AAK Theorem and a Design of Multidimensional BIBO Stable Filters ()
1. Introduction
Multidimensional filters are central elements in digital signal processing and control systems.
Consider a n dimensional (nD) filter S with an analytic transfer function
(1)
The sequence
is the so called impulse response of the filter.
To avoid burst in a filter, stability is a necessary condition. For instance, the filter S is BIBO stable if for any bounded input signal x, the output y is bounded. A fundamental theorem states that S is BIBO stable if and only if
(2)
However, only a rational filter (filter having a rational transfer function) is physically realizable. Under some assumptions, the filter S is rational and BIBO stable if and only if all poles of H lie in the unit polydik
(see [1] [2] for example).
Note that BIBO stability has also been studied in the consensus coordination of control systems [3] [4] .
Given the significance of rational approximation theory in signal processing and systems theory, during the last years several mathematicians have been researching the problem of approximating multivariate functions (see for example [5] [6] [7] ). One can also cite the works of Avilov and all [8] through the method of Padé-type extended to several variables and recently Austin and all [9] by discrete least-squares methods.
We propose in this paper to introduce a possibility to design multidimensional BIBO stable Infinite Impulse Response (IIR) filters as realizable filters thanks to the one dimensional AAK (Adamyan, Arov and Kreĭn) theorem [2] . To our knowledge such approach does not exist in the literature. We are concerned with filters S such that the multidimensional transfer function
may be written as a combination (sum or product) of separable polynomials. We assume that the Hankel operator associated to H has a finite rank.
It is known that the AAK theorem gives the best approximation of a univariate polynomial as a rational function in the Hankel semi-norm. A remarkable contribution of AAK’s result is its significance to engineering. One of its important applications is the problem of system reduction which consists of finding a lower-dimensional linear system to approximate a given high-dimensional linear system in a certain optimal sense. The AAK result provides a wonderful characterization of this problem in sense that the Hankel norm of the error between the given high-order transfer function and the approximant is minimized over all transfer functions of the same (lower) order (see [2] for an overview on the topic). The origins of the AAK method can be founded in the papers by Adamyan-Arov-Kreĭn [10] [11] . One can cite also the following paper [12] .
Outline
This paper is organized as follows. Section 2 provides some explanations on the system reduction problem and preliminaries on AAK theorem for one dimensional (1D) filters. In Section 3, we propose rational optimal approximations of the transfer functions of a multidimensional (nD) filter by our approach based on AAK results.
2. Preliminaries on AAK Theorem on 1D Filters
Consider
, the transfer function of an 1D filter. The Hankel matrix corresponding to
is the infinite matrix
, namely
(3)
The matrix
can be viewed as an operator on the space of square summable sequences
. The operator norm of
is given by
(4)
and one defines the Hankel semi-norm of
by
(5)
If
, then
is the spectral norm of the operator
.
Note that the terms
are called the analytic and the singular part of
.
For a positive integer m, let
(6)
The problem stated as follows:
(7)
is called system reduction for the filter [2] and is equivalent to the extremal problem
(8)
with
(9)
The following theorem makes a connection between the system reduction problem and the approximation of a polynomial as a rational function.
Theorem 1 ( [2] , Kronecker’s theorem). The infinite Hankel matrix
has finite rank m if and only if the singular part
is a strictly proper rational function in z i.e.
(10)
Moreover,
Corollary 1 (Kronecker Nehari result, [2] ). The function
is in
if and only if its corresponding Hankel matrix
is in
.
Definition 1. Let
be a Hankel matrix and
its adjoint.
The s-numbers (or singular values) of
are the eigenvalues of
.
Any pair
of elements in
that satisfies
is called a Schmidt pair of
.
One denotes by
the espace of essentially bounded functions on the unit circle
. Then
Theorem 2 ( [2] , Adamjan, Arov, and Krein theorem). Let
be a given function in
such that the Hankel matrix
associated to f is a compact operator with s-numbers
and let
, the Schmidt pair corresponding to
. Then a solution to the extremal problem
is given by the singular part
of
(11)
A detailed proof of this theorem has been made in [2] .
3. Approximation of the Transfer Function of an (nD) BIBO Stable Filter
Definition 2: We say that a polynomial
is stable if
Let
be the set of rational polynomials in
with stable denominator and such that the degrees of the numerator and denominator do not exceed m.
One has the following result.
Proposition 1: Suppose that
(12)
is the transfer function of a multidimensional (nD) BIBO stable filter such that
1)
,
2) For each
,
is a polynomial in
,
3) The operators
are compact operators with finite ranks
.
Then an optimal approximation of H as a rational function in
is given by
(13)
the elements
being the singular parts of a certain polynomial function
.
Remark 1: If the hypothesis one in the proposition 1 is replaced by
then
Proof. The existence of such approximation of the (nD) transfer function H (in
) given in Equation (12) is certified by the Kronecker-Nehari result applied to each
. The Formula (13) is given by the application of the AAK theorem to each polynomial
and for each
(14)
with
the
s-number of
.
Moreover one has.
Corollary 2 (Lemma 2.1, [2] ). If for each
(15)
and
the principal minor of order m in
, then
(16)
and
(17)
Proposition 2: Suppose that
belongs to
. Then for each
the coefficients of
and
can be computed thanks to corollary 2. More precisely if
and
then
1) The coefficients of
and
except
and
are computed by using (16) and (17);
2) The rank of
,
.
Proof. The item 1 is a consequence of the corollary 2.
The contraposition of the theorem of DeCarlo et al [13] allows to say that
belongs to
. The item 2 is then a consequence of Corollary 1.
4. Conclusion
This paper has proposed an optimal design of a multidimensional BIBO stable filter with a rational transfer function
. The approach is based on the one dimensional AAK theorem. Two propositions have been developed and require that the Hankel operators
associated with the polynomial function H are compact with finite ranks. We hope that in further works, comparisons (based on numerical tests) with other methods that exist in the literature on the approximation of multivariate functions will be made.
Data Availability
No underlying data was collected or produced in this study.
Funding Statement
This study did not receive any funding in any form.