Q (ii) W'= B'= Q, W"= B"= 2P. In neither of the
above
two
independent
of
to factor
the
cases
fundamental
equation
is
found
to
factorizaion.
However,
the
corresponding
0-matrix can be
=0@'0", where 0' and 0" are described with p=p'=p"=l
X'= A'=(Z lZ2 +2Zl
1 +2Z2 +3)//3, Y'= B'=(Zl+Z2 +2)//3, W'= C'= R,
X"=
have two linearly
solutions, thus proving, in view of Theorem 5.3, the impossibility
intended
factored as
the
Al"=(zlz2 -2zl-2z2 +3)//3,
Y"=
B"=(-2z1 Z 2 +Z2 +Z )/13,
W"=
as:
a'=y'=1, n'= (1,1)
C"=
S, r"=y"=l,
n"=(1,1)
Note
of
that
the
Y"(O)= B"(0) = 0; thus, there is no delay free loop at the junction
discrete two-ports. By interchanging the roles of B-polynomial and the
C-polynomial in the above example we can similarly demonstrate the existence of
28
a
discrete lossless two-port for which the r-matrix can be factored as rr'r",,
but the associated e is not factorable as 8=0'0".
(III)
Consider
n=(2,2), y=a=l,
next
a discrete
R=Zl+l,
A
S=2Z 12-2z
Z2
detailed
that
the
2
22
2
2
2
z2-Z2 +3zlz2 ), B=Z lZ2 -3Z1 z2 +z1+Z 1-2zlz
2 -2z
21 z -4,
-2z 2
1 2+2
examination
only
two-port described by A, B, C=RS,
where
2
A---(7+3z -2z2-21
lossless
of
possible
way
the degrees of the polynomials A, B and C reveals
of
factoring
the
transfer
function matrix Z
associated with the two-port is to attempt either (i) n'=(1,0), n"=(1,2), A'=1,
A"=A
or
(ii) n'=(1,2),
fundamental
n"=(1,0),
A'=A,
A"= 1. In both cases, however, the
equation (2.17) fails to yield two linearly independent solutions.
Thus, Z-matrix associated with the discrete two-port under consideration cannot
be factored. On the otherhand, the associated chain matrix e can be factored as
0=e'0", where 0' and 0" are described as:
X'= A'=-(z1+7)//15, Y'= B'=(2z1 -4)//15, W'= C'= R, y'=a'=1, n'= (1,0)
2
2
222
X"= A"=(4z z2 -8zlz2 +4zl-z2 -2z2 +15)//15, Y"= B"=(- Zl 2 +4z2 +2z z +8z
2 2 -z1 )15
W"=C"=S, y"=a"=1, n"= (1,2)
Note
again
that Y"(0)=B"(0)=O ensures that there is no delay free loop at the
junction. By viewing the chain matrix as a hybrid matrix the same example with
minor modifications can be used to show the existence of a discrete lossless
two-port
for which the transfer function matrix Z cannot be factored, although
it is possible to factor the associated hybrid matrix r.
C. Symmetric and (quasi) antimetric two-ports:
A
discrete lossless two-port will be called symmetric or (quasi) antimetric if
(7.5) (or (7.5')) holds true. Note that the former case corresponds to fan type
symmetry
[20], whereas the latter case correponds to circular symmetry [21] in
frequency response.
29
Let
functions
rational
the
nB=-yBz
,
(Co-nCz.
By-yBz-,
C-Cz--,
(7.5) ((7.5'))
)
L1 and L2 be defined as in (7.6) (or (7.6')) for
symmetric (or (quasi) antimetric) filters.
L1 =(B+C)/A
(or (B+jC)/A )
(7.6a) ((7.6'a))
L2 =(B-C)/A
(or (B-jC)/A )
(7.6b) ((7.6'b))
Then it clearly follows from (2.4c) and (7.5) (or (7.5)') that L1L1=L2L2=1, and
IL1 1=1L2 1=1 for Izl=l, wherever L1 or L2 are well defined. Since
L2 in irreducible form have scattering Schur denominators (cf.
consequently,
L1
both
and
Property A.2.1),
it follows that L1 and L2 are both multidimensional discrete
By making use of (7.6) (or (7.6')), Z as in (2.1) can be
in (7.7a) and (7.7b) in the symmetric case and as in (7.7'a) and
functions.
all--pass
as
expressed
(7.7'b) in the (quasi) antimetric case.
(L1+L2)/2,
=22
11
(7.7 a,b)
Z12=721=(L1-L2)/2
E11 =22=(L1+L2)/2,
(7.7' a,b)
(L2-L1)/2
Z21=-12=J
functions L1 and L2 the matrix Z
by using (7.7) (or (7.7')) is a valid transfer function matrix of a
lossless two-port. Thus, any multidimensional discrete lossless
can be equivalently described by means of two multidimensional
Conversly,
obtained
discrete
two-port
for
any
two
discrete
all-pass
all-pass functions L 1, L 2 . We then have the following important result.
Theorem
Let
7.1:
{L1,
L 2 },
{LI,
L}, {L",
L"} be the all-pass functions
(or (quasi) antimetric) discrete lossless two-port
transfer function matrices X, E' and £" respectively. Then Z = 1'," if and only
associated
if L1 = LitL
with
symmetric
and L2 = LTL" hold true.
Expressing £ = 1'," in terms of the corresponding L1 and L2 via (7.7)
(7.7')) and its counterparts for E' and Z", it follows that the
Proof:
(or
factorability condition Z = Z't"is equivalent to L 1 = LiL,
30
L2 = L".
Since
it
can
be
easily
shown
by pursuing methods outlined in [9] that any
rational all-pass function L can, in fact, be expressed as L=(P/P)z- in
irreducible form, where P is a scattering Schur polynomial, Theorem 7.1 conveys
the
important
fact
that
the
factorability of multidimensional symmetric or
(quasi) antimetric discrete lossless two-port transfer function matrices can be
simply expressed in terms of factorability of two scattering Schur polynomials.
31
8. CONCLUSION:
The
work
present
structurally
has
passive
been
motivated
multidimensional
by
the
digital
possibility
filters.
A
of
designing
simple algorithm
involving the examination of solution of a set of linear simultaneous equations
for
the
studying
lossless
synthesizability
two-port
of an arbitrary multidimensional discrete
has been derived via
factorization of the associated chain
matrix e, hybrid matrix r and transfer function Z by introducing
lossless
two-port
multidimensional
the
1-D
not
feasible.
Although
feasible
realizing
for
synthesis
(quasi)
Existence
turn
can
be
considered
as
a
the
Examples
filters
multidimensional
arbitrary
special
passive
(k>l)
discrete lossless
classes
directly in the digital
case
e,r,
synthesis may not be
and Z, the possibility of
of discrete lossless two-ports is by no means
of such subclasses of two-ports such as the symmetric or
antimetric
discrete
lossless
two-ports
have
been
discussed.
of other classes of discrete lossless two-ports admitting synthesis,
in
special
identified.
of
structurally
in
an
for
out.
certain
in
In the special case of one-dimension our algorithm provides new
domain.
albeit
which
literature. It turns out that under a generic situation, synthesis is
of
the
t,
version of the sigma-lossless transfer functions discussed in
methods
ruled
matrix
a generalized
This
classes
topological
is
especially
structures, seems feasible, but remains to be
true
in view of cascade synthesizability of
of two-dimensional continuous time systems arising in studies
lumped-distributed
netwoks
[22].
It
may
be
noted
that
the
cascade
synthesizability
of
lumped-distributed networks can be characterized in terms
of
of
certain
properties
bigradients
(otherwise
matrices
similar
of
polynomial
called
type
has
matrices
resultants).
been
noted
The
in
having
the
occurrence
our
structure
of
of
polynomial
study in the context of
computing a solution to the fundamental equation (cf. equation (7.3)). However,
further
utilizing
investigation
is
needed
to
explore this connection in successfully
the results of lumped-distributed network theory in multidimensional
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A.
Fettweis,
filter
Digital
structures
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to
classical filter
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P.
Dewilde
multiport
and
Deprettere, Orthogonal cascade realizations of real
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digital
Int.
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J.
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Th. & Appl., vol. 8, pp.
245-277, 1980.
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P.
P.
Vaidyanathan
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A
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K. Mitra, Low passband sensitivity digital
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S.
Rao
K.
and
T.
Kailath,
Orthogonal
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S.
Basu
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the factorization of scattering transfer
On
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32, pp. 925-934, Sept. 1985.
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A.
Fettweis
and
S.
Basu,
New
results
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multidimensional
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its
S. Swamy and P. K. Rajan, Symmetry in two-dimensional filters and
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S.
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Part
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New
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II: discrete case, IEEE Trans. on CAS, vol. 34, no.
11, Nov. 1987.
33
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S.
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and
A.
multidimensional
Tan,
On
structurally
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of
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System, pp. 44-48, North-Holland, Aug. 17-18, 1987, Syracuse, New York.
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S.
Basu
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On
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Yves Genin, Paul Van Dooren, Tom Kailath, Jean-Marc Delosme, Martin Morf,
On
E-lossless
transfer functions and related questions, Linear Alg. and
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P. P. Vaidyanathan and S. K. Mitra, A generalized family of multivariable
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Dewilde,
A. Vierra and T. Kailath, On the generalized Szego-Levinson
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Fettweis, Wave Digital Filters: Theory and Practice, Proc. IEEE, Vol.
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34
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Fettweis, Design of recursive quadrant fan filters, AEO, vol. 34, pp.
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35
APPENDIX A
In
this
appendix, we introduce definitions and properties of various kinds of
multidimensional
Schur
polynomials
and
multidimensional
rational
discrete
positive functions associated with passive discrete multidimensional systems.
Proofs
of
these properties and other related definitions can be found in [9],
16].
A.1 DEFINITIONS:
Definition A.1.1: A polynomial A is called widest sense Schur (WSS) if
A#O for Izl<l
Definition
Izt<l
Ijl
[9].
A.1.2:
(i.e.,
A
A
polynomial
is WSS) and A
=
A is called self-reciprocal Schur if A•O for
yA where y is a (necessarily unimodular i.e.,
=1) complex number [9].
Definition
Izl<l,
A.1.3:
A proper polynomial A is called scattering Schur if A•O for
(i.e., A is WSS) and A and A do not have any common (i.e., nonconstant)
proper factor [9].
Definition
Schur
if
A.1.4: A self-reciprocal Schur polynomial A is said to be reactance
the
irreducible
factors of A do not occur with multiplicity larger
than one [9].
Definition
A.1.5:
A
polynomial A is said to be immittance Schur if it is the
product of a scattering Schur polynomial and a reactance Schur polynomial [9].
Definiton
A.1.6:
A
rational
function
F
will be called a discrete positive
function if Re F > 0 for those z0 in Izl<l for which F is holomorphic [9].
Various
other
definitions,
which
are
stated above are available in [9].
36
mathematically equivalent to the ones
Definition
A.1.7: Let Z be a set of k-tuples z -
belong
the same number field K (hereafter always the field of real numbers
or
to
the
almost
field
all
except,
of
complex
values
if
possibly,
variable
may
a
variable
take
Zk), where all z.
numbers). We will say a certain property holds for
finitely
then
(Zl, z2'
is
many
may
of
be equal to any element of the field
them.
The
set
of all values that the
said to be almost complete. The symbol 9 will be
reserved to denote the set Z when the variables are restricted to be unimodular
[6].
Definition
m
A.1.8: We say that Z is a sequentially almost complete set of order
> 1, with m< k, if there exists a permutation il, i2 .... ,ik of the integers
1,2,...,k
exists
any
such
that
all
z
c Z can be generated in the following way: There
an almost complete set K1 c K such that any zi
choice thus made, assuming m > 2, there exists an almost complete set K2 c
K (possibly depending on the particular zi
K2
e K1 may be chosen. For
e K1 selected) such that any zi
c
may be chosen. Again for any choice this made, assuming m > 3, there exists
an almost complete set K3 c K (possibly depending on the particular Zil and zi2
selected) such that any zi
continued
until
we
have
c K 3 may be chosen, etc. If m
3
above
to zi
definition
k this process is
reached Zi . If m <k, once we have reached im there
exists at least one (k-m)-tuple (zi
particular zi
=
...
,
,
Zi ) (possibly depending on the
selected) that may be chosen. Finally, we may extend the
to the situation m = 0 by saying that in this case the set Z
is not empty [6].
Definition A.1.9: We say that Z is sequentially infinite of order m, 1 < m < k,
if
it
can be generated as in Definition A.1.8 except for replacing everywhere
the term "almost complete set" by the term "infinite set" [6].
A.2. PROPERTIES OF VARIOUS CLASSES OF MULTIDIMENSIONAL SCHUR POLYNOMIALS:
Property
A.2.1:
(i) Factors of scattering Schur (SS) polynomials are also SS.
(ii) Conversely, products of SS polynomials are also SS (Theorem 10 in [9]).
37
Property A.2.2:
pairwise
(i) Products
of
reactance
Schur (RS) polynomials that are
relatively prime are RS. (ii) Any factor of a RS polynomial is also a
RS polynomial (Theorem 21 in [9]).
Property
A.2.3:
scattering
Let
Schur
A
if
be
and
a widest
only
if
sense Schur polynomial. Then A is also
the
zeros
of
A
on IzI=1 do not form a
sequentially infinite set of order (k-1) (Theorem 9 in [9]).
A.3. PROPEETIES OF MULTIDIMENSIONAL RATICNAL DISCRETE POSITIVE FUNCTIONS:
Property
A.3.1:
polynomial
If
such
A
that
is
an
scattering
Re(B/A)>O
for
jzj1l
Schur
polynomial
and B a further
whenever A•O, then Re(B/A)>O for
Izl<l. Furthermore, it is impossible to have Re(B/A)=O for some zO on Izl<l
unless B/A is a constant (Lemma 3b in [9]).
Property
(DPF)
A.3.2:
in
The numerator and denominator of a discrete positive function
irreducible
Conversely,
every
rational
IS
polynomial
form
is
is an immitance Schur (IS) polynomial.
the
denominator
(and consequently the
numerator) of a DPF in irreducible rational form (Theorem 27 in [9]).
A.4. OTHER PROPERTIES:
Property
A.4.1:
polynomial
Furthermore,
If
such
A
is
a widest
sense
Schur polynomial and B a further
that
IB/AI<1 for Izl=l whenever A•O, then IB/Al<1 for Izl<l.
is impossible to have IB/AIj= for some z0 in Izl<l unless B/A
it
is a constant (Lemma 3a in [9]).
Property
A.4.2: If A and B are polynomials in k variables, then A and B have a
proper (i.e., nonconstant) common factor if and only if the set Z of zeros that
are
common
to
A
and B is sequentially infinite of order (k-1) (Theorem 4 in
[6]).
38
APPENDIX B
B.1. Proof of Proposition 3.2:
H
Let
gcd(W, X), where W=HW1 , X=HX1 . Since W or X is scattering Schur H is
=
so. It then follows from property (P2) of (2.5) that H is a factor of YY.
also
H=H'H", where H', H" are factors of Y, Y respectively. Since H" divides Y,
Let
H"
divides Y. Due to the scattering Schur property of H' and H" inherited from
H,
H'
and H" are coprime and thus Y=(H'H")Y 1 for some polynomial Y 1 . A direct
of the last equation along with W=HW 1, X=HX1 in XX -pYY--WW yields
substitution
(B.1.la) in the following.
X1X1 - pY1Y1 = W1W
1,
Next,
XcX c - PYcYc = WcW c
let F = gcd(X1 , W1 ), where X1 =FXc, W1
(B.l.la,b)
= FWc and the monomial factors of
degree in Xc and X1 are identical (note that this uniquely defines
of
maximal
Xc
upto a constant multiplier). Clearly, F, being a factor of X1 cannot have a
factor and thus, from (B.1.la), F must divide Y1 Y1 . Let F=F'F", where
monomial
F' divides Y1 and F" divides Y1. The last requirement implies F" divides Y1 . If
then F is scattering Schur since W 1 is also so. On the other hand, if p=l
p=-l
since
Xl=FXc,
then
F
Schur.
Thus,
either
scattering Schur, because X=HX
1
is
both
F'
is also scattering
and F" or both F' and F" are scattering Schur.
F' and F" are coprime, and^Y 1 = (F'F")Yc for some polynomial Yc'
Consequently,
define matrices ~f = Diag(F'/F', H"/H"), 4r = Diag(F"/F", H'/H')
~and as:
m
rI ~ m)
and
c as: [c]11=Xc/Wc' [Jc]21=Yc/Wc'
=c]22=P(XYcZ)/W
[
)/Wc
where m=(ml,m2 ,...mk), mi=ni-degi(HF) it follows in a straightforward manner
Next,
that
if
we
4 4
-rtctf
=
and
each
of the matrices so defined is a generalized lossless
two-port matrix with same p. In particular, (B.l.lb) holds true, where m i is at
least as large as degiX c, degiY c, degiW c for each i=l to k.
Clearly,
Wc
is
definition
of
otherwise,
due
and
also
WcW c
are
implies
coprime
Xc
to
with Xc. On the otherhand, since it follows from the
that X1=FXc, Wc and Xc cannot have a common factor, because
W1=FWc, W1
and X1 would not be coprime. Consequently, XcXc
coprime. If mi=degiW c for mall i=l tomk then the last conclusion
that the polynomials WcWCz
39
and XcX z
are coprime, the proof of
the present theorem is complete. Otherwise, diagonal lossless two-port matrices
from
left
and/or
right
of
needs to be further extracted to satisfy the
c
coprimeness requirement.
For this purpose, note that, due to (B.l.lb), any monomial factor of X c present
in
WcWczm must
also
(total)
divides
Y
zm.
k-tuple
m'=(mj,m ,...m.)
with
factor of YcY zM . Let S=S S2 be such a factor of
a
degree, where the monomial S1 divides Yc' and the monomial S2
maximal
Wc,Wc,
be
Consider polynomials Xc,=Xc/S, Yc,=Yc/Sl1 Wc=Wc and the integer
with
mi = mi-degiS. Then clearly Xc Xc, - PYYc' =
degiX c,, degiY c, and degiWc, upper bounded by m! for all i=l to k
holds
M
must also
true. Thus, any monomial factor of X c,z- present in Wc,wc,z
be
factor
a
of
Yc'Yc'Z-
. Let
T=-T1T 2
be such a factor of maximal (total)
degree, where the monomial T1 divides Yc,, and the monomial T2 divides Yc,Zm .
Next, consider X =Xc, Yo=Yc,/Tl and the integer k-tuple no=(nol, n 2 ...nok),
where noi= ml-degiT. By letting Wo=Wc,=Wc it is then routinely verified that fc
Diag(S 2,
T1 ).%o.Diag(S1 ,
T2),
where %o is a generalized lossless two-port
is
described nby XO' YO' Wo, n and a as in (2.5) such that WoW z
-coprime with XoX z . The proof of the theorem is then completed by setting:
matrix
t1 = Diag(S 2, T1 ).4 f and
t2 = Diag(S1, T2 ).
r.
B.2. Completion of proof of Theorem 5.2:
(i) p = -1: Assume for contradiction that L = 0, which due to (5.3a) implies
that X'X' + Y'Y' =
and thus, IX' 2 + IY'12 - 0 for Izl=l. Consequently, X'I
0, Y' - 0,
Y,
and via (2.17) Y" - 0, which contradicts linear independence of {Xi,
Y"}, i=1,2.
(ii) p
XiX'
=
1: Assume for contradiction that L = 0, which due to (5.3b) implies
= YiYj. Since {Xi, Yj, Y1} is singular, we have XjiX
equations
together
imply
Xj/Xi
= Y
= H2 /H1 ,
=/Yi
polynomials. Clearly, there exists polynomials Xo,
0
Xi = H1X'
Y' = H1Y',
X2 = H
2 Xo ,
40
= YiYi. The last two
where H1 and H2 are coprime
YO' such that
Y2 = H2 Y
°
(B.2.1a,b,c,d)
fundamental equation for the triple {Xi , Yj, Y') and {Xi, Yj,
the
Considering
Y"}, we obtain (B.2.2a,b) where Y" is defined via (B.2.2c).
2
o
= H Y"
Y
Clearly,
11o
'
Yo"
is
Y
a
2
H
(YX' - XY') = pa'z-W'W'Y
o
o'
polynomial,
since
o
-
o
(B.2.2a,b,c)
otherwise its least denominator would
divide both H 1 and H2, i.e., H1 and H2 would not be coprime.
of
follows from (B.2.la), (B.2.lb) and (B.2.2a) that the degrees
it
Furthermore,
the polynomials in the triple {Xo , YO, Y"} cannot exceed the degrees of the
corresponding polynomials in the triple {Xi , Yj, Y"}. Thus, in view of (B.2.2c)
,~~n'
~
'
j, -cn'
*
{Xo, Y', Y"}, and consequently due to Lemma 4.1, {pYz- , Xoz- , -pa X"} is a
to the fundamental equation for some X". This last mentioned equation
o
n'
n'
*~
Xz--p
,
X
along with (B.2.1) and the fundamental equations for {pYi z~n'
*
(cf. Lemma 4.1) and {pY2z- , Xzn-I -pc' X2} yield:
solution
if
Next,
we
define
equation
fundamental
Fo
for
(B.2.3a,b)
x1 = H2Xo
1X= H X"
=
X'X" ,
the
then by eliminating X" between F
{pY'z0o-
triple
Xz0-
,
-pa XO}
o
and the
one obtains
(B.2.4).
Fo = (XoX~'/W')[1 -(pYY x'x)]
From
Since
{Xi,
of t in (2.5) we have IY/XI < 1 on Izl=l, whenever X • 0.
(P2)
Property
Y1}
Yj,
fundamental equation,
implying that IX'/YIl
that
contain
been
assumed~to
be
a
singular
solution to the
we have Xi
= YY
thus via (B.2.a,b),
= Y
= 1 on Izl=l, whenever Y' • 0. Then (B.2.4) yields that
is well defined. Using an argument analogous to
the proof of Theorem 3.1 it then follows (via discrete positive
in
used
nature
has
Izl=l, wherever F
ReFo > 0 for
(B.2.4)
of Fo ) that X o, X" are widest sense Schur polynomials, and thus, cannot
zi
as
it
polynomials,
conclusion
1
{Xi, Yi, Y'
a
factor
then
for
follows
any
i=1
that
H1
to k. Since in (B.2.3) X1 and XI are
and
H2
are
constants. This latter
violates, due to (B.2.1) and (B.2.2a,b), the linear independence of
and {xi, Y½, Yl}.
41
B.3. Proof of Theorem 5.3:
Due
to
Theorem
5.2,
it is only needed to show that one can have at most two
linearly independent solutions. Assume for contradiction that {Xi, Y,
to
YPI,
i=1
3 be three linearly independent solution to the fundamental equation and X'
+ O3X§, Y' = OlY + O2YY + O3Y, Y"
1Y " +
+2X 2Y" +
3Y". Then
and, due to lemma 4.1,
{py'z- , x'zn , -pa' X"} is also a
=
lX
+
{X', Y', Y"}
solution to the fundamental equation. From Lemma 4.3, K as defined in (B.3.1)
is a constant.
K = (X'X'-pY'Y')/(W'W') = (a'Y'z-n',
Y" + X'X")/X
(B.3.1)
Consider next the following cases:
(i) p=l: Clearly, we may choose Ci, i=1 to 3 such that X'(O) = Y"(O) = 0. Since
X(O)O0 due to the scattering Schur property of X, it follows from (B.3.1) that
K=O
and thus, again from (B.3.1) that X'X' - pY'Y'. Next, define F = (X'X")/X.
From
the second equality of (4.1a) and (4.1c), one obtains X" = (XX' pYY')/(W'W').
Eliminating
(X'X'/W'W')[1
-
it
is
finally,
(YY'/XX')].
from the last two equations it follows that F =
From Property
(P2)
of 4 and X'X' = pY'Y', one
= 1 that IY/XI < 1, whenever X•O and IY'/X'l = 1,
Using an argument similar to that used in proving Theorem 3.1,
obtains for Izl
respectively
whenever
X"
X'0O.
then
possible
X'X"
to
is widest
show that
sense
F
Schur,
is a discrete positive function, and
which
is
in contradiction with our
construction that X'(O)=O.
(ii) p=-l:
Property
It
then
implying
We
claim that there exists 0O on Izi = 1, such that W'W' • 0 (cf.
A.2.3)
and ai', i=1 to 3 can be chosen such that X'(z0)
= Y'(z)
= 0.
1
-=O
2
2
follows from (B.3.1) that for z = z0, K = 0, i.e., IX'I +IY'I
= 0
X'
-
0, Y'
independence of {IX,
- 0, and via (2.17), Y" - 0, which contradicts linear
Yi, Y), i=l to 3.
To substantiate the claim we show that there exists z=z
on Izl = 1 with W'W'
0
0 such that XiYI - XjYi • 0 and thus, it is possible for any nonzero o3 to
solve the following linear simultaneus equations for l1 and a 2.
42
For
this,
=
Y'(Z
)
consider
factorization
follows
X'(ZO)
(B.3.2a)
=
lY(O) =+ a2Y(Zo) + a3Y§(ZO) = 0
a
solution
Xi
X,
Yi = Y
(B.3.2b)
X
= X', Y
= yV to the
problem. Then from the necessity part of proof of Theorem 5.2 it
' IpYzX2 =
Y=
X'znn' Y2 = -pa' *X" is also a solution to the
that
fundamental
3+
c X(
3
olXi(Zo) + a2 X(io)
equation
and
that {Xi, Y!, Yij, i=1,2 are linearly
independent.
2
= 1 if XiY2-X Yi = 0 then iX'
+ IY-,2
i.e.,
'121
=
o
o
X' = 0, Y' = 0, and thus, due to the equation corresponding to (2.8a) satisfied
Next,
for
by XI,
Y' we have W' = W' = 0. Since W' is scattering Schur, there exists z
0
some
z
on
Jzl
-o on
JIz=1 with W'•O (cf. Property A.2.3) and, consequently, with XiYi - X2Yj
O0.
.
B.4 Proof of Fact 5.4:
We
have
K
=
Y"(0)/X"(O),
where
X"
and
Y"
are corresponding polynomials
associated with V", and VI is defined in (5.5).
Next,
since
IY"/X"J
two
<
p = 1, it respectively follows from Properties (P2) and (P4) that
1, wherever X"•0 on Izl=1 and that X" is scattering Schur. The last
properties,
modulus
to
an
extended multidimensional version of the maximum
theorem proved in Property A.4.1, imply that either Y"/X" = unimodular
constant
or
Property
(P2)
case,
due
we
IY"/X"l
have,
< 1 for Jzj < 1. In the former case we would have, due to
satisfied
in
by V", that W"-O, which is impossible. In the latter
particular,
JIK
=
JY"(O)/X"(O)J
<
1.
Thus,
VI is a
generalized lossless two-port matrix with p=1. Since it is easily verified that
(,~)-1 also satisfies this property it follows from Fact 2.1 that
f
r
is also a generalized lossless two-port matrix. Finally, the
Y-polynomial
()
f
fact that
associated with V', say Y", satisfies Y"(0) = 0 follows from Y" =
II
t2
r
r
rr
(Y"-K X")//(1-JKJ ) (obtained by considering the
(2,1) entry of the matrix
equation
r = ()f) 1t") and K = Y"(O)/X"(O).
43
~X
Figure
l~~
Y-2
1. Network
e
-
o'.
e
resulting
from the
CX1
where
Y1j
factorization
E2
Y
-
X2
I
I
Y°-
I
Figure 2. Network resulting from the
r * r r
where [X
factorization
r[.
y2t
-
1
X
I
-
Figure
3. Network
resulting
* ._
a where
E
from
Y12 tt
the
Ji
factorization
rX
Lb1
x1ZJ
X2
02
X1
Y2
Y1
X2
Figure 4.
Elementary section with n
where
= 1
rectangular boxes are
Gray-Markel
sections.
03
Y
Yl
01
.
1 _-~
Figure
5.
-
Elementary
X.
'-"
section
with
n1
=
2.