# MIMO Radar Waveform-Filter Design for Extended Target Detection from a View
of Games

Zhou Xu, Chongyi Fan and Xiaotao Huang Manuscript received *** **, 2018;
accepted *** **, 2018. Date of publication *** **, 2018; date of current
version *** **, 2018._(Corresponding author: Chongyi Fan.)_ Z. Xu is with the
College of Electronic Science and Engineering, National University of Defense
Technology, Changsha 410073, and also with the Colledge of Electronic
Countermeasure,National University of Defense Technology, Hefei 230037, China
(e-mail: zhouzhou900521@126.com).C. Fan and X. Huang are with the College of
Electronic Science and Engineering, National University of Defense Technology,
Changsha 410073, China(e-mail: chongyifan@nudt.edu.cn).Color versions of one
or more of the figures in this paper are available online at
http://ieeexplore.ieee.org.Digital Object Identifier **.****/TIP.2018.******.

###### Abstract

This paper studies the Two-Person Zero Sum(TPZS) game between a Multiple-Input
Multiple-Output(MIMO) radar and an extended target with payoff function being
the output Signal-to-Interference-pulse-Noise Ratio(SINR) at the radar
receiver. The radar player wants to maximize SINR by adjusting its transmit
waveform and receive filter. Conversely, the target player wants to minimize
SINR by changing its Target Impulse Response(TIR) from a scaled sphere
centered around a certain TIR. The interaction between them forms a
Stackelberg game where the radar player acts as a leader. The Stackelberg
equilibrium strategy of radar, namely robust or minimax waveform-filter pair,
for three different cases are taken into consideration. In the first case,
Energy Constraint(EC) on transmit waveform is introduced, where we
theoretically prove that the Stackelberg equilibrium is also the Nash
equilibrium of the game, and propose Algorithm 1 to solve the optimal
waveform-filter pair through convex optimization. Note that the EC can’t meet
the demands of radar transmitter due to high Peak Average to power Ratio(PAR)
of the transmit waveform, thus Constant Modulus and Similarity Constraint(CM-
SC) on waveform is considered in the second case, and Algorithm 2 is proposed
to solve this problem, where we theoretically prove the existence of Nash
equilibrium for its Semi-Definite Programming(SDP) relaxation form. And the
optimal waveform-filter pair is solved by calculating the Nash equilibrium
followed by the randomization schemes. In the third case, a more challenge
task with Spectral Compatibility and Similarity Constraint(SC-SC) is
considered. Algorithm 3 is devised to address this problem by leveraging on
Majorization Minimization(MM) method, where the Stackelberg game is
approximated with a sequence of sub-minimax problem, and we prove that the
sub-minimax problem can be efficiently solved by the convex optimization even
though it belongs to the convex non-concave optimization problems. Finally,
numerical results highlight the effectiveness and competitiveness of the
proposed algorithms as well as the optimized waveform-filter pair.

###### Index Terms:

Extended target detection, Waveform-filter design, Two-person zero sum game,
Stackelberg equilibrium, Nash equilibrium, Minimax problem.

##  I Introduction

The past decades have witnessed the great development of Multiple-Input
Multiple-Output(MIMO) radar which improves the parameter estimation and target
detection performance significantly by utilizing waveform diversity[1, 2, 3,
4]. Owing to the fact that targets usually exhibits range-spread
characteristics for high resolution radar, waveform design for extended
targets has drawn increasing attention from researchers during the past
decades[5, 6, 7, 8, 9, 10].

Given that the performance of target detection and parameter estimation
depends on the output Signal-to-Interference-plus-Noise Ratio(SINR), a
preferred waveform optimization criterion is maximizing output SINR at the
receiver[6, 11, 12, 9, 13, 14, 15, 16]. In [6], Bell studies the matched
illumination waveform for the extended target. As stated in [16], the authors
investigate joint design of transmit waveform and receive filter via
maximizing the output SINR. However, the aforementioned waveform design
algorithms need the exact knowledge of Target Impulse Response(TIR) or Power
Spectral Density(PSD), which is often used to characterize scattering behavior
of extended target[17, 18]. Unfortunately, TIR is sensitive to a variety of
factors, and it is impossible to get the exact knowledge of TIR when designing
radar waveform. To this end, robust waveform design is taken into account
leading to the minimax optimization problem[19, 11, 12, 10]. In most cases,
the uncertainty of TIR is modeled as a scaled sphere centered around a known
but imprecise TIR(i.e., see [11, 5, 9, 10] and references therein). Aiming at
maximizing the worst-case SINR under the spherical uncertainty of TIR, [11]
proposes an iterative algorithm to optimize transmit waveform and receive
filter alternatively. And [9] addresses the problem based on the minimax
theorems[20]. Nevertheless, the algorithms proposed in [11] and [9] can only
deal with the energy constrained waveform, which might result in the undesired
waveform for radar with a high Peak-to-Average power Ratio(PAR). In order to
avoid this drawback, [5] imposes a PAR constraint on the waveform and proposes
the alternative optimization algorithm with respect to the covariance of
waveform and filter, where the spherical uncertainty of TIR is approximated by
random samples from its surface. To our knowledge, there is few literature
considering other constraints on transmit waveform, such as similarity or
spectral compatibility constraint[21, 22, 23], for the worst-case situation.

Game theory, modeling the interaction of players, is applied in a broad
variety of fields, such as economics, social science and machine learning[24,
25, 26]. Recently, game theory methods have been introduced to the context of
signal processing for jamming suppression and waveform design[27, 28, 29, 30].
As to the extended target detection problem, it can be modeled as a Two-Person
Zero-Sum game(TPZS) if the target is ”smart” enough and always tries to
prevent being detected by radar. Apart from this, the interaction between
radar and target forms a Stackelberg game if one acts as the leader, for
example, the robust waveform design problem mentioned in [11, 5] can be
regarded as the Stackelberg game where radar acts as the leader. And the
robust waveform can be considered as the equilibrium strategy of radar in the
Stackelberg game.

In this paper, we consider the MIMO radar waveform-filter design problem by
constructing Stackelberg game between radar and the extended target with SINR
being the payoff function. The goal is to obtain the Stackelberg equilibrium
strategy of radar under several practical constraints on waveform, namely,
Energy Constraint(EC), Constant Modulus and Similarity Constraint(CM-SC),
Spectral Compatibility and Similarity Constraint(SC-SC). Furthermore, under
some special conditions, we will prove that the Stackelberg equilibrium is
also the Nash equilibrium for the game and the optimal waveform-filter pair is
calculated by solving Nash equilibrium. Generally speaking, our work makes the
following contributions:

1) The Nash equilibrium under EC on waveform: For the case of EC on waveform,
we theoretically prove that the Stackelberg equilibrium is also the Nash
equilibrium, and propose Algorithm 1 to solve it through convex optimization.
Even though, the relevant researches are reported in [9], the proof procedure
is imperfect since the Stackelberg game in this case belongs to the non-convex
concave minimax problem and can’t be converted to the convex concave form by
diagonal loading. Consequently, we prove it from a totally new view by
utilizing the optimality condition, which has been seldom reported in open
literature.

2) The Nash equilibrium for the (Semi-Definite Programming)SDP relaxation form
under CM-SC on waveform: In the case of CM-SC on waveform, the Nash
equilibrium might not exist for the TPZS game between radar and target. But we
prove that the Nash equilibrium always exists for its SDP relaxation based on
Sion’s minimax theorem[20]. And we devise Algorithm 2 to calculate the Nash
equilibrium approximately. Finally, the optimal waveform-filter is synthesized
from the strategy of Nash equilibrium with randomization.

3) The Stackelberg equilibrium under SC-SC on waveform: Algorithm 3 is
proposed to solve the Stackelberg equilibrium under SC-SC on waveform
leveraging on (Majorization Minimization)MM algorithm. Based on the dual
theorem[31], we prove that the minimax problem at each iteration of MM
algorithm can be solved through convex optimization, even though it is convex
non-concave. Thus, the Stackelberg equilibrium can be calculated in polynomial
time.

4) Analyses and experiments for proposed algorithms: The convergences as well
as computational complexities of the proposed 3 algorithms are analyzed,
respectively. Numerical experiments are carried out to verify the
effectiveness of the proposed algorithms. In addition, some comparisons with
the current algorithms are given. The results highlight the superiority of our
algorithms to some extend.

The remainder of this paper is organized as follows. Section II establishes
the MIMO radar signal model and formulates the TPZS game between the radar and
extended target. Section III is devoted to the algorithms and analyses for
equilibrium strategies under different waveform constraints. Section IV
provides several numerical experiments and analyses of the proposed
algorithms, and exhibits the performance of the waveform-filter pair. Finally,
Section V concludes the paper.

Notations: Throughout this paper, scalars are denoted by italic letters(e.g.,
a, A); vectors are denoted by bold italic lowercase letters(e.g., $\bm{a}$)
and $\bm{a}(i)$ denotes the $i$th element of $\bm{a}$; matrices are denoted by
bold italic capital letters(e.g., $\bm{A}$) and $\bm{A}(i,j)$ denotes the
element in the $i$th row and $j$th column of $\bm{A}$. Superscript
${\bar{\left(\cdot\right)}}$, ${\left(\cdot\right)^{\rm{T}}}$ and
${\left(\cdot\right)^{\rm{H}}}$ denote complex conjugate, transpose and
conjugate transpose, respectively. tr($\cdot$) denotes the trace of a square
matrix. vec($\cdot$) denotes the operator of column-wise stacking a matrix.
$\otimes$ and $\odot$ represent Kronecker product and Hadamard product,
respectively. diag($\bm{a}$) denotes the diagonal matrix with the diagonal
elements formed by $\bm{a}$, while diag($\bm{A}$) denotes the vector with
elements formed by the diagonal elements of $\bm{A}$.
$\left\langle\cdot,\cdot\right\rangle$, $\left\|\cdot\right\|_{2}$,
$\left\|\cdot\right\|_{\infty}$ and $\left\|\cdot\right\|_{F}$ denote the
inner product, $l_{2}$ norm, $l_{\infty}$ norm and Frobenius norm in Euclidean
space, respectively. arg(${a}$) and $\left|{a}\right|$ denote the phase angle
and the modulus of ${a}$, respectively. The representation $\bm{A}\succ
0(\bm{A}\succeq 0)$ means $\bm{A}$ is positive definite(semi-definite).

##  II Signal Model and Problem Formulation

###  II-A Signal model

As outlined in Fig.1, the mission of radar is to detect an extended target at
direction ${\theta_{t}}$, and the extended target tries to prevent being
detected by adjusting its attitude within a certain range, which affects its
returns to radar. Then, the returns received by radar are given by

${\bm{y}}={{\bm{y}}_{t}}+{\bm{n}},$ (1)

if the clutter is neglected, where ${{\bm{y}}_{t}}$ and ${\bm{n}}$ denote the
extended target returns and noise, respectively.

Now, we are going to derive the signal model of ${{\bm{y}}_{t}}$. Let us
consider a colocated MIMO radar with ${N_{T}}$ transmitters and ${N_{R}}$
receivers, assuming that the waveform transmitted by the nth transmitter with
L samples is denoted by
${{\bm{s}}_{n}}={[{{\bm{s}}_{n}}(1),{{\bm{s}}_{n}}(2),...,{\bm{s}}_{n}(L)]^{\rm{T}}}$,
then the transmitting matrix for MIMO array can be represented as
${\bm{S}}={[{{\bm{s}}_{1}},{{\bm{s}}_{2}},...,{{\bm{s}}_{{N_{T}}}}]^{\rm{T}}}\in{\mathbb{C}}^{{N_{T}}\times
L}$. In the far field, the signal incident on the target can be written as
${{\bm{a}}^{\rm{T}}}({\theta_{t}}){\bm{S}}$, where ${{\bm{a}}({\theta_{t}})}$
is the transmit array steering vector at ${\theta_{t}}$.

Denote by $\bm{t}\in{\mathbb{C}}^{Q}$ the TIR of target, and the returns
modulated by it can be represented as

${\bm{T}}{\left({{{\bm{a}}^{\rm{T}}}({\theta_{t}}){\bm{S}}}\right)^{\rm{T}}}={\bm{T}}{{\bm{S}}^{\rm{T}}}{\bm{a}}({\theta_{t}}),$
(2)

where ${\bm{T}}=\sum\limits_{i=1}^{Q}{{\bm{t}}(i){{\bm{J}}_{i-1}}}$ is the TIR
matrix with ${{\bm{J}}_{i}}$ being the $(Q+L-1)\times L$ shift matrix given
by[5]

${{\bm{J}}_{i}}(m,n)=\left\\{\begin{array}[]{l}1\;,\quad m-n=i\\\ 0,\;\quad
m-n\neq i\end{array}\right..$ (3)

Consequently, one get the signal matrix $\bm{Y}_{t}$ at the MIMO receiver with

${{\bm{Y}}_{t}}={\bm{b}}({\theta_{t}}){{\bm{a}}^{\rm{T}}}({\theta_{t}}){\bm{S}}{{\bm{T}}^{\rm{T}}},$
(4)

where ${\bm{b}}({\theta_{t}})$ denotes the receive steering vector at
${\theta_{t}}$. Let ${\bm{s}}={\rm{vec}}({\bm{S}})$, the vectorization form of
$\bm{Y}_{t}$ is given by

${{\bm{y}}_{t}}=\left({{\bm{T}}\otimes\left({{\bm{b}}({\theta_{t}}){{\bm{a}}^{\rm{T}}}({\theta_{t}})}\right)}\right){\bm{s}}=\bm{G}(\bm{t})\bm{s}.$
(5)

After a couple of matrix operations, it is easy to get another expression form
of (5) as follow,

${{\bm{y}}_{t}}={\bm{H}}({\bm{s}}){\bm{t}},$ (6)

where
${\bm{H}}({\bm{s}})=\left[{{{\bm{h}}_{1}}({\bm{s}}),{{\bm{h}}_{2}}({\bm{s}}),...,{{\bm{h}}_{Q}}({\bm{s}})}\right]$
with
${{\bm{h}}_{i}}({\bm{s}})=\left({{{\bm{J}}_{i-1}}\otimes\left({{\bm{b}}({\theta_{t}}){{\bm{a}}^{\rm{T}}}({\theta_{t}})}\right)}\right){\bm{s}}={\bm{A}_{i}}\bm{s}$.

As to the noise, it is assumed to be complex Gaussian with
${\bm{n}}\sim\mathcal{CN}({\bf{0}},{{\bm{R}}_{c}})$.

Figure 1: Extended target detection diagram

###  II-B Games between radar and target

The radar processor extract the desired information at the receiver with a
linear filter $\bm{w}$ achieving the output SINR as

${\rm{SINR}}=\frac{{{{\left|{{{\bm{w}}^{\rm{H}}}{{\bm{y}}_{t}}}\right|}^{2}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}.$
(7)

For a given probability of false alarm $P_{fa}$, analytical form of the
detection probability $P_{d}$ is given by [32, 33, 34]

${P_{d}}{\rm{=}}Q\left({\sqrt{2{\rm{SINR}}}{\rm{,}}\sqrt{-2\log({P_{fa}})}}\right),$
(8)

where $Q\left({\cdot{\rm{,}}\cdot}\right)$ denotes the Marcum-Q function.

Consider the TPZS game [24, 25] between the radar and the target with a common
payoff function $\rm{SINR}$ with respect to ($\bm{s}$,$\bm{w}$,$\bm{t}$). The
radar player tries to maximize the SINR to capture the target with high
probabilities by choosing the waveform-filter pair ($\bm{s}$,$\bm{w}$) from
its strategy set $\Psi$. Alternatively, the target player tries to minimize
the SINR to avoid being detected by choosing the TIR from its strategy set
$\Omega$.

Assume that the radar player goes first and the target player is able to
perceive the radar’s strategy. Then, the interaction between radar and target
results in the following Stackelberg game where the radar player acts as the
leader,

${{\cal
P}_{r}}\left\\{\begin{array}[]{l}\underline{{\rm{SINR}}}=\mathop{{\rm{max}}}\limits_{{\bm{s}},{\bm{w}}}\mathop{{\rm{min}}}\limits_{\bm{t}}{\rm{}}\frac{{{{\left|{{{\bm{w}}^{\rm{H}}}{{\bm{y}}_{t}}}\right|}^{2}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}\\\
s.t.\quad\left({{\bm{s}},{\bm{w}}}\right)\in\Psi,{\bm{t}}\in\Omega\end{array}\right..$
(9)

Conversely, assume that the target player goes first and the radar player is
able to perceive the target’s strategy. Then, the interaction between radar
and target results in the following Stackelberg game where the target player
acts as the leader,

${{\cal
P}_{t}}\left\\{\begin{array}[]{l}\overline{{\rm{SINR}}}=\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{{\bm{s}},{\bm{w}}}{\rm{}}\frac{{{{\left|{{{\bm{w}}^{\rm{H}}}{{\bm{y}}_{t}}}\right|}^{2}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}\\\
s.t.\quad\left({{\bm{s}},{\bm{w}}}\right)\in\Psi,{\bm{t}}\in\Omega\end{array}\right..$
(10)

The following weak minimax inequality

$\underline{{\rm{SINR}}}\leq\overline{{\rm{SINR}}}$ (11)

always holds, which can be regarded as the information gain for the player who
plays second. However, in some certain conditions, the strong minimax
inequality[20]

$\underline{{\rm{SINR}}}=\overline{{\rm{SINR}}}$ (12)

holds, which means that it makes no difference for the player who goes first
or second. And the quantity
${{\rm{SINR}}}^{*}=\underline{{\rm{SINR}}}=\overline{{\rm{SINR}}}$ is called
the value of the TPZS game.

It is worth pointing out that the problem ${{\cal P}_{r}}$ is also referred as
robust or minimax waveform-filter design in some literatures[11, 12, 5].
Moreover, denote by $(\bm{s}^{*},\bm{w}^{*})$ the Stackelberg equilibrium
strategy of radar for ${{\cal P}_{r}}$, and $\bm{t}^{*}$ the Stackelberg
equilibrium strategy of target for ${{\cal P}_{t}}$, respectively. Then, the
strategy pair $(\bm{s}^{*},\bm{w}^{*},\bm{t}^{*})$ is also a Nash equilibrium
[25] for the TPZS game, if the equality (12) holds. To this end, in the
following paper, we focus on solving the optimal solution
$(\bm{s}^{*},\bm{w}^{*})$, namely Stackelberg or Nash equilibrium strategies
of radar in the TPZS game in different cases.

##  III Equilibrium Strategies and Optimal Waveform-Filter Pair

This section is devoted to the equilibrium strategies as well as the optimal
waveform-filter pair of radar for the TPZS game under different constraints on
waveform. In particular, we assume that the target strategies are bounded in a
scaled sphere centered around a previous known $\bm{t}_{0}$, namely,

$\Omega=\left\\{{{\bm{t}}|{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}\right\\}.$ (13)

As to the radar player, we assume that the filter $\bm{w}$ is unconstrained
and the waveform belongs to the following three sets: $\bullet$ EC set

${\Psi_{E}}=\left\\{{{{\bm{s}}}|{{\left\|{\bm{s}}\right\|}_{2}^{2}}\leq
e_{t},}\right\\},$ (14)

where $e_{t}$ is the total available power;

$\bullet$ CM-SC set

${\Psi_{C}}=\left\\{{{\bm{s}}|\left|{{\bm{s}}(i)}\right|=\sqrt{\frac{{{{e_{t}}}}}{{{{N_{T}}L}}}},{{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|}_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}}}\right\\},$
(15)

where $\bm{s}_{0}$ is a reference signal with good ambiguity properties, and
$\delta\leq 2$ controls the similarly between $\bm{s}$ and $\bm{s}_{0}$;

$\bullet$ SC-SC set

${\Psi_{S}}=\left\\{{{{\bm{s}}}|{{\left\|{\bm{s}}\right\|}_{2}^{2}}\leq
e_{t},{{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|}_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}},{{\bm{s}}^{\rm{H}}}{{\bm{R}}_{I}}{\bm{s}}\leq{e_{I}}}\right\\},$
(16)

where $e_{I}$ denotes the maximum allowed energy allocated on the specified
frequency bands, and $\bm{R}_{I}$ is the corresponding spectrum compatibility
matrix defined as
${{\bm{R}}_{I}}=\sum\limits_{k=1}^{K}{{\alpha_{k}}{{\bf{I}}_{{N_{T}}}}\otimes{\bm{R}}_{I}^{k}}$
with

${\bm{R}}_{I}^{k}(m,n)=\left\\{\begin{array}[]{l}f_{2}^{k}-f_{1}^{k},\;\;m=n\\\
\frac{{{e^{j2\pi f_{2}^{k}(m-n)}}-{e^{j2\pi
f_{1}^{k}(m-n)}}}}{{j2\pi(m-n)}},m\neq n\end{array}\right.,$ (17)

where $[f_{1}^{k},f_{2}^{k}]$ denotes the $k$th normalized frequency interval
of transmission and ${\alpha_{k}}\geq 0$ is the weight for the $k$th frequency
band[35, 36].

Under this assumption, the waveform-filter pair design problem belongs to a
class of non-convex concave minimax problems. Nevertheless, as will be
illustrated later, some of these problems can be solved or approximately
solved by constructing the Nash equilibrium between the two players. To this
end, let’s provide the basic theorem in our paper.

###### Theorem 1

Let $\Omega$ be a convex set and $\bm{w}$ be unconstrained, then
${\cal{P}}_{r}$ is equivalent111By ”equivalent”, we mean that the optimal
solutions can be constructed from each other to

${\tilde{\cal
P}_{r}}\left\\{\begin{array}[]{l}\mathop{{\rm{max}}}\limits_{\bm{s}}\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{\bm{w}}{\rm{}}\frac{{{{\left|{{{\bm{w}}^{\rm{H}}}{{\bm{y}}_{t}}}\right|}^{2}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}\\\
s.t.\quad{\bm{s}}\in\Psi,{\bm{t}}\in\Omega\end{array}\right.$ (18)

###### Proof:

See Appendix A. ∎

As an immediate consequence of Theorem 1, we can reformulate ${\cal{P}}_{r}$
and ${\cal{P}}_{t}$ by solving the inner maximization problem with
$\bm{w}=\bm{R}_{c}^{-1}\bm{y}_{t}$ as

${{\cal
P}_{r}}\left\\{\begin{array}[]{l}\underline{{\rm{SINR}}}=\mathop{{\rm{max}}}\limits_{\bm{s}}\mathop{{\rm{min}}}\limits_{\bm{t}}{\rm{}}{\bm{y}}_{t}^{\rm{H}}{\bm{R}}_{c}^{-1}{{\bm{y}}_{t}}\\\
s.t.\quad{\bm{s}}\in\Psi,{\bm{t}}\in\Omega\end{array}\right.$ (19)

and

${{\cal
P}_{t}}\left\\{\begin{array}[]{l}\overline{{\rm{SINR}}}=\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{\bm{s}}{\rm{}}{\bm{y}}_{t}^{\rm{H}}{\bm{R}}_{c}^{-1}{{\bm{y}}_{t}}\\\
s.t.\quad{\bm{s}}\in\Psi,{\bm{t}}\in\Omega\end{array}\right..$ (20)

###  III-A EC on waveform

In this subsection, we discuss the optimal solutions of ${\cal{P}}_{r}$ and
${\cal{P}}_{t}$ with the case $\Psi=\Psi_{E}$. Substituting $\Psi_{E}$ into
${\cal{P}}_{r}$ and ${\cal{P}}_{t}$, we recast the game as

${{\cal
P}_{r}^{E}}\left\\{\begin{array}[]{l}\underline{{\rm{SINR}}}_{E}=\mathop{{\rm{max}}}\limits_{\bm{s}}\mathop{{\rm{min}}}\limits_{\bm{t}}{\rm{}}{\bm{y}}_{t}^{\rm{H}}{\bm{R}}_{c}^{-1}{{\bm{y}}_{t}}\\\
s.t.\quad\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}},{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq
r\end{array}\right.$ (21)

and

${{\cal
P}_{t}^{E}}\left\\{\begin{array}[]{l}\overline{{\rm{SINR}}}_{E}=\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{\bm{s}}{\rm{}}{\bm{y}}_{t}^{\rm{H}}{\bm{R}}_{c}^{-1}{{\bm{y}}_{t}}\\\
s.t.\quad\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}},{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq
r\end{array}\right..$ (22)

And the following proposition demonstrates the existence of Nash equilibrium
for the game.

###### Proposition 1

Let $\bm{s}_{E}^{*}$ and $\bm{t}_{E}^{*}$ are the optimal solutions of
${{\cal{P}}_{r}^{E}}$ and ${{\cal{P}}_{t}^{E}}$, respectively. Then, the
strategy pair $(\bm{s}_{E}^{*},\bm{t}_{E}^{*})$ is Nash equilibrium for the
TPZS game and $\underline{{\rm{SINR}}}_{E}=\overline{{\rm{SINR}}}_{E}$.

###### Proof:

See Appendix B. ∎

According to Proposition 1, one can get the idea that the optimal waveform can
be obtained by solving either ${\cal{P}}_{r}^{E}$ or ${\cal{P}}_{t}^{E}$.
Obviously, solving ${\cal{P}}_{t}^{E}$ is a much better choice since it can be
converted to a convex optimization problem. Note that ${\cal{P}}_{t}^{E}$ is
equivalent to the following problem by substituting (5) into it

${\cal
P}_{t}^{E}\left\\{\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{\bm{s}}{\rm{}}{{\bm{s}}^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}\\\
s.t.\quad\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}},{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq
r\end{array}\right.,$ (23)

and the analytical form of the optimal solution with respect to $\bm{s}$ is

${\bm{s}_{E}^{*}}=\sqrt{e_{t}}{\cal{M}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}})}\right),$
(24)

where ${\cal{M}}(\bm{A})$ denotes the normalized principle eigenvector of
$\bm{A}$. Then, it reduces to

$\tilde{\cal
P}_{t}^{E}\left\\{\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{\bm{t}}{\rm{}}{\lambda_{\max}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}})}\right)e_{t}\\\
s.t.\quad{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq
r\end{array}\right.,$ (25)

where $\lambda_{\max}(\bm{A})$ denotes the maximum eigenvalue of $\bm{A}$.

Based on Schur complement theorem[31], we recast $\tilde{\cal P}_{t}^{E}$ as
follow

$\hat{\cal
P}_{t}^{E}\left\\{\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{\bm{t},\mu}{\rm{}}e_{t}\mu\\\
s.t.\quad\left[{\begin{array}[]{*{20}{c}}{\mu{\bf{I}}}&{{\bm{G}}{{({\bm{t}})}^{\rm{H}}}}\\\
{{\bm{G}}({\bm{t}})}&{{{\bm{R}}_{c}}}\end{array}}\right]\succeq
0,{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq r\end{array}\right.,$
(26)

where $\mu$ is the auxiliary variable.

It is worth pointing out that $\bm{G}(\bm{t})$ is a affine function with
respect to $\bm{t}$ and the SDP problem can be solved efficiently in
polynomial time by CVX[37]. Once the optimal solution $\bm{t}_{E}^{*}$ is
obtained, we can obtain the optimal waveform-filter pair
$(\bm{s}_{E}^{*},\bm{w}_{E}^{*})$ with

${\bm{s}}_{E}^{*}=\sqrt{e_{t}}{\cal{M}}\left({{\bm{G}}{{({\bm{t}}_{E}^{*})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}_{E}^{*})}\right),{\bm{w}}_{E}^{*}={\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{E}^{*}){\bm{t}}_{E}^{*}.$
(27)

The overall procedure for waveform-filter design under EC is summarized in
Table I as Algorithm 1. The convergence of Algorithm 1 is guaranteed by the
convexity of ${\cal{\hat{P}}}_{t}^{E}$. As to the computational complexities,
it requires at most
$O\left({{Q{\left({L{N_{T}}+L{N_{R}}+Q{N_{R}}}\right)}^{3.5}}}\right)$ to
solve ${\cal{\hat{P}}}_{t}^{E}$[38] and
$O\left({{{\left({L{N_{T}}}\right)}^{3}}}\right)+O\left({\left({L{N_{R}}+Q{N_{R}}}\right)^{2}}\right)$
operations to calculate ($\bm{s}_{E}^{*},\bm{w}_{E}^{*}$) with (27).
Therefore, the total computational complexities will not exceed
$O\left({{Q{\left({L{N_{T}}+L{N_{R}}+Q{N_{R}}}\right)}^{3.5}}}\right)$.

TABLE I: Algorithm for waveform-filter optimization with EC Input:
$\bm{a}(\theta_{t})$, $\bm{b}(\theta_{t})$, $e_{t}$, ${{\bm{R}}_{c}}$,
${{\bm{t}}_{0}}$ and $r$.
---
Step 1: Solve ${\cal{\hat{P}}}_{t}^{E}$ and obtain $\bm{t}_{E}^{*}$.
Step 2: Synthesize ${\bm{s}_{E}^{*}}$ and ${\bm{w}_{E}^{*}}$ with (27).
Output: ${\bm{s}_{E}^{*}}$ and ${\bm{w}_{E}^{*}}$.

###  III-B CM-SC on waveform

In this subsection, we study ${\cal{P}}_{r}$ and ${\cal{P}}_{t}$ with the case
that $\Psi=\Psi_{C}$. Now, we recast the game as

${\cal{P}}_{r}^{C}\left\\{{\begin{array}[]{*{20}{l}}\underline{{\rm{SINR}}}_{C}={\mathop{{\rm{max}}}\limits_{\bm{s}}\mathop{{\rm{min}}}\limits_{\bm{t}}{{\bm{s}}^{\rm{H}}}{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}}\\\
{s.t.\;\;\;{\kern
1.0pt}\left|{\bm{s}}(i)\right|=\sqrt{\frac{{{e_{t}}}}{{{N_{T}}L}}},{{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|}_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}}}\\\
{\;\;\;{\kern 1.0pt}\;\;\;{\kern
1.0pt}{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}\end{array}}\right.$ (28)

and

${\cal{P}}_{t}^{C}\left\\{{\begin{array}[]{*{20}{l}}\overline{{\rm{SINR}}}_{C}={\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{\bm{s}}{{\bm{s}}^{\rm{H}}}{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}}\\\
{s.t.\;\;\;{\kern
1.0pt}\left|{\bm{s}}(i)\right|=\sqrt{\frac{{{e_{t}}}}{{{N_{T}}L}}},{{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|}_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}}}\\\
{\;\;\;{\kern 1.0pt}\;\;\;{\kern
1.0pt}{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}\end{array}}\right..$ (29)

Note that the objective function in ${\cal{P}}_{r}^{C}$ and
${\cal{P}}_{r}^{C}$ is convex with respectively to $\bm{t}$, but non-concave
with respect to $\bm{s}$, thus, it is always impossible to find a satisfactory
solution in polynomial time. To this end, we study its relaxation form by
letting $\bm{R}_{s}=\bm{s}\bm{s}^{\rm{H}}$ and dropping the rank as well as
the similarity constraint, which leads to the following SDP problems

$\tilde{{\cal{P}}}_{r}^{C}\left\\{\begin{array}[]{l}\mathop{{\rm{max}}}\limits_{{{\bm{R}}_{s}}}\mathop{{\rm{min}}}\limits_{\bm{t}}{\rm{trace}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{R}}_{s}}}\right)\\\
s.t.\;\;\;{\kern
1.0pt}{{\bm{R}}_{s}}(i,i)=\frac{{{e_{t}}}}{{{N_{T}}L}},\bm{R}_{s}\succeq 0,\\\
\;\quad\quad{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq
r\end{array}\right.$ (30)

and

$\tilde{{\cal{P}}}_{t}^{C}\left\\{\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{{{\bm{t}}}}\mathop{{\rm{max}}}\limits_{\bm{R}_{s}}{\rm{trace}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{R}}_{s}}}\right)\\\
s.t.\;\;\;{\kern
1.0pt}{{\bm{R}}_{s}}(i,i)=\frac{{{e_{t}}}}{{{N_{T}}L}},\bm{R}_{s}\succeq 0,\\\
\;\quad\quad{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq
r\end{array}\right..$ (31)

Now, we turn to the properties of $\tilde{{\cal{P}}}_{r}^{C}$ and
$\tilde{{\cal{P}}}_{t}^{C}$. For a fixed $\bm{t}$, it is easy to verify that
${z^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)={\rm{trace}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{R}}_{s}}}\right)$
is linear with respect to $\bm{R}_{s}$. On the other hand, based on (6) we can
derive another expression of ${z^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)$,
given by

${z^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)={{\bm{t}}^{\rm{H}}}{\bm{H}}{({\bm{s}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}){\bm{t}}={{\bm{t}}^{\rm{H}}}{\bm{U}}({{\bm{R}}_{s}}){\bm{t}}$
(32)

with

${\bm{U}}({{\bm{R}}_{s}})(i,j)={{\bm{s}}^{\rm{H}}}{\bm{A}}_{i}^{\rm{H}}{\bm{R}}_{c}^{-1}{\bm{A}}_{i}{\bm{s}}={\rm{trace}}\left({{\bm{A}}_{i}^{\rm{H}}{\bm{R}}_{c}^{-1}{\bm{A}}_{i}{{\bm{R}}_{s}}}\right).$
(33)

As a result, for a fixed $\bm{R}_{s}$,
${z^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)$ is a convex quadratic form
with respect to $\bm{t}$. According to Sion’s theorem in [20], we have

$\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{\bm{t}}\mathop{{\rm{max}}}\limits_{{{\bm{R}}_{s}}}{\rm{trace}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{R}}_{s}}}\right)\\\
=\mathop{\max}\limits_{{{\bm{R}}_{s}}}\mathop{{\rm{min}}}\limits_{\bm{t}}{\rm{trace}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{R}}_{s}}}\right)\end{array},$
(34)

where $\bm{R}_{s}$ and $\bm{t}$ belong to the feasible set of $\tilde{\cal
P}_{r}^{P}$.

Combining (30), (31) and (34), we know that the Nash equilibrium exists for
the game modeled by $\tilde{\cal P}_{r}^{C}$ and $\tilde{\cal P}_{t}^{C}$.
Further, both $\tilde{\cal P}_{r}^{C}$ and $\tilde{\cal P}_{t}^{C}$ belong to
the convex-concave minimax problems, and the Nash equilibrium can be
approximately solved by the following iterative first order method[39]. In
particular, we start the algorithm with an initial point
($\bm{R}_{s}^{(0)}$,$\bm{t}^{(0)}$) and let

${\tilde{z}^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)={z^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)-\beta\left\|{{{\bm{R}}_{s}}-{\bm{R}}_{s}^{(0)}}\right\|_{F}^{2},$
(35)

where $\beta\geq 0$ is a proximal parameter to make sure
${\tilde{z}^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)$ is strongly concave
with respect to ${\bm{R}}_{s}$. Then, we carry out the following iteration[40,
41, 39, 19]

$\left\\{{\begin{array}[]{*{20}{l}}{{\bm{R}}_{s}^{(t+1)}=\arg\mathop{\max}\limits_{\left\\{{{{\bm{R}}_{s}}|{{\bm{R}}_{s}}(i,i)=\frac{{{e_{t}}}}{{{N_{T}}L}},{{\bm{R}}_{s}}\succeq
0}\right\\}}{\tilde{z}^{C}}\left({{{\bm{R}}_{s}},{{\bm{t}}^{(t)}}}\right)}\\\
{{\bm{t}}^{(t+1)}={{\rm{Proj}_{\Omega}}\left({{\bm{t}}^{(t)}-\eta{\nabla_{\bm{t}}}{{\tilde{z}}^{C}}\left({{\bm{R}}_{s}^{(t+1)},{{\bm{t}}^{(t)}}}\right)}\right)}}\end{array}}\right.,$
(36)

until the gap

${g^{C}}(t,t+1)=\left|{{{\tilde{z}}^{C}}\left({{\bm{R}}_{s}^{\left(t\right)},{{\bm{t}}^{(t)}}}\right)-{{\tilde{z}}^{C}}\left({{\bm{R}}_{s}^{(t+1)},{{\bm{t}}^{(t+1)}}}\right)}\right|$
(37)

is small enough, where $\rm{Proj}_{\Omega}(\cdot)$ denotes the projection on
$\Omega$ and $\eta$ is the iteration step size.

Note that ${\tilde{z}^{C}}\left({{{\bm{R}}_{s}},{\bm{t}}}\right)$ is strongly
concave, thus $\bm{R}_{s}^{(t+1)}$ can be solved by CVX in polynomial time. In
addition, since $\Omega$ is a scaled sphere centered by $\bm{t}_{0}$,
$\rm{Proj}_{\Omega}(\bm{t})$ is given by

${\rm{Pro}}{{\rm{j}}_{\Omega}}\left({\bm{t}}\right)=\left\\{\begin{array}[]{l}{\bm{t}},\;{\bm{t}}\in\Omega\\\
{{\bm{t}}_{0}}+\frac{{r\left({{\bm{t}}-{{\bm{t}}_{0}}}\right)}}{{{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}}},\;{\bm{t}}\notin\Omega\end{array}\right..$
(38)

Once the equilibrium strategy
$\left({{\bm{R}}_{s,C}^{*},{\bm{t}}_{C}^{*}}\right)$ is obtained, the
following task is to synthesize practical waveform-filter pair
$\left({{\bm{s}}_{C}^{*},{\bm{w}}_{C}^{*}}\right)$ complying with CM-SC from
$\left({{\bm{R}}_{s,C}^{*},{\bm{t}}_{C}^{*}}\right)$.

An inspection of the similarity constraint reveals that the phase of constant
modulus signal needs to meet the following condition

$\arg{\bm{s}}(i)\in\left[{\arg{{\bm{s}}_{0}}(i)-\varphi,\arg{{\bm{s}}_{0}}(i)+\varphi}\right]$
(39)

with $\varphi=\rm{accos}\left({1-\frac{{{\delta^{2}}}}{2}}\right)$.

In order to get the feasible $\bm{s}$ of ${\cal P}_{r}^{C}$, the randomization
schemes[22] are used. More precisely, we generate $M$ random vectors
$\bm{\xi}_{m}$ from
${\cal{CN}}\left({{\bf{0}},{\bm{R}}_{s,C}^{*}\odot\left({{{{\bm{\bar{s}}}}_{0}}{\bm{s}}_{0}^{\rm{T}}}\right)}\right)$,
and adjust its phase with

${{\bm{s}}^{(m)}}(i)={{\bm{s}}_{0}}(i){e^{\frac{{j\left({\arg({{\bm{\xi}}_{m}}(i))-\pi}\right)\varphi}}{{2\pi}}}}.$
(40)

Thus, $\bm{s}^{(m)}$ meets CM-SC due to
$\frac{{\left({\arg({{\bm{\xi}}_{m}}(i))-\pi}\right)\varphi}}{{2\pi}}\in\left[{-\varphi,\varphi}\right)$.
Finally, the optimal waveform is selected from
$\left\\{{{{\bf{s}}^{(m)}}}\right\\}_{m=1}^{M}$ as the one who performs best.
In more detail, denote by $\bm{t}_{C}^{(m)}$ and ${{{\rm{SINR}}}^{(m)}}$ the
optimal sultion and optimal value for the following problem

$\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{\bm{t}}{{\bm{t}}^{\rm{H}}}{\left({{\bm{H}}({{\bm{s}}^{(m)}})}\right)^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({{\bm{s}}^{(m)}}){\bm{t}}\\\
s.t.\;\;\;{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}}\leq r\end{array}$
(41)

respectively, and we pick the optimal signal ${{\bm{s}}^{(k)}}$ achieving the
maximum value ${{{\rm{SINR}}}^{(k)}}$. To this end, we get the optimal
waveform-filter pair with

${\bm{s}}_{C}^{*}={{\bm{s}}^{(k)}},{\bm{w}}_{C}^{*}={\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{C}^{*}){\bm{t}}_{C}^{(k)}.$
(42)

The overall procedure for waveform-filter design under CM-SC is summarized in
Table II as Algorithm 2. The convergence of Algorithm 2 is determined by the
iteration steps described by (36). In fact, it converges if the step size
$\eta$ is properly chosen and the choice of $\eta$ can be found in [39].
However, it goes beyond the scope of our paper. The computational complexities
consist of two parts, namely, iteration algorithm and waveform-filter pair
synthesis. One can see that the computational complexities of the iteration
algorithm is proportion to the number of iterations. At each iteration, it
requires at most $O\left({{{\left({L{N_{T}}}\right)}^{9}}}\right)$[38] to
optimize $\bm{R}_{s}^{(t+1)}$ and $O\left({{{{Q}}^{2}}}\right)$ to compute
$\bm{t}^{(t+1)}$. As to the synthesis stage, it requires
$O\left({{M\left({\left({L{N_{T}}}\right)}^{2}+Q^{3}\right)}}\right)$[42, 43]
to generate $\left\\{{{{\rm{SINR}}^{(m)}}}\right\\}_{m=1}^{M}$ and
$O\left({\left({L{N_{R}}+Q{N_{R}}}\right)^{2}}\right)$ to calculate
$\bm{w}_{C}^{*}$. Therefore, the total computational complexities are
dominated by its highest order
$O\left({{{\left({L{N_{T}}}\right)}^{9}}}\right)$.

TABLE II: Algorithm for waveform-filter optimization with CM-SC Input:
$\bm{a}(\theta_{t})$, $\bm{b}(\theta_{t})$, $\bm{s}_{0}$, $e_{t}$, $\delta$,
${{\bm{R}}_{c}}$, ${{\bm{t}}_{0}}$, $r$, $\beta$, $\eta$, $\epsilon^{C}$ and
$M$.
---
Step 1: Initialize $t=0$, $\bm{t}^{(0)}=\bm{t}_{0}$ and
$\bm{R}_{s}^{(0)}=\bm{s}_{0}\bm{s}_{0}^{\rm{H}}$.
Step 2: Solve optimization problems described in (36) alternatively.
Step 3: Verify inequality $g^{C}(t,t+1)\leq\epsilon^{C}$. If true,
${\bm{R}}_{s,C}^{*}={\bm{R}}_{s}^{(t+1)}$,
${\bm{t}}_{C}^{*}={\bm{t}}^{(t+1)}$ and go to Step 4, otherwise, $t=t+1$ and
go
to Step 2.
Step 4: Generate $\left\\{{{{\bm{\xi}}_{m}}}\right\\}_{m=1}^{M}$ from
${\cal{CN}}\left({0,{\bm{R}}_{s,C}^{*}\odot\left({{{{\bm{\bar{s}}}}_{0}}{\bm{s}}_{0}^{\rm{T}}}\right)}\right)$
and
calculate $\left\\{{{{\bm{s}}^{(m)}}}\right\\}_{m=1}^{M}$ with (40).
Step 5: For each ${{\bm{s}}^{(m)}}$, optimize (41) and record the
corresponding
optimal value ${\rm{SINR}}^{(m)}$ and solution $\bm{t}_{C}^{(m)}$.
Step 6: Pick the maximal value in
$\left\\{{{{\rm{SINR}}^{(m)}}}\right\\}_{m=1}^{M}$, for example,
${\rm{SINR}}^{(k)}$. Finally, synthesize $\bm{s}_{C}^{*}$ and $\bm{w}_{C}^{*}$
with (42).
Output: ${\bm{s}_{C}^{*}}$ and ${\bm{w}_{C}^{*}}$.

###  III-C SC-SC on waveform

This subsection is devoted to the Stackelberg equilibrium of ${\cal{P}}_{r}$
with $\Psi=\Psi_{S}$. Leveraging on the MM tools[44], we construct a sequence
of Stackelberg games[30], which belongs to the convex non-concave minimax
problems. Further, we prove that the Stackelberg games can be equivalently
solved by optimizing a convex problem. Thus, the optimal waveform can be
obtained without solving the target strategies in ${\cal{P}}_{t}$.

Similarly to the former cases, we recast ${\cal{P}}_{r}$ as follows

${\cal{P}}_{r}^{S}\left\\{{\begin{array}[]{*{20}{l}}{\mathop{{\rm{max}}}\limits_{\bm{s}}\mathop{{\rm{min}}}\limits_{\bm{t}}{{\bm{s}}^{\rm{H}}}{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}}\\\
{s.t.\;\;\;{\kern
1.0pt}\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}},{{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|}_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}}}\\\
{\;\;\;{\kern 1.0pt}\;\;\;{\kern
1.0pt}{{\bm{s}}^{\rm{H}}}{{\bm{R}}_{I}}{\bm{s}}\leq{e_{I}},{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}\end{array}}\right..$ (43)

Note that the feasibility of ${\cal{P}}_{r}^{S}$ is well discussed in [35,
23], and it is beyond the scope of our paper, where $e_{I}$ is carefully set
to make sure that ${\cal{P}}_{r}^{S}$ is always feasible.

The key point of MM algorithm for solving the maximization problem is to find
a proper minorizer of objective. And the following proposition provides a
minorizer for solving ${\cal{P}}_{r}^{S}$.

###### Proposition 2

Let

${z^{S}}({\bm{s}})=\mathop{{\rm{min}}}\limits_{{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}{{\bm{s}}^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}},$

then

$\begin{array}[]{l}{{\tilde{z}}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})=\mathop{{\rm{min}}}\limits_{{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}2{\mathop{\rm
Re}\nolimits}\left\\{{{{\left({{{\bm{s}}^{(l)}}}\right)}^{\rm{H}}}{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}}\right\\}-\\\
\quad\quad\quad\quad\quad\quad\quad\quad{\left({{{\bm{s}}^{(l)}}}\right)^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{s}}^{(l)}}\end{array}$

is a minorizer of ${z^{S}}({\bm{s}})$ at $\bm{s}^{(l)}$.

###### Proof:

See Appendix C. ∎

Leveraging on the MM algorithm, the Stackelberg equilibrium described by
${\cal{P}}_{r}^{S}$ can be solved by sequentially optimizing
$\left\\{{{\cal{P}}_{r}^{S,(l)}}\right\\}_{l=1}^{\infty}$ with

${\cal{P}}_{r}^{S,(l)}\left\\{{\begin{array}[]{*{20}{l}}{{{\bm{s}}^{(l+1)}}=\arg\mathop{{\rm{max}}}\limits_{\bm{s}}{{\tilde{z}}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})}\\\
{s.t.\;\;\;{\kern
1.0pt}\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}},{{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|}_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}}}\\\
{\;\;\;{\kern 1.0pt}\;\;\;{\kern
1.0pt}{{\bm{s}}^{\rm{H}}}{{\bm{R}}_{I}}{\bm{s}}\leq{e_{I}}}\end{array}}\right.,$
(44)

which is also a Stackelberg game but easier to solve.

Now, we turn to solving ${\cal{P}}_{r}^{S,(l)}$. It is worth pointing out that
the objective ${{\tilde{z}}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})$ is a linear
function with respect to $\bm{s}$. Substituting (6) into
${{\tilde{z}}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})$, and after some algebraic
manipulations, we obtain another form of it, given by

${\tilde{z}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})=\mathop{{\rm{min}}}\limits_{\left\\{{{\bm{t}}|{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}\right\\}}{{\bm{t}}^{\rm{H}}}{\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}}){\bm{t}}$
(45)

with

$\begin{array}[]{l}{\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}})={\bm{H}}{({{\bm{s}}^{(l)}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}{({\bm{s}})}+{\bm{H}}{({\bm{s}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}{({{\bm{s}}^{(l)}})}\\\
\quad\quad\quad\quad\quad\;-{\bm{H}}{({{\bm{s}}^{(l)}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}{({{\bm{s}}^{(l)}})}.\end{array}$
(46)

Unfortunately,
${{\bm{t}}^{\rm{H}}}{\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}}){\bm{t}}$ is not convex
with respect to $\bm{t}$ due to ${\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}})\nsucceq
0$, even though it is a quadratic form. Thus, the iteration algorithm
mentioned in (36) is invalid when solving ${\cal{P}}_{r}^{S,(l)}$. To this
end, we have to devise another algorithm to address ${\cal{P}}_{r}^{S,(l)}$,
which is based on the following proposition.

###### Proposition 3

${\cal{P}}_{r}^{S,(l)}$ is equivalent to the following problem

$\hat{\cal{P}}_{r}^{S,(l)}\left\\{\begin{array}[]{l}\mathop{\max}\limits_{{\bm{s}},\lambda,\gamma}\;\gamma\\\
s.t.\left[{\begin{array}[]{*{20}{c}}{{\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}})+\lambda{\bf{I}}}&{\lambda{{\bm{t}}_{0}}}\\\
{\lambda{\bm{t}}_{0}^{\rm{H}}}&{\lambda{\bm{t}}_{0}^{\rm{H}}{{\bm{t}}_{0}}-\lambda{r^{2}}-\gamma}\end{array}}\right]\succeq
0\vspace{2pt}\\\
\quad\quad{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}},{\kern
1.0pt}\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}\vspace{2pt}}\\\
\quad\quad{{\bm{s}}^{\rm{H}}}{{\bm{R}}_{I}}{\bm{s}}\leq{e_{I}},\lambda\geq
0\end{array}\right.,$

where $\lambda$ and $\gamma$ are auxiliary variables.

###### Proof:

See Appendix D. ∎

Obviously, $\hat{\cal{P}}_{r}^{S,(l)}$ is a SDP problem since
${\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}})$ is linear with respect to $\bm{s}$.
Based on Proposition 3, we sequentially solve ${\cal{P}}_{r}^{S,(l)}$ until
converges, and get the optimal waveform $\bm{s}_{S}^{*}$.

Once the optimal waveform $\bm{s}_{S}^{*}$ is obtained, we need to calculate
the optimal filter $\bm{w}_{S}^{*}$ to construct the Stackelberg equilibrium
strategy of radar. In oder to achieve this goal, we first get the target
player’s strategy in the Stackelberg game by solving

$\begin{array}[]{l}{\bm{t}}_{{S}}^{\bm{*}}=\arg\min{{\bm{t}}^{\rm{H}}}{\bm{H}}{({\bm{s}}_{S}^{*})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{S}^{*}){\bm{t}}\\\
s.t.\quad\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|_{2}\leq r\end{array},$ (47)

then we calculate $\bm{w}_{S}^{*}$ with

${\bm{w}}_{S}^{*}={\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{S}^{*}){\bm{t}}_{S}^{*}.$
(48)

The overall procedure for waveform-filter design under SC-SC is summarized in
Table III as Algorithm 3. The convergence of Algorithm 3 is ensured by the MM
algorithm[45] owing to the fact that ${z^{S}}({\bm{s}})$ is upper bounded by
$e_{t}{\lambda_{\max}}\left({{\bm{G}}{{({{\bm{t}}_{0}})}^{H}}{\bm{R}}_{c}^{-1}{\bm{G}}({{\bm{t}}_{0}})}\right)$.
As to the computational complexities, it is proportional to the number of
iterations of MM algorithm when optimizing $\bm{s}_{S}^{*}$. At each
iteration, it requires at most
$O\left({{{\left({Q+L{N_{T}}}\right)}^{0.5}}\left({{{\left({L{N_{T}}}\right)}^{4}}+{{\left({QL{N_{T}}}\right)}^{2}}+{Q^{3}}L{N_{T}}}\right)}\right)$.
Additionally, it requires at most
$O\left({{Q^{3}}+{{\left({L{N_{R}}+Q{N_{R}}}\right)}^{2}}}\right)$ to
calculate $\bm{w}_{S}^{*}$. Therefore, the total computational complexities
are dominated by the highest order
$O\left({{{\left({Q+L{N_{T}}}\right)}^{0.5}}\left({{{\left({L{N_{T}}}\right)}^{4}}+{{\left({QL{N_{T}}}\right)}^{2}}+{Q^{3}}L{N_{T}}}\right)}\right)$.

TABLE III: Algorithm for waveform-filter optimization with CM-SC Input:
$\bm{a}(\theta_{t})$, $\bm{b}(\theta_{t})$, $\bm{s}_{0}$, $e_{t}$, $\delta$,
${{\bm{R}}_{c}}$, ${{\bm{t}}_{0}}$, $r$, $[f_{1}^{k},f_{2}^{k}]$,
$\alpha_{k}$, $e_{I}$ and $\epsilon^{S}$.
---
Step 1: Initialize $l=0$ and $\bm{s}^{(0)}$ with any feasible waveform;
let $\tilde{z}^{S}(\bm{s}^{(0)},\bm{s}^{(-1)})={z}^{S}(\bm{s}^{(0)})$.
Step 2: Solve optimization problem ${\cal{P}}_{r}^{S,(l)}$.
Step 3: Verify
$\tilde{z}^{S}(\bm{s}^{(l+1)},\bm{s}^{(l)})-\tilde{z}^{S}(\bm{s}^{(l)},\bm{s}^{(l-1)})\leq\epsilon^{S}$.
If true, go to Step 4, otherwise, $l=l+1$ and go to Step 2.
Step 4: Let $\bm{s}_{S}^{*}=\bm{s}^{(l+1)}$; solve (47) and obtain
$\bm{w}_{S}^{*}$ with (48).
Output: ${\bm{s}_{S}^{*}}$ and ${\bm{w}_{S}^{*}}$.

##  IV Numerical experiments

In this section, several numerical experiments are conducted to assess the
performance of the proposed algorithms. Unless otherwise specified, in the
following experiments, we assume a colocated MIMO radar system with $N_{T}=2$
transmitters and $N_{R}=4$ receivers, where the inter-element space is wave-
length for transmitters and half wave-length for receivers. The carrier
frequency is 3GHz, and the code length is $L=16$. Meanwhile, the target T is
assumed at $\theta_{t}=$30∘ azimuth with a prescribed

${{\bm{t}}_{\rm{0}}}{\rm{=}}{\left[{{\rm{0}}{\rm{.2}}{e^{\frac{{j\pi}}{4}}}{\rm{,0}}{\rm{.3}}{e^{\frac{{j\pi}}{3}}}{\rm{,0}}{\rm{.8,0}}{\rm{.3}}{e^{-\frac{{j\pi}}{6}}}{\rm{,0}}{\rm{.2}}{e^{-\frac{{j\pi}}{3}}},0.1{e^{-\frac{{j\pi}}{3}}}}\right]^{\rm{T}}}\in\mathbb{Q}^{6}.$

As to the noise, we also assume that it is complex Gaussian distribution with
${\bm{n}}\sim{\cal{CN}}\left({{\bf{0}},{{\bm{R}}_{n}}}\right)$, where
${{\bm{R}}_{n}}\left({m,n}\right)={0.8^{\left|{m-n}\right|}}$. All numerical
experiments are analyzed using Matlab 2014a version and performed in a
standard PC (with CPU Core i5 3.0 GHz and 16 GB RAM).

###  IV-A Experiments for EC on waveform

Now, we consider the algorithm for EC on waveform, namely, Algorithm 1. Given
that the algorithm in [11] are proposed to address the EC on waveform, we also
give comparisons with it in terms of detection probability and running time.

Fig.2 depicts the detection probability versus the transmit energy $e_{t}$ for
different $r$ by substituting $\underline{{\rm{SINR}}}_{E}$ or
$\overline{{\rm{SINR}}}_{E}$ into (8) with $P_{fa}=10^{-6}$. As expected, the
detection probability monotonically increases with respect to $e_{t}$ for a
fixed $r$. Meanwhile, it monotonically decreases with respect to $r$ for a
fixed $e_{t}$, since a larger $r$ means a ”smarter” target, which is
unfavorable from the view of radar player. Moreover, one can observe from the
figure that Algorithm 1 achieves almost the same results as the algorithm in
[11] does, which demonstrates the effectiveness of the proposed Algorithm 1.
Next we compare the running time of the two algorithms, and illustrate the
results in Table IV. According to the data in Table IV, we find that less
running time is needed for Algorithm 1 compared with its counterpart in [11].
The reason can be explained that Algorithm 1 solves the optimal waveform-
filter directly based on Theorem 1 without outer iteration, thus only the
inner iteration is needed for optimizing $\hat{{\cal{P}}}_{t}^{E}$. By
contrast, the running time of algorithm in [11] depends on not only inner
iteration of optimization $\bm{t}$, but also the number of outer iterations.

Fig.3 illustrates modulus of the optimal waveform with $r=0.8$ and $e_{t}=1$
for two algorithms. One can see serious fluctuation of transmit waveform,
which will not meet the demands of radar transmitter given that the
transmitter always operates at saturation situation. These results inspire us
to find more suitable waveform to fulfill the detection task.

TABLE IV: Running time for Algorithm 1 and the algorithm in [11] with $e_{t}=1$ | Algorithm 1 | Algorithm in [11]
---|---|---
$r=0.1$ | 2.03s | 5.37s
$r=0.3$ | 1.83s | 4.44s
$r=0.5$ | 1.83s | 4.35s
$r=0.8$ | 1.84s | 4.39s
Figure 2: Detection probability versus $e_{t}$ for different $r$ Figure 3:
Modulus of waveform

###  IV-B Experiments for CM-SC on waveform

In this subsection, we consider the algorithm for CM-SC on waveform, namely,
Algorithm 2. In the following experiments, the parameters $\beta$, $\eta$ and
$\epsilon^{C}$ are set to be 0.05, 0.002 and 0.001, respectively. Moreover, we
conduct $M=100$ trails when synthesizing waveform with randomization. As to
the reference waveform, the orthogonal LFM is chosen with
$\bm{s}_{0}={\rm{vec}}(\bm{S}_{0})$, where ${{\bm{S}}_{\rm{0}}}$ is defined
by[22, 14, 46]

${{\bm{S}}_{\rm{0}}}{\rm{(}}n{\rm{,}}l{\rm{)=}}\sqrt{\frac{{{e_{t}}}}{{{N_{T}}L}}}{e^{\frac{{j\pi(2n(l-1)+{{(l-1)}^{2}})}}{L}}}.$
(49)

Fig.4 depicts the value of $\tilde{z}^{C}(\bm{R}_{s},\bm{t})$ versus the
number of iterations for different $r$. An inspection of Fig.4 reveals that
Algorithm 2 converges after several iterations and number of iterations
increases with respect to $r$ due to the expansive of $\Omega$. More
precisely, the algorithm converges after 3 iterations for $r=0.1$, but 52
iterations for $r=0.8$.

Next, we investigate the detection performance of the optimal waveform-filter
pair for different $\delta$ by substituting $\underline{{\rm{SINR}}}_{C}$ into
(8) with $P_{fa}=10^{-6}$, where $r$ is set to be 0.8. We also give
comparisons with the algorithm in [5] realizing that the algorithm in [5] can
be applied to the similarity constraint by modifying its randomization
schemes. The corresponding results are shown in Fig.5. As expected, the
detection probability increases with respect to $e_{t}$ as well as $\delta$,
which is consistent with our intuition, since a higher transmit energy means a
higher SINR and a larger $\delta$ means more freedom of waveform. It is
observed from the figure that our algorithm shows its superiority over the
algorithm in [5] in terms of detection probability with the same $\delta$.

To be honest, compared with the algorithm in [5], our algorithm suffer heavier
computational burden, even though it achieves a higher SINR. The relevant
results are illustrated in Table V. It can be seen that the running time of
our algorithm increases with respect to $r$. We explain the reason that
solving (36) is time consuming at each iteration and the number of iterations
is also large for $r=0.8$(see Fig.4). While the running time of algorithm in
[5] is relatively stable with respect to $r$, since $\Omega$ is approximated
by randomly sampling. Nevertheless, it is worth pointing out that the number
of iterations can be reduced if the parameters $\beta$ and $\eta$ are
carefully chosen, and interested readers may refer to [39, 40, 41].

In addition, Fig.6 depicts the properties of pulse compression for different
$\delta$. These results display that larger $\delta$ suffers from higher
sidelobe levels, even though a higher detection probability is achieved. In
particular, the highest sidelobe level for $\delta=1$ is about -6dB, while it
is about -13dB for $\delta=0.1$. Moreover, the sidelobe level achieved by
Algorithm 2 is slightly higher than its counterpart, which also demonstrates
the trade-off between better SINR and low side lobes.

Figure 4: $\tilde{z}^{C}(\bm{R}_{s},\bm{t})$ versus number of iterations for different $r$ TABLE V: Running time for Algorithm 2 and the algorithm in [5] with $e_{t}=1$ and $\delta=1$ | Algorithm 2 | Algorithm in [5]
---|---|---
$r=0.1$ | 19.81s | 15.80s
$r=0.3$ | 28.74s | 15.83s
$r=0.5$ | 41.42s | 15.98s
$r=0.8$ | 110.63s | 16.11s
Figure 5: Detection probability versus $e_{t}$ for different $\delta$ |
---|---
(a) | (b)
|
(c) | (d)
Figure 6: Pulse compression for different $\delta$: (a) $\delta=2$, (b)
$\delta=1$, (c) $\delta=0.5$, (d) $\delta=0.2$

###  IV-C Experiments for SC-SC on waveform

At last, we consider the algorithm for SC-SC on waveform, namely, Algorithm 3.
In the following experiments, the parameters $r$ and $\epsilon^{S}$ are set to
be 0.8 and 0.001, respectively. Moreover, the orthogonal LFM waveform is also
used as the reference waveform $\bm{s}_{0}$. As to the spectral compatibility
parameters, transmission power is limited in two frequency intervals. The
first one is $[f_{1}^{1},f_{2}^{2}]=[0.30,0.40]$ with $\alpha_{1}=0.6$, and
the second one is $[f_{1}^{2},f_{2}^{2}]=[0.60,0.80]$ with $\alpha_{2}=0.4$.
In order to find a feasible waveform and make sure ${\cal{P}}_{r}^{S}$ is
always feasible, we first solve the following optimization problem

$\begin{array}[]{l}{{\bm{s}}^{(0)}}={\kern
1.0pt}\arg\mathop{\min}\limits_{\bm{s}}{{\bm{s}}^{\rm{H}}}{{\bm{R}}_{I}}{\bm{s}}\\\
s.t.\;\;\;{\kern
1.0pt}\left\|{\bm{s}}\right\|_{2}^{2}\leq{e_{t}},{\left\|{{\bm{s}}-{{\bm{s}}_{0}}}\right\|_{\infty}}\leq\frac{{\delta\sqrt{{e_{t}}}}}{{\sqrt{{N_{T}}L}}}\end{array},$
(50)

then set ${e_{I}}\geq{\kern
1.0pt}{\left({{{\bm{s}}^{(0)}}}\right)^{\rm{H}}}{{\bm{R}}_{I}}{{\bm{s}}^{(0)}}$.
Given the fact that there is no available algorithm solving the minimax
waveform-filter design problem for extended target under SC-SC, we only
carried out some experiments to test Algorithm 3 in this subsection.

Now, we turn to the convergence of Algorithm 3. Fig.7 depicts the iteration
curve of ${{\tilde{z}}^{S}}({\bm{s}^{(l)}},{{\bm{s}}^{(l-1)}})$ with fixed
$e_{t}=20$dB and $\delta=1$. Obviously, Algorithm 3 converges in several
iterations, and SINR monotonically increases with respect to $e_{I}$.
Furthermore, Fig.8 shows some properties of the transmit waveform. In
particular, Fig.8(a) depicts the PSD of transmit waveform. One can see from
Fig.8(a) that the transmit waveform forms deep notches in the limited
frequency bands, and the smaller $\delta$ results in deeper notches. While,
the corresponding pulse compression results are given in Fig.8(b). It is
interesting that the parameter $e_{I}$ also affects the sidelobe level under
the same $\delta$. These phenomena are consistent with the results in [46,
47].

Finally, we randomly pick up some samples from $\Omega$ and calculate the
actually output SINR to verify that the designed waveform-filter pair is the
Stackelberg equilibrium strategy for radar. The corresponding results are
shown in Fig.9, and we are glad to see that the actually achieved SINR is
significantly higher than the SINR(worst-case) optimized by Algorithm 3, which
demonstrates the effectiveness of our algorithm.

Figure 7: ${{\tilde{z}}^{S}}({\bm{s}^{(l)}},{{\bm{s}}^{(l-1)}})$ versus the
number of iterations for different $e_{I}$
---
(a)
(b)
Figure 8: Waveform properties for different $e_{I}$: (a) Power Spectral
Density, (b) Pulse Compression Figure 9: Actually achieved SINR with random
samples

##  V Conclusion

In this paper, we study the joint design problem of transmit waveform and
receive filter for extended target detection from the view of TPZS games,
where the SINR is used as the payoff function. We assume that the radar player
aims at maximizing SINR by choosing its waveform and filter from the strategy
set, while the target player is smart enough to minimize SINR by changing its
TIR from the strategy set to prevent being detected. The interaction between
the radar and target is modeled as a Stackelberg game where radar acts as the
leader, and the goal is to find the optimal strategy of radar from Stackelberg
equilibrium. The strategy set of target is constrained in a scaled sphere
centered by a prescribed TIR. As to the strategy set of radar, the following
three cases are studied: 1) EC on Waveform; 2) CM-SC on waveform; 3) SC-SC on
waveform. We resort to Algorithm 1, Algorithm 2 and Algorithm 3 to address the
three cases mentioned above, respectively. The conclusions are drawn as
follows:

1) Under the EC on waveform, the Stackelberg equilibrium is also the Nash
equilibrium, which means that the play who acts as the leader in the game
makes no difference on the equilibrium strategies. Thus, in Algorithm 1, the
equilibrium strategy is be solved from the game where target acts as the
leader, which can be convert to a convex optimization problem and solved in
polynomial time. Algorithm 1 achieves the same performance as the algorithm in
[11] in terms of detection probability, but shows superiority on running time.

2) When considering CM-SC on waveform, the Stackelberg equilibrium may not be
the Nash equilibrium, but the Nash equilibrium can be approximately
constructed from its relaxation form by optimizing the covariance of transmit
waveform. Algorithm 2 focuses on solving the Nash equilibrium of the
relaxation form, and synthesize the Stackelberg equilibrium strategy of the
original game with randomization process. Even though it is time consuming
compared with the algorithm in [5], it achieves higher detection probability.
Moreover, the tighter similarity constraint on waveform results in better
pulse compression properties but lower detection probability.

3) As to SC-SC on waveform, the Stackelberg equilibrium are solved leveraging
on the MM algorithm, which is the core idea of Algorithm 3. Accordingly, the
optimal waveform forms deep notches in the limited frequency bands and show
relatively satisfactory pulse compression properties. Interestingly, the
spectral compatibility parameter not only affects the its PSD, but also its
pulse compression properties. Thus, a suitable trade-off among the output
SINR, the similarity level and the spectral compatibility property should be
made in practice. Moreover, the Stackelberg equilibrium strategy for radar is
essentially the robust waveform-filter.

Our future researches may include the equilibrium strategies in the presence
of signal dependent clutter. Besides, researches on more practical strategy
set of radar and target will be also interesting.

##  Appendix A Proof of Theorem 1

For a fixed $\bm{s}^{0}$, the optimization problem in ${\tilde{\cal P}_{r}}$
can be recast as

$F(\bm{s}^{0})=\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}\mathop{{\rm{max}}}\limits_{\bm{w}}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{\bm{y}}_{t}^{0}{{\left({{\bm{y}}_{t}^{0}}\right)}^{\rm{H}}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bf{w}}}}$
(51)

where ${{\bm{y}}_{t}^{0}}={\bm{H}}({\bm{s}^{0}}){\bm{t}}$. On the other hand,
the optimization problem in ${{\cal P}_{r}}$ can be recast as

$G(\bm{s}^{0})=\mathop{{\rm{max}}}\limits_{\bm{w}}\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{\bm{y}}_{t}^{0}{{\left({{\bm{y}}_{t}^{0}}\right)}^{\rm{H}}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bf{w}}}}.$
(52)

Further, by solving the inner maximization problem of $F(\bm{s}^{0})$, we have

$F({{\bm{s}}^{0}})=\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}{\left({{\bm{y}}_{t}^{0}}\right)^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{y}}_{t}^{0}=\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}{{\bm{t}}^{\rm{H}}}{\left({{\bm{H}}({{\bm{s}}^{0}})}\right)^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({{\bm{s}}^{0}}){\bm{t}}$
(53)

Note that $F(\bm{s}^{0})\geq G(\bm{s}^{0})$ always holds due to the weak
minimax inequality, which means

$G(\bm{s}^{0})\leq\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}{{\bm{t}}^{\rm{H}}}{\left({{\bm{H}}({{\bm{s}}^{0}})}\right)^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({{\bm{s}}^{0}}){\bm{t}}..$
(54)

Denote by $\bm{t}_{s_{0}}^{*}$ the optimal solution of (53), and let
$\bm{w}_{s_{0}}^{*}={\bm{R}}_{c}^{-1}{\bm{H}}({{\bm{s}}^{0}}){\bm{t}_{s_{0}}^{*}}$,
then we have

$G({{\bm{s}}^{0}})\geq\mathop{{\rm{max}}}\limits_{{\bm{w}}=\bm{w}_{s_{0}}^{*}}\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{\bm{y}}_{t}^{0}{{\left({{\bm{y}}_{t}^{0}}\right)}^{\rm{H}}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}.$
(55)

The inequality holds due to the shrinkage of feasible set on $\bm{w}$.

The right hand in (55) is equal to $F({{\bm{s}}^{0}})$ according to Theorem 1
in [48] given that $\Omega$ is convex.

Combining (54) and (55), one can obtain the following strong minimax
inequality

$\mathop{{\rm{max}}}\limits_{\bm{w}}\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{\bm{y}}_{t}^{0}{{\left({{\bm{y}}_{t}^{0}}\right)}^{\rm{H}}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}=\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}\mathop{{\rm{max}}}\limits_{\bm{w}}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{\bm{y}}_{t}^{0}{{\left({{\bm{y}}_{t}^{0}}\right)}^{\rm{H}}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}.$
(56)

Recall that (56) holds for any $\bm{s}^{0}$, then we have

$\mathop{{\rm{max}}}\limits_{{\bm{s}}\in\Psi,{\bm{w}}}\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{{\bm{y}}_{t}}{\bm{y}}_{t}^{\rm{H}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}}=\mathop{{\rm{max}}}\limits_{{\bm{s}}\in\Psi}\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}\mathop{{\rm{max}}}\limits_{{\bm{w}}}{\rm{}}\frac{{{{\bm{w}}^{\rm{H}}}{{\bm{y}}_{t}}{\bm{y}}_{t}^{\rm{H}}{\bm{w}}}}{{{{\bm{w}}^{\rm{H}}}{{\bm{R}}_{c}}{\bm{w}}}},$
(57)

and the optimal solutions for ${\cal{P}}_{r}$ and $\tilde{\cal{P}}_{r}$ are
the same.

Thus, we complete the proof of Theorem 1.

##  Appendix B Proof of Proposition 1

The proof starts with $\tilde{\cal P}_{t}^{E}$ described in (25), which is
equivalent to the following problem

$\left\\{{\begin{array}[]{*{20}{l}}{{{\overline{{\rm{SINR}}}}_{E}}=\mathop{{\rm{min}}}\limits_{\bm{G(t)}}{\lambda_{\max}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}})}\right){e_{t}}}\\\
{s.t.\;\;\;{\kern
1.0pt}{\bm{G}}({\bm{t}})\in\tilde{\Omega}}\end{array}}\right.,$ (58)

where $\tilde{\Omega}=\left\\{{{\bm{G}}({\bm{t}})|\bm{t}\in\Omega}\right\\}$
is also convex since $\bm{G(t)}$ is an affine function with respect to
$\bm{t}$.

Denote by $\bm{t}_{E}^{*}$ the optimal solution of $\tilde{\cal P}_{t}^{E}$,
then $\bm{G}\left(\bm{t}_{E}^{*}\right)$ is an optimal solution of (58), which
meets the following optimality condition

$\begin{array}[]{l}\left\langle{{{\left.{{\nabla_{{\bm{G}}({\bm{t}})}}{\lambda_{\max}}\left({{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}})}\right)}\right|}_{{\bm{G}}({\bm{t}}_{E}^{*})}},{\bm{G}}({\bm{t}})-{\bm{G}}({\bm{t}}_{E}^{*})}\right\rangle\\\
\geq 0,{\rm{}}\forall{\bm{G}}({\bm{t}})\in\tilde{\Omega}\end{array}.$ (59)

After some algebraic manipulations [49], (59) reduces to

${\left({{\bm{s}}_{E}^{*}}\right)^{\rm{H}}}{\bm{G}}{({\bm{t}}_{E}^{*})^{\rm{H}}}{\bm{R}}_{c}^{-1}\left({{\bm{G}}({\bm{t}})-{\bm{G}}({\bm{t}}_{E}^{*})}\right){\bm{s}}_{E}^{*}\geq
0,\forall{\bm{G}}({\bm{t}})\in\tilde{\Omega}$ (60)

with ${{\bm{s}}_{E}^{*}}$ defined in (24).

On the other hand,

$\begin{array}[]{l}{\underline{{\rm{SINR}}}_{E}}=\mathop{{\rm{max}}}\limits_{{\bm{s}}\in{\Psi_{E}}}\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{{\bm{t}}^{\rm{H}}}{\bm{H}}{({\bm{s}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}){\bm{t}}\\\
\quad\quad\quad\;\;\geq\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{{\bm{t}}^{\rm{H}}}{\bm{H}}{({\bm{s}}_{E}^{*})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{E}^{*}){\bm{t}}\end{array}.$
(61)

Next, we are going to prove that the value of right hand in inequality (61) is
equal to ${\overline{{\rm{SINR}}}_{E}}$, i.e., $\bm{t}_{E}^{*}$ meets the
optimality condition of
$\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{{\bm{t}}^{\rm{H}}}{\bm{H}}{({\bm{s}}_{E}^{*})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{E}^{*}){\bm{t}}$.

Denote by $\tilde{\bm{t}}_{E}^{*}$ the optimal solution of
$\mathop{{\rm{min}}}\limits_{{\bm{t}}\in\Omega}{{\bm{t}}^{\rm{H}}}{\bm{H}}{({\bm{s}}_{E}^{*})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{E}^{*}){\bm{t}}$,
then we get the following optimality condition

${\left({{\bm{t}}-{\bm{\tilde{t}}}_{E}^{*}}\right)^{\rm{H}}}{\bm{H}}{({\bm{s}}_{E}^{*})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{H}}({\bm{s}}_{E}^{*}){\bm{\tilde{t}}}_{E}^{*}\geq
0,\forall{\bm{t}}\in\Omega.$ (62)

Note that $\bm{G(t)}$ is the one-to-one mapping with respect to $\bm{t}$.
Therefore, combining (61) and $\bm{G}{(\bm{t})\bm{s}}=\bm{H}{(\bm{s})\bm{t}}$,
we get that $\bm{t}_{E}^{*}$ also meets the optimality condition (62), namely,

$\underline{\rm{SINR}}_{E}\geq\overline{\rm{SINR}}_{E}.$ (63)

Furthermore, the weak minimax inequality
$\underline{\rm{SINR}}_{E}\leq\overline{\rm{SINR}}_{E}$ always holds. To this
end, we have

$\underline{\rm{SINR}}_{E}=\overline{\rm{SINR}}_{E},$ (64)

and the strategy pair
$(\bm{s}_{E}^{*},{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}_{E}^{*}){\bm{s}}_{E}^{*},\bm{t}_{E}^{*})$
is Nash equilibrium.

Thus we complete the proof of Proposition 1.

##  Appendix C Proof of Proposition 2

For any $\bm{t}$, we have

$\begin{array}[]{l}{{\bm{s}}^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}=2{\mathop{\rm
Re}\nolimits}\left\\{{{{\left({{{\bm{s}}^{(l)}}}\right)}^{\rm{H}}}{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}}\right\\}-\\\
\quad\quad\quad\quad\quad\quad\quad\quad{\left({{{\bm{s}}^{(l)}}}\right)^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{s}}^{(l)}}+\\\
\quad\quad\quad\quad\quad\quad\quad\quad{\left({{\bm{s}}-{{\bm{s}}^{(l)}}}\right)^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}})\left({{\bm{s}}-{{\bm{s}}^{(l)}}}\right)\\\
\end{array}.$ (65)

Given the fact that
${\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}})\succeq 0$,
we get the following inequality

$\begin{array}[]{l}{{\bm{s}}^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}\geq
2{\mathop{\rm
Re}\nolimits}\left\\{{{{\left({{{\bm{s}}^{(l)}}}\right)}^{\rm{H}}}{\bm{G}}{{({\bm{t}})}^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){\bm{s}}}\right\\}-\\\
\quad\quad\quad\quad\quad\quad\quad\quad{\left({{{\bm{s}}^{(l)}}}\right)^{\rm{H}}}{\bm{G}}{({\bm{t}})^{\rm{H}}}{\bm{R}}_{c}^{-1}{\bm{G}}({\bm{t}}){{\bm{s}}^{(l)}}\end{array},$
(66)

with equality if and only if $\bm{s}=\bm{s}^{(l)}$.

Consequently, ${{\tilde{z}}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})$ is a minorizer of
${z^{S}}({\bm{s}})$ at $\bm{s}^{(l)}$.

Thus we complete the proof of Proposition 2.

##  Appendix D Proof of Proposition 3

Let us start with

${\tilde{z}^{S}}({\bm{s}},{{\bm{s}}^{(l)}})=\mathop{{\rm{min}}}\limits_{\left\\{{{\bm{t}}|{{\left\|{{\bm{t}}-{{\bm{t}}_{0}}}\right\|}_{2}}\leq
r}\right\\}}{{\bm{t}}^{\rm{H}}}{\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}}){\bm{t}}$

described in (45). And the dual problem is given by

$\begin{array}[]{l}\mathop{\max}\limits_{\lambda,\gamma}\;\gamma\\\
s.t.\left[{\begin{array}[]{*{20}{c}}{{\bm{U}}({\bm{s}},{{\bm{s}}^{(l)}})+\lambda{\bf{I}}}&{\lambda{{\bm{t}}_{0}}}\\\
{\lambda{\bm{t}}_{0}^{\rm{H}}}&{\lambda{\bm{t}}_{0}^{\rm{H}}{{\bm{t}}_{0}}-\lambda{r^{2}}-\gamma}\end{array}}\right]\succeq
0\\\ \quad\lambda\geq 0\end{array}.$ (67)

Note that it is a single constraint quadratic problem and strong duality
holds, even though it is non-convex[31]. Therefore, we reformulate it as the
dual form and get $\hat{{\cal{P}}}_{r}^{S,(l)}$ by substituting the dual form
into ${\cal{P}}_{r}^{S,(l)}$.

Thus we complete the proof of Proposition 3.

## Acknowledgment

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