Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures

Orthogonal vs. Non-Orthogonal Reducibility
of Matrix-Valued Measures\star\star$\star$This paper is a contribution to the
Special Issue on Orthogonal Polynomials, Special Functions and Applications.
The full collection is available at
http://www.emis.de/journals/SIGMA/OPSFA2015.html

Erik KOELINK † and Pablo ROMÁN †‡

E. Koelink and P. Román

† IMAPP, Radboud Universiteit, Heyendaalseweg 135, 6525 GL Nijmegen, The
Netherlands<EMAIL_ADDRESS>http://www.math.ru.nl/~koelink/

‡ CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad
Universitaria,
‡ Córdoba, Argentina<EMAIL_ADDRESS>http://www.famaf.unc.edu.ar/~roman/

Received September 23, 2015, in final form January 21, 2016; Published online
January 23, 2016

A matrix-valued measure $\Theta$ reduces to measures of smaller size if there
exists a constant invertible matrix $M$ such that $M\Theta M^{*}$ is block
diagonal. Equivalently, the real vector space $\mathscr{A}$ of all matrices
$T$ such that $T\Theta(X)=\Theta(X)T^{*}$ for any Borel set $X$ is non-
trivial. If the subspace $A_{h}$ of self-adjoints elements in the commutant
algebra $A$ of $\Theta$ is non-trivial, then $\Theta$ is reducible via a
unitary matrix. In this paper we prove that $\mathscr{A}$ is $*$-invariant if
and only if $A_{h}=\mathscr{A}$, i.e., every reduction of $\Theta$ can be
performed via a unitary matrix. The motivation for this paper comes from
families of matrix-valued polynomials related to the group
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ and its quantum analogue. In both cases
the commutant algebra $A=A_{h}\oplus iA_{h}$ is of dimension two and the
matrix-valued measures reduce unitarily into a $2\times 2$ block diagonal
matrix. Here we show that there is no further non-unitary reduction.

matrix-valued measures; reducibility; matrix-valued orthogonal polynomials

33D45; 42C05

##  1 Introduction

The theory of matrix-valued orthogonal polynomials was initiated by Krein in
1949 and, since then, it was developed in different directions. From the
perspective of the theory of orthogonal polynomials, one wants to study
families of truly matrix-valued orthogonal polynomials. Here is where the
issue of reducibility comes into play. Given a matrix-valued measure, one can
construct an equivalent measure by multiplying on the left by a constant
invertible matrix and on the right by its adjoint. If the equivalent measure
is a block diagonal matrix, then all the objects of interest (orthogonal
polynomials, three-term recurrence relation, etc.) reduce to block diagonal
matrices so that we could restrict to the study of the blocks of smaller size.
An extreme situation occurs when the matrix-valued measure is equivalent to a
diagonal matrix in which case we are, essentially, dealing with scalar
orthogonal polynomials.

Our interest in the study of the reducibility of matrix-valued measures was
triggered by the families of matrix-valued orthogonal polynomials introduced
in [2, 9, 10, 11]. In [10] the study of the spherical functions of the group
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ leads to a matrix-valued measure $\Theta$
and a sequence of matrix-valued orthogonal polynomials with respect to
$\Theta$. From group theoretical considerations, we were able to describe the
symmetries of $\Theta$ and pinpoint two linearly independent matrices in the
commutant of $\Theta$, one being the identity. The proof that these matrices
actually span the commutator required a careful computation. It then turns out
that it is possible to conjugate $\Theta$ with a constant unitary matrix to
obtain a $2\times 2$ block diagonal matrix. An analogous situation holds true
for a one-parameter extension of this example [9]. In [2] from the study of
the quantum analogue of $\mathrm{SU}(2)\times\mathrm{SU}(2)$ we constructed
matrix-valued orthogonal polynomials which are matrix analogues of a subfamily
of Askey–Wilson polynomials. The weight matrix can also be unitarily reduced
to a $2\times 2$ block diagonal matrix in this case, again arising from
quantum group theoretic considerations.

In [12], the authors study non-unitary reducibility for matrix-valued measures
and prove that a matrix-valued measure $\Theta$ reduces into a block diagonal
measure if the real vector space $\mathscr{A}$ of all matrices $T$ such that
$T\Theta(X)=\Theta(X)T^{*}$ for any Borel set $X$ is not trivial, in contrast
to the reducibility via unitary matrices that occurs when the commutant
algebra of $\Theta$ is not trivial.

The aim of this paper is to develop a criterion to determine whether unitary
and non-unitary reducibility of a weight matrix $W$ coincide in terms of the
$*$-invariance of $\mathscr{A}$. Every reduction of $\Theta$ can be performed
via a unitary matrix if and only if $\mathscr{A}$ is $*$-invariant, in which
case $\mathscr{A}=A_{h}$ where $A_{h}$ is the Hermitian part of the commutant
of $\Theta$, see Section 2. We apply our criterion to our examples [2, 9, 10]
and we conclude that there is no further reduction than the one via a unitary
matrix. We expect that a similar strategy can be applied to more general
families of matrix-valued orthogonal polynomials as, for instance, the
families related to compact Gelfand pairs given in [7]. We also discuss an
example where $\mathscr{A}$ and $A_{h}$ are not equal.

It is worth noting that unitary reducibility strongly depends on the
normalization of the matrix-valued measure. Indeed, if the matrix-valued
measure is normalized by $\Theta(\mathbb{R})=I$, then the real vector space
$\mathscr{A}$ is $*$-invariant and by our criterion, unitary and non-unitary
reduction coincide. This is discussed in detail in Remark 3.7.

##  2 Reducibility of matrix-valued measures

Let $M_{N}(\mathbb{C})$ be the algebra of $N\times N$ complex matrices. Let
$\mu$ be a $\sigma$-finite positive measure on the real line and let the
weight function $W\colon\mathbb{R}\to M_{N}(\mathbb{C})$ be strictly positive
definite almost everywhere with respect to $\mu$. Then

$\displaystyle\Theta(X)=\int_{X}W(x)d\mu(x),$ (2.1)

is a $M_{N}(\mathbb{C})$-valued measure on $\mathbb{R}$, i.e., a function from
the $\sigma$-algebra $\mathscr{B}$ of Borel subsets of $\mathbb{R}$ into the
positive semi-definite matrices in $M_{N}(\mathbb{C})$ which is countably
additive. Note that any positive matrix measure can be obtained as in (2.1),
see for instance [4, Theorem 1.12] and [5]. More precisely, if
$\tilde{\Theta}$ is a $M_{N}(\mathbb{C})$-valued measure, and
$\tilde{\Theta}_{\mathrm{tr}}$ denotes the scalar measure defined by
$\tilde{\Theta}_{\mathrm{tr}}(X)=\mathrm{Tr}(\tilde{\Theta}(X))$, then the
matrix elements $\tilde{\Theta}_{ij}$ of $\tilde{\Theta}$ are absolutely
continuous with respect to $\tilde{\Theta}_{\mathrm{tr}}$ so that, by the
Radon–Nikodym theorem, there exists a positive definite function $V$ such that

$\displaystyle
d\tilde{\Theta}_{i,j}(x)=V(x)_{i,j}\,d\tilde{\Theta}_{\mathrm{tr}}(x).$

Note that we do not require the normalization $\Theta(\mathbb{R})=I$ as in
[5]. A detailed discussion about the role of the normalization in the
reducibility of the measure is given at the end of Section 3.

Going back to the measure (2.1), we have
$d\Theta_{\mathrm{tr}}(x)=\mathrm{Tr}(W(x))\,d\mu(x)$ so that
$\Theta_{\mathrm{tr}}$ is absolutely continuous with respect to $\mu$. Note
that $\mathrm{Tr}(W(x))>0$ a.e. with respect to $\mu$ so that $\mu$ is
absolutely continuous with respect to $\Theta_{\mathrm{tr}}$. The unicity of
the Radon–Nikodym theorem implies $W(x)=V(x)\mathrm{Tr}(W(x))$, i.e., $W$ is a
positive scalar multiple of $V$.

We say that two $M_{N}(\mathbb{C})$-valued measures $\Theta_{1}$ and
$\Theta_{2}$ are equivalent if there exists a constant nonsingular matrix $M$
such that $\Theta_{1}(X)=M\Theta_{2}(X)M^{*}$ for all $X\in\mathscr{B}$, where
$*$ denotes the adjoint. A $M_{N}(\mathbb{C})$-valued measure matrix $\Theta$
reduces to matrix-valued measures of smaller size if there exist positive
matrix-valued measures $\Theta_{1},\ldots,\Theta_{m}$ such that $\Theta$ is
equivalent to the block diagonal matrix
$\mathrm{diag}(\Theta_{1}(x),\Theta_{2}(x),\ldots,\Theta_{m}(x))$. If $\Theta$
is equivalent to a diagonal matrix, we say that $\Theta$ reduces to scalar
measures. In [12, Theorem 2.8], the authors prove that a matrix-valued measure
$\Theta$ reduces to matrix-valued measures of smaller size if and only if the
real vector space

$\displaystyle\mathscr{A}=\mathscr{A}(\Theta)=\big{\\{}T\in
M_{N}(\mathbb{C})\,|\,T\Theta(X)=\Theta(X)T^{*}\,\,\,\forall\,X\in\mathscr{B}\big{\\}},$

contains, at least, one element which is not a multiple of the identity, i.e.,
$\mathbb{R}I\subsetneq\mathscr{A}$, where $I$ is the identity. Note that our
definition of $\mathscr{A}$ differs slightly from the one considered in [12].
If $W$ is a weight matrix for $\Theta$, then we require that $T\in\mathscr{A}$
satisfies $TW(x)=W(x)T^{*}$ almost everywhere with respect to $\mu$.

If there exists a subspace $V\subset\mathbb{C}^{N}$ such that
$\Theta(X)V\subset V$ for all $X\in\mathscr{B}$, since $\Theta(X)$ is self-
adjoint for all $X\in\mathscr{B}$, it follows that $\Theta(X)V^{\perp}\subset
V^{\perp}$ for all $X\in\mathscr{B}$. If $\iota_{V}\colon V\to\mathbb{C}^{N}$
is the embedding of $V$ into $\mathbb{C}^{N}$, then
$P_{V}=\iota_{V}\iota_{V}^{*}\in M_{N}(\mathbb{C})$ is the orthogonal
projection on $V$ and satisfies

$\displaystyle P_{V}\Theta(X)=\Theta(X)P_{V},\qquad\text{for all
}X\in\mathscr{B}.$

Hence, the projections on invariant subspaces belong to the commutant algebra

$\displaystyle A=A(\Theta)=\big{\\{}T\in
M_{N}(\mathbb{C})\,|\,T\Theta(X)=\Theta(X)T\,\,\,\forall\,X\in\mathscr{B}\big{\\}}.$

Since $\Theta(X)$ is self-adjoint for all $X\in\mathscr{B}$, $A$ is a unital
$*$-algebra over $\mathbb{C}$. We denote by $A_{h}$ the real subspace of $A$
consisting of all Hermitian matrices. Then it follows that $A=A_{h}\oplus
iA_{h}$. If $\mathbb{C}I\subsetneq A$, then there exists $T\in A_{h}$ such
that $T\notin\mathbb{C}I$. The eigenspaces of $T$ for different eigenvalues
are orthogonal invariant subspaces for $\Theta$. Therefore $\Theta$ is
equivalent via a unitary matrix to matrix-valued measures of smaller size.

######  Remark 2.1.

Let $S\in A$ and $T\in\mathscr{A}$. Then we observe that $S^{*}\in A$ and
therefore $STS^{*}\Theta(x)=\Theta(x)ST^{*}S^{*}=\Theta(x)(STS^{*})^{*}$ for
all $X\in\mathscr{B}$. Hence there is an action from $A$ into $\mathscr{A}$
which is given by

$\displaystyle S\cdot T=STS^{*}.$

######  Lemma 2.2.

$\mathscr{A}$ does not contain non-zero skew-Hermitian elements.

###### Proof.

Suppose that $S\in\mathscr{A}$ is skew-Hermitian. Then $S$ is normal and thus
unitarily diagonalizable, i.e., there exists a unitary matrix $U$ and a
diagonal matrix $D=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$,
$\lambda_{i}\in i\mathbb{R}$, such that $S=UDU^{*}$, see for instance [8,
Chapter 4]. Since $S\in\mathscr{A}$, we get

$\displaystyle
DU^{*}\Theta(X)U=U^{*}\Theta(X)UD^{*}=-U^{*}\Theta(X)UD,\qquad\text{ for all
}X\in\mathscr{B}.$

The $(i,i)$-th entry of the previous equation is given by

$\displaystyle\lambda_{i}(U^{*}\Theta(X)U)_{i,i}=-(U^{*}\Theta(X)U)_{i,i}\lambda_{i}.$

Take any $X_{0}\in\mathscr{B}$ such that $\Theta(X_{0})$ is strictly positive
definite. Since $U$ is unitary, $U^{*}\Theta(X_{0})U$ is strictly positive
definite and therefore $(U^{*}\Theta(X_{0})U)_{i,i}>0$ for all $i=1,\ldots,N$,
which implies that $\lambda_{i}=0$. ∎

######  Theorem 2.3.

$\mathscr{A}\cap\mathscr{A}^{*}=A_{h}$.

###### Proof.

Observe that if $T\in A_{h}$, then $T\Theta(X)=\Theta(X)T=\Theta(X)T^{*}$ for
all $X\in\mathscr{B}$, and thus $A_{h}\subset\mathscr{A}$. Since $T$ is self-
adjoint, we also have $T=T^{*}\in\mathscr{A}^{*}$.

On the other hand, let $T\in\mathscr{A}\cap\mathscr{A}^{*}$. Then
$T^{*}\in\mathscr{A}^{*}\cap\mathscr{A}\subset\mathscr{A}$, and since
$\mathscr{A}$ is a real vector space, $(T-T^{*})\in\mathscr{A}$. The matrix
$(T-T^{*})$ is skew-Hermitian and therefore by Lemma 2.2 we have
$(T-T^{*})=0$. Hence $T$ is self-adjoint and $T\in A_{h}$. ∎

######  Corollary 2.4.

If $T\in\mathscr{A}\cap\mathscr{A}^{*}$, then $T=T^{*}$.

###### Proof.

The corollary follows directly from the proof of Theorem 2.3. ∎

######  Corollary 2.5.

$\mathscr{A}$ is $*$-invariant if and only if $\mathscr{A}=A_{h}$.

###### Proof.

If $\mathscr{A}=A_{h}$ then $\mathscr{A}$ is trivially $*$-invariant. On the
other hand, if we assume that $\mathscr{A}$ is $*$-invariant then the
corollary follows directly from Theorem 2.3. ∎

######  Remark 2.6.

Corollary 2.4 says that if $\mathscr{A}$ is $*$-invariant, then it is
pointwise $*$-invariant, i.e., $T=T^{*}$ for all $T\in\mathscr{A}$.

######  Remark 2.7.

Suppose that there exists $X\in\mathscr{B}$ such that
$\Theta(X)\in\mathbb{R}_{>0}I$, then every $T\in\mathscr{A}$ is self-adjoint
and Corollary 2.5 holds true trivially. Since $T\Theta(X)=\Theta(X)T^{*}$ for
all $X\in\mathscr{B}$, if there is a point $x_{0}\in\mathrm{supp}(\mu)$ such
that

$\displaystyle\lim_{\delta\to
0}\left\|\frac{1}{\mu((x_{0}-\delta,x_{0}+\delta))}\Theta((x_{0}-\delta,x_{0}+\delta))-I\right\|=0,$

then it follows that $T=T^{*}$ and so Corollary 2.5 holds true. This is the
case, for instance, for the examples given in [3, 1], where $W(x_{0})=I$ for
some $x_{0}\in\mathrm{supp}(\mu)$. For Examples 4.2 and 4.3, in general, there
is no $x_{0}\in[-1,1]$ for which $W(x_{0})=I$.

##  3 Reducibility of matrix-valued orthogonal polynomials

Let $M_{N}(\mathbb{C})[x]$ denote the set of $M_{N}(\mathbb{C})$-valued
polynomials in one variable $x$. Let $\mu$ be a finite measure and $W$ be a
weight matrix as in Section 2. In this section we assume that all the moments

$\displaystyle M_{n}=\int x^{n}W(x)\,d\mu(x),\qquad n\in\mathbb{N},$

exist and are finite. Therefore we have a matrix-valued inner product on
$M_{N}(\mathbb{C})$

$\displaystyle\langle P,Q\rangle=\int P(x)W(x)Q(x)^{*}\,d\mu(x),\qquad P,Q\in
M_{N}(\mathbb{C})[x],$

where $*$ denotes the adjoint. By general considerations, e.g., [5, 6], it
follows that there exists a unique sequence of monic matrix-valued orthogonal
polynomials $(P_{n})_{n\in\mathbb{N}}$, where
$P_{n}(x)=\sum\limits_{k=0}^{n}x^{k}P^{n}_{k}$ with $P^{n}_{k}\in
M_{N}(\mathbb{C})$ and $P^{n}_{n}=I$, the $N\times N$ identity matrix. The
polynomials $P_{n}$ satisfy the orthogonality relations

$\displaystyle\langle P_{n},P_{m}\rangle=\delta_{nm}H_{n},\qquad H_{n}\in
M_{N}(\mathbb{C}),$

where $H_{n}>0$ is the corresponding squared norm. Any other family
$(Q_{n})_{n\in\mathbb{N}}$ of matrix-valued orthogonal polynomials with
respect to $W$ is of the form $Q_{n}(x)=E_{n}P_{n}(x)$ for invertible matrices
$E_{n}$. The monic orthogonal polynomials satisfy a three-term recurrence
relation of the form

$\displaystyle xP_{n}(x)=P_{n+1}(x)+B_{n}P_{n}(x)+C_{n}P_{n-1}(x),\qquad n\geq
0,$ (3.1)

where $P_{-1}=0$ and $B_{n}$, $C_{n}$ are matrices depending on $n$ and not on
$x$.

######  Lemma 3.1.

Let $T\in\mathscr{A}$. Then we have

  1. $(1)$

The operator $T\colon M_{N}(\mathbb{C})[x]\to M_{N}(\mathbb{C})[x]$ given by
$P\mapsto PT$ is symmetric with respect to $\Theta$.

  2. $(2)$

$TP_{n}=P_{n}T$ for all $n\in\mathbb{N}$.

  3. $(3)$

$TH_{n}=H_{n}T^{*}$ for all $n\in\mathbb{N}$.

  4. $(4)$

$TM_{n}=M_{n}T^{*}$ for all $n\in\mathbb{N}$.

  5. $(5)$

$TB_{n}=B_{n}T$ and $TC_{n}=C_{n}T$ for all $n\in\mathbb{N}$.

###### Proof.

Let $P,Q\in M_{N}(\mathbb{C})[x]$. Then

$\displaystyle\langle PT,Q\rangle=\int P(x)TW(x)Q(x)^{*}\,d\mu(x)=\int
P(x)W(x)T^{*}Q(x)^{*}\,d\mu(x)$ $\displaystyle\hphantom{\langle
PT,Q\rangle}{}=\int P(x)W(x)(Q(x)T)^{*}\,d\mu(x)=\langle P,QT\rangle,$

so that $T$ is a symmetric operator. This proves (1). It follows directly from
(1) that the monic matrix-valued orthogonal polynomials are eigenfunctions for
the operator $T$, see, e.g., [6, Proposition 2.10]. Thus, for every
$n\in\mathbb{N}$ there exists a constant matrix $\Lambda_{n}(T)$ such that
$P_{n}T=\Lambda_{n}(T)P_{n}$. Equating the leading coefficients of both sides
of the last equation, and using that $P_{n}$ is monic, yields
$T=\Lambda_{n}(T)$. This proves (2).

The proof of (3) follows directly from (2) and the fact that
$T\in\mathscr{A}$. We have

$\displaystyle TH_{n}=\int P_{n}(x)TW(x)P_{n}(x)^{*}\,d\mu(x)=\int
P_{n}(x)W(x)T^{*}P_{n}(x)^{*}\,d\mu(x)$ $\displaystyle\hphantom{TH_{n}}{}=\int
P_{n}(x)W(x)(P_{n}(x)T)^{*}d\mu(x)=\int
P_{n}(x)W(x)(TP_{n}(x))^{*}\,d\mu(x)=H_{n}T^{*}.$

The proof of (4) is analogous to that of (3), replacing the polynomials
$P_{n}$ by $x^{n}$. Finally, we multiply the three-term recurrence relation
(3.1) by $T$ on the left and on the right and we subtract both equations.
Using that $TP_{n}=P_{n}T$ and $TP_{n+1}=P_{n+1}T$ we get

$\displaystyle TB_{n}P_{n}+TC_{n}P_{n-1}=B_{n}TP_{n}+C_{n}TP_{n-1}.$

The coefficient of $x^{n}$ is $TB_{n}=B_{n}T$ and therefore we also have
$TC_{n}=C_{n}T$. ∎

Corollary 2.5 provides a criterion to determine whether the set of Hermitian
elements of the commutant algebra $A$ is equal to $\mathscr{A}$. However, for
explicit examples, it might be cumbersome to verify the $*$-invariance of
$\mathscr{A}$ from the expression of the weight. Our strategy now is to view
$\mathscr{A}$ as a subset of a, in general, larger set whose $*$-invariance
can be established more easily and that implies the $*$-invariance of
$\mathscr{A}$. Motivated by Lemma 3.1 we consider a sequence
$(\Gamma_{n})_{n}$ of strictly positive definite matrices such that if
$T\in\mathscr{A}$, then $T\Gamma_{n}=\Gamma_{n}T^{*}$ for all $n$. Then for
each $n\in\mathbb{N}$ and $\mathcal{I}\subset\mathbb{N}$ we introduce the
$*$-algebras

$\displaystyle A^{\Gamma}_{n}=A(\Gamma_{n})=\\{T\in
M_{N}(\mathbb{C})\,|\,T\Gamma_{n}=\Gamma_{n}T\\},\qquad
A^{\Gamma}_{\mathcal{I}}=\bigcap_{n\in\mathcal{I}}A^{\Gamma}_{n},$

and the real vector spaces

$\displaystyle\mathscr{A}^{\Gamma}_{n}=\mathscr{A}(\Gamma_{n})=\\{T\in
M_{N}(\mathbb{C})\,|\,T\Gamma_{n}=\Gamma_{n}T^{*}\\},\qquad\mathscr{A}^{\Gamma}_{\mathcal{I}}=\bigcap_{n\in\mathcal{I}}\mathscr{A}^{\Gamma}_{n}.$
(3.2)

It is clear from the definition that
$\mathscr{A}\subset\mathscr{A}^{\Gamma}_{n}$ for all $n\in\mathbb{N}$.

######  Remark 3.2.

For any subset $\mathcal{I}\subset\mathbb{N}$, the sequence $(\Gamma_{n})_{n}$
induces a discrete $M_{N}(\mathbb{C})$-valued measure supported on
$\mathcal{I}$

$\displaystyle
d\Gamma_{\mathcal{I}}(x)=\sum_{n\in\mathcal{I}}\Gamma_{n}\delta_{n,x}.$

Theorem 2.3 applied to the measure $d\Gamma_{\mathcal{I}}$ yields that
$\mathscr{A}^{\Gamma}_{\mathcal{I}}\cap(\mathscr{A}^{\Gamma}_{\mathcal{I}})^{*}$
is the subset of Hermitian matrices in $A^{\Gamma}_{\mathcal{I}}$.

######  Theorem 3.3.

If $\mathscr{A}^{\Gamma}_{\mathcal{I}}$ is $*$-invariant for some non-empty
subset $\mathcal{I}\subset\mathbb{N}$, then $\mathscr{A}=A_{h}$. In
particular, the statement holds true if $\mathscr{A}^{\Gamma}_{n}$ is
$*$-invariant for some $n\in\mathbb{N}$.

###### Proof.

If $T\in\mathscr{A}$, then $T\in\mathscr{A}^{\Gamma}_{n}$ for all
$n\in\mathcal{I}$. Since $\mathscr{A}^{\Gamma}_{\mathcal{I}}$ is
$*$-invariant, then $T^{*}\in\mathscr{A}^{\Gamma}_{n}$ for all
$n\in\mathcal{I}$. If we apply Corollary 2.4 to the measure in Remark 3.2, we
obtain $T=T^{*}$. Therefore $T\in A_{h}\subset\mathscr{A}$ and thus
$\mathscr{A}$ is $*$-invariant. Hence the theorem follows from Corollary 2.5.
∎

######  Remark 3.4.

Two obvious candidates for sequences $(\Gamma_{n})_{n}$ are given in Lemma
3.1, namely the sequence of squared norms $(H_{n})_{n}$ and the sequence of
even moments $(M_{2n})_{n}$.

######  Remark 3.5.

Let $\Theta$ be a $M_{N}(\mathbb{C})$-valued measure, not necessarily with
finite moments and take a positive definite matrix $\Gamma$ such that
$T\Gamma=\Gamma T^{*}$ for all $T\in\mathscr{A}(\Theta)$. Let $S$ be a
positive definite matrix such that $\Gamma=S^{2}$. We can now consider the
$M_{N}(\mathbb{C})$-valued measure $S^{-1}\Theta S^{-1}$. By a simple
computation, we check that $T\in\mathscr{A}(\Theta)$ if and only if
$S^{-1}TS\in\mathscr{A}(S^{-1}\Theta S^{-1})$. This gives

$\displaystyle\mathscr{A}\big{(}S^{-1}\Theta
S^{-1}\big{)}=S^{-1}\mathscr{A}(\Theta)S.$

Moreover, if $T\in\mathscr{A}(\Theta)$, then $T\Gamma=\Gamma T^{*}$ implies
that $S^{-1}TS=ST^{*}S^{-1}=(S^{-1}TS)^{*}$. Hence $S^{-1}TS$ is self-adjoint
for all $T\in\mathscr{A}(\Theta)$. Then we have by Corollary 2.5

$\displaystyle A\big{(}S^{-1}\Theta
S^{-1}\big{)}_{h}=\mathscr{A}\big{(}S^{-1}\Theta S^{-1}\big{)}.$

On the other hand, if $U\in A_{h}(\Theta)$, then

$\displaystyle
S^{-1}USS^{-1}\Theta(X)S^{-1}=S^{-1}U\Theta(X)S^{-1}=S^{-1}\Theta(X)S^{-1}SUS^{-1}$
$\displaystyle\hphantom{S^{-1}USS^{-1}\Theta(X)S^{-1}}{}=S^{-1}\Theta(X)S^{-1}\big{(}S^{-1}US\big{)}^{*},$

for all $X\in\mathscr{B}$. Therefore
$S^{-1}A(\Theta)_{h}S\subset\mathscr{A}(S^{-1}\Theta S^{-1})=A(S^{-1}\Theta
S^{-1})_{h}$. In general this is an inclusion, see Example 4.1.

######  Remark 3.6.

Suppose that $\Theta$ is a $M_{N}(\mathbb{C})$-valued measure with finite
moments and that $M_{2n}\in\mathbb{R}_{>0}I$, respectively
$H_{n}\in\mathbb{R}_{>0}I$, for some $n\in\mathbb{N}$. Then it follows from
Lemma 3.1 and (3.2) that $T=T^{*}$ for all $T\in\mathscr{A}^{M}_{2n}$,
respectively $T\in\mathscr{A}^{N}_{n}$. Then Theorem 3.3 says that
$\mathscr{A}=A_{h}$.

######  Remark 3.7.

Let $\Theta$ be a $M_{N}(\mathbb{C})$-valued measure such that the first
moment $M_{0}$ is finite. Then there exists a positive definite matrix $S$
such that $M_{0}=S^{2}$. The measure $\widetilde{\Theta}=S^{-1}\Theta S^{-1}$
satisfies

$\displaystyle\widetilde{\Theta}(\mathbb{R})=S^{-1}\Theta(\mathbb{R})S^{-1}=S^{-1}M_{0}S^{-1}=I.$

Therefore by Remark 3.6 we have that $\mathscr{A}(S^{-1}\Theta
S^{-1})=A(S^{-1}\Theta S^{-1})_{h}$. Observe that the normalization
$\widetilde{\Theta}(\mathbb{R})=I$ is assumed in [5] so that in the setting of
that paper the real subspace of Hermitian elements in the commutant coincides
with the real vector space $\mathscr{A}(\Theta)$.

##  4 Examples

In this section we discuss three examples of matrix-valued weights that
exhibit different features. The first example is a slight variation of [12,
Example 2.6].

######  Example 4.1.

Let $\mu$ be the Lebesgue measure on the interval $[0,1]$, and let $W$ be the
weight

$\displaystyle W(x)=\begin{pmatrix}x^{2}+x&x\\\
x&x\end{pmatrix}=\begin{pmatrix}1&\frac{\sqrt{6}}{3}\\\
0&1\end{pmatrix}\begin{pmatrix}x^{2}&0\\\
0&x\end{pmatrix}\begin{pmatrix}1&0\\\ \frac{\sqrt{6}}{3}&1\end{pmatrix}.$

A simple computation gives that $A$ and $\mathscr{A}$ are given by

$\displaystyle A=\mathbb{C}I,\qquad\mathscr{A}=\mathbb{R}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\mathbb{R}\begin{pmatrix}1&-\frac{\sqrt{6}}{3}\\\
0&0\end{pmatrix}.$

Observe that $\mathscr{A}$ is clearly not $*$-invariant since
$\left(\begin{smallmatrix}1&-\frac{\sqrt{6}}{3}\\\
0&0\end{smallmatrix}\right)^{*}\notin\mathscr{A}$. Now we consider the
sequence $(M_{2n})_{n}$ of even moments. The first moment is given by
$M_{0}=\left(\begin{smallmatrix}\frac{2}{3}&\frac{\sqrt{6}}{6}\\\
\frac{\sqrt{6}}{6}&\frac{1}{2}\end{smallmatrix}\right)$ and the algebras
$A^{M}_{0}$ and $\mathscr{A}^{M}_{0}$ are

$\displaystyle A^{M}_{0}=\mathbb{C}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\mathbb{C}\begin{pmatrix}\frac{\sqrt{6}}{6}&1\\\
1&0\end{pmatrix},$
$\displaystyle\mathscr{A}^{M}_{0}=\mathbb{R}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\mathbb{R}\begin{pmatrix}\frac{\sqrt{6}}{6}&1\\\
1&0\end{pmatrix}+\mathbb{R}\begin{pmatrix}1&-\frac{\sqrt{3}}{6}\\\
0&0\end{pmatrix}+i\mathbb{R}\begin{pmatrix}1&-\frac{2\sqrt{6}}{3}\\\
\frac{\sqrt{6}}{2}&-1\end{pmatrix}.$ (4.1)

This gives the inclusions
$\mathbb{R}I\subsetneq(A^{M}_{0})_{h}\subsetneq\mathscr{A}^{M}_{0}$. It is
also clear from (4.1) that
$\mathscr{A}^{M}_{0}\cap(\mathscr{A}^{M}_{0})^{*}=(A^{M}_{0})_{h}$.

Now we proceed as in Remark 3.7, we take the positive definite matrix $S$ such
that $M_{0}=S^{2}$. Here
$S=\frac{1}{15}\left(\begin{smallmatrix}\sqrt{6}+9&3\sqrt{6}-3\\\
3\sqrt{6}-3&\frac{3}{2}\sqrt{6}+6\end{smallmatrix}\right)$. Then

$\displaystyle
S^{-1}W(x)S^{-1}=\frac{1}{25}\begin{pmatrix}(33+12\sqrt{6})x^{2}+(28-8\sqrt{6})x&-(6+9\sqrt{6})x^{2}+(4+6\sqrt{6})x\\\
-(6+9\sqrt{6})x^{2}+(4+6\sqrt{6})x&(42-12\sqrt{6})x^{2}+(22+8\sqrt{6})x\end{pmatrix}.$

We finally have that

$\displaystyle\mathscr{A}(S^{-1}\Theta
S^{-1})=\mathbb{R}I+\mathbb{R}E+\mathbb{R}F,\quad\mathscr{A}(S^{-1}M_{0}S^{-1})=\mathbb{R}I+\mathbb{R}E+\mathbb{R}F+\mathbb{R}G,$

where

$\displaystyle E=S^{-1}\begin{pmatrix}\frac{\sqrt{6}}{6}&1\\\
1&0\end{pmatrix}S=\begin{pmatrix}\frac{\sqrt{6}}{6}&1\\\
1&0\end{pmatrix},\qquad G=iS^{-1}\begin{pmatrix}1&-\frac{2\sqrt{6}}{3}\\\
\frac{\sqrt{6}}{2}&-1\end{pmatrix}S=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},$
$\displaystyle F=S^{-1}\begin{pmatrix}1&-\frac{\sqrt{3}}{6}\\\
0&0\end{pmatrix}S=\frac{1}{25}\begin{pmatrix}11+4\sqrt{6}&-(2+3\sqrt{6})\\\
-(2+3\sqrt{6})&14-4\sqrt{6}\end{pmatrix}.$

Then we have the following inclusions:

$\displaystyle\mathbb{R}I=S^{-1}A(\Theta)_{h}S\subsetneq A\big{(}S^{-1}\Theta
S^{-1}\big{)}_{h}=\mathscr{A}(S^{-1}\Theta S^{-1}),$

and

$\displaystyle S^{-1}\big{(}A^{M}_{0}\big{)}_{h}S\subsetneq
A\big{(}S^{-1}M_{0}S^{-1}\big{)}_{h}=\mathscr{A}\big{(}S^{-1}M_{0}S^{-1}\big{)}.$

######  Example 4.2.

Our second example is a family of matrix-valued Gegenbauer polynomials
introduced in [9]. For $\ell\in\frac{1}{2}\mathbb{N}$ and $\nu>0$, let
$d\mu(x)=(1-x^{2})^{\nu-1/2}dx$ where $dx$ is the Lebesgue measure on $[-1,1]$
and let $W^{(\nu)}$ be the $(2\ell+1)\times(2\ell+1)$ matrix

$\displaystyle\big{(}W^{(\nu)}(x)\big{)}_{m,n}=\sum_{t=\max(0,n+m-2\ell)}^{m}\alpha_{t}^{(\nu)}(m,n)C_{m+n-2t}^{(\nu)}(x),$
$\displaystyle\alpha_{t}^{(\nu)}(m,n)=(-1)^{m}\frac{n!m!(m+n-2t)!}{t!(2\nu)_{m+n-2t}(\nu)_{n+m-t}}\frac{(\nu)_{n-t}(\nu)_{m-t}}{(n-t)!(m-t)!}\frac{(n+m-2t+\nu)}{(n+m-t+\nu)}$
$\displaystyle\hphantom{\alpha_{t}^{(\nu)}(m,n)=}{}\times(2\ell-m)!(n-2\ell)_{m-t}(-2\ell-\nu)_{t}\frac{(2\ell+\nu)}{(2\ell)!},$

where $n,m\in\\{0,1,\dots,2\ell\\}$ and $n\geq m$. The matrix $W^{(\nu)}$ is
extended to a symmetric matrix,
$\left(W^{(\nu)}(x)\right)_{m,n}=\left(W^{(\nu)}(x)\right)_{n,m}$. In [9,
Proposition 2.6] we proved that $A$ is generated by the identity matrix $I$
and the involution $J\in M_{2\ell+1}(\mathbb{C})$ defined by $e_{j}\mapsto
e_{2\ell-j}$

Now we will use Theorem 3.3 to prove that $A_{h}=\mathscr{A}$. This says that
there is no further non-unitary reduction of the weight $W$. As a sequence of
positive definite matrices we take the squared norms of the monic polynomials,
$(\Gamma_{n})_{n}=(H_{n})_{n}$, that were explicitly calculated in [9, Theorem
3.7] and are given by the following diagonal matrices

$\displaystyle\bigl{(}H^{(\nu)}_{n}\bigr{)}_{i,k}=\delta_{i,k}\sqrt{\pi}\frac{\Gamma(\nu+\frac{1}{2})}{\Gamma(\nu+1)}\frac{\nu(2\ell+\nu+n)}{\nu+n}\frac{n!(\ell+\frac{1}{2}+\nu)_{n}(2\ell+\nu)_{n}}{(\nu+k)_{n}(2\ell+2\nu+n)_{n}(2\ell+\nu-k)_{n}}$
$\displaystyle\hphantom{\bigl{(}H^{(\nu)}_{n}\bigr{)}_{i,k}=}{}\times\frac{k!(\ell+\nu)_{n}(2\ell-k)!(n+\nu+1)_{2\ell}}{(2\ell+\nu+1)_{n}(2\ell)!(n+\nu+1)_{k}(n+\nu+1)_{2\ell-k}}.$

For any $n\in\mathbb{N}$ we choose $\mathcal{I}=\\{n,n+1\\}$. If we take
$T\in\mathscr{A}^{\Gamma}_{\mathcal{I}}$, i.e.,
$TH^{(\nu)}_{n}=H^{(\nu)}_{n}T^{*}$ and
$TH^{(\nu)}_{n+1}=H^{(\nu)}_{n+1}T^{*}$, it follows that

$\displaystyle
T_{i,j}=\frac{\big{(}H^{(\nu)}_{n}\big{)}_{j,j}}{(H^{(\nu)}_{n})_{i,i}}\overline{T}_{j,i}=\frac{j!(2\ell-j)!(\nu+i)_{n}(2\ell+\nu-i)_{n}(n+\nu+1)_{i}(n+\nu+1)_{2\ell-i}}{i!(2\ell-i)!(\nu+j)_{n}(2\ell+\nu-j)_{n}(n+\nu+1)_{j}(n+\nu+1)_{2\ell-j}}\overline{T}_{j,i}.$

It follows directly from this equation that $T_{i,i}\in\mathbb{R}$ and
$T_{i,2\ell-i}=\overline{T}_{2\ell-i,i}$. Now we observe that

$\displaystyle
T_{i,j}=\frac{\big{(}H^{(\nu)}_{n}\big{)}_{j,j}}{\big{(}H^{(\nu)}_{n}\big{)}_{i,i}}\overline{T}_{j,i}=\frac{\big{(}H^{(\nu)}_{n}\big{)}_{j,j}}{(H^{(\nu)}_{n})_{i,i}}\frac{\big{(}H^{(\nu)}_{n+1}\big{)}_{i,i}}{\big{(}H^{(\nu)}_{n+1}\big{)}_{j,j}}T_{i,j}$
$\displaystyle\hphantom{T_{i,j}}{}=\frac{(\nu+j+n)(\nu+j+n+1)(2\ell+n+\nu-j)(2\ell+n+\nu+1-j)}{(\nu+i+n)(\nu+i+n+1)(2\ell+n+\nu-i)(2\ell+n+\nu+1-i)}T_{i,j}.$
(4.2)

Equation (4.2) implies that $T_{i,j}=0$ unless $j=i$ or $j=2\ell-i$. Hence $T$
is self-adjoint and thus $\mathscr{A}^{\Gamma}_{\mathcal{I}}$ is $*$-invariant
and from Theorem 3.3 we have $\mathscr{A}=A_{h}$. We conclude that
$\mathscr{A}$ is the real span of $\\{I,J\\}$, and so there is no further non-
unitary reduction.

######  Example 4.3.

Our last example is a $q$-analogue of the previous example for $\nu=1$. This
sequence of matrix-valued orthogonal polynomials matrix analogues of a
subfamily of Askey–Wilson polynomials and were obtained by studying matrix-
valued spherical functions related to the quantum analogue of
$\mathrm{SU}(2)\times\mathrm{SU}(2)$. For any $\ell\in\frac{1}{2}\mathbb{N}$
and $0<q<1$, we have the measure $d\mu(x)=\sqrt{1-x^{2}}\,dx$ supported on
$[-1,1]$ and a $(2\ell+1)\times(2\ell+1)$ weight matrix $W$ which is given in
[2, Theorem 4.8], we omit the explicit expression here and we give, instead,
the explicit expression for the squared norms $H_{n}$ of the monic orthogonal
polynomials

$\displaystyle(H_{n})_{i,j}=\delta_{i,j}\frac{q^{-2\ell}2^{-2n}(q^{2},q^{4\ell+4};q^{2})^{2}_{n}(1-q^{4\ell+2})^{2}}{(q^{2i+2},q^{4\ell-2i+2};q^{2})^{2}_{n}(1-q^{2n+2i+2})(1-q^{4\ell-2i+2n+2})}.$

The expression for $H_{n}$ is obtained combining Theorem 4.8 and Corollary 4.9
of [2]. The commutant algebra is generated by $\\{I,J\\}$ as in the previous
example, see [2, Proposition 4.10]. We take $(\Gamma_{n})_{n}=(H_{n})_{n}$,
$\mathcal{I}=\\{n,n+1\\}$ for any $n\in\mathbb{N}$ and observe that
$TH_{n}=H_{n}T^{*}$ and $TH_{n+1}=H_{n+1}T^{*}$ implies

$\displaystyle
T_{i,j}=\frac{\big{(}H^{(\nu)}_{n}\big{)}_{j,j}}{\big{(}H^{(\nu)}_{n}\big{)}_{i,i}}\overline{T}_{j,i}=\frac{\big{(}H^{(\nu)}_{n}\big{)}_{j,j}}{\big{(}H^{(\nu)}_{n}\big{)}_{i,i}}\frac{\big{(}H^{(\nu)}_{n+1}\big{)}_{i,i}}{\big{(}H^{(\nu)}_{n+1}\big{)}_{j,j}}T_{i,j}$
$\displaystyle\hphantom{T_{i,j}}{}=\frac{(1-q^{2n+2j+2})(1-q^{2n+2j+4})(1-q^{4\ell+2n-2i+2})(1-q^{4\ell+2n-2i+4})}{(1-q^{2n+2i+2})(1-q^{2n+2i+4})(1-q^{4\ell+2n-2j+2})(1-q^{4\ell+2n-2j+4})}T_{i,j}.$

As in the previous example, it follows that $T$ is self-adjoint and therefore,
$\mathscr{A}=A_{h}$. Hence there is no further non-unitary reduction for $W$.

######  Remark 4.4.

Theorem 3.3 can be used to determine the irreducibility of a weight matrix. In
fact, with the commutant algebras already determined in [9] and [2], Theorem
3.3 implies that the restrictions of the weight matrices of Examples 4.2 and
4.3 to the eigenspaces of the matrix $J$ are irreducible. For some explicit
cases see [10, Section 8].

### Acknowledgements

We thank I. Zurrián for pointing out a similar example to Example 4.1 to the
first author. The research of Pablo Román is supported by the Radboud
Excellence Fellowship. We would like to thank the anonymous referees for their
comments and remarks, that have helped us to improve the paper.

## References

  * [1] Aldenhoven N., Koelink E., de los Ríos A.M., Matrix-valued little $q$-Jacobi polynomials, J. Approx. Theory 193 (2015), 164–183, arXiv:1308.2540.
  * [2] Aldenhoven N., Koelink E., Román P., Matrix-valued orthogonal polynomials for the quantum analogue of $({\rm SU}(2)\times{\rm SU}(2),\mathrm{diag})$, arXiv:1507.03426.
  * [3] Álvarez-Nodarse R., Durán A.J., de los Ríos A.M., Orthogonal matrix polynomials satisfying second order difference equations, J. Approx. Theory 169 (2013), 40–55.
  * [4] Berg C., The matrix moment problem, in Coimbra Lecture Notes on Orthogonal Polynomials, Editors A. Branquinho, A. Foulquié Moreno, Nova Sci. Publ., New York, 2008, 1–57.
  * [5] Damanik D., Pushnitski A., Simon B., The analytic theory of matrix orthogonal polynomials, Surv. Approx. Theory 4 (2008), 1–85, arXiv:0711.2703.
  * [6] Grünbaum F.A., Tirao J., The algebra of differential operators associated to a weight matrix, Integral Equations Operator Theory 58 (2007), 449–475.
  * [7] Heckman G., van Pruijssen M., Matrix-valued orthogonal polynomials for Gelfand pairs of rank one, Tohoku Math. J., to appear, arXiv:1310.5134.
  * [8] Horn R.A., Johnson C.R., Matrix analysis, Cambridge University Press, Cambridge, 1985.
  * [9] Koelink E., de los Ríos A.M., Román P., Matrix-valued Gegenbauer polynomials, arXiv:1403.2938.
  * [10] Koelink E., van Pruijssen M., Román P., Matrix-valued orthogonal polynomials related to $({\rm SU}(2)\times{\rm SU}(2),{\rm diag})$, Int. Math. Res. Not. 2012 (2012), 5673–5730, arXiv:1012.2719.
  * [11] Koelink E., van Pruijssen M., Román P., Matrix-valued orthogonal polynomials related to $({\rm SU}(2)\times{\rm SU}(2),{\rm diag})$, II, Publ. Res. Inst. Math. Sci. 49 (2013), 271–312, arXiv:1203.0041.
  * [12] Tirao J., Zurrián I., Reducibility of matrix weights, arXiv:1501.04059.


# Bolidozor - Distributed radio meteor detection system

Jakub Kakona 1, Pavel Kovar 1, Martin Kakona 2

1 Dept. of Radio Engineering,
Czech Technical University in Prague,
Technicka 2,
166 27 Praha 6,
Czech Republic

2 Dept. of Dosimetry and Application of Ionizing Radiation,
Czech Technical University in Prague,
Brehova 7,
115 19 Praha 1,
Czech Republic kakonjak<EMAIL_ADDRESS>
###### Abstract

Most of the meteor radioastronomical radars are backscatter radars which cover
only a small area of the atmosphere. Therefore a daytime meteor flux models
are based on sparse data collected by only a few radar systems. To solve this
issue, a radar system with a wide coverage is required. We present a new
approach of open-source multi-static radio meteor detection system which could
be distributed over a large area. This feature allows us to detect meteor
events taking place over a larger area as well and gather more uniform data
about meteor flux and possibly about meteor trajectories. Therefore, a new
network which detects atmospheric meteor impacts and produce scientifically
valuable data about each detected event is needed.

##  1 Introduction

Meteors, the products of meteoroids travelling in space, are studied for many
decades. Results of such research help us to understand the evolution of not
only the planetary system but the interplanetary medium [2] as well. As the
density of the interplanetary medium is low, large statistical and long term
continuous data set is necessary to describe its properties.

###  1.1 Optical observation methods

Historically, the observations were based mainly on optical methods such as
camera fireball network [3]. Results of these observations were not
continuous, a fact which was caused by a non-uniform coverage and sensitivity
of detection systems. There are several reasons for data absence, one of them
being a weather dependency of the observation method. It results from the fact
that optical method needs mainly clear air and low light background [4].
Therefore, optical stations cannot be operated under inappropriate weather
conditions or during the daytime.

###  1.2 Radio observation methods

To overcome these issues, alternative observation methods were developed.
Apart form the infrasonic and seismic methods which are suitable for detection
of large meteor impacts, there exist two radio waves observation methods which
use the scattering ability of the ionized meteor trail [6].

The oldest known radio method is a backscatter radar. It is an ordinary type
of radar which expects meteor trails to be reflective targets. Currently,
several radars of this type are in operation to study meteors SkiYMet [5],
CMOR [7].

However, all of these monostatic or bistatic radars have very small detection
coverage, usually limited to the radar antenna field of view. Therefore this
radar types observe only several spatially limited areas of the Earth’s
atmosphere. But the benefits of using this method are obvious - radio meteor
detection capability is not dependent on the current weather and can work even
during the daylight or nights with full Moon [9].

Besides the above-mentioned scientific radar systems a multistatic radio
meteor detection networks have evolved [10]. These forward scattering
multistatic systems have great advantage of a large detection coverage.
Unfortunately, the current spread of this technology is not sufficient to
entirely cover the meteor flux of the Earth’s atmosphere. Therefore a new
optimized and low cost Bolidozor network was developed. Bolidozor uses
multistatic forward scattering approach which allows an efficient use of the
radar transmitter energy to maximize the information value collected from the
meteor reflection.

##  2 Forward scattering Method

The general principle of meteor observing by the forward scattering of radio
waves off their trails is illustrated in figure 1. A radio receiver with
operating frequency range of 30-200MHz is located at proper distance (about
500-2000 km) from a transmitter. A curvature of the Earth over this distance
ensures no possibility of direct radio wave contact. When a meteoroid enters
the atmosphere, its meteor trail may reflect the radio waves emitted by
transmitter to the receiver. The signal can be received until the ionized
meteor trail recombines. Reflections can last from a tenths of a second to a
few minutes, depending on their frequency and ionization intensity. The
received signal characteristics are related to physical parameters of the
meteoric event [8].

Figure 1: The method of radio meteor detection based on the forward scattering
radar system

Bolidozor network currently uses GRAVES [12] transmitter located in France,
which transmits a CW (continuous wave) signal at a frequency of 143.05 MHz. A
radiation of the transmitter is directed mainly to the south hemisphere, but
due to the imperfections of its antenna system, the signal from meteor
reflections can be observed in almost all European countries. Therefore the
transmitter is suitable for receivers’ network operations with the aim of
meteor trail detection and the development of algorithms for the calculation
of meteor trajectory.

Figure 2: Bolidozor stations network

The use of such high frequency beacon has one main advantage over the previous
experiments. Previous attempts used longer wavelengths in frequency range
20-50 MHz. Such long wavelengths were used to obtain a higher sensitivity to
finer meteor trails. According to a simplified formula (1), where $T$ is
exponential time constant and $D$ is ambipolar diffusion coefficient and
$\lambda$ is wavelength [11], we have longer meteor echos from meteors with
the same ionisation energy observed by longer wavelengths compared to meteor
reflections observed by shorter wavelengths. But shorter wavelengths allow us
to detect finer details of meteor trails.

$T=\frac{\lambda^{2}}{16\pi^{2}D}$ (1)

The head echo of meteor is usually called as overdense due to its relatively
high plasma frequency compared to used observation frequency ($F_{obs}$) in
the front of the meteoroid shock wave that is created in the air. This
condition is expressed by the equation (2). However, if we use a frequency
near to the plasma frequency ($f_{pe}$) of the meteor trail we can distinguish
the head echo and meteor trail reflection because the Doppler shift is applied
on the part of reflected signal. This situation is shown in the figure 3 where
head echoes are marked by sloped dotted lines. Static meteor trail reflections
are marked by straight vertical lines.

Figure 3: Example of meteor reflections for multiple stations - Spectrograms
show the time aligned signal evolution over vertical axis. (The oldest data
being at the bottom) Horizontal axis corresponds to frequency. (Highest
frequency on right)

$F_{obs}<<f_{pe}=\frac{\sqrt{\frac{n_{e}e^{2}}{m\epsilon_{0}}}}{2\pi}$ (2)

  * •

$n_{e}$ is the number density of electrons,

  * •

$e$ is the elementary electrical charge,

  * •

$m$ is the effective mass of the electron,

  * •

$\epsilon_{0}$ is the permittivity of vacuum.

Obviously not all parts of the meteor trail are observable from one station as
the signal can be scattered to different directions non-specularly. But if we
use multiple receiving stations, we greatly increase the statistical
sensitivity because several stations could be located on the reflection spots.
Therefore it is very useful to have stations in a form of cooperative
detection network. The network of stations has many advantages over a single
transmitter - single receiver configuration. For example it brings robustness
which allows operability even in case that a part of the system is under a
maintenance and therefore not functional.

There also exist signal processing advantages, especially if we want to
compute meteor parameters such as its velocity and trajectory from the meteor
radio observation. All physical parameters we could determine from the single
reflection are signal intensity and frequency shift in a given time which
corresponds to an ionisation intensity and bi-static velocity. Therefore we
must combine information about one meteor event from multiple stations to
obtain points in space corresponding to the meteor trajectory. We should
alight the events according to time stamps. Therefore, a precise time
synchronization between stations is required. The exact required precision
depends on the network geometry. But if we want to work in 300 m distance
resolution, which is typical for the current optical methods, we need the time
precision on approximately the microseconds scale.

##  3 Network Architecture

Bolidozor network consists of several types of stations: the reception
stations, data servers and processing stations. Interconnection of these
network nodes is displayed in figure 4.

Network data are generated exclusively by Radio Meteor Detection Station
(RMDS) [16] with current version RMDS02D and several other network data source
stations are planed (Video meteor stations, infrasonic station etc.).

Real-time transfer of all data volume is impractical because of stations’
internet connection bandwidth limits and extensive data server load. Moreover,
the stations are usually situated in remote sites or observatories without
fast internet connection. Therefore only the meteor events are recorded and
stored locally on station’s SDcard storage media. This design minimizes the
network connectivity bandwidth demands and connection reliability.

The storage is large enough to handle several days of data records. The data
from station storage is periodically transferd to the central data storage
server (space.astro.cz). Process is controlled by standard utility Rsync which
is commanded by python script.

The central data server currently contains around 7 TB of data (since 2014)
and increases in size by approximately 1GB of data per station per day [17].
All 9 million of data records are available for public via the HTTP requests.
The server is backuped by CESNET Storage Department.

Current centralized storage system is not optimal because expanding the
storage capacity is quite complicated and expensive. Another decentralised
approach based on purely data storage stations connected to the network was
considered, but temporary rejected for its system’s complexity and current
unavailability of suitable tools.

Figure 4: Bolidozor network interconnection schema.

Stored data are processed by several separate data processing servers.
Statistical processing is continuously performed on the stored data using the
following technologies and programming languages: R, RapidMiner, Python [18]
and Jupyter. One example of an existing data processing facility is the system
meteor1.astrozor.cz which downloads metadata files, searches for meteor events
and prepares the outputs for rmob.org site. Data processing system is
continuously improved. The first implementation used was RapidMiner framework,
but recent advantages in open-source scientific tools development allow us to
use more sophisticated technologies like Jupyter [19]. More interactive
computational tools like [20] emerged recently.

Several tools are prepared for simplification of potential data retrieving
from the storage server. For example a python module bzbrowser was created.
This module helps with the effective data downloading from the server so that
anyone can use Bolidozor data for data mining.

###  3.1 Detection Station Software

Each detection node runs our software configuration on ARM based computer with
Linux OS. The signal from receiver is captured by a modified sdr-widget server
from GHPSDR3 project [15]. SDR-widget server accepts socket connections from
other processing and displaying software clients. The main software component
is radio-observer code which detects the meteor events in data stream and
creates data records as FITS files.

Figure 5: Example of snapshot spectrogram with several meteor reflections.
(Vertical axis correnponds to time (60 s range), horizontal axis to frequency
(600 Hz range)

We are currently using several types of FITS recordings.

  * •

A ’snapshot’ format continually records spectrograms of every minute. These
spectrograms record only a small part of spectra where meteor echoes usually
appear - an example of snapshot record is shown in figure 5. The purpose of
snapshots is an illustrative recording of events which are not detected by
radio-observer software. This could be useful for a new event type discovery
or station debugging. e.g. if noise interference appears.

  * •

A more detailed format is represented by a ’met FITS’ which contains
spectrogram of the meteor reflections. The most detailed output file is the
"raw FITS" which contains signal samples directly from the analog to digital
converter. The RAW FITS is intended primarily for post-processing.

All FITS files contain metadata information in FITS’s header, but for
effective searching in data structure two metadata files are created in CSV
format. First is meta.csv which contains information about events detected by
radio-observer software e.g. meteor echo duration, frequency of maximal energy
etc. Another output is freq.csv which contains recordings about local
oscillator drifts and its corrections.

###  3.2 Trajectory estimation possibility

Every meteor trajectory in atmosphere create its own Doppler shift footprint.
This process could be easily modelled by a numerical calculation of Doppler
shifts in points of trajectory. For a simplicity, the presented model expects
constant velocity along a straight line of the meteor path which is divided to
equidistant time samples. A numerical difference of path distances between
transmitter, meteor and receiver is calculated. Then velocity and Doppler
shift value is obtained for every point of trajectory. The resulting figure of
Doppler shifts calculated along the model meteor path visible to every
existing station is shown in figure 6.

Figure 6: Doppler shifts calculated for meteor ground path displayed in the
figure 2.

This signal model can be easily confirmed from meteor database where several
meteor events are detected on multiple stations. If we plot such meteor event
in time aligned spectrogram we obtain an image similar to the figure 3.
Precise meteor trajectory estimation methods are intensively investigated at
the moment. One of the difficulties is a suboptimal geometry situation and low
inter-station events correlation. Therefore expanding of the network is
another task which runs in parallel.

###  3.3 Station Hardware

Bolidozor network main objective is a calculation of meteor trajectories from
echoes. Therefore, we needed to develop a high performance receiver with
unique parameters, especially with a high quality of time synchronization.
Tight time synchronization requirements between nodes increase the complexity
of the receiver system. To simplify the development process we used MLAB open
source electronic prototyping platform. Therefore the whole station has an
open hardware design [13].

Figure 7: The Bolidozor detection station version RMDS02D.

The use of modular concept allows us on-the-fly modifications and testing of
new station’s features. After verifying the new hardware features, other
stations could be easily updated by adding or replacing MLAB modules. The
current station hardware is shown in figure 7.

The station receiver consist of several subsystems shown in figure 8. At the
input, the RF signal from a low noise amplifier placed near the antenna is
conducted by a coaxial cable to the sampled mixer module. The mixer module is
fed by the high stability local oscillator signal that is modulated by the
time stamps. The local oscillator signal is synthesised by the TCXO
(temperature compensated oscillator). Frequency errors of the oscillator are
measured with the help of the GPS and used for a long term error compensation
of the local oscillator frequency. Precise measurement of the oscillator
frequency is executed by a frequency measurement circuit which completes the
measurement once every ten seconds in absolute UTC time. Frequency measurement
intervals of local oscillator are controlled by GPS PPS output. Measured
oscillator parameters (frequency and temperature) are processed by a
frequency-guard software running on the station computer. The software
determines if some correction of a local oscillator is necessary and
compensates the oscillator parameters. Therefore the resulting Allan variance
of the local oscillator signal could be very low.

The time marks in the meteor data are coded by PPS signal from GPS receiver
which triggers a divider connected to the receiver and to the frequency
counter, the clock signal to the signal mixer is interrupted by PPS impulses.
Interruption of clock signal generates wide band spike which appears as
horizontal line in spectrogram of the RF signal. Interruptions of signal could
then be used as precise time stamps in post-processing algorithms. GPS Time
marks linked to UTC have much better time resolution compared to the one used
in current digitalization unit.

Low frequency output of the mixer is directly digitized by SDR-widget
compatible hardware [14]. The signal is currently digitized with 24bit
resolution with sampling rate of 96 kHz. But parameters will be further
improved by altering the station configuration. Actual precision are
considered as adequate for a current stage of development process.

Figure 8: Block schematics of RMDS station.

###  3.4 Receiver parameters

The minimum detectable signal at receiver input is approximately -120 dBm for
143 MHz with currently used software configuration where FFT resolution is
around 4 Hz. However, we have placed a LNA01A low noise amplifier before the
receiver directly at the antenna output. Gain of LNA01A is around 25 dB and
noise figure of this amplifier is around 0.6 dB. Such design allows us the use
of small omnidirectional antennas for detection of lager meteors. We currently
use a 1/4 lambda Ground Plane antenna for easy deployment to as many stations
as possible, but another types of antennas with better parameters are
currently in development process.

##  4 Conclusion

A new type of radio meteor detection system is being build, based on a new
idea of distributed scientific measurement systems. At the moment Bolidozor
network produces large volume of valuable radioastronomy data ready for
further processing. We currently have a large database of multi-station meteor
reflections. Unfortunately, we have yet not reached suitable network geometry
to obtain meteor position and trajectory by the known methods [21]. To
overcome this issue, the further evolution of Bolidozor network should be
focused on the research of the new trajectory estimation algorithms. We are
also working on the network extension with the aim of extension of the service
area as well as increasing the stations density.

The Bolidozor network depends primarily on volunteers that run and maintain
the network nodes, therefore their work should be appreciated accordingly. We
would like to thank the Czech Astronomical society for maintenance of our
central data server space.astro.cz Thanks to Center of machine perception of
Czech Technical University for financial support of Bolidozor system.

Thanks to SDR-widget development team for open source hardware and related
software. The study is supported by the SGS grant No SGS16/166/OHK3/2T/13 of
the Czech Technical University in Prague.

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