# Schwarz symmetrizations in parabolic equations on complete manifolds

Haiqing Cheng  School of Mathematical Sciences, Soochow University, Suzhou,
215006, China<EMAIL_ADDRESS>,  Tengfei Ma  School of Mathematical Sciences,
Soochow University, Suzhou, 215006, China<EMAIL_ADDRESS>and  Kui
Wang  School of Mathematical Sciences, Soochow University, Suzhou, 215006,
China<EMAIL_ADDRESS>
###### Abstract.

In this article, we prove a sharp estimate for the solutions to parabolic
equations on manifolds. Precisely, using symmetrization techniques and
isoperimetric inequalities on Riemannian manifold, we obtain a Bandle’s
comparison on complete noncompact manifolds with nonnegative Ricci curvature
and compact manifolds with positive Ricci curvature respectively. Our results
generalize Bandle’s result [6] to Riemannian setting, and Talenti’s comparison
for elliptic equation on manifolds by Colladay-Langford-McDonald [12] and
Chen-Li [9] to parabolic equations.

###### Key words and phrases:

Bandle’s comparison, Parabolic equations, Manifolds

###### 2020 Mathematics Subject Classification:

53C21, 53C42

The research of the third author is supported by NSFC No.11601359

##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary,
$\Omega^{\sharp}\subset\mathbb{R}^{n}$ be a round ball with the same volume as
$\Omega$, $f(x)$ and $g(x)$ be nonnegative functions on $\Omega$, and
$f^{\sharp}$, $g^{\sharp}$ be the Schwarz rearrangement of $f$ and $g$, see
Definition 2.1 below. Let $u(x)$ and $v(x)$ be solutions to

$\displaystyle\begin{cases}u_{t}(x,t)-\Delta
u(x,t)=f(x),&(x,t)\in\Omega\times(0,\infty),\\\ u(x,0)=g(x),&x\in\Omega,\\\
u(x,t)=0,&(x,t)\in\partial\Omega\times(0,\infty),\end{cases}$

and

$\displaystyle\begin{cases}v_{t}(x,t)-\Delta
v(x,t)=f^{\sharp}(x),&(x,t)\in\Omega^{\sharp}\times(0,\infty),\\\
v(x,0)=g^{\sharp}(x),&x\in\Omega^{\sharp},\\\
v(x,t)=0,&(x,t)\in\partial\Omega^{\sharp}\times(0,\infty).\end{cases}$

Then for all $(a,t)\in(0,|\Omega|)\times[0,+\infty)$, it holds

(1.1) $\displaystyle\int_{0}^{a}u^{*}(s,t)\,ds\leq\int_{0}^{a}v^{*}(s,t)\,ds,$

where $u^{*}$ is the decreasing rearrangement of $u$, see (2.1) below.
Inequality (1.1) was proved by Bandle [6] for strong solutions, by Vázquez
[24] and Mossino-Rakotoson [20] for weak solutions. The elliptic version of
(1.1) is known as Talenti’s comparison [22, 23]. The proof of (1.1) is mainly
based on the isoperimetric inequality and symmetrization techniques. Talent
and Bandle’s comparisons play an important role in mathematical studies, since
there provide sharp estimates for solutions to elliptic and parabolic
equations, Faber-Krahn type inequality for eigenvalues [22, 3], and the bounds
on the exit times of Brownian motion[8, 12]. Bandle’s comparison (1.1) was
generalized to nonlinear parabolic equations with Dirichlet boundary condition
(see for instance [4, 5] and references therein), and to parabolic equations
with Neumann boundary condition [14] as well. We also refer the reader to the
excellent books [16, 17] for related topics.

Recently, Talenti’s comparison for Dirichlet boundary was generalized to the
compact manifolds with positive Ricci curvature by Colladay, Langford and
McDonald [12], and to complete noncompact manifold with nonnegative Ricci
curvature and positive asymptotic volume ratio by Chen and Li [9]. Also
Talenti’s comparison for Robin boundary was proved in both Euclidean space [3,
2] and complete manifolds [10, 11]. To the best of our knowledge, Bandle’s
comparison, however, did not receive as much attention as Talenti’s comparison
on manifolds. The purpose of the present paper is to study the Bandle’s
comparison for solutions to parabolic equation on manifolds. In particular, we
will establish a sharp comparison between the solutions to (1.3) and (1.4) on
complete and noncompact manifolds with nonnegative Ricci curvature and
positive asymptotic volume ratio, and compact manifolds with positive Ricci
curvature, see Theorem 1.1 below.

Let $(M,g)$ be a complete Riemannian manifold of dimension $n$, which is
either compact with $\operatorname{Ric}\geq(n-1)\kappa$ for $\kappa>0$ or
noncompact with nonnegative Ricci curvature and positive positive asymptotic
volume ratio. Denote by

(1.2)
$\displaystyle\theta:=\begin{cases}\lim_{r\to\infty}\frac{|B_{p}(r)|}{\omega_{n}r^{n}},&\quad\kappa=0,\\\
\frac{|M|}{|M_{\kappa}|},&\quad\kappa>0,\end{cases}$

where $B_{p}(r)$ is the geodesic ball centered at $p$ with radius $r$ in $M$,
$M_{\kappa}$ is the $n$ dimensional space form of constant sectional curvature
$\kappa$, $\omega_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$, and
$|M|$ denotes the volume of $M$. It follows from the volume comparison that
$\theta\leq 1$. Let $\Omega\subset M$ be a bounded domain with smooth
boundary, $f(x)$ and $g(x)$ be smooth and nonnegative function not identically
zero on $\Omega$, we consider the following Cauchy problem

(1.3) $\displaystyle\begin{cases}u_{t}(x,t)-\Delta
u(x,t)=f(x),&\quad(x,t)\in\Omega\times(0,\infty),\\\
u(x,t)=0,&\quad(x,t)\in\partial\Omega\times(0,\infty),\\\ u(x,0)=g(x),&\quad
x\in\Omega.\end{cases}$

Let $\Omega^{\sharp}$ be a geodesic ball in $M_{\kappa}$ satisfying
$\theta|\Omega^{\sharp}|=|\Omega|$, $f^{\sharp}$ and $g^{\sharp}$ defined on
$\Omega^{\sharp}$ are the Schwarz rearrangement of $f$ and $g$ respectively.
Denote by $v(x)$ the solution to the following Schwarz rearrangement system of
(1.3)

(1.4) $\displaystyle\begin{cases}v_{t}(x,t)-\Delta
v(x,t)=f^{\sharp}(x),&\quad(x,t)\in\Omega^{\sharp}\times(0,\infty),\\\
v(x,t)=0,&\quad(x,t)\in\partial\Omega^{\sharp}\times(0,\infty),\\\
v(x,0)=g^{\sharp}(x),&\quad x\in\Omega^{\sharp}.\end{cases}$

The main result of this paper is the following theorem, which gives a sharp
Bandle’s comparison on Riemannian manifolds.

######  Theorem 1.1.

Let $u(x)$ and $v(x)$ be the solutions to (1.3) and (1.4) respectively. Then

(1.5) $\displaystyle U(a,t)\leq
V(a,t),\qquad(a,t)\in(0,|\Omega^{\sharp}|]\times(0,\infty).$

Where

(1.6) $\displaystyle U(a,t)=\frac{1}{\theta}\int_{0}^{\theta
a}u^{*}(s,t)ds,\quad V(a,t)=\int_{0}^{a}v^{*}(s,t)ds.$

Moreover if inequality (1.5) holds as an equality for all
$(s,t)\in(0,|\Omega^{\sharp}|]\times(0,\infty)$, then $M$ is isometric to
$M_{\kappa}$, $\Omega$ is isometric to $\Omega^{\sharp}$, and $u(x,t)=v(x,t)$
on $\Omega^{\sharp}\times(0,\infty)$.

######  Remark 1.2.

In this paper, we mainly focus on Bandle’s comparison (1.1) on manifolds and
we then set up Theorem 1.1 in smooth case. In fact, Theorem 1.1 remains valid
for weak solutions to (1.3), see [20] and [4].

The following corollary is a simple consequence of (1.5), which gives a sharp
$L^{p}$ norm estimate of $u$ by $v$ on space variables for all $t$.

######  Corollary 1.3.

Let $u(x)$ and $v(x)$ be the solutions to (1.3) and (1.4) respectively. Then
we have

(1.7)
$\displaystyle\Big{(}\frac{1}{\theta}\int_{\Omega}u^{p}(x,t)\,dx\Big{)}^{1/p}\leq\Big{(}\int_{\Omega^{\sharp}}v^{p}(x,t)\,dx\Big{)}^{1/p}$

for all $t>0$ and $p\in[1,\infty]$.

We mention finally that Talenti’s comparison for Poisson equation with Robin
boundary has recently been confirmed by Alvino, Nitsch, and Trombetti in
dimension two, as well as an integral estimate for higher dimensions [3]. It
is a natural question to extend Bandle’s comparison to heat equation with
Robin boundary, and we leave this to a future study.

This paper is organized as follows. In Section 2 we recall the Schwarz
rearrangements and isoperimetric inequalities on manifolds. In Section 3 we
prove Theorem 1.1.

##  2\. Preliminaries

Let $\Omega$ be a bounded smooth domain in $M$ and $\Omega^{\sharp}$ be a
geodesic ball in $M_{\kappa}$ (i.e. $\mathbb{R}^{n}$ or $\mathbb{S}^{n}$) with
volume $|\Omega|/\theta$, where $\theta$ is a constant defined by (1.2). We
recall the definitions and properties of the Schwarz rearrangement of
nonnegative functions on manifolds, see also [9, Section 2] and [12, Section
2].

######  Definition 2.1.

Let $h(x)$ be a nonnegative measurable function on $\Omega$. Denote by
$\Omega_{h,s}=\\{x\in\Omega:h(x)>s\\}$ and $\mu_{h}(s)=|\Omega_{h,s}|$, the
decreasing rearrangement $h^{*}$ of $h$ is defined by

(2.1) $\displaystyle
h^{*}(s)=\begin{cases}\operatorname{ess}\sup_{x\in\Omega}\,h(x),&\quad s=0,\\\
\inf\\{t\geq 0:\mu_{h}(t)<s\\},&\quad s>0,\end{cases}$

for $s\in[0,|\Omega|]$. The Schwarz rearrangement of $h$ is defined by

(2.2) $\displaystyle h^{\sharp}(x)=h^{*}(\theta\omega_{n}r^{n}(x)),\quad
x\in\Omega^{\sharp},$

where $r(x)$ is the distance function from the center of $\Omega^{\sharp}$ in
$M_{\kappa}$, and $\omega_{n}$ is the volume of unit ball in $\mathbb{R}^{n}$.

It follows directly from (2.1) and (2.2) that

(2.3) $\displaystyle\mu_{h}(s)=\theta\mu_{h^{\sharp}}(s)$

for $s\geq 0$. Meanwhile, the Fubini’s theorem gives

(2.4)
$\displaystyle\int_{\Omega}h^{p}(x)\,dx=\int_{0}^{|\Omega|}(h^{*})^{p}(s)\,ds=\theta\int_{\Omega^{\sharp}}(h^{\sharp})^{p}(x)\,dx,$

for $h\in L^{p}(\Omega)$, $p\geq 1$. Moreover for any nonnegative functions
$f(x)$ and $g(x)$, the following inequality, known as Hardy-Littlewood
inequality, holds true

(2.5)
$\displaystyle\int_{\Omega}f(x)g(x)\,dx\leq\int_{0}^{|\Omega|}f^{*}(s)g^{*}(s)\,ds.$

Taking $g(x)$ as the characteristic function of $\Omega_{h,s}$ in above
inequality yields

(2.6)
$\displaystyle\int_{\Omega_{h,s}}f(x)\,dx\leq\int_{0}^{\mu_{h}(s)}f^{*}(\eta)\,d\eta.$

To prove Theorem 1.1, we require the following isoperimetric inequality on
manifolds with Ricci curvature bounded from below.

######  Theorem 2.2.

With $M$, $\theta$, $\Omega$ and $\Omega^{\sharp}$ as above, there holds

(2.7) $\displaystyle|\partial\Omega|\geq\theta|\partial\Omega^{\sharp}|,$

where $|\partial\Omega|$ denotes the $(n-1)$-dimensional area of
$\partial\Omega$. The equality holds if only if $\Omega$ is isometric to
$\Omega^{\sharp}$.

When $\operatorname{Ric}\geq n-1$, inequality (2.7) was shown by Lévy and
Gromov [15], see also Theorem 2.1 of [21]. When $M$ is noncompact and
$\operatorname{Ric}\geq 0$, inequality (2.7) was shown in dimension three by
Agostiniani, Fogagnolo and Mazzieri [1] and in all dimensions by Brendle [7].
Meanwhile, Brendle proved in [7] that (2.7) also holds true when $\Omega$ is a
compact minimal submanifold of $M$ of dimension $n+2$ with nonnegative
sectional curvature.

##  3\. Proof of Theorem 1.1

In this section, we will prove the main theorem. For simplicity, we write
$\Omega_{u}(a,t)$ and $\Omega_{u}^{\sharp}(a,t)$ as $\Omega_{a,t}$ and
$\Omega^{\sharp}_{a,t}$ respectively for short. Let

$\displaystyle\Phi(s)=|\partial B_{s}|$

for $s\in(0,|\Omega^{\sharp}|]$, where $B_{s}$ is a round geodesic ball in
$M_{\kappa}$ with volume $s$. It can be easily checked that
$\Phi(s)=n\omega_{n}^{1/n}s^{(n-1)/n}$ if $\kappa=0$, and $\Phi(s)$ is
monotone increasing in $s$.

######  Lemma 3.1.

Under the hypotheses of Theorem 1.1, we have

(3.1)
$\displaystyle\theta^{2}\Phi^{2}(\frac{\mu_{u}(s,t)}{\theta})\leq-\partial_{s}\mu_{u}(s,t)\int_{\Omega_{s,t}}f(x)-u_{t}(x,t)\,dx,$

and

(3.2)
$\displaystyle\Phi^{2}(\mu_{v}(s,t))=-\partial_{s}\mu_{v}(s,t)\int_{\Omega^{\sharp}_{s,t}}f^{\sharp}(x)-v_{t}(x,t)\,dx,$

for a.e. $t>0$.

###### Proof.

By the Sard’s theorem, for each $t>0$ we have

(3.3) $\displaystyle\partial\Omega_{s,t}=\\{x\in\Omega:u(x,t)=s\\}.$

for almost every $s\geq 0$. Observing from isoperimetric inequality (2.7) that

(3.4) $\displaystyle\theta\Phi(\frac{|\Omega_{s,t}|}{\theta})\leq$
$\displaystyle|\partial\Omega_{s,t}|,$

and from the Hölder inequality that

(3.5)
$\displaystyle|\partial\Omega_{s,t}|^{2}\leq\int_{\partial\Omega_{s,t}}\frac{1}{|\nabla
u|}\,dA\int_{\partial\Omega_{s,t}}|\nabla u|\,dA,$

then we get

(3.6)
$\displaystyle\theta^{2}\Phi^{2}(\frac{|\Omega_{s,t}|}{\theta})\leq\int_{\partial\Omega_{s,t}}\frac{1}{|\nabla
u|}\,dA\int_{\partial\Omega_{s,t}}|\nabla u|\,dA,$

here and thereafter $dA$ denotes the induced measure on $(n-1)$ dimensional
surface in $\Omega$. Because $u(x,t)$ is vanishing on $\partial\Omega$, so
$\Omega_{s,t}$ is a compact set in $\Omega$, therefore by the coarea formula
we have

$\displaystyle\partial_{s}\mu_{u}(s,t)=-\int_{\partial\Omega_{s,t}}\frac{1}{|\nabla
u|}\,dA$

then (3.6) becomes to

(3.7)
$\displaystyle\theta^{2}\Phi^{2}(\frac{|\Omega_{s,t}|}{\theta})\leq\partial_{s}\mu_{u}(s,t)\int_{\partial\Omega_{s,t}}|\nabla
u|\,dA.$

For $t>0$, noticing from Definition 2.1 and equality (3.3) that

$|\nabla u|=-\frac{\partial u}{\partial\nu}$

on $\partial\Omega_{s,t}$ for a.e. $s>0$, where $\nu$ is the unit outer normal
to $\partial\Omega_{s,t}$, we then get

$\displaystyle\int_{\partial\Omega_{s,t}}|\nabla
u|\,dA=\int_{\Omega_{s,t}}-\Delta
u(x)\,dx=\int_{\Omega_{s,t}}f(x)-u_{t}(x,t)\,dx.$

So inequality (3.1) follows from the above equality and inequality (3.7).

If $v(x,t)$ is the solution to system (1.4), $v(x,t)$ is radial and decreasing
along the radial direction on $\Omega^{\sharp}$ for all $t$, hence
$\Omega^{\sharp}_{s,t}$ is a round ball. Therefore both (3.4) and (3.5) hold
as equalities with $\theta=1$ if we replace $u$ by $v$, hence equality (3.2)
holds true. $\square$

Now we turn to prove the main theorem.

###### Proof of Theorem 1.1.

Recall

$U(a,t)=\frac{1}{\theta}\int_{0}^{\theta a}u^{*}(s,t)ds$

for $a\in[0,|\Omega^{\sharp}|)$, then the first derivative of $a$ gives

$U^{\prime}(a,t)=u^{*}(\theta a,t),$

and from Lemma 1.1 of [6] it holds

(3.8) $\displaystyle U_{t}(a,t)=\frac{\partial}{\partial
t}\Big{(}\frac{1}{\theta}\int_{\Omega_{u^{*},t}}u(x,t)\,dx\Big{)}=\frac{1}{\theta}\int_{\Omega_{u^{*},t}}u_{t}(x,t)\,dx,$

where $u^{*}$ is evaluated at $(\theta a,t)$ and $U^{\prime}=\partial_{a}U$.
Observing that

$\mu_{u}(u^{*}(\theta a,t))=\theta a,$

for a.e. $a>0$ (c.f. [20, page 66]), and differentiating above identity in $a$
yields

$\mu_{u}^{\prime}(u^{*}(\theta a,t))U^{\prime\prime}(a,t)=\theta,$

then

$\displaystyle
U^{\prime\prime}(a,t)=\frac{\theta}{\mu_{u}^{\prime}(u^{*})}\geq-\frac{\int_{\Omega_{u^{*},t}}f(x)-u_{t}(x,t)\,dx}{\theta\Phi^{2}(a)},$

where we used (3.1) in the inequality. Recall from (2.6) that

(3.9)
$\displaystyle\int_{\Omega_{u^{*},t}}f(x)\,dx\leq\int_{0}^{a\theta}f^{*}(s)\,ds=\theta\int_{0}^{a}(f^{\sharp})^{*}(s)\,ds,$

then we deduce that

(3.10) $\displaystyle
U_{t}(a,t)-\Phi^{2}(a)U^{\prime\prime}(a,t)-\int_{0}^{a}(f^{\sharp})^{*}(s)\,ds\leq
0$

for $(a,t)\in(0,|\Omega^{\sharp}|)\times(0,\infty)$ in the viscosity sense,
where we used equality (3.8).

Note that $v(x,t)$ is radial function decreasing along the radial direction,
and $f^{\sharp}$ is radial as well, then the inequality (3.9) holds as
equality if we replace $u$ by $v$, $\Omega$ by $\Omega^{\sharp}$ and $f(x)$ by
$f^{\sharp}$ according to equality case of Hardy-Littlewood inequality, i.e.

(3.11)
$\displaystyle\int_{\Omega^{\sharp}_{v^{*},t}}f(x)\,dx=\int_{0}^{a}(f^{\sharp})^{*}(s)\,ds.$

Combining equality (3.2) and equality (3.11), we find similarly as (3.10) that

(3.12) $\displaystyle
V_{t}(a,t)-\Phi^{2}(a)V^{\prime\prime}(a,t)-\int_{0}^{a}(f^{\sharp})^{*}(s)\,ds=0,$

for $(a,t)\in(0,|\Omega^{\sharp}|)\times(0,\infty)$. It then follows from
inequality (3.10) and equality (3.12) that

(3.13) $\displaystyle(U-V)_{t}(a,t)-\Phi^{2}(a)(U-V)^{\prime\prime}(a,t)\leq
0.$

On the other hand, by the initial condition of $u$ and $v$ we have

(3.14)
$\displaystyle\begin{split}U(a,0)-V(a,0)=&\frac{1}{\theta}\int_{0}^{\theta
a}u^{*}(s,0)ds-\int_{0}^{a}v^{*}(s,0)ds\\\ =&\frac{1}{\theta}\int_{0}^{\theta
a}g^{*}(s)ds-\int_{0}^{a}(g^{\sharp})^{*}(s)\,ds\\\ =&0.\end{split}$

Moreover, by direct calculations we have

(3.15) $\displaystyle U(0,t)=V(0,t)=0,$

and

(3.16) $\displaystyle
U^{\prime}(|\Omega^{\sharp}|,t)-V^{\prime}(|\Omega^{\sharp}|,t)=u^{*}(|\Omega|,t)-v^{*}(|\Omega^{\sharp}|,t)=0.$

We then conclude that differential inequality (3.13) holds with initial
condition (3.14) and boundary conditions (3.15) and (3.16). Therefore (1.5)
follows by the standard maximum principle for viscosity solutions to parabolic
equation, see [13, Theorem 3.3] and [18, Section 2].

Now we consider the equality case of (1.5). If $U(a,t)=V(a,t)$, then
$u^{*}(\theta a,t)=v^{*}(a,t)$, hence
$\mu_{u}(u^{*},t)=\theta\mu_{v}(v^{*},t)$ for all $a>0$. Then we estimate that

(3.17)
$\displaystyle\begin{split}|\partial\Omega_{u^{*},t}|^{2}&\leq-\mu_{u}^{\prime}(u^{*},t)\big{(}\int_{\Omega_{u^{*},t}}f(x,t)-u_{t}(x,t)\
dx\big{)}\\\ &\leq-\mu_{u}^{\prime}(u^{*},t)\big{(}\int_{0}^{\theta
a}f^{*}(s,t)\ ds-\theta U_{t}(a,t)\big{)}\\\
&=-\theta^{2}\mu_{v}^{\prime}(v^{*},t)\big{(}\int_{0}^{a}(f^{\sharp})^{*}(s,t)\
ds-V_{t}(a,t)\big{)}\\\
&=\theta^{2}|\partial\Omega^{\sharp}_{v^{*},t}|^{2}.\end{split}$

where in the first step we used inequality (3.1), in the second inequality we
used (3.8) and (3.9). On the other hand, the isoperimetric inequality (2.7)
gives

$\displaystyle|\partial\Omega_{u^{*},t}|^{2}\geq\theta^{2}|\partial\Omega^{\sharp}_{v^{*},t}|^{2},$

so inequality (3.17) holds as equality. Thus we conclude from the equality
case in (2.7) that $\Omega_{u^{*},t}$ is a geodesic ball in $M_{\kappa}$,
$\theta=1$ and $u(x,t)=v(x,t)$ on $\Omega^{\sharp}\times[0,\infty)$, hence
$f(x)=f^{\sharp}(x)$ and $g(x)=g^{\sharp}(x)$. $\square$

###### Proof of Corollary 1.3.

The proof of this lemma is in the same spirit as in Theorem 3.2 of [6]. For
the reader’s convenience, we give the details. For $p\geq 1$, the Fubini’s
theorem gives

$\displaystyle\frac{1}{\theta}\int_{\Omega}u^{p}(x,t)\,dx=\frac{1}{\theta}\int_{0}^{|\Omega|}(u^{*})^{p}(s,t)ds=\int_{0}^{|\Omega^{\sharp}|}(u^{*})^{p}(\theta
s,t)\,ds$

and

$\displaystyle\int_{\Omega^{\sharp}}v^{p}(x,t)\,dx=\int_{0}^{|\Omega^{\sharp}|}(v^{*})^{p}(s,t)\,ds.$

Then we estimate by integration by parts that

(3.18)
$\displaystyle\begin{split}&\frac{1}{\theta}\int_{\Omega}u^{p}(x,t)\,dx-\int_{\Omega^{\sharp}}v^{p}(x,t)\,dx\\\
=\quad&\int_{0}^{|\Omega^{\sharp}|}(u^{*})^{p}(\theta
s,t)-(v^{*})^{p}(s,t)\,ds\\\
\leq\quad&\int_{0}^{|\Omega^{\sharp}|}p(u^{*})^{p-1}(u^{*}-v^{*})\,ds\\\
=\quad&(U-V)p(u^{*})^{p-1}\Big{|}_{0}^{|\Omega^{\sharp}|}-\int_{0}^{|\Omega^{\sharp}|}(U-V)p(p-1)(u^{*})^{p-2}\partial_{s}u^{*}\,ds,\end{split}$

where in the inequality we used the elementary inequality $x^{p}-y^{p}\leq
px^{p-1}(x-y)$ for $x,y\in\mathbb{R}$ when $p\geq 1$. By (3.15), (1.5) and
$\partial_{s}u^{*}\leq 0$, we see

$\displaystyle(U-V)p(u^{*})^{p-1}\Big{|}_{0}^{|\Omega^{\sharp}|}=(U-V)p(u^{*})^{p-1}\Big{|}_{s=|\Omega^{\sharp}|}\leq
0,$

and

$\displaystyle(U-V)p(p-1)(u^{*})^{p-2}\partial_{s}u^{*}\geq 0,$

therefore it follows from (3.18) that

$\displaystyle\frac{1}{\theta}\int_{\Omega}u^{p}(x,t)\,dx-\int_{\Omega^{\sharp}}v^{p}(x,t)\,dx\leq
0$

for all $p\geq 1$, hence for $p=\infty$. We complete the proof of the
corollary. $\square$

In Euclidean space, the argument in the proof of (1.5) can be adapted to prove
Bandle’s comparison for the heat kernels. This is outlined in Theorem 2.2 of
[6]. It’s not hard to adapt our argument to prove Bandle’s comparison for the
heat kernels on compact manifolds with positive Ricci curvature, and complete
and noncompact manifolds with nonnegative Ricci curvature and positive
asymptotic volume ratio. Hence one can easily obtain the Faber-Krahn
inequality for the first Dirichlet eigenvalue on such manifolds via heat
kernel comparison, which has already been proved by Mattia-Lorenzo [19] and
Chen-Li [9] respectively.

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# Green Interference Based Symbiotic Security in Integrated Satellite-
terrestrial Communications

Zhisheng Yin, , Nan Cheng, , Tom H. Luan, , Yilong Hui, , and Wei Wang  Z. Yin
and Tom H. Luan are with State Key Lab. of ISN and School of Cyber
Engineering, Xidian University, Xi’an, 710071, China (e-mail:
<EMAIL_ADDRESS>tom.luan@xidian.edu.cn). N. Cheng and Y. Hui are with
State Key Lab. of ISN and School of Telecommunications Engineering, Xidian
University, Xi’an, 710071, China (e-mail<EMAIL_ADDRESS>ylhui@xidian.edu.cn). W. Wang is with the College of Electronic Information
Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing,
211106, China (e-mail: wei_wang@nuaa.edu.cn). Corresponding author:
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>This work was supported in
part by the Fundamental Research Funds for the Central Universities of
Ministry of Education of China under Grant XJS221501, the National Natural
Science Foundation of Shaanxi Province under Grant 2022JQ-602, National
Natural Science Foundation of China (No. 62071356 and 6210010668), Natural
Science Foundation of Jiangsu Province (No. BK20200440), and Natural Sciences
and Engineering Research Council (NSERC) of Canada.

###### Abstract

In this paper, we investigate secure transmissions in integrated satellite-
terrestrial communications and the green interference based symbiotic security
scheme is proposed. Particularly, the co-channel interference induced by the
spectrum sharing between satellite and terrestrial networks and the inter-beam
interference due to frequency reuse among satellite multi-beam serve as the
green interference to assist the symbiotic secure transmission, where the
secure transmissions of both satellite and terrestrial links are guaranteed
simultaneously. Specifically, to realize the symbiotic security, we formulate
a problem to maximize the sum secrecy rate of satellite users by cooperatively
beamforming optimizing and a constraint of secrecy rate of each terrestrial
user is guaranteed. Since the formulated problem is non-convex and
intractable, the Taylor expansion and semi-definite relaxation (SDR) are
adopted to further reformulate this problem, and the successive convex
approximation (SCA) algorithm is designed to solve it. Finally, the tightness
of the relaxation is proved. In addition, numerical results verify the
efficiency of our proposed approach.

###### Index Terms:

Satellite-terrestrial, Green interference, Symbiotic security, Secrecy rate.

##  I Introduction

In the emerging 6G, integrated satellite-terrestrial networks can provide a
scenario of Internet access with expanded coverage and seamless
connectivity[1, 2]. The use cases of integrated satellite-terrestrial networks
have attracted a lot of attention in applications such as Internet-of-things
(IoT), vehicle-to-everything (V2X), holographic communications, etc[3, 4, 5,
6]. With the increasing number of connective devices and abundant service
requirements, the spectrum resource of space information networks is
increasingly scarce[7]. To address this issue, the spectrum sharing within
satellite-terrestrial networks and frequency reuse among multiple beams of
satellite are widely adopted to improve the spectrum utilization[8]. However,
the co-channel/inter-beam interference caused by the spectrum sharing and
frequency reuse degrades the overall communication performance[9, 10, 11], and
the security vulnerability is opportunistic to penetrate crossing satellite
and terrestrial networks[12, 13]. Particularly, due to the openness of
wireless channels and the broad broadcasting coverage, the integrated
satellite-terrestrial communications are vulnerable to eavesdropping threats
and the very large and complex geographical areas give harbour to attackers
and eavesdroppers (Eves), which results in serious security issues.

To combat the eavesdropping threats, physical layer security has been widely
investigated in conventional wireless communications and also actively
explored in satellite and aerial networks recently. As a supplement of
cryptography protocol in the upper layer, physical layer security approaches
are aimed to ensure secrecy capacity by exploiting the randomness difference
of wireless channels[14]. Particularly, several related works of physical
layer security in satellite-terrestrial communications have been reported. To
achieve the secure transmissions of satellite communications, additional
communication facilities generally work as an assistance to enhance the main
channel capacity or degrade the eavesdropping channel capacity. By exploiting
terrestrial base stations (BSs) to serve as cooperative relays of satellite
communications, an opportunistic user-relay selection criteria is proposed to
improve the secrecy performance in hybrid satellite-terrestrial relay networks
[12]. An unmanned aerial vehicle (UAV) relay and an aerial Eve are
particularly considered in hybrid satellite-terrestrial networks, a 3D mobile
UAV relaying strategy is designed and the secrecy capacity of satellite-
terrestrial communications is analyzed in [15]. A multi-user cooperation
scheme is proposed to improve the secrecy rate of satellite communications,
where the inter-user-interference serves as the green interference to degrade
the Eves[13]. In addition, the mutual interference between satellite and
terrestrial networks generally degrades the system performance [4], however
the green interference from terrestrial network can be designed by beamforming
(BF) optimization to minimize the total transmission power while guaranteeing
the secrecy rate constraint of satellite users and the common quality of
service of terrestrial users[16]. However, aforementioned related works in
satellite-terrestrial networks only consider the security of satellite or
terrestrial link independently and generally external resources are consumed
to assist with the concerned secure link.

Figure 1: Symbiotic secure transmission model in integrated satellite-
terrestrial downlink communications.

In this paper, we consider the downlink secure transmission in integrated
satellite-terrestrial networks as shown in Fig. 1, where the multi-beam
satellite shares spectrum resource with terrestrial networks and the full
frequency reuse is considered among multi-beam. Specifically, a BS coexists in
each satellite beam and an Eve is considered to wiretap the satellite and
terrestrial link simultaneously. Particularly, the channel similarity between
satellite links and limited resource at satellite challenges the
implementation of physical layer security in integrated satellite-terrestrial
communications. To address this issue and hold the security requirement of
terrestrial segment, we conduct a work to guarantee secure transmissions in
both satellite and terrestrial communications in this paper and the satellite-
terrestrial symbiotic security is realized. The main contributions of this
paper are as follows.

  * •

A framework of satellite-terrestrial symbiotic secure transmission is first
proposed, where the green interference from the internal system without
additional assistance is considered to implement the symbiotic security, where
the co-channel interference caused by the spectrum sharing within satellite-
terrestrial networks and the inter-beam interference due to frequency reuse
among satellite beams serve as the green interference. By carefully designing
such green interference with BF optimization to confuse the wiretap channel,
both satellite and terrestrial links can benefit from each other in achieving
the secure transmission. Due to the inherent green interference in integrated
satellite-terrestrial communications, the symbiotic security can be realized
without consuming external resources.

  * •

To realize the symbiotic security, we formulate a problem to maximize the sum
secrecy rate of satellite users (SUs) in multiple beams by jointly optimizing
BFs of satellite and all BSs, and meanwhile a predefined secrecy rate of
terrestrial user (TU) in each beam is ensured. To solve this intractable non-
convex problem, the Taylor expansion and semi-definite relaxation (SDR) are
first adopted to reformulate this problem with constructing convex
approximations, and secondly a successive convex approximation (SCA) algorithm
is conduct to solve it.

  * •

To prove the tightness of relaxation, we introduce an equivalent power
minimization problem and by which the rank-one constraints of BF vectors of
satellite and BSs are proved. In addition, we develop a reference benchmark
which jointly optimizes satellite BF and the power allocation of BSs, where
the BS can utilize partial power to create artificial noise (AN) for confusing
Eve. Finally, simulations are carried out to verify the efficiency of our
proposed approach.

The reminder of this work is organized as follows. In Section III, the system
model of symbiotic secure transmission in integrated satellite-terrestrial
downlink communications is presented and we formulate the problem to maximize
the sum secrecy rate of SUs with constrained secrecy rate of TUs. Then a
series of reformations and a cooperative BF optimization are conduct to solve
this non-convex problem in Section IV. In Section V, for the reference
propose, the joint optimization approach of the satellite BF and the power
allocation of BSs is given, and simulation results are carried out to evaluate
the secrecy performance of our proposed approaches. Finally, this paper is
concluded in Section VI.

##  II Related Works

By utilizing multi-antenna technology, spatial modulation based physical layer
security approaches can be summarized as precoding, friendly jamming, and
transmitting selection[17, 18, 19]. In terrestrial networks, abundant channel
uncertainties can be utilized to distinguish with the main and wiretap
channels, and recently intelligent reflecting surface (IRS) is introduced to
enhance the physical layer security recently[20, 21]. Particularly, secure BF
schemes in multiple-input–single-output (MISO) and multiple-input–multiple-
output (MIMO) systems are widely investigated [22, 23, 24], and artificial
noise (AN) aided secure BF approach is also generally adopted to confuse
Eves[25, 26]. In addition, non-orthogonal multiple access (NOMA) based
physical layer security approaches have also attracted some attentions[27,
28].

Whereas for satellite communications, a two-way physical layer security
protocol is proposed in optical links and the random nature of Poisson channel
is exploited to ensure secret communications[29], where a nano-satellite is
particularly considered as the Eve in the uplink. A threshold-based user
scheduling scheme is proposed to address a case of multiple eavesdroppers[30].
By jamming from satellite and full-duplex receiver, a joint BF and power
allocation scheme is proposed to ensure the secrecy outage probability of
multi-beam satellite communications[31]. Particularly, a comprehensive survey
on physical layer security in satellite communications is conducted in [32].
However, the inherent difference between satellite and terrestrial
communications has not been clearly pointed out when executing physical layer
security respectively. Few works highlight the resource limitation at
satellite and the distinctive channel characteristic of satellite channels,
i.e., channel similarity[33, 18, 19, 34].

Considering a coexistence of satellite and terrestrial networks, current
investigations show that either the secure transmission of satellite
communication or terrestrial communication is considered. To guarantee secure
satellite communications, the signal-to-interference-plus-noise (SINR) of Eve
for wiretapping satellite signals is decreased by terrestrial BS’s cooperative
BF, and the SINR of legitimate satellite user is increased[35], where the
quality of service (QoS) of terrestrial link is guaranteed. The BS provides
interference resource to assist secure satellite link in cognitive satellite-
terrestrial communications[36]. A secrecy-energy efficiency is considered
while constrains the cellular users’ rate requirements in a cognitive
satellite-terrestrial network[37]. In addition, by exploiting the BS and a
cooperative terminal as green interference resources, the Eve within the
signal beampattern region of BS is degraded, where the interference level of
satellite link is constrained[38]. However, the work considering secure
transmissions of both satellite and terrestrial communications has not been
investigated, which motivates we conduct an investigation on physical layer
security in integrated satellite-terrestrial communications.

TABLE I: Summary of Main Notations and Definitions Notation | Definition
---|---
$N$ | number of satellite beams
$M$ | number of BS transmit antennas
${\rm{S}}{{\rm{U}}_{k}}$ | satellite user in the $k^{th}$ beam
${\rm{T}}{{\rm{U}}_{k}}$ | terrestrial user in the $k^{th}$ beam
${\bf{v}}\in\mathbb{C}^{(M-1)\times 1}$ | AN vector
${\bf{h}}_{su}\in{\mathbb{C}^{N\times 1}}$ | channel vector from satellite to SU
${\bf{h}}_{tu}\in{\mathbb{C}^{N\times 1}}$ | channel vector from satellite to TU
${\bf{h}}_{e}\in{\mathbb{C}^{N\times 1}}$ | channel vector from satellite to Eve
${\bf{g}}_{su}\in\mathbb{C}^{M\times 1}$ | channel vector from BS to SU
${\bf{g}}_{tu}\in\mathbb{C}^{M\times 1}$ | channel vector from BS to TU
${\bf{g}}_{e}\in\mathbb{C}^{M\times 1}$ | channel vector from BS to Eve
${{\bf{w}}_{k}}\in{\mathbb{C}^{N\times 1}}$ | satellite BF vector for the $k^{th}$ beam
${\bf{f}}_{k}\in\mathbb{C}^{M\times 1}$ | BS BF vector in the $k^{th}$ beam
$\Delta$ | norm-bounded channel estimate error
${\hat{h}_{e}}$ | eavesdropping channel estimation by the BS
${{{\bf{\hat{g}}}}_{e}}$ | eavesdropping channel estimation by satellite

##  III System Model and Problem Formulation

We consider the downlink secure transmission in integrated satellite-
terrestrial communications shown in Fig. 1, where a multi-beam satellite is
assumed and a BS with terrestrial network exists in each beam. Full frequency
reuse is adopted among satellite beams and the spectrum is shared with the
BSs. Particularly, in each beam, we consider a legitimate satellite user (SU)
and terrestrial user (TU) coexisting in the common coverage of satellite and
the BS. Therefore, when a passive Eve hides in such common area, it could
wiretap the SU or TU possibly.

In this section, the channel and signal models of satellite and terrestrial
communications are first presented, where the imperfect channel state
information (CSI) for the Eve is assumed. Based on the signal models, the
received SINRs of SUs, TUs and Eves are given. Further, we conduct a symbiotic
security problem that the secure transmissions of both satellite and
terrestrial links are guaranteed simultaneously without using additional
resources. In addition, the notations are defined in Table I.

###  III-A Channel and Signal Models

For the satellite-to-ground channel, the free space path loss (FSPL), rain
attenuation, and satellite beam gain are generally considered to construct the
channel model[39], which is given by

${\bf{h}}=\sqrt{{C_{L}}b\beta}\exp\left({-j{\bm{\theta}}}\right),$ (1)

where $C_{L}$ denotes the FSPL, $b$ denotes the beam gain, $\beta$ denotes the
channel gain due to rain attenuation, and $\bm{\theta}$ is the phase vector
with uniform distribution over $\left[{\left.{0,2\pi}\right)}\right.$.
Specifically,

${C_{L}}={\left({{\lambda\mathord{\left/{\vphantom{\lambda{4\pi}}}\right.\kern-1.2pt}{4\pi}}}\right)^{2}}/\left({{d^{2}}+{h^{2}}}\right),$
(2)

where $\lambda$ denotes signal wavelength, $d$ denotes the distance from the
beam center to the center of satellite coverage, and $h$ accounts for the
height of satellite. The beam gain is defined by

$b=G{\left({\frac{{{J_{1}}\left(u_{0}\right)}}{{2u_{0}}}-36\frac{{{J_{3}}\left(u_{0}\right)}}{{u_{0}^{2}}}}\right)^{2}},$
(3)

where ${G}$ denotes the maximum satellite antenna gain,
${u_{0}}=2.07123\frac{{\sin\left({{\alpha}}\right)}}{{\sin\left({\alpha_{3{\rm{dB}}}}\right)}}$
with ${{\alpha}}$ being the elevation angle between the beam center and
${\rm{S}}{{\rm{U}}}$ and ${\alpha_{3{\rm{dB}}}}$ being the 3 dB angle of
satellite beam. Additionally, ${J_{1}}\left(\cdot\right)$ and
${J_{3}}\left(\cdot\right)$ are the first-kind Bessel functions of order 1 and
3, respectively. $\beta$ is modeled as a log-normal random variable, i.e.,
$\ln\left({{\beta_{dB}}}\right)\sim\mathcal{N}\left({u,{\delta^{2}}}\right)$
with ${{\beta_{dB}}}$ being the dB form of $\beta$. Particularly,
${{\bf{h}}_{su}}\in\mathbb{C}^{N\times 1}$,
${{\bf{h}}_{tu}}\in\mathbb{C}^{N\times 1}$,
${{\bf{h}}_{e}}\in\mathbb{C}^{N\times 1}$ are assumed to be the channel
vectors from satellite to SU, TU, and Eve, respectively.

Whereas, we adopt the channel model for terrestrial links as
${\bf{g}}=\sqrt{\alpha}{{\bf{g}}_{0}}$, where $\alpha$ denotes the large-scale
fading, $\alpha={C_{0}}{r^{-4}}$ with ${C_{0}}$ being the channel power gain
at the reference distance of 1 m and $r$ denoting the distance from BS to the
destination, and ${{\bf{g}}_{0}}$ denotes the small-scale fading which
undergoes Nakagami-$m$ fading with fading severity $m$ and average power
$\Omega$. Particularly, ${\bf{g}}_{su}\in\mathbb{C}^{M\times 1}$,
${\bf{g}}_{gu}\in\mathbb{C}^{M\times 1}$, and
${\bf{g}}_{e}\in\mathbb{C}^{M\times 1}$ denote the channel vectors between BS
and SU, TU, and Eve, respectively.

In addition, the imperfect CSI of Eve is modeled as
${{\bf{h}}_{e}}={{{\bf{\hat{h}}}}_{e}}+\Delta{{\bf{h}}_{e}}$ and
${{\bf{g}}_{e}}={{{\bf{\hat{g}}}}_{e}}+\Delta{{\bf{g}}_{e}}$, where
${\bf{\hat{h}}}_{e}$ and ${{{\bf{\hat{g}}}}_{e}}$ denote the estimations of
eavesdropping channels by satellite and the BS, with ${\Delta{{\bf{h}}_{e}}}$
and $\Delta{{\bf{g}}_{e}}$ being the norm-bounded estimate errors
correspondingly. To facilitate the easy, we assume
$\left\|{\Delta{{\bf{h}}_{e}}}\right\|=\left\|{\Delta{{\bf{g}}_{e}}}\right\|\leq\Delta$
in this work.

$R_{s}^{su,k}={\log_{2}}\left({\frac{{\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}{{\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}}\right)-{\log_{2}}\left({\frac{{\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}{{\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}}\right).$
(13)

$R_{s}^{tu,k}={\log_{2}}\left({\frac{{{\rm{Tr}}\left({{{\bf{G}}_{tu,k}}{{\bf{F}}_{k}}}\right)+\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1}}{{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1}}}\right)-{\log_{2}}\left({\frac{{{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right)+\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}1}}{{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}1}}}\right).$
(14)

$\sum\limits_{k=1}^{N}{R_{s}^{su,k}}={\rm{lo}}{{\rm{g}}_{2}}(\mathop{\Pi}\limits_{k=1}^{N}\frac{{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}{{\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}})-{\rm{lo}}{{\rm{g}}_{2}}(\mathop{\Pi}\limits_{k=1}^{N}\frac{{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}{{\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1}}).$
(15)

Assuming that $x_{su,i}$ and $x_{tu,k}$ denote the expected signal of SU and
TU respectively. Without loss of generality, the signal received by
$\text{SU}_{k}$ in the $k^{th}$ beam can be represented as

${y_{su,k}}={\bf{h}}_{su,k}^{\dagger}\sum\limits_{i}^{N}{{{\bf{w}}_{i}}{x_{su,i}}}+{\bf{g}}_{su,k}^{\dagger}{{\bf{f}}_{k}}{x_{tu,k}}+{n_{su,k}},$
(4)

where ${\bf{h}}_{su,k}$ denotes the channel vector from satellite to
$\text{SU}_{k}$, ${\bf{g}}_{su,k}$ denotes the channel vector from the BS to
$\text{TU}_{k}$, ${{\bf{w}}_{k}}\in\mathbb{C}^{N\times 1}$ and
${{\bf{f}}_{k}}\in\mathbb{C}^{M\times 1}$ denote the BF vectors of satellite
and BS in the $k^{th}$ beam, ${n_{su,k}}$ is the noise received by
$\text{SU}_{k}$. The signal received by $\text{TU}_{k}$ can be written as

${y_{tu,k}}={\bf{g}}_{tu,k}^{\dagger}{{\bf{f}}_{k}}{x_{tu,k}}+{\bf{h}}_{tu,k}^{\dagger}\sum\limits_{i}^{N}{{{\bf{w}}_{i}}{x_{su,i}}}+{n_{tu,k}},$
(5)

where ${{\bf{f}}_{k}}\in\mathbb{C}^{M\times 1}$ denotes the BF at BS.

Besides, the received signal by the Eve in the $k^{th}$ beam is given by

${y_{e,k}}={\bf{h}}_{e,k}^{\dagger}\sum\limits_{i=1}^{N}{{{\bf{w}}_{i}}{x_{su,i}}}+{\bf{g}}_{e,k}^{\dagger}{{\bf{f}}_{k}}{x_{tu,k}}+{n_{e,k}}.$
(6)

###  III-B Problem Formulation

From (6), it can be see that the Eve wiretaps signals of SU and TU. However,
(4) and (5) indicate that the SU and TU receive interference from BS and
satellite respectively, which can be designed to unequally degrade the
legitimate users and Eve in each beam. Since such interference can be regarded
as the green interference to assist the implementation of physical layer
security, the secrecy rate of SU and TU can be obtained as follows.

Based on (4) and (5), the SINR of $\text{SU}_{k}$ can be calculated as

${\gamma_{su,k}}{\rm{=}}\frac{{{{\left\|{{\bf{h}}_{su,k}^{\dagger}{{\bf{w}}_{k}}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i\neq
k}^{N}{{{\left\|{{\bf{h}}_{su,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}+{{\left\|{{\bf{g}}_{su,k}^{\dagger}{{\bf{f}}_{k}}}\right\|}^{\rm{2}}}{\rm{+}}\delta_{su,k}^{2}}},$
(7)

and the SINR of $\text{TU}_{k}$ is obtained as

${\gamma_{tu,k}}{\rm{=}}\frac{{{{\left\|{{\bf{g}}_{tu,k}^{\dagger}{{\bf{f}}_{k}}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i=1}^{N}{{{\left\|{{\bf{h}}_{tu,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}+\delta_{tu,k}^{2}}},$
(8)

where $\delta_{su,k}^{2}$ and $\delta_{tu,k}^{2}$ denote the noise power
received by $\text{SU}_{k}$ and $\text{TU}_{k}$.

Accordingly, the SINR of Eve for wiretapping $\text{SU}_{k}$ can be written as

${\gamma_{se,k}}{\rm{=}}\frac{{{{\left\|{{\bf{h}}_{e,k}^{\dagger}{{\bf{w}}_{k}}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i\neq
k}^{N}{{{\left\|{{\bf{h}}_{e,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}+{{\left\|{{\bf{g}}_{e,k}^{\dagger}{{\bf{f}}_{k}}}\right\|}^{\rm{2}}}{\rm{+}}\delta_{e,k}^{2}}},$
(9)

and the SINR of Eve for wiretapping $\text{TU}_{k}$ is given by

${\gamma_{te,k}}{\rm{=}}\frac{{{{\left\|{{\bf{g}}_{e,k}^{\dagger}{{\bf{f}}_{k}}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i=1}^{N}{{{\left\|{{\bf{h}}_{e,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}{\rm{+}}\delta_{e,k}^{2}}},$
(10)

where $\delta_{e,k}^{2}$ denotes the noise power received by the Eve.

The secrecy rate of $\text{SU}_{k}$ and $\text{TU}_{k}$ can be respectively
given by

$R_{s}^{su,k}={\left[{{{\log}_{2}}\left({1+{\gamma_{su,k}}}\right)-{{\log}_{2}}\left({1+{\gamma_{se,k}}}\right)}\right]^{+}},$
(11)
$R_{s}^{tu,k}={\left[{{{\log}_{2}}\left({1+{\gamma_{tu,k}}}\right)-{{\log}_{2}}\left({1+{\gamma_{te,k}}}\right)}\right]^{+}}.$
(12)

Based on (7–10), the secrecy rates in (11) and (12) can be further represented
in (13) and (14) as shown at the bottom of this page, where
${{\bf{H}}_{su,i}}={\bf{h}}_{su,k}{\bf{h}}_{su,k}^{\dagger}$,
${{\bf{H}}_{tu,i}}={\bf{h}}_{tu,k}{\bf{h}}_{tu,k}^{\dagger}$,
${{\bf{H}}_{e,i}}={\bf{\hat{h}}}_{e,i}{\bf{\hat{h}}}_{e,i}^{\dagger}$,
${{\bf{W}}_{i}}={{\bf{w}}_{i}}{\bf{w}}_{i}^{\dagger}$,
${{\bf{G}}_{e,k}}={\bf{\hat{g}}}_{e,k}{\bf{\hat{g}}}_{e,k}^{\dagger}$,
${{\bf{G}}_{tu,k}}={\bf{g}}_{tu,k}{\bf{g}}_{tu,k}^{\dagger}$,
${{\bf{G}}_{su,k}}={\bf{g}}_{su,k}{\bf{g}}_{su,k}^{\dagger}$,
${{\bf{F}}_{k}}={{\bf{f}}_{k}}{\bf{f}}_{k}^{\dagger}$.

From (9) and (10), we can see the SINR of Eve in each beam which targets SU is
degraded by the green interference from BS, and meanwhile the SINR of Eve
targeting TU is degraded by the green interference from satellite. Further, by
using (13), the sum secrecy rate of legitimate SUs in satellite multi-beam is
given in (15).

To guarantee the secrecy performance of both satellite link and terrestrial
link simultaneously, our focus is on optimizing the BFs of multi-beam
satellite and BS to maximize the sum secrecy rate of legitimate SUs in
satellite multi-beam and satisfy a predefined secrecy constraint of legitimate
TU in each beam.

Thus, the problem can be mathematically formulated as

$\displaystyle{\mathcal{P}1:}\quad$
$\displaystyle\mathop{\max}\limits_{{\left\\{{{{\bf{w}}_{k}},{{\bf{f}}_{k}}}\right\\}_{k=1}^{N}}}\quad\sum\limits_{k=1}^{N}{R_{s}^{su,k}}$
(16a) s.t. $\displaystyle R_{s}^{tu,k}\geq{Q}_{tu,k},k=1,2,...,N,$ (16b)
$\displaystyle\sum\limits_{k=1}^{N}{{{\left\|{{{\bf{w}}_{k}}}\right\|}^{\rm{2}}}}\leq{P_{S}},$
(16c)
$\displaystyle{\left\|{{{\bf{f}}_{k}}}\right\|^{\rm{2}}}\leq{P_{B}},k=1,2,...,N.$
(16d)

In ${\mathcal{P}1}$, (16b) guarantees the secrecy requirements of TUs within
satellite beams, where ${Q}_{tu,k}$ is a predefined secrecy rate threshold for
TU in the $k^{th}$ beam; (16c) and (16d) represent the power constraints of
satellite and BS, respectively.

Obviously, the problem ${\mathcal{P}1}$ has non-convex objective function and
constraints. A series of reformulations are conducted in the following section
to convert such intractable problem into a solvable alternative, and a joint
satellite-terrestrial BF optimizing approach is proposed to solve this
problem. To facilitate the simplification, the noise power is normalized,
i.e., $\delta_{su,k}^{2}=\delta_{tu,k}^{2}=\delta_{e,k}^{2}=1$.

##  IV Joint Satellite-terrestrial BF Optimizing

In this section, we aim to solve the secrecy rate maximization problem by
reformulating the objection function and its non-convex constraints simplify
this problem as a solvable form. Particularly, a Taylor expansion and SDR are
first adopted to reformulate the problem ${\mathcal{P}1}$ and then a
successive convex approximation approach is carried out to solve the
reformulated problem. Finally, the tightness of the relaxation is proved.

###  IV-A Taylor Expansion and SDR Reformulation

We introduce exponential variables to make the following changes.

$\displaystyle{e^{{s_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1,$
(17) $\displaystyle{e^{{\mu_{k}}}}=\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1,$
(18)
$\displaystyle{e^{{q_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1,$
(19) $\displaystyle{e^{{v_{k}}}}=\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1.$
(20)

By substituting (17–20) into (16a), the objection function in ${\mathcal{P}1}$
can be equivalently represented as

$\mathop{\max}\limits_{\left\\{{{s_{k}},{\mu_{k}},{q_{k}},{v_{k}}}\right\\}_{k=1}^{N}}\sum\limits_{k=1}^{N}{\left({{s_{k}}-{\mu_{k}}-{q_{k}}+{v_{k}}}\right)},$
(21)

which is a convex problem because the criterion is a sum of affine functions
(composed with ${{s_{k}}-{\mu_{k}}}$ and ${v_{k}}-{q_{k}}$). Particularly, the
constrains of ${s_{k}},\mu_{k},q_{k},v_{k}$ in (21) hold the following bounds.

$\displaystyle{e^{{s_{k}}}}\leq\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1,$
(22) $\displaystyle{e^{{\mu_{k}}}}\geq\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1,$
(23)
$\displaystyle{e^{{q_{k}}}}\geq\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1,$
(24) $\displaystyle{e^{{v_{k}}}}\leq\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1.$
(25)

It can be verified that all the inequalities from (22) to (25) hold with
equalities at the optimal points by the monotonicity of objective function.
However, it can be observed that the constraints in (23) and (24) are still
non-convex.

In addition, keeping (14) in mind, the constraint in (16b) is also non-convex
due to the non-convex fractional programming. Similarly, we make the following
changes.

${e^{{\tau_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}{\rm{+Tr}}\left({{{\bf{G}}_{tu,k}}{{\bf{F}}_{k}}}\right)+1,$
(26)
${e^{{\eta_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1,$
(27)
${e^{{\alpha_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}1.$
(28)

By substituting (26–28, 19) into (14), the secrecy constraint of TU in (16b)
can be equivalently reformulated as

${\eta_{k}}+{q_{k}}-{\tau_{k}}-{\alpha_{k}}\leq-\frac{{{Q_{tu,k}}}}{{{{\log}_{2}}e}},$
(29)

with the successive constraints as follows

$\displaystyle{e^{{\tau_{k}}}}$
$\displaystyle\leq\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}{\rm{+Tr}}\left({{{\bf{G}}_{tu,k}}{{\bf{F}}_{k}}}\right)+1,$
(30) $\displaystyle{e^{{\eta_{k}}}}$
$\displaystyle\geq\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1,$
(31) $\displaystyle{e^{{\alpha_{k}}}}$
$\displaystyle\leq\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}1,$
(32)

and (24) is also satisfied, where (31) and (24) are non-convex.

To address the non-convex constraints in (23), and (24), (31), we adopt Taylor
expansion to make these constraints conservatively convex approximating at
$\left({\left\\{{{{\tilde{\mu}}_{k}}}\right\\},\left\\{{{{\tilde{q}}_{k}}}\right\\},\left\\{{{{\tilde{\eta}}_{k}}}\right\\}}\right)$.
By using the first-order Taylor expansion of ${e^{{\mu_{k}}}},{e^{{q_{k}}}}$,
and ${e^{{\eta_{k}}}}$, the restrictive approximations for (23), and (24),
(31) can be given as

$\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1\leq{e^{{{\tilde{\mu}}_{k}}}}\left({{\mu_{k}}-{{\tilde{\mu}}_{k}}+1}\right),$
(33)

${\gamma_{se,k}}{\rm{=}}\frac{{{{\left\|{{\bf{h}}_{e,k}^{\dagger}{{\bf{w}}_{k}}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i\neq
k}^{N}{{{\left\|{{\bf{h}}_{e,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}+{\ell_{k}}{P_{B}}{{\left\|{{\bf{g}}_{e,k}^{\dagger}{\bf{f}}_{k}^{mrt}}\right\|}^{\rm{2}}}{\rm{+}}\left({1-{\ell_{k}}}\right){P_{B}}{{\left\|{{\bf{g}}_{e,k}^{\dagger}{{\bm{{\rm
O}}}_{k}}{{\bf{v}}_{k}}}\right\|}^{2}}{\rm{+}}\delta_{su,k}^{2}}},$ (40)

$\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right){\rm{+}}1\leq{e^{{{\tilde{q}}_{k}}}}\left({{q_{k}}-{{\tilde{q}}_{k}}+1}\right),$
(34)
$\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1{\rm{}}\leq{e^{{{\tilde{\eta}}_{k}}}}\left({{\eta_{k}}-{{\tilde{\eta}}_{k}}+1}\right),$
(35)

where ${{\tilde{\mu}}_{k}}$, ${{\tilde{q}}_{k}}$, and ${{{\tilde{\eta}}_{k}}}$
begin with an initial values.

###  IV-B Successive Convex Approximation Approach

By defining the renewed optimization variables as a set, i.e.,

$\Im=\left\\{{\left\\{{{{\bf{W}}_{k}}}\right\\},\left\\{{{{\bf{F}}_{k}}}\right\\},\left\\{{{s_{k}}}\right\\},\left\\{{{\mu_{k}}}\right\\},\left\\{{{q_{k}}}\right\\},\left\\{{{v_{k}}}\right\\},\left\\{{{\tau_{k}}}\right\\},\left\\{{{\eta_{k}}}\right\\}}\right\\},$

thus the primal problem ${\mathcal{P}1}$ can be reformulated as

$\displaystyle{\mathcal{P}2:}\quad$
$\displaystyle\mathop{\max}\limits_{{\Im}}\quad\sum\limits_{k=1}^{N}{\left({{s_{k}}-{\mu_{k}}-{q_{k}}+{v_{k}}}\right)}$
(36a) s.t.
$\displaystyle\sum\limits_{k=1}^{N}{{\rm{Tr}}\left({{{\bf{W}}_{k}}}\right)}\leq{P_{S}},k=1,2,...,N,$
(36b)
$\displaystyle{{\rm{Tr}}\left({{{\bf{F}}_{k}}}\right)}\leq{P_{B}},k=1,2,...,N,$
(36c) $\displaystyle{{\bf{F}}_{k}}\succeq 0,{\bf{W}}_{k}\succeq 0,$ (36d)
$\displaystyle(\ref{Eq-16}),(\ref{Eq-19}),(\ref{Eq24}-\ref{Eq26-6}),(\ref{Eq26-8}),(\ref{Eq-27}-\ref{Eq-32}).$
(36e)

To solve ${\mathcal{P}2}$, we adopt the cvx tool and carry out a SCA based
approach, where ${{\tilde{\mu}}_{k}}$, ${{\tilde{q}}_{k}}$, and
${{{\tilde{\eta}}_{k}}}$ can be updated by each iteration of SCA.
Particularly, the details of SCA-based joint satellite-terrestrial BF
optimization can be seen in algorithm table 1. In addition, the main
computational complexity is solving the convex approximation problem in each
iteration. Considering the SDP for solving ${{\bf{w}}_{n}^{\star}}$ and
${{\bf{f}}_{n}^{\star}}$ can be calculated by
${\cal{O}}(max\\{m,n\\}^{\text{4}}n^{\text{1/2}})$, where $m$ and $n$ are the
constraint order and the dimension of equality constraints for SDP,
respectively. Thus, the total complexity can be calculated as
$t\cdot({\cal{O}}((8N+2)^{\text{4}})+log(1/\epsilon))$.

Input: $\\{{Q}_{tu,k}\\},\varepsilon,P_{S},P_{B}$.

Result: {${{\bf{w}}_{n}^{\star}},n=1,...,N\\}$,
$\\{{{\bf{f}}_{k}^{\star}},k=1,...,N$}.

1 Initialization:
$\left\\{{\tilde{\eta}_{k}^{0}}\right\\},\left\\{{\tilde{\mu}_{k}^{0}}\right\\},\left\\{{\tilde{q}_{k}^{0}}\right\\}$.

2 Set step $t=0$;

3 repeat

4

5 Using the CVX solver SDP to solve $\mathcal{P}2$;

Output:
$\left\\{{s_{k}^{\star}}\right\\},\left\\{{\eta_{k}^{\star}}\right\\},\left\\{{\mu_{k}^{\star}}\right\\},\left\\{{q_{k}^{\star}}\right\\},\left\\{{v_{k}^{\star}}\right\\}$,
$\left\\{{{\bf{W}}_{k}^{\star}}\right\\},\left\\{{{\bf{F}}_{k}^{\star}}\right\\}$;

6

7 Obtain
$R_{s,sum}^{t}=\sum\limits_{k=1}^{N}{s_{k}^{\star}-}\mu_{k}^{\star}-q_{k}^{\star}+v_{k}^{\star}$
.

8
$\left\\{{\tilde{\eta}_{k}^{t}=\eta_{k}^{\star};\tilde{\mu}_{k}^{t}=\mu_{k}^{\star};\tilde{q}_{k}^{t}=q_{k}^{\star}}\right\\}$.

9

10until  _$\left|{R_{s,sum}^{t}-R_{s,sum}^{t-1}}\right| <\epsilon$_;

11Obtain $\left\\{{{\bf{w}}_{n}^{\star}}\right\\}$ and
$\left\\{{{\bf{f}}_{n}^{\star}}\right\\}$ by the singular value decomposition
(SVD) of $\left\\{{{\bf{W}}_{k}^{\star}}\right\\}$ and
$\left\\{{{\bf{F}}_{k}^{\star}}\right\\}$.

12 Procedure End

Algorithm 1 SCA-based joint satellite-terrestrial BF optimization

In addition, to analyze the tightness of SDR from ${\mathcal{P}1}$ to
${\mathcal{P}2}$, the rank-one of $\\{{{\bf{W}}_{k}}$} and
$\\{{{\bf{F}}_{k}}\\}$ is proved as follows.

######  Proposition 1.

We consider a power minimization problem, which can be expressed as

$\displaystyle{\mathcal{P}3:}\quad$
$\displaystyle\mathop{\min}\limits_{\Im}\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{W}}_{i}}}+{{{\bf{F}}_{i}}}\right)}$
(37a) s.t.
$\displaystyle\sum\limits_{k=1}^{N}{\left({{s_{k}}-{\mu_{k}}-{q_{k}}+{v_{k}}}\right)}\geq{\varphi^{\circ}},$
(37b) $\displaystyle(\ref{p2-222}-\ref{p2-5}),$ (37c)

where $\varphi^{\circ}$ denotes the optimal objective value of
${\mathcal{P}2}$. It can be obtained that any feasible solutions of
${\mathcal{P}3}$ is optimal for ${\mathcal{P}2}$.

###### Proof.

Assuming that ${\Im^{\star}}$ is the optimal solution for solving
${\mathcal{P}3}$, then the following condition is restrictively satisfied

${\varphi^{\circ}}\leq\sum\limits_{k=1}^{N}{\left({s_{k}^{\star}-{\mu_{k}^{\star}}-{q_{k}^{\star}}+{v_{k}^{\star}}}\right)},$
(38)

with ${\bf{W}}_{k}^{\star}$ and ${\bf{F}}_{k}^{\star}$ satisfying constraints
in (36b-36e).

From (38), it can be found that the maximum ${\varphi^{\circ}}$ is reached
when $\sum\limits_{k=1}^{N}{\left({{s_{k}}-{\mu_{k}}-{q_{k}}+{v_{k}}}\right)}$
has the optimal solution at ${\Im^{\star}}$. Thus, ${\Im^{\star}}$ is the
optimal solution for ${\mathcal{P}2}$ with objective value ${\varphi^{\circ}}$
and the proof is concluded. ∎

Based on Proposition 1, we indirectly prove the rank-one condition in
${\mathcal{P}2}$ by proving that in ${\mathcal{P}3}$ by the following theorem.

######  Theorem 1.

For any feasible solutions of ${{{\bf{W}}_{k}}}$ and ${{{\bf{F}}_{k}}}$ from
${\mathcal{P}3}$, $Rank\left({{{\bf{W}}_{k}}}\right)=1$ and
$Rank\left({{{\bf{F}}_{k}}}\right)=1$.

###### Proof.

Please see the appendix A. ∎

##  V Joint Optimization of Satellite BF and BS PA

For the reference propose, an alternative approach that jointly optimizing
satellite BF and the PA of BS is given in this section, which is as a
benchmark in this paper.

We consider the BSs can use a partial transmission power to generate AN for
confusing Eve deliberately. The SINRs of SU, TU, and Eve can be respectively
rewritten as

${\gamma_{su,k}}{\rm{=}}\frac{{{{\left\|{{\bf{h}}_{su,k}^{H}{{\bf{w}}_{k}}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i\neq
k}^{N}{{{\left\|{{\bf{h}}_{su,i}^{H}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}+{\ell_{k}}{P_{B}}{{\left\|{{\bf{g}}_{su,k}^{H}{\bf{f}}_{k}^{mrt}}\right\|}^{\rm{2}}}{\rm{+}}\delta_{su,k}^{2}}},$
(39)
${\gamma_{tu,k}}{\rm{=}}{\ell_{k}}{P_{B}}{\left\|{{\bf{g}}_{tu,k}^{\dagger}{\bf{f}}_{k}^{mrt}}\right\|^{\rm{2}}}/\big{(}{\sum\limits_{i=1}^{N}{{{\left\|{{\bf{h}}_{tu,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}+\delta_{tu,k}^{2}}\big{)},$
(41)
${\gamma_{te,k}}{\rm{=}}\frac{{{\ell_{k}}{P_{B}}{{\left\|{{\bf{g}}_{e,k}^{\dagger}{\bf{f}}_{k}^{mrt}}\right\|}^{\rm{2}}}}}{{\sum\limits_{i=1}^{N}{{{\left\|{{\bf{h}}_{e,i}^{\dagger}{{\bf{w}}_{i}}}\right\|}^{\rm{2}}}}{\rm{+}}\left({1-{\ell_{k}}}\right){P_{B}}{{\left\|{{\bf{g}}_{e,k}^{\dagger}{{\bm{{\rm
O}}}_{k}}{{\bf{v}}_{k}}}\right\|}^{2}}{\rm{+}}\delta_{tu,k}^{2}}},$ (42)

where ${\ell_{k}}$ is the PA coefficient of BS,
${\bf{f}}_{k}^{mrt}={{{{\bf{g}}_{tu,k}}}{\left/{\vphantom{{{{\bf{g}}_{tu,k}}}{\left\|{{{\bf{g}}_{tu,k}}}\right\|}}}\right.}{\left\|{{{\bf{g}}_{tu,k}}}\right\|}}$
denotes the MRT-based BF vector of BS for transmitting useful signal to TU,
${{\bf{v}}_{k}}\in\mathbb{C}^{(M-1)\times 1}$ is the AN vector. and
${{\bf{O}}_{k}}=\left({{\bf{I}}-{\bm{\hbar}}\left({{{\bm{\hbar}}^{\dagger}}{\bm{\hbar}}}\right){{\bm{\hbar}}^{\dagger}}}\right){{\bf{g}}_{e,k}}$
with $\bm{\hbar}=[{{{\bf{g}}_{tu}}},{{{\bf{g}}_{su}}}]$ is the projection
matrix into the null of two legitimate channels, i.e., ${{{\bf{g}}_{tu}}}$ and
${{{\bf{g}}_{su}}}$. In addition, we make
${\bf{\Lambda}_{k}}={{\bf{O}}_{k}}{{\bf{v}}_{k}}{\bf{v}}_{k}^{\dagger}{\bf{O}}_{k}^{\dagger},$
and

${{\bf{A}}_{k}}=\left({1-{\ell_{k}}}\right){P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}\bf{{\Lambda}_{k}}}\right).$
(43)

Based on (39–43), the secrecy rate of SU and TU are obtained in(44) and (45)
as shown at the top of this page.

$R_{s}^{su,k}={\log_{2}}\left({\frac{{\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}1}}{{\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}1}}}\right)-{\log_{2}}\left({\frac{{\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1}}{{\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1}}}\right).$
(44)

$R_{s}^{tu,k}={\log_{2}}\left({\frac{{{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{tu,k}}{\bf{F}}_{k}^{mrt}}\right)+\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1}}{{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1}}}\right)-{\log_{2}}\left({\frac{{{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right)+\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1}}{{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1}}}\right).$
(45)

Hence, another optimization problem targeting the primal objective and
constraints in ${\mathcal{P}1}$ can be formulated as

$\displaystyle{\mathcal{P}4:}\quad$
$\displaystyle\mathop{\max}\limits_{\left\\{{{{\bf{w}}_{k}}}\right\\}_{k=1}^{N},{\ell_{k}}}\quad\sum\limits_{k=1}^{N}{R_{s}^{su,k}}$
(47a) s.t. $\displaystyle R_{s}^{tu,k}\geq{Q}_{tu,k},k=1,2,...,N,$ (47b)
$\displaystyle\sum\limits_{k=1}^{N}{{{\left\|{{{\bf{w}}_{k}}}\right\|}^{\rm{2}}}}\leq{P_{S}},$
(47c) $\displaystyle 0\leq{\ell_{k}}\leq 1.$ (47d)

From (44–47d), it is observed that the problem ${\mathcal{P}4}$ is also
intractable since its non-convex objective function and constraints.
Similarly, the Taylor expansion is adopted to reformulate the problem
${\mathcal{P}4}$ and changes by the corresponding exponential variables are
made as follows.

${e^{{{s^{\prime}}_{k}}}}=\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}1,$
(48) ${e^{{{\mu^{\prime}}_{k}}}}=\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}1,$
(49)
${e^{{{q^{\prime}}_{k}}}}=\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1,$
(50) ${e^{{{v^{\prime}}_{k}}}}=\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1,$
(51)
${e^{{{\tau^{\prime}}_{k}}}}={\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{tu,k}}{\bf{F}}_{k}^{mrt}}\right)+\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1,$
(52)
${e^{{{\eta^{\prime}}_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1,$
(53)
${e^{{{\alpha^{\prime}}_{k}}}}=\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1.$
(54)

By defining the renewed optimization variables as a set, i.e.,

$\Im^{\prime}=\left\\{{\left\\{{{{\bf{W}}_{k}}}\right\\},{\ell_{k}},\left\\{{{{s^{\prime}}_{k}}}\right\\},\left\\{{{{\mu^{\prime}}_{k}}}\right\\},\left\\{{{{q^{\prime}}_{k}}}\right\\},\left\\{{{{v^{\prime}}_{k}}}\right\\},\left\\{{{{\tau^{\prime}}_{k}}}\right\\},\left\\{{{{\eta^{\prime}}_{k}}}\right\\}}\right\\},$

and using the above replacements in (48–54), the problem ${\mathcal{P}4}$ can
be reformulated as

$\displaystyle{\mathcal{P}5:}\quad\mathop{\max}\limits_{\Im^{\prime}}\sum\limits_{k=1}^{N}{\left({{{s^{\prime}}_{k}}-{{\mu^{\prime}}_{k}}-{{q^{\prime}}_{k}}+{{v^{\prime}}_{k}}}\right)}$
(55a)
$\displaystyle\text{s.t.}\quad{{\eta^{\prime}}_{k}}+{{q^{\prime}}_{k}}-{{\tau^{\prime}}_{k}}-{{\alpha^{\prime}}_{k}}\leq-\frac{{{Q_{tu,k}}}}{{{{\log}_{2}}e}},$
(55b)
${e^{{{s^{\prime}}_{k}}}}\leq\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}1,$
(55c) $\displaystyle\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}1$
$\displaystyle\leq{e^{{{\tilde{\mu}^{\prime}}_{k}}}}\left({{{\mu^{\prime}}_{k}}-{{\tilde{\mu}^{\prime}}_{k}}+1}\right),$
(55d)
$\displaystyle\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1$
$\displaystyle\leq{e^{{{\tilde{q}^{\prime}}_{k}}}}\left({{{q^{\prime}}_{k}}-{{\tilde{q}^{\prime}}_{k}}+1}\right),$
(55e) ${e^{{{v^{\prime}}_{k}}}}\leq\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{\bf{F}}_{k}^{mrt}}\right){\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1,$
(55f)
${e^{{{\tau^{\prime}}_{k}}}}\leq{\ell_{k}}{P_{B}}{\rm{Tr}}\left({{{\bf{G}}_{tu,k}}{\bf{F}}_{k}^{mrt}}\right)+\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1,$
(55g)
$\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}+1\leq{e^{{{\tilde{\eta}^{\prime}}_{k}}}}\left({{{\eta^{\prime}}_{k}}-{{\tilde{\eta}^{\prime}}_{k}}+1}\right),$
(55h)
${e^{{{\alpha^{\prime}}_{k}}}}\leq\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}{\rm{+}}{{\bf{A}}_{k}}{\rm{+}}1,$
(55i)
$\displaystyle\sum\limits_{k=1}^{N}{{\rm{Tr}}\left({{{\bf{W}}_{k}}}\right)}\leq{P_{S}},$
$\displaystyle 0\leq{\ell_{k}}\leq 1.$ (55k)

From (50), we can see that the problem ${\mathcal{P}5}$ can also be solved by
a SCA-based optimization algorithm. Finally, we conduct the solving procedure
in the algorithm table 2.

Input: $\\{{Q}_{tu,k}\\},\varepsilon,P_{S},P_{B}$.

Result: {${{\bf{w}}_{n}^{\star}},n=1,...,N\\}$, $\\{{\ell_{k}},k=1,...,N$}.

1 Initialization:
$\left\\{{\tilde{\eta}^{\prime}}_{k}\right\\},\left\\{{\tilde{\mu}^{\prime}}_{k}\right\\},\left\\{{\tilde{q}^{\prime}}_{k}\right\\}$.

2 Set step $t=0$;

3 Calculate
${\bf{f}}_{k}^{mrt}={{{{\bf{g}}_{tu,k}}}{\left/{\vphantom{{{{\bf{g}}_{tu,k}}}{\left\|{{{\bf{g}}_{tu,k}}}\right\|}}}\right.}{\left\|{{{\bf{g}}_{tu,k}}}\right\|}}$;

4 Obtain
${\bf{F}}_{k}^{mrt}={\bf{f}}_{k}^{mrt}{\left({{\bf{f}}_{k}^{mrt}}\right)^{\dagger}}$;

5 repeat

6

7 Using the CVX solver SDP to solve $\mathcal{P}2$;

Output:
$\left\\{{{s^{\prime}}_{k}^{\star}}\right\\},\left\\{{{\eta^{\prime}}_{k}^{\star}}\right\\},\left\\{{{\mu^{\prime}}_{k}^{\star}}\right\\},\left\\{{{q^{\prime}}_{k}^{\star}}\right\\},\left\\{{{v^{\prime}}_{k}^{\star}}\right\\}$,
$\left\\{{{\bf{W}}_{k}^{\star}}\right\\},\left\\{{\ell_{k}}\right\\}$;

8

9 Obtain
$R_{s,sum}^{t}=\sum\limits_{k=1}^{N}{{s^{\prime}}_{k}^{\star}-}{\mu^{\prime}}_{k}^{\star}-{q^{\prime}}_{k}^{\star}+{v^{\prime}}_{k}^{\star}$
.

10 Update $\eta_{k}$, $\mu_{k}$, and $q_{k}$ by $\eta_{k}^{\star}$,
$\mu_{k}^{\star}$, and $q_{k}^{\star}$, respectively.

11

12until  _$\left|{R_{s,sum}^{t}-R_{s,sum}^{t-1}}\right| <\epsilon$_;

13Obtain $\left\\{{{\bf{w}}_{n}^{\star}}\right\\}$ by the singular value
decomposition (SVD) of $\left\\{{{\bf{W}}_{k}^{\star}}\right\\}$.

14 Procedure End

Algorithm 2 SCA-based joint satellite BF and the PA of BS optimization

##  VI Performance Evaluation

TABLE II: System Parameters Setting System Parameters | Numerical Value
---|---
_Satellite-to-ground channel parameters_ |
Satellite height | 600 Km
Carrier frequency | 2 GHz
Maximum beam gain | 46.6 dB
3 dB angle (for all beams) | 0.4∘
Rain attenuation parameters |  ${\mu_{{\zeta_{dB}}}}=$ -3.152, ${\delta^{2}}=$ 1.6
_Terrestrial BS channel parameters_ |
Channel power gain | -38.46 dB
Nakagami-$m$ channel parameters |  $m=2$, $\Omega=1$

In this section, we conduct extensive numerical simulations to evaluate the
secrecy performance. The main system parameters are set in Table I.
Specifically, the height of satellite orbit is 600 Km and the maximum beam
gain $G_{max}$ is set to 46.6 dB. The 3 dB angles of all beams are set to
${\alpha_{i,3dB}}=0.4^{\circ}$. The rain attenuation parameters of downlink
satellite-terrestrial channels are set to ${\mu_{{\zeta_{dB}}}}=-3.152$ and
${\delta^{2}}=1.6$. The carrier frequency of satellite downlink transmission
is 2 GHz. The channel power gain of BS downlink at the reference distance of 1
m is $-38.46$ dB. The horizontal distance from BS to SU, TU, and Eve is 100 m,
100 m, 120m, respectively. For convenience, the same secrecy rate constraint
of TUs in different satellite beams is preset as $Q$ for the simulation.
Particularly, the impact of BS transmission power ($P_{B}$), satellite
transmission power ($P_{S}$), number of BS transmit antennas ($M$), and
secrecy rate constraint of TUs ($Q$) on the maximum secrecy rate performance
of SU are evaluated as follows.

Figure 2: Secrecy rate performance of SUs vs. BS transmission power.
(${P_{S}}=20$ dB, $Q=0.5$ bit/s/Hz, $M=4$.)

Fig. 2 shows the impact of BS transmission power on the maximum sum secrecy
rate of SUs. From Fig. 2, it can be seen that the maximum secrecy rate of SU
is monotonically increasing of the BS transmission power. This is because the
more green interference from BS downlink degrades the eavesdropping of
satellite link as the BS transmission power increases. Particularly, it can be
found that the objective value in (37a) increases as the BS transmission power
(${{\rm{Tr}}\left({{{\bf{F}}_{i}}}\right)}$) increases. According to the
Proposition 1, the objective function in the problem ${\mathcal{P}2}$ in (36a)
has the same monotonicity as that in (37a), which indicates the maximum sum
secrecy rate of SUs increases as the BS transmission power. In addition, our
proposed cooperative BFs optimization approach outperforms these two
benchmarks, e.g., the approach that jointly optimizes both satellite BF and
the PA of BS, and the approach that optimizes satellite BF with ZF-based BS
BF. This is because the BS power focuses more on the main channel of TU by
MRT-based BF while the AN damages the Eve poorly. Whereas the ZF-based BS BF
approach doesn’t concentrate the green interference and the AN from BS to
damage the Eve, which improves the secrecy performance of SU slightly only by
the satellite BF since the similarity of satellite channels.

In Fig. 3, the impact of satellite transmission power on the maximum sum
secrecy rate of SUs is evaluated. From Fig. 3, it can be seen that the maximum
sum secrecy rate of SUs is monotonically increasing of the satellite
transmission power. This is because the more green interference from satellite
degrades the eavesdropping of TU as the satellite transmission power
increases, thus the more BS power resource can serve as the green interfere
for SU to degrade the Eve. Similarly, the objective value in (37a) increases
as the satellite transmission power
(${{\rm{Tr}}\left({{{\bf{W}}_{i}}}\right)}$) increases. According to the
Proposition 1, the maximum sum secrecy rate of SUs increases as the satellite
transmission power. In addition, our proposed cooperative BFs optimization
approach outperforms the approach that jointly optimizes both satellite BF and
the PA of BS and the approach that optimizes satellite BF with ZF-based BS BF.
Particularly, the approach that optimizes satellite BF with ZF-based BS BF
outperforms the approach that jointly optimizes both satellite BF in low
satellite transmission power region, which indicates that the main channel of
TU is damaged slightly by satellite with low transmission power and the BS
tends to suppress the Eve rather than to enhance the main channel of TU for
guaranteeing the secrecy rate constraint.

Figure 3: Secrecy rate performance of SU vs. satellite transmission power.
(${P_{B}}=30$ dB, $Q=0.5$ bit/s/Hz, $M=4$.) Figure 4: Secrecy rate performance
of SU vs. the number of BS transmit antennas. (${P_{B}}=30$ dB, ${P_{S}}=20$
dB, $Q=0.5$ bit/s/Hz, $M=4$.) Figure 5: Secrecy rate performance of SU vs.
secrecy rate constraint of TUs ($Q$). (${P_{S}}=20$ dB, ${P_{B}}=20$ dB,
$M=4$.)

Fig. 4 shows the impact of BS transmit antenna number on the maximum sum
secrecy rate of SUs. From Fig. 4, it can be seen that the maximum sum secrecy
rate of SUs is monotonically increasing of the number of BS transmit antennas.
This is because the green interference from BS downlink can focus more
centralized to degrade the eavesdropping of satellite link as the BS
transmission power increases. Besides, it is observed that the impact of
channel estimation error declines gradually as the number of BS transmit
antenna increases, which happens since the antenna diversity could benefit
from the channel estimation error. In addition, for the fixed BF schemes at
the BS side, the signal or the AN can concentrate more on its direction as the
number of BS transmit antenna increases.

Fig. 5 shows the impact of secrecy rate constraint of TUs on the maximum sum
secrecy rate of SU. From Fig. 5, it can be seen that the maximum sum secrecy
rate of SUs is monotonically decreasing of the secrecy rate constraint of TUs.
This is because the BS transmission power focus more on the main channel of TU
to guarantee the secrecy rate constraint of TUs as the constraint increases,
while the green interference from the BS for damaging the Eve is degraded.
Also, it can be seen our proposed cooperative BFs optimization approach
outperforms the approach that jointly optimizes both satellite BF and the PA
of BS and the approach that optimizes satellite BF with ZF-based BS BF, which
verifies the efficiency of our proposed approach.

##  VII Conclusion

In this paper, a symbiotic secure transmission scheme based on cooperative BF
optimization in integrated satellite-terrestrial communications has been
proposed. Particularly, the co-channel interference induced by spectrum
sharing within satellite-terrestrial networks and the inter-beam interference
due to frequency reuse among satellite beams serve as the green interference
has been used to ensure secure transmissions of both satellite and terrestrial
links simultaneously. Specifically, the problem to maximize the sum secrecy
rate of SUs is formulated and the BFs optimizing is conducted cooperatively,
where the secrecy rate constraint of each TU is guaranteed. Furthermore, the
Taylor expansion and SDR have been adopted to reformulate this problem, and a
SCA based joint satellite-terrestrial BFs optimization approach has been
proposed to solve this problem. The tightness of the relaxation has also been
proved. In addition, numerical results have verified the efficiency of our
proposed cooperative BFs optimization approach and revealed that the inherent
green interference from internal system without additional assistance can
assist the implement of symbiotic secure transmissions in integrated
satellite-terrestrial networks.

##  Appendix A Proof of Theorem 1

###### Proof.

The Lagrangian function of ${\mathcal{P}2}$ can be obtained as shown at the
bottom of this page.

$\begin{array}[]{*{20}{l}}{{\cal
L}\left({\left\\{{{\xi_{k}},{\varsigma_{k}},{\lambda_{k}},{\rho_{k}},{\tau_{k}},{\zeta_{k}},{\upsilon_{k}},{\vartheta_{k}},{\phi_{k}},{\beta_{k}},{\kappa_{k}},{{\bf{W}}_{k}},{{\bf{F}}_{k}},{{\bf{U}}_{k}},{{\bf{L}}_{k}}}\right\\}}\right)=}\\\
{\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{W}}_{i}}+{{\bf{F}}_{i}}}\right)}-\sum\limits_{k=1}^{N}{{\xi_{k}}\left({{s_{k}}-{\mu_{k}}-{q_{k}}+{v_{k}}-{\varphi^{\circ}}}\right)}+\sum\limits_{k=1}^{N}{{\lambda_{k}}\left({{e^{{s_{k}}}}-\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}-{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right)-1}\right)}}\\\
{+\sum\limits_{k=1}^{N}{{\rho_{k}}\left({{e^{{v_{k}}}}-\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}-{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right)-1}\right)}+\sum\limits_{k=1}^{N}{{\tau_{k}}\left({\sum\limits_{i\neq
k}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{su,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{su,k}}{{\bf{F}}_{k}}}\right)-{e^{{{\tilde{\mu}}_{k}}}}\left({{\mu_{k}}-{{\tilde{\mu}}_{k}}}\right)}\right)}}\\\
{+\sum\limits_{k=1}^{N}{{\zeta_{k}}\left({\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}+{\rm{Tr}}\left({{{\bf{G}}_{e,k}}{{\bf{F}}_{k}}}\right)-{e^{{{\tilde{q}}_{k}}}}\left({{q_{k}}-{{\tilde{q}}_{k}}}\right)}\right)}+\sum\limits_{k=1}^{N}{{\upsilon_{k}}\left({\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}-{e^{{{\tilde{\eta}}_{k}}}}\left({{\eta_{k}}-{{\tilde{\eta}}_{k}}}\right)}\right)}}\\\
{+\sum\limits_{k=1}^{N}{{\vartheta_{k}}\left({{e^{{\tau_{k}}}}-\sum\limits_{i}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{tu,i}}{{\bf{W}}_{i}}}\right)}{\rm{-Tr}}\left({{{\bf{G}}_{tu,k}}{{\bf{F}}_{k}}}\right)-1}\right)}+\sum\limits_{k=1}^{N}{{\phi_{k}}\left({{\eta_{k}}+{q_{k}}-{\tau_{k}}-{\alpha_{k}}+\frac{{{Q_{tu,k}}}}{{{{\log}_{2}}e}}}\right)-\sum\limits_{k=1}^{N}{{{\bf{U}}_{k}}{{\bf{W}}_{k}}}}}\\\
{+\sum\limits_{k=1}^{N}{{\beta_{k}}\left({\sum\limits_{k=1}^{N}{{\rm{Tr}}\left({{{\bf{W}}_{k}}}\right)}-{P_{S}}}\right)}+\sum\limits_{k=1}^{N}{{\varsigma_{k}}\left({{e^{{\alpha_{k}}}}-\sum\limits_{i=1}^{N}{{\rm{Tr}}\left({{{\bf{H}}_{e,i}}{{\bf{W}}_{i}}}\right)}-1}\right)+\sum\limits_{k=1}^{N}{{\kappa_{k}}\left({{\rm{Tr}}\left({{{\bf{F}}_{k}}}\right)-{P_{B}}}\right)}-\sum\limits_{k=1}^{N}{{{\bf{L}}_{k}}{{\bf{F}}_{k}}}}}.\end{array}$
(56)

Based on (56), we take the partial derivative of
$\mathcal{L}\left(\cdot\right)$ with respect to ${{{\bf{W}}_{k}}}$ and apply
KKT conditions as follows

$\begin{split}&{\bf{D}}-{{\bf{H}}_{su,k}}\sum\limits_{k=1}^{N}{{\lambda_{k}}}-{{\bf{U}}_{k}}={\bf{0}},\end{split}$
(57) ${{\bf{U}}_{k}}{{\bf{W}}_{k}}={\bf{0}},$ (58)
${{\bf{W}}_{k}}\succeq{\bf{0}},$ (59)

where

$\displaystyle{\bf{D}}=$
$\displaystyle{{\bf{I}}_{N}}\big{(}{1+\sum\limits_{i=1}^{N}{{\beta_{i}}}}\big{)}+{{\bf{H}}_{e,k}}\big{(}{\sum\limits_{i=1}^{N}{{\zeta_{i}}}-\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}-\sum\limits_{i=1}^{N}{{\varsigma_{i}}}}\big{)}$
$\displaystyle+{{\bf{H}}_{tu,k}}\big{(}{\sum\limits_{i=1}^{N}{{\upsilon_{i}}}-\sum\limits_{i=1}^{N}{{\vartheta_{i}}}}\big{)}+{{\bf{H}}_{su,k}}\sum\limits_{i\neq
k}^{N}{{\tau_{i}}}.$ (60)

By using (57–58), we have the reformulation as follows

${\bf{D}}{{\bf{W}}_{k}}={{\bf{H}}_{su,k}}{{\bf{W}}_{k}}\sum\limits_{k=1}^{N}{{\lambda_{k}}}.$
(61)

From (A) and (61), we can find that

$N-2\leq rank\left({\bf{D}}\right)\leq N,$ (62)

By (57), we have ${{\bf{E}}={\bf{D}}}$ by denoting
${\bf{E}}={{\bf{H}}_{su,k}}\sum\limits_{k=1}^{N}{{\lambda_{k}}}+{{\bf{U}}_{k}}$,
and thus $rank\left({\bf{E}}\right)\geq N-2$. With (58), we have

$\displaystyle rank\left({{\bf{E}}{{\bf{W}}_{k}}}\right)$
$\displaystyle=rank\big{(}{{{\bf{H}}_{su,k}}{{\bf{W}}_{k}}\sum\limits_{k=1}^{N}{{\lambda_{k}}}}\big{)}$
$\displaystyle\leq rank\left({{{\bf{H}}_{su,k}}}\right)=1$ (63)

and $rank\left({\bf{E}}\right)+rank\left({{{\bf{W}}_{k}}}\right)\leq
rank\left({{\bf{E}}{{\bf{W}}_{k}}}\right)+N.$

If $rank\left({{\bf{E}}{{\bf{W}}_{k}}}\right)=1$, we have
$rank\left({{{\bf{W}}_{k}}}\right)=N$ and $rank\left({\bf{E}}\right)\leq 1$,
which conflicts with the precondition $rank\left({\bf{E}}\right)\geq N-2$.
Consequently, the provision of $rank\left({{\bf{E}}{{\bf{W}}_{k}}}\right)=0$
should be satisfied. Then, $rank\left({{{\bf{W}}_{k}}}\right)\leq 2$ is
achieved.

Further, we reformulated (57) as

${\bf{\Theta}}-{\bf{E}}-{{\bf{H}}_{e,k}}\big{(}{\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}+\sum\limits_{i=1}^{N}{{\varsigma_{i}}}}\big{)}-\sum\limits_{i=1}^{N}{{\vartheta_{i}}{{\bf{H}}_{tu,k}}}={\bf{0}},$
(64)

where
${\bf{\Theta}}={{\bf{I}}_{N}}\big{(}{1+\sum\limits_{i=1}^{N}{{\beta_{i}}}}\big{)}+{{\bf{H}}_{e,k}}\sum\limits_{i=1}^{N}{{\zeta_{i}}}+\sum\limits_{i\neq
k}^{N}{{\tau_{i}}{{\bf{H}}_{su,k}}}+\sum\limits_{i=1}^{N}{{\upsilon_{i}}{{\bf{H}}_{tu,k}}}$,
and it can be observed that ${\bf{\Theta}}\succ{\bf{0}}$ due to
${{\beta_{i}}}\geq 0,{{\zeta_{i}}}\geq 0,{\tau_{i}}\geq 0$, and
${\upsilon_{i}}\geq 0$.

By post-multiplying ${{{\bf{W}}_{k}}}$ at both sides of (64) and using
$rank\left({{\bf{E}}{{\bf{W}}_{k}}}\right)=0$, (64) can be rewritten as

${\bf{\Theta}}{{\bf{W}}_{k}}=\big{(}{{\bf{E}}+{{\bf{H}}_{e,k}}(\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}+\sum\limits_{i=1}^{N}{{\varsigma_{i}}})+\sum\limits_{i=1}^{N}{{\vartheta_{i}}{{\bf{H}}_{tu,k}}}}\big{)}{{\bf{W}}_{k}}.$
(65)

In (65), since ${\bf{\Theta}}\succ{\bf{0}}$, we have

$rank\left({{\bf{\Theta}}{{\bf{W}}_{k}}}\right)=rank\left({{{\bf{W}}_{k}}}\right),$
(66)

and thus (A) is achieved.

$\displaystyle
rank\big{(}{{{\bf{W}}_{k}}}\big{)}=rank\big{(}{\big{(}{{{\bf{H}}_{e,k}}(\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}+\sum\limits_{i=1}^{N}{{\varsigma_{i}}})+\sum\limits_{i=1}^{N}{{\vartheta_{i}}{{\bf{H}}_{tu,k}}}}\big{)}{{\bf{W}}_{k}}}\big{)}$
$\displaystyle\leq
rank\big{(}{{{\bf{H}}_{e,k}}{{\bf{W}}_{k}}(\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}+\sum\limits_{i=1}^{N}{{\varsigma_{i}}})}\big{)}+rank\big{(}{{{\bf{H}}_{tu,k}}{{\bf{W}}_{k}}\sum\limits_{i=1}^{N}{{\vartheta_{i}}}}\big{)}.$
(67)

Particularly,

$\displaystyle rank\big{(}{{{\bf{H}}_{e,k}}{{\bf{W}}_{k}}(\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}+\sum\limits_{i=1}^{N}{{\varsigma_{i}}})}\big{)}\leq
rank\left({{{\bf{H}}_{e,k}}}\right)=1,$ (68) $\displaystyle
rank\big{(}{{{\bf{H}}_{tu,k}}{{\bf{W}}_{k}}\sum\limits_{i=1}^{N}{{\vartheta_{i}}}}\big{)}\leq
rank\left({{{\bf{H}}_{tu,k}}}\right)=1.$ (69)

However, if $rank\left({{{\bf{W}}_{k}}}\right)=2$, it can be observed from
(68) that $rank\big{(}{{{\bf{H}}_{e,k}}{{\bf{W}}_{k}}(\sum\limits_{i\neq
k}^{N}{{\rho_{i}}}+\sum\limits_{i=1}^{N}{{\varsigma_{i}}})}\big{)}=1$ and
$rank\big{(}{{{\bf{H}}_{tu,k}}{{\bf{W}}_{k}}\sum\limits_{i=1}^{N}{{\vartheta_{i}}}}\big{)}=1$,
which indicates an incompatible result that
$rank\big{(}{{{\bf{W}}_{k}}}\big{)}=N$.

Therefore, $rank\left({{{\bf{W}}_{k}}}\right)\leq 1$ is kept, and
$rank\left({{{\bf{W}}_{k}}}\right)=0$ can not be a solution and which should
be discarded. Finally, $rank\left({{{\bf{W}}_{k}}}\right)=1$ is proved.

Similarly, for the proof of $Rank\left({{{\bf{F}}_{k}}}\right)=1$, we take the
partial derivative of $\mathcal{L}\left(\cdot\right)$ with respect to
${{{\bf{F}}_{k}}}$ and apply KKT conditions as follows

${\bf{A}}-{\vartheta_{k}}{{\bf{G}}_{tu,k}}-{{\bf{L}}_{k}}={\bf{0}},$ (70)
${{\bf{L}}_{k}}{{\bf{F}}_{k}}={\bf{0}},$ (71) ${{\bf{L}}_{k}}\succeq{\bf{0}},$
(72)

where

${\bf{A}}=(1+\sum\limits_{k=1}^{N}{{\kappa_{k}}}){{\bf{I}}_{N}}+\left({{\tau_{k}}-{\lambda_{k}}}\right){{\bf{G}}_{su,k}}+\left({{\zeta_{k}}-{\rho_{k}}}\right){{\bf{G}}_{e,k}},$
(73)

and it is observed that $N-2\leq rank\left({\bf{A}}\right)\leq N$. Due to
${{\bf{G}}_{tu,k}}={\bf{h}}_{tu,k}^{\dagger}{\bf{h}}_{tu,k}$,
$rank\left({{{\bf{G}}_{tu,k}}}\right)=1$. Thus, similar to the way of proof
for $Rank\left({{{\bf{W}}_{k}}}\right)=1$ above,
$Rank\left({{{\bf{F}}_{k}}}\right)=1$ can also be proved. ∎

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|  Zhisheng Yin (M’20) received his Ph.D. degree from the School of
Electronics and Information Engineering, Harbin Institute of Technology,
Harbin, China, in 2020, and the B.E. degree from the Wuhan Institute of
Technology, the B.B.A. degree from the Zhongnan University of Economics and
Law, Wuhan, China, in 2012, and the M.Sc. degree from the Civil Aviation
University of China, Tianjin, China, in 2016. From Sept. 2018 to Sept. 2019,
Dr. Yin visited in BBCR Group, Department of Electrical and Computer
Engineering, University of Waterloo, Canada. He is currently an Associate
Professor with School of Cyber Engineering, Xidian University, Xi’an, China.
His research interests include space-air-ground integrated networks, wireless
communications, digital twin, and physical layer security.
---|---
|  Nan Cheng (M’16) received the Ph.D. degree from the Department of
Electrical and Computer Engineering, University of Waterloo in 2016, and B.E.
degree and the M.S. degree from the Department of Electronics and Information
Engineering, Tongji University, Shanghai, China, in 2009 and 2012,
respectively. He worked as a Post-doctoral fellow with the Department of
Electrical and Computer Engineering, University of Toronto, from 2017 to 2019.
He is currently a professor with State Key Lab. of ISN and with School of
Telecommunications Engineering, Xidian University, Shaanxi, China. His current
research focuses on B5G/6G, space-air-ground integrated network, big data in
vehicular networks, and self-driving system. His research interests also
include performance analysis, MAC, opportunistic communication, and
application of AI for vehicular networks.
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|  Tom H. Luan (M’14, SM’17) received the B.Eng. degree from Xi’an Jiaotong
University, Xi’an, China, in 2004, the M.Phil. degree from The Hong Kong
University of Science and Technology, Hong Kong, in 2007, and the Ph.D. degree
from the University of Waterloo, Waterloo, ON, Canada, in 2012. He is
currently a Professor with the School of Cyber Engineering, Xidian University,
Xi’an. He has authored or coauthored more than 40 journal articles and 30
technical articles in conference proceedings. His research interests include
content distribution and media streaming in vehicular ad hoc networks, peer-
to-peer networking, and the protocol design and performance evaluation of
wireless cloud computing and edge computing. Dr. Luan was the recipient of one
U.S. patent. He was a TPC Member of the IEEE Global Communications Conference,
the IEEE International Conference on Communications, and the IEEE
International Symposium on Personal, Indoor and Mobile Radio Communications,
and the Technical Reviewer for multiple IEEE Transactions, including the IEEE
Transactions on Mobile Computing, the IEEE Transactions on Parallel and
Distributed Systems, the IEEE Transactions on Vehicular Technology, the IEEE
Transactions on Wireless Communications, and the IEEE Transactions on
Intelligent Transportation Systems.
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|  Yilong Hui (M’18) received the Ph.D. degree in control theory and control
engineering from Shanghai University, Shanghai, China, in 2018. He is
currently an Associate Professor with the State Key Laboratory of Integrated
Services Networks, and with the School of Telecommunications Engineering,
Xidian University, China. He has published over 50 scientific articles in
leading journals and international conferences. His research interests include
wireless communication, mobile edge computing, vehicular networks, intelligent
transportation systems and autonomous driving. He was the recipient of the
Best Paper Award of International Conference WiCon2016 and IEEE Cyber-
SciTech2017.
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|  Wei Wang (M’19) received the B.Eng. degree in Information Countermeasure
Technology and the M.Eng. degree in Signal and Information Processing from
Xidian University in 2011 and 2014, respectively, and the PhD degree in
Electrical and Electronic Engineering from Nanyang Technological University
(NTU), Singapore, in 2018. From Sep. 2018 to Aug. 2019, he was a postdoctoral
fellow at the Department of Electrical and Computer Engineering, University of
Waterloo, Canada. Currently, he is a Professor at Nanjing University of
Aeronautics and Astronautics. His research interests include wireless
communications, space-air-ground integrated networks, wireless security, and
blockchain. He was awarded the Chinese government award for outstanding self-
financed students abroad in 2018, and the Young Elite Scientist Sponsorship
Program, China Association for Science and Technology in 2021.
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