# Preformation Probability and Kinematics of Clusters Emission yielding Pb-
daughters

Joshua T. Majekodunmi1<EMAIL_ADDRESS>M. Bhuyan2,3<EMAIL_ADDRESS>K.
Anwar1 N. Abdullah1 Raj Kumar4<EMAIL_ADDRESS>1Institute of Engineering
Mathematics, Universiti Malaysia Perlis, Arau, 02600, Perlis, Malaysia
2Center for Theoretical and Computational Physics, Department of Physics,
Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia
3Institute of Research and Development, Duy Tan University, Da Nang 550000,
Vietnam  4School of Physics and Materials Science, Thapar Institute of
Engineering and Technology, Patiala, Punjab 147004, India

###### Abstract

In the present study, the newly established preformation formula is applied
for the first time to study the kinematics of the cluster emission from
various radioactive nuclei, especially those decaying to the double-shell
closure 208Pb nucleus and its neighbours as daughters. The recently proposed
universal cluster preformation formula has been established based on the
concepts that underscore the influence of the mass and charge asymmetry
($\eta_{A}$ and $\eta_{Z}$), cluster mass $A_{c}$ and the Q-value, paving the
way to quantify the energy contribution during the preformation as well as the
tunnelling process separately. The cluster-daughter interaction potential is
obtained by folding the relativistic mean-field (RMF) densities with the
recently developed microscopic R3Y using the NL$3^{*}$ and the
phenomenological M3Y NN potentials to compare their adaptability. The
penetration probabilities are calculated from the WKB approximation. With the
inclusion of the new preformation probability $P_{0}$, the predicted half-
lives from the R3Y and M3Y interactions are in good agreement with the
experimental data. Furthermore, a careful inspection reflects slight
differences in the decay half-lives, which arise from their respective barrier
properties. The $P_{0}$ for the systems with the double magic shell closure
208Pb daughter are found to be relatively higher with an order of $\approx
10^{2}$ than those with neighbouring Pb-daughter nuclei. By exploring the
contributions of the decay energy, the recoil effect of the daughter nucleus
is appraised, unlike several other conjectures. Thus, the centrality of the
Q-value in the decay process is demonstrated and re-defined within the
preformed cluster-decay model. Besides, we have introduced a simple and
intuitive set of criteria that governs the estimation of recoil energy in the
cluster radioactivity.

###### pacs:

21.65.Mn, 26.60.Kp, 21.65.Cd

##  I INTRODUCTION

Clustering is one of the notable dynamical attributes of an atomic nucleus
exhibiting regular patterns, despite the complexities associated with the
nuclear many-body systems. Its prediction dates back to the theoretical
investigation of Sandulescu et al. in 1980 sand80  in which the shell closure
effect of one of the reactants was used to reproduce a cold reaction based on
the fragmentation theory gupt94 . The findings were subsequently validated by
Rose and Jones rose84  where cluster radioactivity was established as a highly
asymmetric spontaneous disintegration of radioactive nuclei in which the
emitted particle is heavier than 4He and yet smaller than the lightest fission
fragments. So far, the emission of 14C, 18,20O, 23F, 22,24-26Ne, 28,30Mg and
32,34Si clusters from various trans-lead nuclei (221Fr - 242Cm) have been
observed bone07 . The emitted clusters in this region are usually associated
with the double magic nucleus ${}^{208}_{82}$Pb or nuclei in its vicinity as
daughters. Considering the kinematics of cluster emission, earlier studies
qian12 ; deng15  have revealed that like $\alpha$-decay and spontaneous
emission, the rate at which clusters are emitted from odd-parent nuclei is
confronted with more structural hindrance as compared to the rate of cluster
emissions from its neighbouring even-even isotopes. Moreover, it has been
shown that there is a substantial difference between the observed kinetic
energy and Q-value of the cluster decay, suggesting a considerable recoil
effect hoos05 .

From the theoretical viewpoint, cluster radioactivity follows the description
of the Gamow model of $\alpha$-decay which hinges on the quantum tunnelling
effect. This description can be grouped into two main categories based on
their treatment of cluster emission namely fission and $\alpha$-like models.
The fission models e.g the analytic super asymmetric fission model (ASAFM) of
Poenaru et al. poen85 ; poen86  assume that the parent nucleus undergoes
continuous deformation until it penetrates the confining interaction barrier
and thus attains the saddle configuration. This approach takes no cognizance
of the preformation of the cluster within the parent nucleus before its
emission. In other words, preformation probability ($P_{0}$) is taken as
unity. In contrast, the $\alpha$-like models like the preformed cluster-decay
model (PCM) mali89 ; gupt88 ; wei17 , which is rooted in the quantum
mechanical fragmentation theory (QMFT), assume that clusters are composed of
several nucleons pre-born within the parent nucleus before tunnelling through
the potential barrier. Thus, realistic values of $P_{0}$ can be calculated and
as a result, the experimental half-lives can be accurately reproduced. The
literature blen88 ; sant21c ; ni10 ; bala14 ; deng14  is replete with
different expressions to estimate $P_{0}$ of which the predictive power of
most are either restricted to a certain region of the nuclear chart or fitted
with some arbitrary constants with no clear link/relevance to the kinematics
and underlying concept of cluster emission.

Deliberate attention has been given to this concept in the derivation of our
newly proposed preformation formula josh22L  and is extended in the present
study to reproduce the experimentally measured cluster decay half-lives.
Besides, by exploring the mechanism and kinematics in the decay channel, we
contemplate that a certain amount of energy must be expended during cluster
formation just before its emission, unlike previous studies ropk14 ; xu16
where much emphasis was given to the kinetic energy but the recoil energy was
assumed to be negligible. Assumption such as Levinger’s approximation levi53
where the recoil energy is considered too small becomes invalid for natural
radioactivity in which heavy ions are emitted stra01 . Hence, this study is
aimed at investigating the systematic contribution of the decay energy into
three distinct parts, accounting for cluster preformation, emission or
tunnelling and the residual energy with which the daughter nucleus recoils.
Also, the relative separation between the decay fragments denoted $\Delta$R is
used to account for the neck-formation effect which decides the first turning
point for the barrier penetration within the PCM kuma12c  which is employed
here for the present investigation. This barrier is formed by the interplay of
the Coulomb and nuclear potential.

The Coulomb potential can be simply estimated as the ratio of the product of
the charges of the decay fragments (emitted cluster and the daughter nucleus)
to the sum of their radii. However, obtaining the nuclear potential usually
involves the use of either the phenomenological quen78 ; horn75  or
microscopic approaches schu16 ; vaut72 . Besides the fundamental approaches
epel09 ; ekst13 , the R3Y nucleon-nucleon (NN) potential sing12 ; sing10
which stems from the relativistic mean-field (RMF) Lagrangian using the
NL$3^{*}$ parameter set is employed in the present study along with the
phenomenological M3Y NN potential satc79 . The RMF theory is apt to take care
of the ground and excited-state properties of the atomic nuclei Bisw20 ;
Itag20 ; Tani20 . The Q-values are also calculated from RMF (NL$3^{*}$) and
are compared with the macroscopic-microscopic WS3 liu11  and those obtained
from the experimental binding energy data wang17 . The WKB approximation is
used to estimate the penetration probability $P$. The paper is presented in
the following manner: Section II gives a brief description of the theoretical
framework which includes the relativistic mean-field (RMF) and the folding
procedure for R3Y and M3Y NN potentials. The preformed cluster-decay model
(PCM) and the new $P_{0}$ formula yielding a set of new equations are also
presented. Section III details the presentation of the results and their
corresponding discussions. Finally, the conclusion and summary of this work
are given in Section IV.

##  II Theoretical formalism

The relativistic mean-field (RMF) Lagrangian is built from the interaction
between the nucleonic field and the three mesonic fields, isoscalar–scalar
$\sigma$, isoscalar–vector $\omega$, and isovector–vector $\rho$ as well as
the photon field $A_{\mu}$ together with their respective coupling constants
($g_{\sigma}$, $g_{\omega}$, $g_{\rho}$), is given by jos22a ; ring96 ; bhu15
; bhu20

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\overline{\psi}_{i}\left\\{i\gamma^{\mu}\partial_{\mu}-M\right\\}\psi_{i}+\frac{1}{2}\partial^{\mu}\sigma\partial_{\mu}\sigma$
(1)
$\displaystyle-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}-g_{\sigma}\overline{\psi}_{i}\psi_{i}\sigma$
$\displaystyle-\frac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\frac{1}{2}m^{2}_{\omega}\omega^{\mu}\omega_{\mu}-g_{\omega}\overline{\psi}_{i}\gamma^{\mu}\psi_{i}\omega_{\mu}$
$\displaystyle-\frac{1}{4}\vec{B}^{\mu\nu}.\vec{B}_{\mu\nu}+\frac{1}{2}m^{2}_{\rho}\vec{\rho}^{\mu}.\vec{\rho}_{\mu}-g_{\rho}\overline{\psi}_{i}\gamma^{\mu}\vec{\tau}\psi_{i}.\vec{\rho}^{\mu}$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-e\overline{\psi}_{i}\gamma^{\mu}(\frac{1-\tau_{3i}}{2})\psi_{i}A_{\mu}.$

Parameters $g_{2}$, $g_{3}$ and $\frac{e^{2}}{4\pi}$ are the coupling
constants of the non-linear terms. The third component of the isospin is
$\tau_{3i}$. $M$ is the mass of nucleons while the masses of $\sigma$,
$\omega$ and $\rho$-mesons are $m_{\sigma}$, $m_{\omega}$, $m_{\rho}$ and
respective fields $\omega^{\mu}$, $\vec{\rho}_{\mu}$ and $A_{\mu}$. It is
worth noting that the contribution of the $\pi$-meson has been omitted in Eq.
(1) in the mean-field calculation as a result of its pseudoscalar nature
ring96 ; sero86 . A detailed description of the field tensors for
$\omega^{\mu}$, $\vec{\rho}_{\mu}$ and $A_{\mu}$ fields can be found in Ref
sing22  and the references therein. The field tensors are treated as classical
fields and thus, the Dirac equation is obtained for the nucleons and
simplified as

$[-i\alpha.\nabla+\beta(M^{*}+g_{\sigma}\sigma)+g_{\omega}\omega+g_{\rho}\tau_{3}\rho_{3}]\psi_{i}=\epsilon_{i}\psi_{i}.\\\
$ (2)

Similarly, the Klein-Gordon equations for the participating mesons are
simplified as

$\displaystyle(-\nabla^{2}+m^{2}_{\sigma})\sigma(r)$ $\displaystyle=$
$\displaystyle-g_{\sigma}\rho_{s}(r)-g_{2}\sigma^{2}(r)-g_{3}\sigma^{3}(r),$
$\displaystyle(-\nabla^{2}+m^{2}_{\omega})V(r)$ $\displaystyle=$
$\displaystyle g_{\omega}\rho(r),$
$\displaystyle(-\nabla^{2}+m^{2}_{\rho})\rho(r)$ $\displaystyle=$
$\displaystyle g_{\rho}\rho_{3}(r).$ (3)

This equations are solved self consistently using the NL3∗ parameter set.
Within the limit of one-meson exchange for a heavy and static baryonic medium,
the microscopic R3Y NN potential is obtained as

$\displaystyle V_{eff}^{R3Y}(r)$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{\omega}}{4\pi}\frac{e^{-m_{\omega}r}}{r}+\frac{g^{2}_{\rho}}{4\pi}\frac{e^{-m_{\rho}r}}{r}-\frac{g^{2}_{\sigma}}{4\pi}\frac{e^{-m_{\sigma}r}}{r}$
(4)
$\displaystyle+\frac{g^{2}_{2}}{4\pi}re^{-2m_{\sigma}r}+\frac{g^{2}_{3}}{4\pi}\frac{e^{-3m_{\sigma}r}}{r}+J_{00}(E)\delta(s),$

where $J_{00}(E)\delta(s)$ is the zero-range pseudopotential denoting the
exchange effect. Eq. (4) is similar to the phenomenological prescription of
Reid-Elliott satc79  called M3Y NN potential which is constructed to reproduce
the G-matrix element. The M3Y NN potential takes the form

$V_{eff}^{M3Y}(r)=7999\frac{e^{-4r}}{4r}-2134\frac{e^{-2.5r}}{2.5r}+J_{00}(E)\delta(s).$
(5)

The double folding technique satc79  is employed to estimate the nuclear
interaction potential $V_{n}(R)$ and expressed as

$V_{n}(R)=\int dr_{c}\int
dr_{d}\rho_{c}(\vec{r}_{c})\rho_{d}(\vec{r}_{d})V_{eff}(\vec{r}_{cd}=\vec{R}+\vec{r}_{d}-\vec{r}_{c}),$
(6)

where $\rho_{c}$ and $\rho_{d}$ are the nuclear matter densities of the
cluster and daughter nuclei. $V_{n}(R)$ given by Eq. (6) combines with the
Coulomb potential $V_{C}(R)=\frac{Z_{c}Z_{d}}{R}e^{2}$ to obtain the total
interaction potential

$V(R)=V_{n}(R)+V_{C}(R),$ (7)

which is used to estimate the WKB penetration probability (as illustrated in
Fig. 1) and hence, the cluster decay half-lives using the preformed cluster-
decay model (PCM) kuma12c . The penetration probability of clusters across the
tunnelling path is given as

$P=P_{a}W_{i}P_{b}$ (8)

which involves a three step process, shown and discussed in Fig. 2.

$\displaystyle P_{a}$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{2}{\hbar}\int^{R_{i}}_{R_{a}}\\{2\mu[V(R)-V(R_{i})]\\}^{1/2}dR\right),$
(9) and $\displaystyle P_{b}$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{2}{\hbar}\int_{R_{i}}^{R_{b}}\\{2\mu[V(R_{i})-Q]\\}^{1/2}dR\right).$
(10)

###  II.1 Preformed cluster-decay model (PCM)

The decay half-life $T_{1/2}$ within the preformed cluster-decay model (PCM)
can be defined in term of the decay constant $\lambda$, penetration
probability $P$, and preformation probability $P_{0}$ with the expression

$T_{1/2}=\frac{\ln 2}{\lambda},\hskip 14.22636pt\lambda=\nu_{0}P_{0}P.$ (11)

The assault frequency $\nu_{0}$ has nearly constant value of $10^{21}$ s-1 and
can be calculated as

$\nu_{0}=\frac{\mbox{ velocity }}{R_{0}}=\frac{\sqrt{2E_{c}/\mu}}{R_{0}},$
(12)

where $R_{0}$ denotes the radius of the parent nucleus and $E_{c}$ is the
kinetic energy of the emitted cluster. The Q-values are estimated from the
ground state binding energies from RMF, AME2016 wang17 , WS3 liu11  mass
tables using the expression

$Q=BE_{p}-(BE_{d}+BE_{c}),$ (13)

where $BE_{p}$, $BE_{d}$ and $BE_{c}$ are the binding energies of the parent,
daughter nuclei and the emitted cluster respectively.

Instead of the primitive cluster-mass dependent preformation formula of
Blendowske and Walliser blen88 , here, we have given a close attention to
study the relationships among various theoretically established
properties/factors that influences cluster preformation such as the cluster
mass $A_{c}$ sing11 , mass and charge asymmetries
$\eta_{A}=(A_{d}-A_{c})/(A_{d}+A_{c})$ and
$\eta_{Z}=(Z_{d}-Z_{c})/(Z_{d}+Z_{c})$ (since the emission of the same cluster
from different parent nuclei as well as different clusters from the same
parent nucleus is an experimentally observed fact bone07 ; gupt94 ; bone99 ),
the relative separation between the centers of the fragments
$r_{B}=1.2(A_{c}^{1/3}+A_{d}^{1/3})$ deli09 ; qian12  and the Q-value isma14 .
Hence, we have proposed a new $P_{0}$ formula josh22L ,

$\log P_{0}=-\frac{aA_{c}\eta_{A}}{r_{B}}-Z_{c}\eta_{Z}+bQ+c,$ (14)

where $a$, $b$ and $c$ are the fitting parameters in Ref.josh22L . The measure
of accuracy is evaluated using the $\chi^{2}$ expression

$\chi^{2}=\sum^{n}_{i=1}\frac{\left[\log_{10}^{Expt.}T_{1/2}-\log_{10}^{cal}T_{1/2}\right]^{2}}{\log_{10}^{cal}T_{1/2}}$

for 14 even-even nuclei and 5 odd-A nuclei whose values are also given in
Table 1 of Ref. josh22L .As explained in the $3^{rd}$ footnote of the table,
it is worth mentioning that only the experimentally measured systems yielding
Pb-daughters are preferentially considered in this fundamental study. Besides,
it is interesting to note that the third term on the right-hand side of Eq.
(14) opens a new window to probe the contributions of the decay energy. In
other words, the term $bQ$ gives a quantitative description of the energy
contributed in cluster formation. Thus, for the first time, the Q-value is
presented in terms of its usage/disbursement in the kinematics of cluster
emission as

$\displaystyle Q=\overbrace{\underbrace{bQ}_{\begin{subarray}{c}\text{energy
}\\\ \text{contributed in}\\\ \text{cluster
formation}\end{subarray}}+\underbrace{\kappa\sqrt{Q}}_{\begin{subarray}{c}\text{energy
}\\\ \text{contributed in}\\\ \text{cluster
emission}\end{subarray}}}^{E_{c}}+\underbrace{E_{d}}_{\begin{subarray}{c}\text{recoil}\\\
\text{energy of}\\\ \text{daughter nucleus}\end{subarray}}$ (15)

where the $\kappa\sqrt{Q}$ is the energy contributed in cluster emission.
Further, following the work of Gupta et al. sing11 , the kinetic energy of the
emitted cluster is expressed as

$\displaystyle E_{c}=\frac{A_{d}}{A}Q=bQ+\kappa\sqrt{Q}.$ (16)

On little simplification, we get

$\kappa=\sqrt{Q}\left(\frac{A_{d}}{A}-b\right).$ (17)

The quantity $\kappa$ in Eq.(17) refers to the tunneling factor. Detailed
explanation and implication of the newly derived Eq.s (14)-(17) are typified,
analysed and discussed in the subsequent section.

##  III CALCULATIONS AND DISCUSSIONS

The decay properties of 14C, 18,20O, 23F, 22-26Ne, 28-30Mg and 34Si clusters
emitted from various heavy nuclei leading to the formation of daughters of Pb-
isotopes. The effect of double-shell closure in terms of 208Pb daughter is
analysed using the theoretical formalism discussed in the previous section.
The relativistic mean-field theory (RMF) is employed here, being an efficient
tool to reproduce the ground state properties of the decaying parent nuclei.
The RMF-based R3Y (NL$3^{*}$) and the phenomenological M3Y NN interactions are
folded with their respective RMF densities to deduce the nuclear interaction
potential. As a representative case, Fig. 1 illustrates the individual
contributions of the nuclear and Coulomb potentials which collectively forms
the total interaction potential $V$ $(=V_{c}+V_{n})$ for 228Th
$\rightarrow^{20}$O + 208Pb for the cases of R3Y (NL$3^{*}$) and M3Y
interactions.

Figure 1:  A schematic representation of the total nucleus-nucleus interaction
potential V (MeV) and the respective contributions of the Coulomb and double-
folded R3Y (NL$3^{*}$) and M3Y nuclear potentials as a function of the radial
separation $R$ (fm) for a representative case of 228Th $\rightarrow^{20}$O +
208Pb. The inset shows a magnified view of the barrier height and position.
Figure 2:  Total interaction potential for energetically favoured reaction
228Th $\rightarrow^{20}$O + 208Pb for both M3Y and R3Y NN potentials,
satisfying the criterion for cluster penetration $V(R_{a})>Q_{val}$ at
$\Delta$R = 1.0 fm (for relative comparison). The three step penetration
process of interaction potential is also shown.

###  III.1 Barrier Characteristics from M3Y and R3Y NN-Potentials

From Fig. 1, it is observed that the total interaction potential $V$ and
nuclear potential $V_{n}(R)$ displays similar characteristics for both R3Y and
M3Y NN potentials. However, the difference lies in their qualitative
description which is apparent in the central region bhuy18 ; jos22a  but
decreases proportionately with the radial separation $R$. Moreover, the R3Y
and M3Y NN interactions are characterized by different barrier properties and
hence the barrier height of the R3Y could be relatively lower (at about 3.6
MeV), being more attractive as shown in the inset.

The cluster penetration process for the energetically favoured reactions,
taking 228Th $\rightarrow^{20}$O + 208Pb as a representative case is shown in
Fig. 2. The three-step procedure involved is initiated by barrier penetration
at the first turning point $R_{a}$ up to the point $R_{i}$ and followed by a
de-excitation (given in the excitation model of Greiner and Scheid grei86  for
heavy cluster emissions as $W_{i}=1$) from $V(R_{i})$ and thereafter, the
cluster penetrates from $R_{i}$ to point $R_{b}$ such that $V(R_{b})=Q_{val}$.
It is imperative to note that this process is highly influenced by the decay
energy Q-value which must be positive. Although this description is not
altogether new, the contribution of the Q-value to the energy contributed
during cluster preformation (captured by the proposed preformation formula) is
separately estimated/analysed for the first time. The figure further stresses
the disparity in the barrier properties of the M3Y and R3Y NN potential as
mentioned earlier. Here, for relative comparison, the neck-length parameter is
kept at $\Delta$R = 1.0 fm for both NN potentials. Consequently, there is a
significant difference between their respective barrier peak/height $V_{B}$
and its derivatives such as the barrier lowering parameter $\Delta V_{B}$ and
the driving potential $V(R_{a})-Q_{val}$.

Figure 3: The barrier lowering parameter $\Delta V_{B}$ (MeV) of 24Ne emission
of 230,232,233,235U isotopes at different neck-length parameter $\Delta$R
values.

The barrier lowering parameter $\Delta V_{B}$ is an inherent feature of the
preformed cluster-decay model (PCM) which encapsulates various modifications
in the barrier region especially those occasioned by the neck-length values
shar19 . Fig. 3 illustrates the profile of $\Delta V_{B}$ with respect to the
neutron number N of 230,232,233,235U isotopes at varied neck-length parameter
$\Delta$R values lying within the proximity potential limit bloc77 ; niyt15 ;
shar18 . Considering the predictions from M3Y (open symbols) and R3Y (solid
symbols) potentials separately, it is clear that the barrier lowering
parameter is largely influenced by the neck configuration, and hence its
modification increases with increasing $\Delta$R. Thus, $\Delta V_{B}$
dictates the cluster tunnelling path and it is usually negative since the
penetration point is always below $V_{B}$.

Figure 4:  The driving potential $V(R_{a})-Q_{AME}$ (MeV) of 24Ne emission as
a function of the neutron number of 230,232,233,235U isotopes at different
neck-lengths ($\Delta$R).

Similarly, the difference between the interaction potential and the energy
available for the cluster decay process is referred to as the driving
potential $V_{d}$ $(=V(R_{a})-Q_{AME}$). It is worth mentioning that the
Q-value ($Q_{AME}$) used here are calculated from the experimental binding
energy data wang17  only for the sake of accuracy. In several studies, the
minima in $V_{d}$ are usually used to indicate the most probable decay
channels, here, variations are made in order to decide on the most appropriate
neck length, especially for the recently developed R3Y interaction. We have
earlier demonstrated that $\Delta$R = 0.5 fm kuma12c ; jos21c  is suitable
enough for M3Y interaction in cluster decay studies. Nonetheless, this
$\Delta$R value is not energetically favourable in the case of R3Y due to its
unique barrier characteristics (see Ref.jos22b  for elaborate details on the
range of predictability of both R3Y and M3Y interactions). Despite the change
in $\Delta$R, a regular pattern is maintained in the profile of the driving
potential for both interactions. Other than these variations, $\Delta$R is
fixed at 1.0 fm for R3Y in the remaining part of this paper for the sake of
relative comparison.

Figure 5:  Variation of the preformation properties: (a) recoil energy of the daughter nuclei (b) cluster emission energy and (c) weighted Q-values of various Radium and Uranium isotopes as a function of the neutron number of the daughters formed from ${}^{A}U\rightarrow^{24}Ne+^{A-24}Pb$. Table 1:  The decay half-lives and preformation analysis of various experimentally observed clusters bone07  forming different Pb-daughters. The Q-values are calculated using the binding energies from experimental data $(Q_{AME})$wang17 . Columns (8-13) are exclusively devoted to the cluster preformation and emission details from the proposed Eqs. (14) - (17). Parent | Cluster | Daughters | $Q_{AME}$ | $\log_{10}T_{1/2}$ | $P_{0}$ | bQ | $\kappa$ | $\kappa\sqrt{Q}$ | $E_{c}$ | $E_{d}$
---|---|---|---|---|---|---|---|---|---|---
nuclei | cluster | nuclei | (MeV) | Expt. | M3Y | R3Y | Eq. (14) | (MeV) |  | (MeV) | (MeV) | (MeV)
221Ra |  14C |  207Pb | 32.40 | 13.39 | 13.95 | 14.22 | $9.19\times 10^{-18}$ | 3.86 | 4.65 | 26.49 | 30.34 | 2.05
222Ra |  14C |  208Pb | 33.05 | 11.01 | 11.12 | 11.94 | $8.82\times 10^{-16}$ | 1.22 | 5.17 | 29.74 | 30.96 | 2.08
223Ra |  14C |  209Pb | 31.83 | 15.06 | 14.71 | 15.40 | $8.22\times 10^{-18}$ | 3.79 | 4.62 | 26.04 | 29.83 | 2.00
224Ra |  14C |  210Pb | 30.53 | 15.86 | 16.58 | 17.10 | $6.71\times 10^{-18}$ | 1.13 | 4.98 | 27.50 | 28.63 | 1.91
226Ra |  14C |  212Pb | 28.20 | 21.19 | 20.31 | 20.60 | $5.68\times 10^{-18}$ | 1.04 | 4.78 | 25.41 | 26.45 | 1.75
226Th |  18O |  208Pb | 45.73 | $\textgreater 15.30$ | 17.31 | 17.63 | $9.84\times 10^{-20}$ | 1.69 | 5.97 | 40.39 | 42.09 | 3.64
228Th |  20O |  208Pb | 44.72 | 20.72 | 20.54 | 22.28 | $3.78\times 10^{-21}$ | 1.65 | 5.85 | 39.15 | 40.80 | 3.92
230U |  22Ne |  208Pb | 61.39 | 19.57 | 19.91 | 19.48 | $4.46\times 10^{-23}$ | 2.27 | 6.80 | 53.25 | 55.52 | 5.87
231Pa |  23F |  208Pb | 51.89 | 26.02 | 25.67 | 25.34 | $5.65\times 10^{-26}$ | 6.17 | 5.63 | 40.55 | 46.72 | 5.17
230U |  24Ne |  206Pb | 61.35 | $\textgreater 18.20$ | 23.68 | 23.18 | $2.47\times 10^{-26}$ | 2.27 | 6.73 | 52.68 | 54.95 | 6.40
232U |  24Ne |  208Pb | 62.31 | 21.08 | 20.58 | 20.94 | $2.92\times 10^{-24}$ | 2.31 | 6.79 | 53.56 | 55.86 | 6.45
233U |  24Ne |  209Pb | 60.49 | 24.84 | 24.39 | 24.55 | $2.30\times 10^{-26}$ | 7.20 | 6.05 | 47.06 | 54.26 | 6.23
234U |  24Ne |  210Pb | 58.83 | 25.92 | 26.11 | 26.11 | $1.97\times 10^{-26}$ | 2.18 | 6.60 | 50.62 | 52.79 | 6.03
235U |  24Ne |  211Pb | 57.36 | 27.62 | 28.29 | 26.44 | $1.06\times 10^{-26}$ | 6.85 | 5.90 | 44.65 | 51.51 | 5.86
234U |  26Ne |  208Pb | 59.41 | 25.92 | 25.02 | 26.61 | $1.63\times 10^{-25}$ | 2.20 | 6.57 | 50.61 | 52.81 | 6.60
236U |  26Ne |  210Pb | 56.69 | $\textgreater 25.9$ | 30.10 | 31.66 | $1.16\times 10^{-27}$ | 2.10 | 6.42 | 48.35 | 50.45 | 6.25
236Pu |  28Mg |  208Pb | 79.67 | 21.67 | 21.48 | 21.40 | $5.99\times 10^{-27}$ | 2.95 | 7.54 | 67.27 | 70.22 | 9.45
238Pu |  30Mg |  208Pb | 76.80 | 25.70 | 25.10 | 25.18 | $4.59\times 10^{-28}$ | 2.84 | 7.33 | 64.27 | 67.12 | 9.68
242Cm |  34Si |  208Pb | 96.51 | 23.24 | 23.70 | 25.46 | $3.80\times 10^{-30}$ | 3.57 | 8.08 | 79.38 | 82.95 | 13.56

###  III.2 The Recoil Energy ($E_{d}$)

A careful inspection of the $4^{th}$ column of Table 1 of Ref. josh22L  and
its footnote shows that the constant parameter ‘$c$’ (see Eq. (14)) is highly
susceptible to shell and/or sub-shell effect and thus is higher for all the
systems having 208Pb-daughter shell closure. In other words, ‘$c$’ can be used
to indicate the signature of shell and/or sub-shell closure over an isotopic
chain. The shell effect is displayed in the last column of Table 1 such that
the same cluster from different isotopes of the particular nucleus is
characterized by a unique Q-value as well as the recoil energy. In such cases,
the recoil energy of the daughters for double magic ${}^{208}Pb$ is relatively
higher as compared to those of the neighbouring isotopes. Furthermore, in Fig.
5 (upper panel) displays the variation of the recoil energy of the daughter
nuclei w.r.t. their corresponding neutron number $N_{d}$ for different
Q-values calculated from the RMF, $Q_{RMF}$ (dash line with blue star) and
compared with the binding energies of Wang et al. wang17  $Q_{AME}$ (solid
line with black square), as well as the WS3 given by Liu et al. liu11
$Q_{WS3}$ (dotted line with red circle). The profile shows a regular pattern
for the three sets of Q-values which directly influences their magnitude.
Nonetheless, in all cases, the peaks are observed for magic and/or close shell
neutron number $N=126$.

Classically, one can correlate the recoil energy of the daughter nuclei
($E_{d}$) with the mass of the parent nuclei and the cluster in the outgoing
channel. Here, we introduce a simple and intuitive systematic that governs the
quantitative estimate of the recoil energy of the daughter nuclei (last column
of Table 1) based on the three distinct possibilities:
Firstly, we consider a case in which the same cluster is emitted from
different parent nuclei. From these systems, one can observe that the heavier
parent produces relatively lower recoil energy provided that the daughters
formed are not magic nuclei. For example, from the table, the reaction
systems, namely, 223Ra $\rightarrow$ 14C+209Pb, 224Ra $\rightarrow$ 14C+210Pb
yields $E_{d}$ = 2.00 and 1.91 MeV, respectively. This observation is found to
be consistent provided $A_{d}$ is not a shell and/or sub-shell closure.
Secondly, we consider emission of different clusters from the same parent
nucleus. It is observed that massive clusters produce relatively higher recoil
energy. For example, the reaction systems, namely, 234U $\rightarrow$
24Ne+210Pb, and 236U $\rightarrow$ 26Ne+210Pb are associated with $E_{d}$ =
6.03 and 6.25 MeV, respectively. Thirdly, we consider the emission of the same
cluster from different parent nuclei in which at least, one of the daughters
formed is a magic nucleus. We notice that the heavier parent can produce a
relatively higher or equal recoil energy. For example, the reaction systems,
namely, 221Ra $\rightarrow$ 14C+207Pb, 222Ra $\rightarrow$ 14C+208Pb, are
yielding $E_{d}$ = 2.05 MeV, =2.08 MeV, respectively. A similar finding is
observed for the reaction systems 230U $\rightarrow$ 24Ne+206Pb, 232U
$\rightarrow$ 24Ne+208Pb with $E_{d}$ = 6.40, 6.45 MeV, respectively.

###  III.3 Cluster Emission Energy ($\kappa\sqrt{Q}$)

Besides, the dominance of the shell effect on the cluster emission energy
$\kappa\sqrt{Q}$ is hinged on the formation of the double magic 208Pb daughter
nucleus in which the highest peak (at $N=126$) is formed as shown in Fig.
5(a). However, the usual peak at $N=128$ is mainly attributed to the lower
value of parameter ‘c’ in Table 1, indicating the formation of non-double
magic daughter nuclei. Thus, since the shell effect is lower in such cases,
the corresponding energy for the cluster tunnelling process is amply
increased. On this account, we define $\kappa$ as the precise
quantity/fractional amount of energy required to liberate a preformed cluster
through the potential barrier. In other words, $\kappa$ is the specific amount
of energy with which the preformed cluster tunnels across the Coulomb-nuclear
interaction barrier. Hence, $\kappa$ can be termed as the tunnelling factor.
From Eq. (17), it is evident that $\kappa$ is largely dependent on the mass of
the parent, and daughter nuclei and hence, the emitted cluster.

Figure 6:  Variation of the preformation probability $P_{0}$ and logarithmic
half-lives $\log_{10}T_{1/2}$ with $Q_{AME}$ wang17  only for
${}^{A}Ra\rightarrow^{14}C+^{A-14}Pb$ and
${}^{A}U\rightarrow^{24}Ne+^{A-24}Pb$.

###  III.4 The Weighted Q-value ($bQ$)

We reiterate that the weighted Q-value ‘$bQ$’ is the share of the decay energy
contributed during the cluster preformation process. The third panel of Fig. 5
depicts the weighted Q-value of 24Ne cluster emission from even-even
230,232,234U and even-odd 233,235U isotopes as a function of the neutron
number of their respective daughter nucleus. By composition, $bQ$ can only be
influenced by the Q-value and parameter $b$. The predictions from the Q-values
estimated from the experimental binding energies and those from the WS3 mass
table relatively good agreements over the RMF deduced Q-values. For example,
the difference in $Q_{AME}$ and $Q_{WS3}$ ranges between 0.1 - 0.5 MeV,
whereas the $Q_{RMF}$ with a difference of about 1.0 - 3.0 MeV. However,
despite the variation of the Q-values, the behaviour of the even(Z)-even(N)
systems are different from those of the even(Z)-odd(N) systems. In other
words, Fig. 5c shows that all the even-even systems have lower $bQ$ values and
follow the same trend, unlike the even-odd systems which are marked with
higher values. As such, it is apparent that parameter $b$ captures the odd-
even staggering effect since the presence of unpaired protons or neutrons in
the open-shell radioactive cluster emitters are uniquely associated with
higher $b$ values (at $N=127$ and $N=129$) and thus influences the
preformation probability. This fully agrees with the systematic study in Ref.
seif15 . Details of the pairing effect for open-shell nuclei, the odd-even
staggering effects as well as their connection with the single-particle
energies and orbital filling have been extensively discussed in Ref. bhu21 ;
an22 ; good21 ; kosz21 . Therefore, it is evident that a careful determination
of $P_{0}$ provides ample information about the nuclear structure and the
kinematics of cluster emission. Using Eq. (14) a detailed calculation of the
cluster preformation is carried out for heavy nuclei decaying to the double-
shell closure 208Pb-daughters and its neighbours. Thus, the credibility of the
estimated $P_{0}$ is graphically illustrated and discussed. Since the
preformation probability is not a direct experimentally observed quantity, one
relies on the theoretical models for its deduction.

###  III.5 Cluster Preformation $P_{0}$ and Half-lives $T_{1/2}$

Figure 6 (upper panels (a) and (b)) displays the calculated preformation
probability of 14C and 24Ne clusters from Ra and U isotopic chains
respectively as a function of the neutron number of their corresponding Pb-
daughter nuclei formed. From the figure, it is apparent that cluster
preformation in heavy nuclei is usually accompanied by the appearance of a
notable dip at $N=126$. This agrees with the statistical analysis of Bonetti
et al. bone74  and Hodgson et al. hodg03  in $\alpha$-particle preformation.
The figure further asserts that the $P_{0}$ cannot maintain a constant
magnitude for different nuclei, unlike the conjecture in Refs. xu06 ; qian11 .
Thus, in both figures, the $-\log P_{0}$ values for similar cluster emissions
from different isotopes of an element have a clear distinction. Taking the
inverse of $-\log P_{0}$ values in Fig. 6(a) and Fig. 6(b), the $P_{0}$ fall
in the range $0<P_{0}<1$. By implication, the treatment of $P_{0}$ as unity
relegates the exclusive properties of the participating nuclei. A detailed
inspection of both figures (and column 8 of Table 1) for
${}^{A}Ra\rightarrow^{14}C+^{A-14}Pb$ and
${}^{A}U\rightarrow^{24}Ne+^{A-24}Pb$ reveals that the cluster preformation
probability $P_{0}$ at the double magic shell closure 208Pb daughter is higher
than those its neighbouring daughter nuclei with about an order of $10^{2}$
and exhibits relatively lower half-lives. This reflects the stability of 222Ra
and 232U parents against 14C and 24Ne cluster decays respectively.

Similarly, the theoretically determined half-lives using the M3Y and R3Y
potentials are compared with the experimental data . As representative cases,
Fig. 6(c) and Fig. 6(d) shows the variation of the logarithmic half-lives
$\log_{10}T_{1/2}$ for 14C and 24Ne cluster decay from Ra and U isotopes
respectively as a function of the neutron number of the daughter ($N_{d}$). In
both instances, the minima in $\log_{10}T_{1/2}$ are identified with the decay
leading to the double magic daughter 208Pb ($Z_{d}=82$ and $N_{d}=126$). In
other words, the shell stabilises at magic daughter nuclei. This infers that
the stability of the cluster emitters could be explained via the shell closure
effect. Besides, the half-lives of cluster emitters appear to increase w.r.t.
the magnitude of the neutron number of the daughter nuclei formed until a
magic number (or its neighbour) is attained. The M3Y and R3Y predictions are
found to be consistent with the experimentally observed half-lives for all the
systems under study. However, the cluster-decay half-lives of certain systems
(like for $N_{d}=124$ in Fig. 6(d)) lack precise experimental measurement, for
which only the lower limits were given (marked with an upward arrow). In that
vein, the prediction of M3Y and R3Y agrees with the experimental lower limit
and can be considered to be more probable from the theoretical point of view
(since the deepest minima are usually associated with the double magic number
$N_{d}=126$). The slight difference in the estimation of the M3Y and R3Y NN
potentials reflects the uniqueness of their barrier characteristics.

##  IV SUMMARY AND CONCLUSIONS

The proposed new preformation probability $P_{0}$ formula captures several
known theoretically established factors (such as the cluster mass $A_{c}$,
mass and charge asymmetry ($\eta_{A}$ and $\eta_{Z}$) and the Q-value)
affecting the mechanism and kinematics of the cluster emissions is applied
here. Also, for the first time, an expression for the contribution of the
decay energy in terms of the cluster preformation, emission and recoil energy
is applied in the cluster radioactivity. Further, we present a new set of
criteria for estimating the recoil energy of daughter nuclei in the cluster
radioactivity. Using the relativistic mean-field (RMF) approach, the cluster
decay half-lives of various nuclei decaying to (or around) 208Pb-daughters are
calculated within the preformed cluster-decay model (PCM). As a result, the
stability of the cluster emitters is closely linked with the pairing and shell
closure effect. Although the barrier properties of the phenomenological M3Y
and microscopic R3Y NN potentials differ qualitatively, their respective
predictions are found to relatively agree with the experimentally measured
half-lives. To gain a comprehensive insight into the cluster decay dynamics,
it is of future interest to systematically investigate the preformation
properties and half-lives of systems yielding non-Pb daughters along with
various proximity potentials with deformation and orientation effects. This
study can also be extended to predict cluster radioactivity in the unknown
territories of the superheavy mass region.

## acknowledgments

The authors would like to acknowledge the support from the Fundamental
Research Grant Scheme (FRGS) under the grant number
FRGS/1/2019/STG02/UNIMAP/02/2 from the Ministry of Education Malaysia
stipulated with the Institute of Engineering Mathematics (IMK), UniMAP as the
beholder, Science and Engineering Research Board (SERB), File No.
CRG/2021/001229, FOSTECT Project Code: FOSTECT.2019B.04, and FAPESP Project
Nos. 2017/05660-0.

## References

  * (1) A. Sandulescu, D. N. Poenaru, and W. Greiner, Sov. J. Part. Nucl.(Engl. Transl.);(United States) 11 (1980).
  * (2) R. K. Gupta and W. Greiner, Int. J. Mod. Phys. E 3, 335 (1994, Suppl.).
  * (3) H. J. Rose and G. A. Jones, Nature 307, 245 (1984).
  * (4) R. Bonetti and A. Guglielmetti, Rom. Rep. Phys. 59, 301 (2007).
  * (5) Y. Qian and Z. Ren, J. Phys. G 39, 015103 (2012).
  * (6) D. Deng, Z. Ren, D. Ni, and Y. Qian, J. Phys. G 42, 075106 (2015).
  * (7) M. A. Hooshyar, I. Reichstein, and F. B. Malik, Nuclear Fission and Cluster Radioactivity: An Energy-Density Functional Approach (Springer-Verlag, Berlin, 2005).
  * (8) D. N. Poenaru, M. Ivascu, A. Sandulescu, and W. Greiner, Phys. Rev. C 32, 572 (1985).
  * (9) D. N. Poenaru, W. Greiner, K. Depta, M. Ivascu, D. Mazilu, and A. Sandulescu, At. Data Nucl. Data Tables 34, 423 (1986).
  * (10) S. S. Malik and R. K. Gupta, Phys. Rev. C 39, 1992 (1989).
  * (11) R. K. Gupta, in Proceedings of the 5th International Conference on Nuclear Reaction Mechanisms, edited by E. Gadioli (Ricerca Scientifica ed Educazione Permanente, Milan, 1988), p. 416.
  * (12) K. Wei and H. F. Zhang, Phys. Rev. C 96, 021601(R) (2017).
  * (13) R. Blendowske and H. Walliser, Phys. Rev. Lett. 61, 1930 (1988).
  * (14) K. P. Santhosh and Tinu Ann Jose, Phys. Rev. C 104, 064604 (2021).
  * (15) D. Ni and Z. Ren, Phys. Rev. C 82, 024311 (2010).
  * (16) M. Balasubramaniam and N. S. Rajeswari, Int. J. Mod. Phys. E 23, 1450018 (2014).
  * (17) D. Deng and Z. Ren, Phys. Rev. C 93, 044326 (2016).
  * (18) T. M. Joshua, R. Kumar, and M. Bhuyan, Submitted to Physical Review Letters (2022).
  * (19) G. Röpke, P. Schuck, Y. Funaki, H. Horiuchi, Zhongzhou Ren, A. Tohsaki, Chang Xu, T. Yamada, and Bo Zhou, Phys. Rev. C 90, 034304 (2014).
  * (20) C. Xu, Z. Ren, G. Röpke, P. Schuck, Y. Funaki, H. Horiuchi, A. Tohsaki, T. Yamada, and B. Zhou, Phys. Rev. C 93, 011306 (2016).
  * (21) J. S. Levinger, Phys. Rev. 90, 11 (1953).
  * (22) G. Stratan, W. Scheid, Int. J. Mod. Phys. E 10, 367 (2001).
  * (23) R. Kumar, Phys. Rev. C 86, 044612 (2012).
  * (24) P. Quentin and H. Flocard, Annu. Rev. Nucl. Part. Sci. 28, 523 (1978).
  * (25) W. H. Hornyak, Nuclear Structure (Academic Press, New York, 1975).
  * (26) N. Schunck and L. Robledo, Rep. Prog. Phys. 79, 116301 (2016).
  * (27) D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626 (1972).
  * (28) E. Epelbaum, H.-W. Hammer, and U.-G. Mei${\ss}$ner, Rev. Mod. Phys. 81, 1773 (2009).
  * (29) A. Ekstr$\ddot{o}$m et al., Phys. Rev. Lett. 110, 192502 (2013).
  * (30) B. B. Singh, M. Bhuyan, S. K. Patra, and R. K. Gupta, J. Phys. G: Nucl. Part. Phys. 39, 069501 (2012).
  * (31) B. B. Singh, M. Bhuyan, S. K. Patra, and R. K. Gupta, arXivpreprint arXiv:1011.5732 (2010).
  * (32) G. R. Satchler and W. G. Love, Phys. Rep. 55, 183 (1979).
  * (33) S. Biswal, M. A. El Sheikh, N. Biswal, N. Yusof, H. Kas-sim, S. K. Patra, and M. Bhuyan, Nucl. Phys. A 1004, 122042 (2020).
  * (34) N. Itagaki, A. Afanasjev, and D. Ray, Phys. Rev. C 101, 034304 (2020).
  * (35) A. Taninah, S. Agbemava, and A. Afanasjev, Bull. Am. Phys. Soc. 65 (2020).
  * (36) Min Liu, Ning Wang, Yangge Deng, Xizhen Wu, Phys. Rev. C 84, 014333 (2011).
  * (37) M. Wang, G. Audi, F. G. Kondev, W. J. Huang, S. Naimi, and X. Xu, Chin. Phys. C 41, 030003 (2017).
  * (38) T. M. Joshua, N. Jain, R. Kumar, K. Anwar, N. Abdullah, and M. Bhuyan, Foundations, 2, 85-104 (2022).
  * (39) P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).
  * (40) M. Bhuyan, Phys. Rev. C 92, 034323 (2015).
  * (41) M. Bhuyan, R. Kumar, S. Rana, D. Jain, S. K. Patra, and B. V. Carlson, Phys. Rev. C 101, 044603 (2020).
  * (42) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).
  * (43) A. Singh, A. Shukla, and M. K. Gaidarov, J. Phys. G: Nucl. Part. Phys, 49, 025101 (2021).
  * (44) B. B. Singh, S. K. Patra, and R. K. Gupta, Phys. Rev. C 82, 014607 (2010); Int. J. Mod. Phys. E 20, 1003 (2011).
  * (45) R. Bonetti and A. Guglielmetti, in Heavy elements and related new phenomena edited by R. K. Gupta and W. Greiner (World Scientific Pub., Singapore, 1999) Vol. 2, p. 643.
  * (46) D. S. Delion, Phys. Rev. C 80, 024310 (2009).
  * (47) M. Ismail, A. Adel, Phys. Rev. C 89, 034617 (2014).
  * (48) M. Bhuyan and R. Kumar, Phys. Rev. C 98, 054610 (2018).
  * (49) M. Greiner and W. Scheid, J. Phys. G: Nucl. Phys. 12, L229 (1986).
  * (50) K. Sharma, G. Sawhney, M. K. Sharma, R. K. Gupta, Eur. Phys. J. A 55, 30 (2019).
  * (51) J. Blocki, J. Randrup, W. J. Swiatecki, and C. F. Tsang, Ann. Phys. (N.Y.) 105, 427 (1977).
  * (52) Niyti, G. Sawhney, M. K. Sharma, and R. K. Gupta, Phys. Rev. C 91, 054606 (2015).
  * (53) K. Sharma, G. Sawhney, M. K. Sharma, and R. K. Gupta, Nucl. Phys. A 972, 1 (2018).
  * (54) T. M. Joshua, K. Anwar, N. Abdullah, N. Jain, S. Rana, R. Kumar, M. Bhuyan, Proceedings of the DAE-BRNS symposium on nuclear physics. V. 65 (2021).
  * (55) J. T. Majekodunmi, M. Bhuyan, D. Jain, K. Anwar, N. Abdullah, and R. Kumar, Phys. Rev. C 105, 044617 (2022).
  * (56) M. Bhuyan, B. Maheshwari, H. A. Kassim, N. Yusof, S. K. Patra, B. V. Carlson and P. D. Stevenson, J. Phys. G: Nucl. Part. Phys. 48, 075105 (2021).
  * (57) R. An, X. Jiang, L.-G. Cao, and F.-S. Zhang, Phys. Rev. C 105, 014325 (2022).
  * (58) T. D. Goodacre et al., Phys. Rev. C 104, 054322 (2021).
  * (59) Koszorús, Á et al., Nat. Phys. 17, 439 (2021).
  * (60) W. Seif, Phys. Rev. C 91, 014322 (2015).
  * (61) R. Bonetti and L. Milazzo-Colli, Phys. Lett. B 49, 17 (1974).
  * (62) P. E. Hodgson and E. Běták, Phys. Rep. 374, 1 (2003).
  * (63) C. Xu and Z. Ren, Phys. Rev. C 74, 014304 (2006); 73, 041301(R) (2006).
  * (64) Y. Qian, Z. Ren, and D. Ni, Phys. Rev. C 83, 044317 (2011).


# Coded Kalman Filtering over MIMO Gaussian Channels with Feedback

Barron Han, Victoria Kostina, Babak Hassibi  Department of Electrical
Engineering
Caltech
Pasadena, CA, USA
Email: {bshan, vkostina<EMAIL_ADDRESS>Oron Sabag  School of Computer
Science and Engineering
The Hebrew University of Jerusalem
Jerusalem, Israel
Email<EMAIL_ADDRESS>
###### Abstract

We consider the problem of remotely stabilizing a linear dynamical system. In
this setting, a sensor co-located with the system communicates the system’s
state to a controller over a noisy communication channel with feedback. The
objective of the controller (decoder) is to use the channel outputs to
estimate the vector state with finite zero-delay mean squared error (MSE) at
the infinite horizon. It has been shown in [1] that for a vector Gauss-Markov
source and either a single-input multiple-output (SIMO) or a multiple-input
single-output (MISO) channel, linear codes require the minimum capacity to
achieve finite MSE. This paper considers the more general problem of linear
zero-delay joint-source channel coding (JSCC) of a vector-valued source over a
multiple-input multiple-output (MIMO) Gaussian channel with feedback. We study
sufficient and necessary conditions for linear codes to achieve finite MSE.
For sufficiency, we introduce a coding scheme where each unstable source mode
is allocated to a single channel for estimation. Our proof for the necessity
of this scheme relies on a matrix-algebraic conjecture that we prove to be
true if either the source or channel is scalar. We show that linear codes
achieve finite MSE for a scalar source over a MIMO channel if and only if the
best scalar sub-channel can achieve finite MSE. Finally, we provide a new
counter-example demonstrating that linear codes are generally sub-optimal for
coding over MIMO channels.

##  I Introduction

Controlling an unstable plant over a noisy communication channel is a hurdle
for emerging technologies such as autonomous vehicles, Internet of Things
devices, and remote surgery systems. This problem setting deviates from
Shannon’s communication problem in two ways that make it more challenging [2].
First, in the control setting, the data to be transmitted correspond to
physical measurements and arrive in a streaming fashion instead of being made
available in its entirety before transmission. Thus, we must design causal
encoders and decoders for this task. Second, typical control systems are
unstable, and their stabilization requires near-instantaneous and accurate
estimates of the plant’s state to produce effective control actions.
Consequently, codes must be low-delay yet highly reliable to perform control
tasks over communication channels. We employ a class of low-delay joint-source
channel codes to address the two objectives.

In a seminal paper, Sahai and Mitter [3] proved that Shannon’s channel
capacity is an insufficient characterization of channel quality when the goal
is to stabilize a system over a channel. They introduced the notion of
_anytime capacity_ , which is in general upper-bounded by channel capacity, as
an alternative measure. While the anytime capacity of a channel provides a
useful converse on the channel quality required to stabilize a specific
system, achievability schemes are generally open. A class of tree codes such
as those studied by Schulman [4] achieve error probabilities that decay
exponentially with the delay since a source symbol was emitted. While tree
codes exist for a large class of discrete channels, they are only known to be
efficiently decodable in limited settings [5].

A noiseless feedback channel connecting the decoder back to the encoder does
not improve the Shannon capacity of the channel [6], but feedback can
significantly simplify code design and improve the reliability-delay trade-
offs for communication [7, 8]. Noiseless feedback channels are reasonable
assumptions when the receiver has access to more power than the transmitter,
which is often the case in control systems since the controller must provide
essentially noiseless control inputs. Coding for bit-streaming sources over
discrete channels with feedback, which is relevant to control systems where
the state has been quantized for digital transmission, has been studied in [9,
5, 10]. This paper considers a setting where measurement and coding are analog
operations applied in discrete time.

Consider the problem of estimating a vector-valued plant, modeled as a Gauss-
Markov source, over a multiple-input multiple-output (MIMO) additive white
Gaussian noise (AWGN) channel with feedback. The causal rate-distortion
function [11] provides a lower bound to the channel capacity necessary for
causally estimating the source subject to a given distortion over this channel
[12]. The causal rate-distortion functions for both scalar and vector Gauss-
Markov sources have been studied in [11] and [13] respectively. The lower
bound to channel capacity provided by the causal rate-distortion function is
known to be tight only when the source is matched to the channel at hand [14].
For example, a scalar Gauss-Markov source is matched to the scalar AWGN
channel [11, 3].

When the criterion is finite MSE, a known converse result is that the Shannon
capacity should be greater than the sum of logs of unstable eigenvalues of the
source [15, Thm. 4.1]. In the case of a scalar source and a scalar channel,
this bound is tight and can be achieved by a linear innovations’ encoder [16,
3, 17, 18]. For a vector source and parallel Gaussian channels with
independent power constraints, [19] also proposes a periodic linear scheme
leading to sufficient conditions for achieving finite MSE. The case of a
vector source with scalar channel was studied in [1], which showed that the
Shannon capacity remains a necessary and sufficient measure even though the
source and channel dimensions are not matched.

This paper considers the general case of vector source and MIMO channels. We
first focus on the fundamental limits of achieving finite MSE using linear
time-invariant codes. The innovations’ encoder that generates channel inputs
as a function of the source estimation error (at the decoder) is optimal for
this general problem [1]. The sequential encoder’s structure implies that the
optimal decoder is a Kalman filter and its MSE can be analyzed with linear
estimation theory.

Our first result is a sufficient condition to achieve finite MSE by
partitioning the vector source to different sub-channels. The analysis is
carried out by showing an equivalence between achieving finite MSE and the
existence of a stabilizing solution to a DARE equation. The sufficient
condition (achievability) is then shown to be necessary for two cases
including the scenario of a scalar and a MIMO channel. In particular, it is
shown that allocating the entire power to the best sub-channel is optimal,
while typical water-filling solutions that distribute the power among the sub-
channels are sub-optimal. Indeed, this example reveals that the Shannon
channel capacity is not the figure of merit if the objective is finite MSE
with linear codes. Motivated by this result, we define the _linear stabilizing
capacity_ (LSC) as an optimization problem, which is in general a lower bound
to the channel capacity. Finite MSE is achievable using linear codes if and
only if there exists a feasible solution. The optimization of the LSC is non-
convex, but we are able to utilize it to show that linear codes are not
optimal by comparing the LSC with rates that can be achieved using non-linear
Shannon-Kotel’nikov mappings for a specific source-channel pair. The general
case of our problem remains open but, based on numerical observation, we
conjecture that the partitioning property is necessary. The equivalence to the
DARE feasibility allows us to extract an algebraic condition that if it holds
true then partitioning schemes achieve the fundamental limits.

The paper is organized as follows. Section II specifies the source and channel
models and defines zero-delay joint source-channel codes with an MSE
performance criterion. Section III presents an optimal linear code structure
and applies it to the MIMO channel setting. It also defines the _linear
stabilizing capacity_. Section IV presents our main contributions on the
sufficient and necessary conditions for finite estimation error of a vector
source over a MIMO Gaussian channel using linear codes and demonstrates that,
in general, linear coding is not optimal.

##  II Problem Setup

###  II-A Notation

We denote by $\\{\mathbf{X}_{t}\\}_{t=0}^{T}$ a discrete time random process
and we denote the vector $X^{t}\triangleq\\{x_{0},x_{1},\ldots,x_{t}\\}$. We
write $\mathbf{X}\sim\mathcal{N}(\mu,\Sigma)$ to say that the random vector
$\mathbf{X}$ has a Gaussian distribution with mean
$\mathbb{E}[\mathbf{X}]=\mu$ and covariance matrix
$\mathrm{Cov}[\mathbf{X}]=\Sigma$. Matrices and vectors are denoted with
uppercase letters, while scalars are denoted with lowercase mathematical font.
Sets are denoted using the calligraphic font.

###  II-B System Model

The setup is depicted in Figure 1. We define its main components: a MIMO AWGN
channel, a Gauss-Markov streaming source, and a zero-delay code.

######  Definition 1 (MIMO AWGN Channel)

The channel accepts a vector input $\mathbf{X}_{t}\in\mathbb{R}^{n}$ and
produces a vector output $\mathbf{Y}_{t}\in\mathbb{R}^{m}$,

$\mathbf{Y}_{t}=H\mathbf{X}_{t}+\mathbf{Z}_{t},\ t\geq 1.$ (1)

$H\in\mathbb{R}^{m\times n}$ is a deterministic, fixed channel gain matrix and
the noise is $\mathbf{Z}_{t}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{N}(0,R).$

We consider a channel with a diagonal channel matrix,
$H=\mathrm{diag}\\{h_{1},\ldots,h_{n}\\}$, and identity noise covariance,
$\mathbf{Z}_{t}\sim\mathcal{N}(0,I)$. This is without loss of generality as
any channel can be diagonalized to this form [20, Thm. 9.1].

The streaming source in Figure 1 is a Gauss-Markov source.

######  Definition 2 (Gauss-Markov source)

The Gauss-Markov source evolves according to the linear dynamical system:

$\mathbf{S}_{t+1}=A\mathbf{S}_{t}+\mathbf{W}_{t},$ (2)

where $A\in\mathbb{R}^{k\times k}$, $\mathbf{W}_{t}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathcal{N}(0,Q)$ and the initial state is
$\mathbf{S}_{0}\sim\mathcal{N}(0,Q)$.

Without loss of generality, we assume that the matrix $A$ can be written as

$A=\left[\begin{array}[]{cc}A_{s}&0\\\ 0&A_{u}\end{array}\right],$ (3)

where $A_{s}$ is stable (eigenvalues on and inside unit circle), $A_{u}$ is
strictly unstable (eigenvalues outside unit circle) and both $A_{s},A_{u}$ are
in Jordan form.

We make the following assumptions about our system:

###### Assumption 1

The pair $(A,Q)$ is controllable.

###### Assumption 2

The source is strictly unstable, so $A_{s}=0$, and $A_{u}$ in (3) has distinct
eigenvalues.

Assumption 1 guarantees that the error covariance is positive definite [21,
Apx. C]. Assumption 2 yields a cleaner analysis since we can take $A$ to be
diagonal without loss of generality. Our results hold in the general case
where $A_{u}$ is in Jordan block form. Assuming that the source is unstable
does not lose generality since the stable part of the source can have a finite
estimation error even if no communication is allowed. We limit the source to
being strictly unstable so that there are no eigenvalues on the unit circle
and a Lyapunov equation of the form $X=AXA^{T}+W$ has a unique solution [21,
Lem. D.1.1]. This assumption is common in classical linear estimation theory
[21, Apx. C].

######  Definition 3 (A zero-delay joint-source channel feedback code)

The feedback code for the source-channel pair in Definitions 1 and 2 consists
of the following:

  1. 1.

An encoder that at time $t$ has access to $\mathbf{S}^{t}$ and
$\mathbf{Y}^{t-1}$ and generates

$\mathbf{X}_{t}=f_{t}(\mathbf{S}^{t},\mathbf{Y}^{t-1}),\ t\geq 1$ (4)

where $f_{t}\colon\mathcal{S}^{t}\times\mathcal{Y}^{t-1}\mapsto\mathbb{R}^{n}$
and $\mathcal{S}=\mathbb{R}^{k},\mathcal{Y}=\mathbb{R}^{m}$. The channel
inputs must satisfy an average power constraint,

$\frac{1}{T}\sum_{t=1}^{T-1}\mathbb{E}[\mathbf{X}_{t}^{T}\mathbf{X}_{t}]\leq
p,$ (5)

over the time horizon $T$.

  2. 2.

A decoder that at time $t$ predicts the next source state,

$\hat{\mathbf{S}}_{t}=g_{t}(\mathbf{Y}^{t-1}).$ (6)

For a given code in Definition 3, we denote the predicted error covariance

$P_{t}\triangleq\mathrm{Cov}(\mathbf{S}_{t}-\hat{\mathbf{S}}_{t}).$ (7)

Note that for an encoder $\\{f_{t}\\}_{t=0}^{\infty}$, the decoder

$\hat{\mathbf{S}}_{t}=\mathbb{E}[\mathbf{S}_{t+1}|\mathbf{Y}^{t}]$ (8)

minimizes the MSE

$D_{t}\triangleq\mathrm{Tr}\left(\mathrm{Cov}(\mathbf{S}_{t}-\hat{\mathbf{S}}_{t})\right).$
(9)

We denote the asymptotic MSE

$D\triangleq\text{limsup}_{t\to\infty}D_{t}.$ (10)

In this paper, we study conditions for there to exist a linear encoder
$\\{f_{t}\\}_{t=0}^{\infty}$ such that $D<\infty$.

This setting is relevant to the control problem where the source in (2) is
modified to include a control input $U_{t}$,

$S_{t+1}=AS_{t}+BU_{t}+W_{t}$ (11)

where $B$ is a constant matrix, and the decoder decides $U_{t}$. Provided that
$U^{t}$ are available at the encoder at time $t$, the classical result [22] of
certainty equivalence holds, implying the system in (11) is stabilizable if
and only if the controller (decoder) can estimate the source with finite MSE,
$D<\infty$ (10).

Figure 1: A MIMO AWGN channel, described in Definition 1, with a noiseless
feedback link is shown. The Gauss-Markov source, described in Definition 2,
produces information at every time, which is encoded and passed through the
channel. The decoder seeks to estimate the source at the next time given all
channel outputs. We display the optimal encoding structure of Lemma 1.

##  III A Linear Code

In this section, we present the optimal linear code and discuss the
performance of linear codes in Definition 3.

###  III-A Optimal Linear Code Structure

A general linear encoder has the form

$\mathbf{X}_{t}=\Xi_{t}(\mathbf{S}^{t})+\Psi_{t}(\mathbf{Y}^{t-1})+\mathbf{M}_{t},$
(12)

where $\Xi_{t},\Psi_{t}$ are linear mappings, and $\mathbf{M}_{t}$ is a
Gaussian random variable that is independent of
$(\mathbf{S}^{t},\mathbf{Y}^{t-1})$. While this encoder involves all past
states and channel outputs from feedback, the next result establishes a
simplified optimal code structure involving only the recent state estimate
error.

######  Lemma 1 (Innovations encoder)

The optimal linear encoder can be written as

$\mathbf{X}_{t}=\tilde{\Gamma}_{t}P_{t}^{-1}(\mathbf{S}_{t}-\hat{\mathbf{S}}_{t})+\mathbf{M}_{t},$
(13)

where $\mathbf{M}_{t}\sim\mathcal{N}(0,\Omega_{t})$ is independent of
$\mathbf{S}_{t}-\hat{\mathbf{S}}_{t}$, and $\hat{\mathbf{S}}_{t}$ is the
optimal decoder’s estimate (8), given recursively by

$\hat{\mathbf{S}}_{t}=A\hat{\mathbf{S}}_{t-1}+\mathrm{K}_{t}(\mathbf{Y}_{t-1}-\tilde{\Gamma}_{t}H\hat{\mathbf{S}}_{t-1}),$
(14)

where
$K_{t}=A\tilde{\Gamma}_{t}^{T}H^{T}\left(H\tilde{\Gamma}_{t}P_{t}^{-1}\tilde{\Gamma}_{t}^{T}H^{T}+I\right)^{-1}$,
$\hat{\mathbf{S}}_{0}=0$, and the error covariance
$P_{t}=\mathrm{Cov}(\mathbf{S}_{t}-\hat{\mathbf{S}}_{t})$. Encoder parameters
$\tilde{\Gamma}_{t},\Omega_{t}$ must satisfy the power constraint (5)

$\frac{1}{T}\sum_{t=1}^{T-1}\mathrm{Tr}(\tilde{\Gamma}_{t}P_{t}^{-1}\tilde{\Gamma}_{t}^{T}+\Omega_{t})\leq
p.$ (15)

Intuitively, the encoder can communicate what is currently unknown to the
decoder with minimal power by transmitting the innovation
$\mathbf{S}_{t}-\hat{\mathbf{S}}_{t}$. The decoder’s prediction,
$\hat{\mathbf{S}}_{t}$, is computed using channel feedback. Compared to [1,
Lem. 1], Lemma 1 above introduces an independent additive term
$\mathbf{M}_{t}$ in (13). Its covariance $\Omega_{t}$ can be chosen at will to
allow more freedom in selecting the channel input distribution.

We make the following assumption on our encoder, which simplifies our analysis
of the infinite horizon estimation error.

###### Assumption 3

The encoder in Definition 3 is time-invariant, meaning that
$\tilde{\Gamma}_{t}=\tilde{\Gamma},\Omega_{t}=\Omega,\forall t$ in (13).

The encoding structure in Lemma 1 reveals a state space model, defined by (13)
and (1), that admits a Kalman filter solution (14). Consequently, the Riccati
recursions in Lemma 2, stated next, give the estimation error at the infinite
horizon. This lemma generalizes Lemma 2 in [1].

######  Lemma 2 (Riccati recursions and convergent behavior)

The prediction error covariance, $P_{t}$, defined in (7), of the optimal
linear code introduced in Lemma 1 evolves according to a Riccati recursion
that either diverges or converges to the stabilizing solution $P$ of the DARE
[21, Sec. E.4]:

$P=APA^{T}+Q-A\tilde{\Gamma}^{T}H^{T}\cdot\\\ \cdot\
(I+H(\tilde{\Gamma}P^{-1}\tilde{\Gamma}^{T}+\Omega)H^{T})^{-1}H\tilde{\Gamma}A^{T},$
(16)

where asymptotically, the power constraint (15) becomes

$\mathrm{Tr}(\tilde{\Gamma}P^{-1}\tilde{\Gamma}^{T}+\Omega)\leq p.$ (17)

The asymptotic MSE is computed as

$D=\mathrm{Tr}(P),$ (18)

where $P$ is the stabilizing solution of (16).

###  III-B The Linear Stabilizing Capacity

We express the channel input covariance (17) as

$\Pi=\tilde{\Gamma}P^{-1}\tilde{\Gamma}^{T}+\Omega,$ (19)

and define a new measure of channel capacity, the _linear stabilizing
capacity_ (LSC), below.

######  Definition 4 (Linear stabilizing capacity)

The linear stabilizing capacity of a MIMO AWGN Channel (Def. 1) for a Gauss-
Markov source (Def. 2) is

$\displaystyle\mathrm{LSC}(p)=\sup_{\begin{subarray}{c}\Pi\succeq 0,\ P\succeq
0,\ \Omega,\ \tilde{\Gamma}\colon\\\ \mathrm{Tr}(\Pi)\leq p,\ \Omega\succeq
0,\ \eqref{DARE},\ \eqref{Pi_input_def}\end{subarray}}\
\frac{1}{2}\log\det(I+H\Pi H)$ (20)

Finite MSE is achievable under power constraint, $p$ (17), with linear
encoders if and only if the constraint set in (20) is feasible. In Section IV,
we will characterize this feasibility condition in terms of the source and
channel parameters.

In general,

$\mathrm{LSC}(p)\leq C(p),$ (21)

where $C(p)$ is the Shannon capacity, expressed as

$\displaystyle C(p)=\sup_{\begin{subarray}{c}\Pi\succeq 0\colon\\\
\mathrm{Tr}(\Pi)\leq p\end{subarray}}\ \frac{1}{2}\log\det(I+H\Pi H).$ (22)

The $\Pi$ that solves (22) is known as the water-filling solution [20, Th.
9.1]. The linear stabilizing capacity (20) adds the additional constraints
(16), (17) and (19), so a solution $\Pi$ of (20) is generally sub-optimal in
(22).

##  IV Main Results

We present sufficient and necessary conditions for finite MSE achievable with
linear encoders in the transmission of a vector-valued source over an
arbitrary rank channel. We also investigate whether linear encoders are
optimal for MIMO channels.

###  IV-A Linear Coding for MIMO Channels

First, we present a sufficient condition for linear codes to achieve finite
MSE.

######  Theorem 1 (MIMO channels - sufficiency)

In zero-delay JSCC (Def. 3) of a $k$-dimensional Gauss-Markov source (Def. 2)
for transmission over an $n$-input MIMO AWGN channel (Def. 1) with power
constraint $p$, finite asymptotic error, $D<\infty$ (10), is achievable if
there exists an $n$-set partition, $\\{\mathcal{S}_{i}\\}_{i=1}^{n}$, of
$\\{1,\ldots,k\\}$, such that

$\sum_{j\in\mathcal{S}_{i}}\log|\lambda_{j}|<C_{i}(\pi_{i}),\ \forall
i=1,\ldots,n$ (23)

where $\lambda_{j}$ are the eigenvalues of $A$ (Def. 2),
$C_{i}(\pi_{i})=\frac{1}{2}\log(1+h_{i}^{2}\pi_{i})$ is the Shannon capacity
of the $i$th channel with power $\pi_{i}\geq 0$ and

$\sum_{i=1}^{n}\pi_{i}=p.$ (24)

Each set, $\mathcal{S}_{i}$, of the partition
$\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ contains the indices of the unstable modes
assigned to channel $i$. Thus, Theorem 1 allocates each unstable mode of the
source to a single channel output and defines a power allocation over the
channels that allows each channel to stabilize its assigned modes
independently. See Figure 2 for an example. We show in Theorem 2 that this
partitioning property is necessary if either the source or the channel is
scalar.

######  Theorem 2 (Scalar source or scalar channel)

If either $a\in\mathbb{R}$ or $h\in\mathbb{R}$, finite asymptotic error,
$D<\infty$ (10), is achievable by linear codes if and only if there exists a
partition $\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ and power allocation vector $\pi$
satisfying (23), (24) in Theorem 1.

_Case I (Scalar channel):_ The vector-source and scalar-channel scenario was
shown in [1]. In this setting $h\in\mathbb{R}$, so the conditions of Theorem 1
reduce to $\mathcal{S}_{1}=\\{1,\ldots,k\\}$, $\pi=p$. We can explicitly
compute

$\mathrm{LSC}(p)=C(p)=\frac{1}{2}\log(1+h^{2}p)$ (25)

_Case II (Scalar source):_ In the scalar source and MIMO channel setting,
$a\in\mathbb{R}$. The sets $\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ as defined in
Theorem 1 satisfy $\mathcal{S}_{i}=\\{1\\}$ and $\mathcal{S}_{l}=\emptyset$
for all $l\neq i$. From (23), finite MSE is achievable by linear encoders if
and only if

$\log|a|<\max_{i}C_{i}(p).$ (26)

Generally, for scalar sources and MIMO channels, $\mathrm{LSC}(p)<C(p)$, which
we show in Theorem 4.

Finally, particularizing Theorem 2 to scalar Gauss-Markov sources that are
transmitted over a scalar AWGN channel recovers the classical result from [11]
that the finite MSE is achievable if and only if $\log|a|<C(p),$ where
$\mathrm{LSC}(p)=C(p)=\frac{1}{2}\log(1+h^{2}p)$ [16, 3, 17].

It is unknown whether the partitioning property of Theorem 1 is necessary for
vector sources and MIMO channels. Towards this goal, we pose the following
matrix-algebraic conjecture, which, if true, implies the necessity of the
partitioning property in Theorem 1.

######  Conjecture 1 (Lyapunov positivity)

Let $J$ be the unique positive solution to the Lyapunov equation

$J=BJB-\Gamma^{T}(I+H\Pi H)^{-1}\Gamma+B\Gamma^{T}\Gamma B$ (27)

where $J\in\mathbb{R}^{k\times k}$, $\Gamma\in\mathbb{R}^{n\times k}$,
$B=\mathrm{diag}(b_{1},\ldots,b_{k}),b_{i}<1\ \forall i$, and
$H=\mathrm{diag}(h_{1},\ldots,h_{n})$. There exists an optimal solution
$\Gamma^{*}$ to

$\inf_{\begin{subarray}{c}\Pi\succeq 0,J\succ 0,\Gamma\colon\\\
\eqref{Jlyap_cond},\ \Pi\text{ is diagonal}\end{subarray}}\mathrm{Tr}(\Pi)$
(28)

that has exactly one non-zero entry per column.

We observe that Conjecture 1 holds numerically for all systems we have
simulated with $n,k\in\\{2,3\\}$.

######  Theorem 3 (Vector source and MIMO channel - necessity)

Assume Conjecture 1 holds. In linear zero-delay JSCC (Def. 3) of a vector
Gauss-Markov source (Def. 2) and MIMO AWGN channel (Def. 1), finite asymptotic
error, $D<\infty$ (10), is achievable if and only if there exists a partition
$\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ and power allocation vector $\pi$ satisfying
(23), (24) in Theorem 1.

If Theorem 3 holds, the necessary and sufficient conditions coincide and (23)
reduces to

$\sum_{i=1}^{n}\ \log|\lambda_{i}|\
\leq\mathrm{LSC}(p)=\sup_{\begin{subarray}{c}\pi\colon\ \pi_{i}\geq 0\\\
\eqref{mimo_cond_part},\ \eqref{powerc}\end{subarray}}\
\sum_{i=1}^{n}C_{i}(\pi_{i}).$ (29)

1

Figure 2: Example assignment of source components into the channels and
associated power allocations for each channel as described in Theorem 1. A
source can only be allocated to a single channel, and every source must be
allocated to a channel. Consequently, the sets
$\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ form a partition of $\\{1,\ldots,k\\}$. The
power allocations must satisfy (24) where $p$ is the total power constraint.

###  IV-B Linear Codes are NOT Optimal for MIMO Channels

The optimality of linear encoding, sometimes referred to as uncoded
transmission, has been studied in [14]. In this section, we establish that
linear encoders are generally sub-optimal for achieving finite MSE over MIMO
channels and present a sufficient optimality condition for linear encoders.

###### Theorem 4

(Linear coding is sub-optimal for MIMO channels) Linear codes are not optimal
for the zero-delay JSCC problem of a Gauss-Markov source over a MIMO AWGN
channel (Def. 1 \- 3). Equivalently, there exist source-channel pairs
$(A,H,p)$ (Def. 1, Def. 2, (5)) such that all linear codes result in
$D=\infty$, while there exist non-linear codes that can achieve $D<\infty$.

###### Proof:

We show this via a counterexample. Consider the setting of a scalar Gauss-
Markov source and an arbitrary rank AWGN channel studied in Theorem 2. Let
$H=I_{2}$ and $p\gg 1$, which correspond to the high SNR regime given two
identical channels. Shannon-Kotel’nikov mappings as studied in [17, Thm. 7.1]
can achieve finite estimation error if

$|a|<C(p)-o(1)$ (30)

as $p\to\infty$, while by Theorem 2, a linear encoder achieves finite
estimation error only if

$|a|<\frac{1}{2}\ C(p).$ (31)

This follows from $\max_{i}C_{i}(p)=\frac{1}{2}\log(1+p)$ in (26) while
$C(p)=2\cdot\frac{1}{2}\log(1+\frac{p}{2})$ since the water-filling solution
places power $\frac{p}{2}$ in each of the parallel channels. Taking the limit
$p\to\infty$, $\max_{i}C_{i}(p)=\frac{1}{2}\ C(p).$ ∎

Next, we will show a sufficient condition for linear codes to achieve finite
MSE with minimum power. To do so, we introduce source channel matching in the
stability sense.

###### Definition 5

(Source-channel matching in the stability sense) The source (Def. 2) and the
channel (Def. 1) are _matched in the stability sense_ if

$\sum_{i=1}^{n}\log\ |\lambda_{i}|\ =\mathrm{LSC}(p)=C(p),$ (32)

where $\mathrm{LSC}(p)$ is the linear stabilizing capacity (Def. 4) and $C(p)$
is given by (22).

If a source (Def. 2) and channel (Def. 1) are matched in the stability sense,
linear codes require the minimum power to stabilize the system. Source-channel
matching in Definition 5 holds if and only if $\Pi$ (20) coincides with the
waterfilling solution to $C(p)$ (22). In that case, condition (23) in Theorem
1 reduces to the converse result from [15, Thm. 4.1] stating that finite MSE
is achievable only if

$\sum_{i=1}^{k}\log\left(|\lambda_{i}|\right)<C(p).$ (33)

Consequently, source-channel matching ensures linear codes perform as well as
the best nonlinear codes.

##  V Conclusion

This paper contributes to a complete characterization of the performance of
linear codes in zero-delay joint-source channel coding with feedback. Theorem
1 showed sufficiency conditions for finite MSE using linear codes for the
general case. We conjecture that these conditions are necessary and prove in
Theorem 2 that our conjecture holds if either the source or channel is scalar.
Finally, we propose a notion of source-channel matching in the stability sense
in Definition 5 and demonstrate that linear encoders are generally not optimal
for MIMO channels in Theorem 4, motivating further research into non-linear
codes. Linear codes remain viable in practice due to their low complexity. An
exciting direction for future work is to explore the entire distortion-
capacity tradeoff for linear coding of Gauss-Markov sources over AWGN
Channels. While converse bounds have been analyzed in [12], achievability
bounds are poorly understood for such problems.

## Acknowledgment

This work was supported in part by the NSF under grants CCF-1751356 and
CCF-1956386, and the Israel Science Foundation (ISF) under grant 1096/23.

## References

  * [1] B. Han, O. Sabag, V. Kostina, and B. Hassibi, “Coded Kalman filtering over Gaussian channels with feedback,” in 2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1–8, Sep. 2023.
  * [2] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 379–423, Oct. 1948.
  * [3] A. Sahai and S. Mitter, “The necessity and sufficiency of anytime capacity for stabilization of a linear system over a noisy communication link—part i: Scalar systems,” IEEE Transactions on Information Theory, vol. 52, pp. 3369–3395, July 2006.
  * [4] L. Schulman, “Coding for interactive communication,” IEEE Transactions on Information Theory, vol. 42, no. 6, pp. 1745–1756, 1996.
  * [5] R. T. Sukhavasi and B. Hassibi, “Linear time-invariant anytime codes for control over noisy channels,” IEEE Transactions on Automatic Control, vol. 61, pp. 3826–3841, Feb. 2016.
  * [6] C. E. Shannon, “The zero error capacity of a noisy channel,” IRE Transactions on Information Theory, vol. 2, pp. 8–19, Sep. 1956.
  * [7] Y. Polyanskiy, H. V. Poor, and S. Verdu, “Feedback in the non-asymptotic regime,” IEEE Transactions on Information Theory, vol. 57, no. 8, pp. 4903–4925, Aug. 2011.
  * [8] M. V. Burnashev, “Data transmission over a discrete channel with feedback,” Problemy Peredaci Informacii, vol. 12, no. 4, pp. 10–30, Oct. 1976.
  * [9] N. Guo and V. Kostina, “Reliability function for streaming over a DMC with feedback,” IEEE Transactions on Information Theory, vol. 69, pp. 2165–2192, Nov. 2023.
  * [10] O. Sabag, P. Tian, V. Kostina, and B. Hassibi, “Reducing the LQG cost with minimal communication,” IEEE Transactions on Automatic Control, vol. 68, no. 9, pp. 5258–5270, 2023.
  * [11] A. Gorbunov and M. S. Pinsker, “Prognostic epsilon entropy of a Gaussian message and a Gaussian source,” Problemy Peredachi Informatsii, vol. 10, pp. 5–25, Aug. 1974.
  * [12] V. Kostina and B. Hassibi, “Rate-cost tradeoffs in control,” IEEE Transactions on Automatic Control, vol. 64, pp. 4525–4540, Apr. 2019.
  * [13] T. Tanaka, K. K. Kim, P. A. Parrilo, and S. K. Mitter, “Semidefinite programming approach to gaussian sequential rate-distortion trade-offs,” IEEE Transactions on Automatic Control, vol. 62, no. 4, pp. 1896–1910, 2017.
  * [14] M. Gastpar, B. Rimoldi, and M. Vetterli, “To code, or not to code: lossy source-channel communication revisited,” IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1147–1158, 2003.
  * [15] S. Yüksel, “Characterization of information channels for asymptotic mean stationarity and stochastic stability of nonstationary/unstable linear systems,” IEEE Transactions on Information Theory, vol. 58, no. 10, pp. 6332–6354, Oct. 2012.
  * [16] S. Tatikonda, A. Sahai, and S. Mitter, “Stochastic linear control over a communication channel,” IEEE Transactions on Automatic Control, vol. 49, pp. 1549–1561, Sep. 2004.
  * [17] A. Khina, E. R. Gårding, G. M. Pettersson, V. Kostina, and B. Hassibi, “Control over Gaussian channels with and without source–channel separation,” IEEE Transactions on Automatic Control, vol. 64, pp. 3690–3705, Apr. 2019.
  * [18] N. Elia, “When Bode meets Shannon: control-oriented feedback communication schemes,” IEEE Transactions on Automatic Control, vol. 49, pp. 1477–1488, Sep. 2004.
  * [19] A. A. Zaidi, S. Yüksel, T. J. Oechtering, and M. Skoglund, “On the tightness of linear policies for stabilization of linear systems over Gaussian networks,” Systems and Control Letters, vol. 88, pp. 32–38, Sep. 2016.
  * [20] A. E. Gamal and Y. Kim, Network Information Theory.  Cambridge University Press, 2011.
  * [21] T. Kailath, A. Sayed, and B. Hassibi, Linear Estimation.  Prentice-Hall information and system sciences series, Prentice Hall, 2000.
  * [22] K. Åström, Introduction to stochastic control theory, vol. 70 of Mathematics in science and engineering.  United States: Academic Press, 1970.

##  Appendix A Proof of Theorem 1 and Conjecture 3

We start with the DARE (16) and power constraint (17) and show that a sequence
of statements are equivalent.

First, we apply the transformation

$\tilde{\Gamma}P^{-1}=\Gamma$ (34)

to obtain a standard DARE.

_Statement 1._ A finite MSE is achievable if and only if there exists a
$P\succeq 0,\Gamma,\Omega\succeq 0$ satisfying (16) and (17).

$\displaystyle\begin{split}&P=APA^{T}+Q-AP\Gamma^{T}H^{T}\\\ \qquad&\ \ \
\left(I+H(\Gamma P\Gamma^{T}+\Omega)H^{T}\right)^{-1}H\Gamma PA^{T}\\\
\end{split}$ (35) $\displaystyle\mathrm{Tr}(\Gamma P\Gamma^{T}+\Omega)\leq p.$
(36)

We will define $\Pi\triangleq\Gamma P\Gamma^{T}+\Omega$ as a redundant
intermediate variable, resulting in Statement 2.

_Statement 2._ A finite MSE is achievable if and only if there exists a
$P\succeq 0,\Gamma,\Omega\succeq 0,\Pi\succeq 0$ satisfying

$\displaystyle\begin{split}&P=APA^{T}+Q\\\ \qquad&\ \ \
-AP\Gamma^{T}H^{T}\left(I+H\Pi H^{T}\right)^{-1}H\Gamma PA^{T}\\\ \end{split}$
(37) $\displaystyle\Pi=\Gamma P\Gamma^{T}+\Omega$ (38)
$\displaystyle\mathrm{Tr}(\Pi)\leq p.$ (39)

We equivalently write (38) as $\Pi\succeq\Gamma P\Gamma^{T}$. Note also that
any $\Omega\neq 0$ hardens the condition of Statement 2 since $P^{+}\succeq P$
where $P^{+}$ is the unique positive solution of

$P^{+}=AP^{+}A^{T}+Q-AP^{+}\Gamma^{T}H^{T}\\\ \left(R+H(\Gamma
P^{+}\Gamma^{T}+\Omega)H^{T}\right)^{-1}H\Gamma P^{+}A^{T}$ (40)

and $P$ is the unique positive solution of

$P=APA^{T}+Q\\\ -AP\Gamma^{T}H^{T}\left(I+H(\Gamma
P\Gamma^{T})H^{T}\right)^{-1}H\Gamma PA^{T}$ (41)

Here, we use $P^{+}$ to denote the solution to (16) when $\Omega\neq 0$ and
$P$ to denote the solution when $\Omega=0$. However, moving forward we will
use $P$ to denote the solution of (41).

_Statement 3._ A finite MSE is achievable if and only if there exists a
$P\succeq 0,\Gamma,\Pi\succeq 0$ satisfying

$\displaystyle P=APA^{T}+Q-AP\Gamma^{T}H^{T}\left(I+H\Pi
H^{T}\right)^{-1}H\Gamma PA^{T}$ (42) $\displaystyle\Pi\succeq\Gamma
P\Gamma^{T}$ (43) $\displaystyle\mathrm{Tr}(\Pi)\leq p.$ (44)

We reapply the transformation (34).

_Statement 4._ A finite MSE is achievable if and only if there exists a
$P\succeq 0,\tilde{\Gamma},\Pi\succeq 0$ satisfying

$\displaystyle P=APA^{T}+Q-A\tilde{\Gamma}^{T}H^{T}\left(I+H\Pi
H^{T}\right)^{-1}H\tilde{\Gamma}A^{T}$ (45)
$\displaystyle\Pi\succeq\tilde{\Gamma}P^{-1}\tilde{\Gamma}^{T}$ (46)
$\displaystyle\mathrm{Tr}(\Pi)\leq p.$ (47)

Finally, we add two redundant constraints. First,

$J=P-\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}\succeq 0$

which holds iff (46) holds by the Schur complement lemma. Then, by
substituting $J$ into (45), we obtain a Lyapunov for $J$,

$AJA^{T}-J+A\Gamma^{T}\Pi^{-1}\Gamma A^{T}+Q-A\Gamma^{T}H^{T}(I+H\Pi
H^{T})^{-1}H\Gamma A^{T}-\Gamma^{T}\Pi^{-1}\Gamma=0.$ (48)

With these redundant conditions, we obtain the statement:

_Statement 5._ A finite MSE is achievable if and only if there exists a
$J\succeq 0,P\succeq 0,\tilde{\Gamma},\Pi\succeq 0$ satisfying

$\displaystyle\begin{split}&AJA^{T}-J+A\Gamma^{T}\Pi^{-1}\Gamma A^{T}+Q\\\
&\qquad\ \ -A\Gamma^{T}H^{T}(I+H\Pi H^{T})^{-1}H\Gamma A^{T}\\\ &\qquad\ \
-\Gamma^{T}\Pi^{-1}\Gamma=0\\\ \end{split}$ (49) $\displaystyle
J=P-\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}$ (50) $\displaystyle
P=APA^{T}+Q-A\tilde{\Gamma}^{T}H^{T}\left(I+H\Pi
H^{T}\right)^{-1}H\tilde{\Gamma}A^{T}$ (51)
$\displaystyle\Pi\succeq\tilde{\Gamma}P^{-1}\tilde{\Gamma}^{T}$ (52)
$\displaystyle\mathrm{Tr}(\Pi)\leq p.$ (53)

As in Statement 3, (52) should be satisfied with equality. We will focus our
analysis on Statement 5.

Left and right multiplying (49) by $A^{-1}$ and $A^{-T}$ to obtain a stable
Lyapunov equation, we have

$A^{-1}JA^{-T}-J-\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}-A^{-1}QA^{-T}+\tilde{\Gamma}^{T}H^{T}(I+H\Pi
H^{T})^{-1}H\tilde{\Gamma}+A^{-1}\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}A^{-T}=0$
(54)

By linearity, we can separate $J=\hat{J}+\tilde{J}$ where

$\displaystyle\hat{J}$ $\displaystyle=A^{-1}JA^{-T}-A^{-1}QA^{-T}$ (55)
$\displaystyle\begin{split}\tilde{J}&=A^{-1}JA^{-T}-\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}\\\
&\qquad+\tilde{\Gamma}^{T}H^{T}(I+H\Pi H^{T})^{-1}H\tilde{\Gamma}\\\
&\qquad+A^{-1}\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}A^{-T}\end{split}$ (56)

The purpose of this step is to separate the terms involving $\tilde{\Gamma}$
so that $\tilde{\Gamma}$ only affects $\tilde{J}$. If there exists a
$\tilde{\Gamma}$ that makes $\tilde{J}$ strictly positive, we can arbitrarily
scale $\tilde{\Gamma}$ so that $\tilde{J}$ is arbitrarily positive. By making
$\tilde{J}$ arbitrarily positive, $J\succeq 0$, so we will limit our
investigation to the positivity of $\tilde{J}$.

Let $\overline{\Gamma}_{i}$ denote the $i$th row of $\overline{\Gamma}$. We
now have the following chain of equalities

$\displaystyle\begin{split}\tilde{J}&=A^{-1}\tilde{J}A^{-T}-\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}\\\
\qquad&\ \ \ +\tilde{\Gamma}^{T}H^{T}(I+H\Pi H^{T})^{-1}H\tilde{\Gamma}\\\
\qquad&\ \ \ +A^{-1}\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}A^{-T}\\\
\end{split}$ (57)
$\displaystyle\begin{split}&=A^{-1}\tilde{J}A^{-T}-\tilde{\Gamma}^{T}(\Pi+\Pi
H^{T}H\Pi)^{-1}\tilde{\Gamma}\\\ \qquad&\ \ \
+A^{-1}\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}A^{-T}\\\ \end{split}$ (58)
$\displaystyle\begin{split}&=A^{-1}\tilde{J}A^{-T}\\\ \qquad&\ \ \
-\tilde{\Gamma}^{T}\Pi^{-\frac{1}{2}}(I+\Pi^{1/2}H^{T}H\Pi^{1/2})^{-1}\Pi^{-\frac{1}{2}}\tilde{\Gamma}\\\
\qquad&\ \ \ +A^{-1}\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}A^{-T}\\\
\end{split}$ (59) $\displaystyle\begin{split}&=A^{-1}\tilde{J}A^{-T}\\\
\qquad&\ \ \
-\tilde{\Gamma}^{T}\Pi^{-\frac{1}{2}}U^{*}(I+\Lambda)^{-1}U\Pi^{-\frac{1}{2}}\tilde{\Gamma}\\\
\qquad&\ \ \ +A^{-1}\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}A^{-T}\\\
\end{split}$ (60)
$\displaystyle\begin{split}&=A^{-1}\tilde{J}A^{-T}-\overline{\Gamma}^{T}(I+\Lambda)^{-1}\overline{\Gamma}\\\
\qquad&\ \ \ +A^{-1}\overline{\Gamma}^{T}\overline{\Gamma}A^{-T}\\\
\end{split}$ (61) $\displaystyle\begin{split}&=A^{-1}\tilde{J}A^{-T}+\\\
\qquad&\ \ \
\sum_{i}\left(-\frac{1}{1+\Lambda_{i}}\overline{\Gamma_{i}}^{T}\overline{\Gamma_{i}}+A^{-1}\overline{\Gamma_{i}}^{T}\overline{\Gamma_{i}}A^{-T}\right),\end{split}$
(62)

where (58) follows from the Matrix Inversion lemma, (60) follows from the
singular value decomposition of

$\Pi^{1/2}H^{T}H\Pi^{1/2}=U\Lambda U^{*}$ (63)

where $U$ is unitary and $\Lambda$ is diagonal, and (61) follows from the
notation

$\overline{\Gamma}\triangleq U\Pi^{-\frac{1}{2}}\tilde{\Gamma}.$ (64)

Let $J_{i}$ be the unique solution to the Lyapunov equation

$\tilde{J}_{i}=A^{-1}\tilde{J}_{i}A^{-T}-\frac{\bar{\Gamma}_{i}^{T}\bar{\Gamma}_{i}}{1+\Lambda_{i}}+A^{-1}\bar{\Gamma}_{i}^{T}\bar{\Gamma}_{i}A^{-T}$
(65)

then it follows that the unique solution to $J$ is given by

$\tilde{J}=\sum J_{i}.$ (66)

From (63),
$\Lambda_{i}=\sigma_{i}\left(\Pi^{\frac{1}{2}}H^{T}H\Pi^{\frac{1}{2}}\right)$,
where $\sigma_{i}(\cdot)$ indicates the $i$th singular value in order.

To connect the definition of $\bar{\Gamma}$ to the condition of Theorem 1, we
have the following definition:

$\displaystyle\begin{split}\textit{Let
}\mathcal{S}_{i}\subset\\{1,\ldots,k\\}\textit{ be the set of indices where
}\\\ \qquad\bar{\Gamma}_{i}\textit{ is non-zero.}\end{split}$ (67)

###  A-A Proof of sufficiency

Leveraging the result for vector sources and scalar channels, see Appendix B,
$\tilde{J}_{i}\succ 0$ if and only if

$\sum_{j\in\mathcal{S}_{i}}\log|a_{j}|<\frac{1}{2}\log(1+h_{i}^{2}\pi_{i}).$
(68)

The sufficiency of the theorem follows by letting $\Pi$ be diagonal, in which
case $\Lambda_{i}=\pi_{i}h_{i}^{2}$. Let $\tilde{J}_{i}^{\mathcal{S}}$ be the
submatrix of $\tilde{J}_{i}$ formed by selecting the column and row indices
that are members of $\mathcal{S}_{i}$, the support of the $i$th row of
$\bar{\Gamma}$. Note that $\tilde{J}_{i}^{\mathcal{S}}$ will be strictly
positive by the MISO theorem and zero elsewhere. Since the union of supports
of the rows of $\bar{\Gamma}$ include every possible index,
$\tilde{J}=\sum_{i}\tilde{J}_{i}\succ 0$ and can be made arbitrarily positive
by scaling $\bar{\Gamma}$. Such a $\Pi$,
$\tilde{\Gamma}=\Pi^{1/2}\bar{\Gamma},J=\tilde{J}+\hat{J},P=J+\tilde{\Gamma}^{T}\Pi^{-1}\tilde{\Gamma}$
is sufficient for Statement 5.

###  A-B Proof of necessity

First, we show that a diagonal $\Pi$ is necessary.

###### Lemma 3

If $\tilde{J}$ can be made arbitrarily positive by some $\Gamma,\Pi$, it can
also be made arbitrarily positive by $\tilde{\Gamma},\tilde{\Pi}$ where
$\tilde{\Pi}$ is diagonal with $\mathrm{Tr}(\tilde{\Pi})\leq\mathrm{Tr}(\Pi)$.

###### Proof:

Fix a $\Lambda$ in (65). We will show that given an arbitrary $\Gamma,\Pi$
satisfying the power constraint, the same $\Lambda$ is achievable by a
diagonal $\tilde{\Pi}$ and associated $\tilde{\Gamma}$ with less power.

Let

$D\triangleq HH^{T},$ (69)

where by assumption, $h_{1}\geq h_{2}\geq\ldots\geq h_{n}$. From (63),

$\Lambda=U^{*}\Pi^{1/2}H^{T}H\Pi^{1/2}U.$ (70)

This implies

$\Pi=U^{*}\Lambda^{\frac{1}{2}}V^{*}D^{-1}V\Lambda^{\frac{1}{2}}U$ (71)

where $V$ is unitary. For a diagonal $\tilde{\Pi}$,

$\Lambda=\tilde{\Pi}^{1/2}H^{T}H\tilde{\Pi}^{1/2}=\tilde{\Pi}HH^{T}$ (72)

Then by isolating $\tilde{\Pi}$ and applying the trace to both sides,

$\mathrm{Tr}(\tilde{\Pi})=\mathrm{Tr}(\Lambda D^{-1}),$ (73)

and

$\mathrm{Tr}(\Pi)=\mathrm{Tr}(U^{*}\Lambda^{\frac{1}{2}}V^{*}D^{-1}V\Lambda^{\frac{1}{2}}U)=\mathrm{Tr}(\Lambda
V^{*}D^{-1}V).$ (74)

Note that the entries of $\Lambda$ are in descending order and the entries of
$D^{-1}$ are listed in ascending order by assumption that $h_{1}\geq
h_{2}\geq\ldots\geq h_{n}$.

We apply Ruhe’s Trace Inequality, which states that if $A,B$ are PSD matrices,
with eigenvalues $a_{1}\geq a_{2}\geq\ldots a_{n}$ and $b_{1}\geq
b_{2}\geq\ldots b_{n}$,

$\sum_{i=1}^{n}a_{i}b_{n-i+1}\leq\mathrm{Tr}(AB).$ (75)

Here, $A=\Lambda$, $B=V^{*}D^{-1}V$, and $\mathrm{Tr}(\tilde{\Pi})$ achieves
the lower bound with equality since $\Lambda$ and $D^{-1}$ are both diagonal
with opposing ordering as in (75). $\mathrm{Tr}(\Pi)$ has a sandwiched unitary
$V$ in (74). Thus,

$\mathrm{Tr}(\tilde{\Pi})\leq\mathrm{Tr}(\Pi)$

as desired. ∎

The above lemma shows that we can take $\Pi$ to be diagonal without loss of
generality since that is what minimizes the power. Then,

$\Lambda_{i}=\pi_{i}H_{i}^{2}.$

and

$\tilde{J}_{i}=A^{-1}\tilde{J}_{i}A^{-T}-\frac{\bar{\Gamma}_{i}^{T}\bar{\Gamma}_{i}}{1+h_{i}^{2}\pi_{i}}+A^{-1}\bar{\Gamma}_{i}^{T}\bar{\Gamma}_{i}A^{-T}$
(76)

To show the partitioning property of Theorem 1, we need to show that the sets
$\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ cover $\\{1,\ldots,k\\}$ and are disjoint.

We show the necessity of $\bigcup_{i=1}^{k}\mathcal{S}_{i}=\\{1,\ldots,k\\}$.
For $\tilde{J}$ to be arbitrarily positive, $\Gamma$ must excite all
directions of $\tilde{J}$. To see this, suppose an index $j$ exists such that
$\bar{\Gamma}_{ij}=0$ for all $i$ in (64). Then, $e_{j}^{T}\tilde{J}e_{j}=0$,
where $e_{j}$ is the $j$th standard basis vector, and $\tilde{J}$ cannot be
positive definite.

What remains to be shown is the necessity of the disjointness of the sets
$\\{\mathcal{S}_{i}\\}_{i=1}^{n}$. By showing Conjecture 1, $\tilde{J}$ (56)
can be made arbitrarily positive by scaling $\tilde{\Gamma}$ of the form
described in Conjecture 1. The sets $\\{\mathcal{S}_{i}\\}_{i=1}^{n}$ are then
disjoint.

We can consider the equivalent (redundant) constraints on $P$ in Statement 5.
In our construction of $\Pi$ in (52), we demonstrated that slack in
$\Pi\succeq\tilde{\Gamma}P^{-1}\tilde{\Gamma}$ only hardens the conditions, so
we can always take $\Pi=\tilde{\Gamma}P^{-1}\tilde{\Gamma}$. In Lemma 3, we
showed that the optimal $\Pi$ is diagonal. Consequently,
$\tilde{\Gamma}^{T}P^{-1}\tilde{\Gamma}$ can also be considered to be
diagonal. We now ask what this diagonality constraint imposes on the structure
of $\tilde{\Gamma}$.

Recall our original DARE:

$P=APA+Q-AP\Gamma^{T}H(I+H\Gamma P\Gamma^{T}H)^{-1}H\Gamma PA^{T}$

where $\Gamma=\tilde{\Gamma}P^{-1}$.

Under the condition $\Gamma P\Gamma^{T}$ is diagonal, the measurement
covariance $I+H\Gamma P\Gamma^{T}H$ is diagonal as well since $H$ is diagonal.
We conjecture that the diagonality of $\Gamma P\Gamma^{T}$ imposes a structure
on $\tilde{\Gamma}$ so that $\mathcal{S}_{i}$ is consistent with the condition
of Theorem 1. This means that $\tilde{\Gamma}$ has only a single non-zero
entry in each column. Equivalently, each mode is assigned to a single channel.
If this matrix-algebraic fact on the structure of $\tilde{\Gamma}$ holds,
Conjecture 3 holds as well.

This concludes the proof.

##  Appendix B Proof of Lemma 3 \- Vector Source over Scalar Channel

This proof was initially presented in [1], but we give it here for
completeness.

The channel is scalar, so $\tilde{J}=\tilde{J}_{1}$ in 76. Further, we can
always set $\pi_{1}=p$ since the optimal encoder uses all the available power.
From (76), we have

$\tilde{J}=A^{-1}\tilde{J}A^{-T}-\frac{\bar{\Gamma}^{T}\bar{\Gamma}_{i}}{1+h^{2}p}+A^{-1}\bar{\Gamma}^{T}\bar{\Gamma}A^{-T}$
(77)

Defining $D_{\Gamma}\triangleq\mbox{diag}(\bar{\Gamma})$, i.e., the diagonal
matrix whose components are the elements of the vector $\bar{\Gamma}^{T}$, we
may now write

$A^{-1}\bar{\Gamma}^{T}=D_{\Gamma}a~{}~{}~{}\mbox{and}~{}~{}~{}\bar{\Gamma}^{T}=D_{\Gamma}1,$

where $a$ is the vector of the diagonal elements of $A^{-1}$ and $1$ is the
all-one vector. (77) becomes

$\tilde{J}=A^{-1}\tilde{J}A^{-T}+D_{\Gamma}\left(aa^{T}-\frac{11^{T}}{1+h^{2}p}\right)D_{\Gamma}.$
(78)

Let $M$ be the solution to the Lyapunov equation

$M=A^{-1}MA^{-T}+11^{T}.$ (79)

Let $(A_{u},1)$ be controllable, which holds by Assumption 2. Then by the
Lyapunov stability theorem, $M\succ 0$. We now claim that

${\tilde{J}}=D_{\Gamma}\left(\frac{h^{2}p}{1+h^{2}p}M-11^{T}\right)D_{\Gamma}.$
(80)

This can be verified by plugging (80) into (78). It follows that
${\tilde{J}}\succ 0$ if and only if $\frac{h^{2}p}{1+h^{2}p}M-11^{T}>0$. But
the latter is equivalent to

$\left[\begin{array}[]{cc}M&1\\\
1^{T}&\frac{h^{2}p}{1+h^{2}p}M-11^{T}\end{array}\right]\succ 0,$ (81)

or

$\frac{h^{2}p}{1+h^{2}p}M-11^{T}>1^{T}M^{-1}1.$ (82)

Assume $M$ satisfies the Lyapunov equation (79). Then

$1^{T}M^{-1}1=1-\left|\mbox{det}(A)\right|^{-2}.$ (83)

This follows since $A^{-1}MA^{-T}=M-11^{T}$ from (79). On the one hand

$\mbox{det}A^{-1}MA^{-T}=\left|\mbox{det}A\right|^{-2}\mbox{det}{M}$ (84)

and on the other

$\mbox{det}(M-11^{T})=\mbox{det}(I-11^{T}M^{-1})\
\mbox{det}M=(1-1^{T}M^{-1}1)\ \mbox{det}M,$ (85)

which yields the desired result.

Then, (82) and (83) imply that $\tilde{J}\succ 0$ if and only if

$\frac{h^{2}p}{1+h^{2}p}>1-\left|\mbox{det}A\right|^{-2},$ (86)

or equivalently

$1+h^{2}p>\left|\mbox{det}A_{u}\right|^{2},$ (87)

which is the capacity condition we are seeking.

Note that when this capacity condition holds,
$\frac{h^{2}p}{1+h^{2}p}M-11^{T}\succ 0$ and therefore ${\tilde{J}}\succ 0$ in
(80). We can arbitrarily scale $\bar{\Gamma}$ and therefore $D_{\Gamma}$ to
make $\tilde{J}$ arbitrarily positive. This demonstrates both sufficiency and
necessity.

##  Appendix C Proof of Lemma 3 \- Scalar Source over MIMO Channel

We also consider a scalar source and arbitrary rank channel in Lemma 3. We can
solve (51) explicitly as

$P=\frac{q-a^{2}\tilde{\Gamma}^{T}H^{T}(I+H\Pi
H^{T})^{-1}H\tilde{\Gamma}}{1-a^{2}}.$ (88)

Plugging this into (50) we obtain

$J=-\frac{q}{a^{2}-1}+\tilde{\Gamma}^{T}\left(\frac{a^{2}}{a^{2}-1}H^{T}(I+H\Pi
H^{T})^{-1}H-\Pi^{-1}\right)\tilde{\Gamma}\succeq 0$ (89)

Finite estimation error can be achieved iff there exists a $\tilde{\Gamma}$
and $\Pi\geq 0$, $\mathrm{Tr}(\Pi)\leq p$ such that $J\geq 0$. The above
statement is equivalent to the statement: finite error is not achievable iff
for all $\tilde{\Gamma},\Pi$ such that $\Pi\geq 0$, $\mathrm{Tr}(\Pi)\leq p$,
we have $J<0$. This is what we set out to show.

Let

$O\triangleq\frac{a^{2}}{a^{2}-1}H^{T}(I+H\Pi H^{T})^{-1}H-\Pi^{-1}.$ (90)

Note that $J<0$ for all $\tilde{\Gamma}$ if and only if $O\preceq 0$.

$\displaystyle-O=\Pi^{-1}-\frac{a^{2}}{a^{2}-1}H^{T}(I+H\Pi H^{T})^{-1}H$
$\displaystyle\geq 0$ (91)

The inequality is equivalent to

$\begin{pmatrix}\Pi^{-1}&\sqrt{\frac{a^{2}}{a^{2}-1}}H^{T}\\\
\sqrt{\frac{a^{2}}{a^{2}-1}}H&I+H\Pi H^{T}\end{pmatrix}\geq 0$ (92)

which is also equivalent to

$\displaystyle I+H\Pi H^{T}-\frac{a^{2}}{a^{2}-1}H\Pi H^{T}$
$\displaystyle\geq 0$ (93) $\displaystyle I-\frac{1}{a^{2}-1}H\Pi H^{T}$
$\displaystyle\geq 0$ (94)

Recall that $H$ is diagonal. Let $h_{max}$ be the entry of $H$ with the
greatest magnitude. Note that $\Pi$ has the best chance of overcoming (94) by
placing all power on the best channel. Thus, if this alignment of $\Pi$ does
not violate (94), the inequality holds for any $\Pi$. Then, the final
inequality holds for any $\Pi\geq 0,\mathrm{Tr}(\Pi)\leq p$ if and only if

$a^{2}\geq 1+h_{max}^{2}p$ (95)

or

$\log a\geq\frac{1}{2}\log(1+h_{max}^{2}p)$ (96)

This is a necessary and sufficient condition for $D<\infty$ to be
unachievable. Taking the inverse of the above, we obtain the desired
condition.