This work was supported in part by the National Natural Science Foundation of
China under Grants 62171450 and 61671453, and Anhui Provincial Natural Science
Foundation under Grant 2108085J30.

(Corresponding author: Bo Tang).

Da Li, Bo Tang, and Lei Xue are with the College of Electronic Engineering,
National University of Defense Technology, Hefei 230037, China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>eeixuelei@163.com).

# Multi-Spectrally Constrained Low-PAPR Waveform Optimization for MIMO Radar
Space-Time Adaptive Processing

Da Li  Bo Tang  Lei Xue  College of Electronic Engineering, National
University of Defense Technology, Hefei 230037, China

###### Abstract

This paper focuses on the joint design of transmit waveforms and receive
filters for airborne multiple-input-multiple-output (MIMO) radar systems in
spectrally crowded environments. The purpose is to maximize the output signal-
to-interference-plus-noise-ratio (SINR) in the presence of signal-dependent
clutter. To improve the practicability of the radar waveforms, both a multi-
spectral constraint and a peak-to-average-power ratio (PAPR) constraint are
imposed. A cyclic method is derived to iteratively optimize the transmit
waveforms and receive filters. In particular, to tackle the encountered non-
convex constrained fractional programming in designing the waveforms (for
fixed filters), we resort to the Dinkelbach’s transform, minorization-
maximization (MM), and leverage the alternating direction method of
multipliers (ADMM). We highlight that the proposed algorithm can iterate from
an infeasible initial point and the waveforms at convergence not only satisfy
the stringent constraints, but also attain superior performance.

MIMO radar, STAP, spectrally crowded environment, waveform optimization, SINR.

##  1 Introduction

Multiple-input-multiple-output (MIMO) radar refers to a radar system with
multiple transmitters and multiple receivers. Different from traditional
phased-array radar, MIMO radar can transmit multiple independent waveforms.
Therefore, MIMO radar can leverage the waveform diversity to improve the
signal-to-interference-plus-noise-ratio (SINR), operate in more flexible
modes, and adapt to the complex environment more intelligently [1]. According
to the array spacing between the transmitters/receivers, MIMO radar can be
categorized into two categories: statistical MIMO radar [2] and coherent MIMO
radar [3]. Statistical MIMO radar has widely separated transmitters/receivers.
Therefore, it can fully utilize the spatial diversity to overcome the target
fluctuations and improve the target localization accuracy [4]. Compared with
statistical MIMO radar, the transmitters/receivers of coherent MIMO radar are
closely spaced. Similar to phased-array radar systems, the transmitters of
coherent MIMO radar share the same viewing angle of the targets. Differently,
the waveform diversity offered by coherent MIMO radar enables a higher number
of degrees of freedom than phased-array radar, resulting in an improved
parameter identifiability [5], better target detection performance [6], and
the capability of supporting multiple functions simultaneously [7].

An airborne early warning (AEW) system (also called AEW and control system),
which refers to a radar system operating at a high altitude, is usually used
to detect target at a long range. When the AEW system is detecting targets at
a low altitude, it might receive strong reflections from, e.g., ground. Owing
to the AEW platform motion, the ground clutter is extended not only in range
and angle, but also in Doppler. Therefore, a weak target is likely to be
obscured by mainlobe clutter from the same angle as the target or by sidelobe
clutter from different angles but with the same Doppler frequency. These
unfavorable factors deteriorate the target detection performance, especially
for the slowly moving targets [8]. To boost the target detection performance
in the presence of strong clutter, space time adaptive processing (STAP)
techniques have been proposed [8, 9, 10]. Through collecting waveforms from
multiple antennas and multiple pulses, the adaptive multi-dimensional filters
of STAP can form deep notches along the clutter ridge and thus suppress the
clutter power to a low level.

Considering the superiority of MIMO radar and STAP, researchers proposed the
concept of MIMO-STAP for future AEW systems and extensive studies have been
devoted to this area (see, e.g., [11, 6, 12, 13, 14, 15] and the references
therein). The results showed that for detection of slowly-moving targets,
MIMO-STAP achieved better performance than conventional STAP methods. However,
these studies mainly focused on the design of receivers for MIMO-STAP
transmitting orthogonal waveforms. To further enhance the detection
performance, there have been ever-increasing interest in jointly optimizing
transmit waveforms and receive filters for MIMO-STAP [16, 17, 18, 19, 20, 21].
In [16, 19, 20], the authors considered the maximization of SINR under several
practical constraints on the sought waveforms, including the constant-envelope
constraint and the similarity constraint. A number of algorithms were
developed therein to tackle the joint design problems efficiently. In [18],
the authors extended the algorithm in [16] to design finite-alphabet
waveforms. In [17, 21], the authors focused on the robust design for MIMO-STAP
under circumstance of prior knowledge mismatch. It was shown that the
synthesized waveforms based on maximizing the worst-case SINR exhibited
increased robustness.

Note that an operating AEW system not only detects targets from hundreds miles
away, but also might communicates with friendly aircrafts/ships to perform
command and control. Therefore, if the radar and the communication systems
onboard share the same frequency band, they will interfere each other.
Moreover, in a spectrally crowded environment, in which the radar has to
operate with many nearby radiators simultaneously, the possibly severe mutual
interference will degrade the system performance significantly. One possible
way to improve the radar performance in spectrally crowded environments is by
transmitting intelligent waveforms [22]. In [23, 24, 25, 26, 27, 28, 29, 30],
the authors considered the waveform design under a spectral constraint. It was
shown that the spectrally constrained waveforms formed notches in the
stopbands (i.e., the frequency bands that the nearby radiators operate in),
thus enhancing the spectral compatibility of the radar system.

In this paper, we consider the joint design of transmit waveforms and receive
filters for MIMO-STAP of AEW systems in spectrally crowded environments.
Considering that multiple nearby radiators might be present and to guarantee
the quality of service of these radiators, we impose a multi-spectral
constraint on the waveforms. Moreover, to minimize the distortion due to the
nonlinear effects in high power amplifier, a peak-to-average-power ratio
(PAPR) constraint is imposed. We assume that the operating frequency band of
the nearby radiators are known _a priori_ (see also similar assumptions in
[23, 26, 27, 29, 30]). Indeed, such prior knowledge can be obtained by
cognitive methods in [31, 32, 33]. Motivated by [16, 18], we develop two
cyclic optimization methods to jointly design the waveforms and the filters.
For the challenging non-convex waveform design problem (for fixed filters), we
use Dinkelbach’s transform [34] to transform the fractional objective function
into a quadratic function. Then we resort to the coordinate-descent (CD)
method to split the quadratic problem into multiple subproblems, and use the
alternating direction method of multipliers (ADMM) to deal with the resulting
quadratically constrained quadratic programming (QCQP) problem (we call it the
DK-ADMM). Alternatively, we also use the minorization-maximization (MM)
technique to construct a quadratic surrogate of the objective, and leverage
the CD and ADMM to design the transmit waveforms (we call it MM-ADMM). We
highlight that the proposed iterative algorithm in this paper can start from
an infeasible point (i.e., a waveform not satisfying the constraints) and the
performance of the devised waveforms is insensitive to the initial points.
Moreover, the proposed algorithm can achieve better target detection
performance than the competing algorithms.

The rest of this paper is organized as follows: Section II establishes the
signal model and formulates the waveform design problem. Section III develops
a cyclic method to optimize the receive filters and transmit waveforms.
Section IV provides numerical examples to demonstrate the performance of the
proposed algorithm. Finally, conclusions are drawn in Section V.

_Notations_ : See Table 1.

Table 1: List of Notations Symbol | Meaning
---|---
${\boldsymbol{M}}$ | Matrix
${\boldsymbol{x}}$ | Vector
$x$ | Scalar
${\boldsymbol{I}}_{N}$ |  $N\times N$ identity matrix
$(\cdot)^{\ast}$, $(\cdot)^{\top}$, $(\cdot)^{\dagger}$ | Conjugate, transpose, conjugate transpose
$(\cdot)^{1/2}$ | Squared root of a positive semi-definite matrix
$\rm{tr}(\cdot)$ | Trace of a square matrix
$\left|\cdot\right|$, $\left\|\cdot\right\|_{2}$, $\left\|\cdot\right\|_{\textrm{F}}$ | Magnitude, Euclidian norm (of a vector), and
| Frobenius norm (of a matrix)
${\mathbb{E}}\\{\cdot\\}$ | Expectation of a random variable
${\mathbb{R}}$, ${\mathbb{C}}$ | Domain of the real and complex numbers
$\rm{vec}(\cdot)$ | Vectorization
${\boldsymbol{A}}\otimes{\boldsymbol{B}}$ | Kronecker product
$\rm{Re}(\cdot)$ | The real part of a (complex-valued) matrix
$\mathcal{U}(\cdot)$ | Uniform distribution
${\boldsymbol{A}}\succ{\boldsymbol{0}}$ $({\boldsymbol{A}}\succeq{\boldsymbol{0}})$ |  ${\boldsymbol{A}}$ is positive definite (semi-definite)

##  2 Signal Model and Problem Formulation

Figure 1: Geometry of an airborne MIMO STAP radar.

###  2.1 Signal Model

As shown in Figure 1, the considered AEW MIMO radar system has $N_{t}$
transmit antennas and $N_{r}$ receive antennas. Let
${\boldsymbol{s}}_{n}\in\mathbb{C}^{L}$ be the (discrete-time) baseband
waveform of the $n$th transmitter, where $L$ is the code length. Let
${\boldsymbol{S}}=[{\boldsymbol{s}}_{1},{\boldsymbol{s}}_{2},\cdots,{\boldsymbol{s}}_{N_{t}}]^{\top}\in\mathbb{C}^{N_{t}\times
L}$ denote the transmit waveform matrix. Assume that the airborne MIMO radar
system transmits a burst of $M$ pulses in a coherent processing interval (CPI)
with the pulse repetition frequency (PRF) denoted $f_{r}$. For a down-looking
airborne MIMO radar system, the received signal includes the target returns,
the signal-dependent clutter, and the receiver noise. Next we present the
signal model associated with these components (we refer to [16, 18] for more
details).

  1. 1.

Target
Assume that the transmit waveforms are narrowband. Under the far-field
assumption, the target return from the $m$th pulse ($m=1,2,...,M$) can be
expressed as

${\boldsymbol{Y}}_{t,m}=\alpha_{t}e^{j(m-1)w_{t}}{\boldsymbol{b}}(\theta_{t}){\boldsymbol{a}}^{\top}(\theta_{t}){\boldsymbol{S}},$
(1)

where $\alpha_{t}$ is the target amplitude, $w_{t}=2\pi f_{t}$, $f_{t}$ is the
normalized target Doppler frequency, $\theta_{t}$ is the target direction of
arrival (DOA), ${\boldsymbol{a}}(\theta_{t})$ and
${\boldsymbol{b}}(\theta_{t})$ are the transmit array steering vector and the
receive array steering vector at $\theta_{t}$, respectively. Let
${\boldsymbol{y}}_{t,m}=\textrm{vec}({\boldsymbol{Y}}_{t,m})$,
${\boldsymbol{s}}=\textrm{vec}({\boldsymbol{S}})$, and
${\boldsymbol{A}}(\theta_{t})={\boldsymbol{b}}(\theta_{t}){\boldsymbol{a}}^{\top}(\theta_{t})$.
Then

${\boldsymbol{y}}_{t,m}=\alpha_{t}e^{j(m-1)w_{t}}({\boldsymbol{I}}_{L}\otimes{\boldsymbol{A}}(\theta_{t})){\boldsymbol{s}}.$
(2)

Let
${\boldsymbol{y}}_{t}=[{\boldsymbol{y}}_{t,1}^{\top},\cdots,{\boldsymbol{y}}_{t,M}^{\top}]^{\top}\in\mathbb{C}^{LMN_{r}}$.
Then ${\boldsymbol{y}}_{t}$ can be expressed as

${\boldsymbol{y}}_{t}=\alpha{\boldsymbol{V}}(w_{t},\theta_{t}){\boldsymbol{s}},$
(3)

where
${\boldsymbol{V}}(w_{t},\theta_{t})={\boldsymbol{d}}(w_{t})\otimes{\boldsymbol{I}}_{L}\otimes{\boldsymbol{A}}(\theta_{t})$
with ${\boldsymbol{d}}(w_{t})=[1,\cdots,e^{j(M-1)w_{t}}]^{\top}$ being the
temporal steering vector at the Doppler frequency $f_{t}$.

  2. 2.

Clutter
The clutter refers to signal-dependent interference due to unwanted
reflections, e.g., from ground, sea, etc. The clutter can be much stronger
than the target echoes, due to the large number of clutter patches in the iso-
range rings (including the range ring that the target is present and the
neighborhood range rings), as shown in Figure 1. Additionally, the clutter is
distributed in Doppler domain owing to the motion of AEW platform [8]. Assume
that there are $2P+1$ clutter rings under consideration, and we split each
clutter ring into $N_{c}$ clutter patches uniformly. Assume that the target is
at the $r$th range cell, the clutter associated with the $m$th pulse, the
$(r+p)$th range cell, and the $k$th patch in azimuth, can be modeled by

$\displaystyle{\boldsymbol{Y}}_{c,m,p,k}=$
$\displaystyle\alpha_{c,p,k}e^{j2\pi(m-1)f_{c,p,k}T_{r}}$
$\displaystyle\times{\boldsymbol{b}}(\theta_{c,p,k}){\boldsymbol{a}}^{\top}(\theta_{c,p,k}){\boldsymbol{S}}{\boldsymbol{J}}_{p},$
(4)

where $\alpha_{c,p,k}$, $f_{c,p,k}$, $\theta_{c,p,k}$ are the amplitude, the
Doppler frequency, and the DOA of the $k$th clutter patch in the $(r+p)$th
range cell, respectively,
${\boldsymbol{J}}_{p}={\boldsymbol{J}}_{-p}^{\top}\in\mathbb{C}^{L\times L}$
is the shift matrix expressed as

${\boldsymbol{J}}_{p}(m,n)=\begin{cases}1,\textrm{if}\ m-n+p=0,\\\
0,\textrm{if}\ m-n+p\neq 0.\end{cases}$ (5)

Let
${\boldsymbol{y}}_{c,p,k}=[\textrm{vec}^{\top}({\boldsymbol{Y}}_{c,1,p,k}^{\top}),\cdots,\textrm{vec}^{\top}({\boldsymbol{Y}}_{c,M,p,k}^{\top})]^{\top}$,
then the $k$th clutter patch in the $(r+p)$th range cell can be expressed as

${\boldsymbol{y}}_{c,p,k}=\alpha_{c,p,k}{\boldsymbol{V}}(w_{c,p,k},\theta_{c,p,k}){\boldsymbol{s}},$
(6)

where
${\boldsymbol{V}}(w_{c,p,k},\theta_{c,p,k})={\boldsymbol{d}}(w_{c,p,k})\otimes{\boldsymbol{J}}_{p}^{\top}\otimes{\boldsymbol{A}}(\theta_{c,p,k})$,
and $w_{c,p,k}=2\pi f_{c,p,k}$. By considering the clutter from the nearest
$2P+1$ range cells, the clutter model can be established by

${\boldsymbol{y}}_{c}=\sum_{p=-P}^{P}\sum_{k=1}^{N_{c}}{\boldsymbol{y}}_{c,p,k}.$
(7)

Assume that the signals associated with different clutter patches are
uncorrelated. Then the clutter covariance matrix, defined by
${\boldsymbol{R}}_{c}({\boldsymbol{s}})=\mathbb{E}({\boldsymbol{y}}_{c}{\boldsymbol{y}}_{c}^{\dagger})$,
can be expressed as

${\boldsymbol{R}}_{c}({\boldsymbol{s}})=\sum_{p=-P}^{P}\sum_{k=1}^{N_{c}}\sigma_{c,p,k}^{2}{\boldsymbol{v}}_{c,p,k}({\boldsymbol{s}}){\boldsymbol{v}}_{c,p,k}^{\dagger}({\boldsymbol{s}}),$
(8)

where $\sigma_{c,p,k}^{2}=\mathbb{E}(|\alpha_{c,p,k}|^{2})$ denotes the
average power of the $k$th clutter patch in the $p$th range ring, and
${\boldsymbol{v}}_{c,p,k}({\boldsymbol{s}})={\boldsymbol{V}}(w_{c,p,k},\theta_{c,p,k}){\boldsymbol{s}}$.

  3. 3.

Noise
Assume that the receiver noise is white, with power of $\sigma^{2}$. Then the
noise covariance matrix can be written as:

${\boldsymbol{R}}_{\textrm{u}}=\mathbb{E}({\boldsymbol{y}}_{\textrm{u}}{\boldsymbol{y}}_{\textrm{u}}^{\dagger})=\sigma^{2}{\boldsymbol{I}}_{LMN_{r}},$
(9)

where ${\boldsymbol{y}}_{\textrm{u}}$ is the vector of receiver noise.

###  2.2 Design Metric

In radar systems, the target detection performance is closely related to the
SINR. Through maximizing the output SINR, the clutter can be suppressed to a
low level and then the detection performance is improved. In this paper, we
aim to maximize the output SINR through jointly designing the transmit
waveforms and the receive filters. Let
${\boldsymbol{w}}=[{\boldsymbol{w}}_{1}^{\top},\cdots,{\boldsymbol{w}}_{N_{r}}^{\top}]^{\top}$
denote the receive filter, with ${\boldsymbol{w}}_{j}\in\mathbb{C}^{ML}$
representing the filter in the $j$th receiver, $j=1,\cdots,N_{r}$. The output
SINR of the MIMO-STAP radar is defined as follows

$\displaystyle\textrm{SINR}({\boldsymbol{w}},{\boldsymbol{s}})$
$\displaystyle=\frac{|{\boldsymbol{w}}^{\dagger}{\boldsymbol{y}}_{t}|^{2}}{{\boldsymbol{w}}^{\dagger}\mathbb{E}({\boldsymbol{y}}_{c}{\boldsymbol{y}}^{\dagger}_{c}+{\boldsymbol{y}}_{\textrm{u}}{\boldsymbol{y}}_{\textrm{u}}^{\dagger}){\boldsymbol{w}}}$
(10)
$\displaystyle=\frac{|\alpha_{t}|^{2}|{\boldsymbol{w}}^{\dagger}{\boldsymbol{v}}_{t}({\boldsymbol{s}})|^{2}}{{\boldsymbol{w}}^{\dagger}{\boldsymbol{R}}_{v}({\boldsymbol{s}}){\boldsymbol{w}}},$

where
${\boldsymbol{v}}_{t}({\boldsymbol{s}})={\boldsymbol{V}}(w_{t},\theta_{t}){\boldsymbol{s}}$,
and
${\boldsymbol{R}}_{v}({\boldsymbol{s}})={\boldsymbol{R}}_{c}({\boldsymbol{s}})+{\boldsymbol{R}}_{\textrm{u}}$.

###  2.3 Transmit Waveform Constraints

Now we briefly discuss the constraints that the transmit waveforms should
satisfy.

  1. 1.

Energy Constraint
Since the energy of transmit waveforms is limited, the energy constraint is
enforced on the sought waveforms:

$\textrm{tr}({\boldsymbol{S}}{\boldsymbol{S}}^{\dagger})=e_{t},$ (11)

where $e_{t}$ is the total transmit energy. Note that
${\boldsymbol{s}}=\textrm{vec}({\boldsymbol{S}})$. Then we can rewrite the
energy constraint as follows

${\boldsymbol{s}}^{\dagger}{\boldsymbol{s}}=e_{t}.$ (12)

Note also that practical radar systems use almost identical radio frequency
amplifiers (RFA), meaning that the transmit energies across different antennas
are usually uniform [35]. Thus, the following uniform transmit energy
constraint is included:

${\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},n=1,\cdots,N_{t}.$
(13)

  2. 2.

PAPR Constraint
To allow the RFA to operate in a saturated condition as well as avoid
nonlinear effects, transmit waveform with low PAPR are desirable [36, 37].
Therefore, we also impose the PAPR constraint on the waveforms, that is,

${\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\
\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,$ (14)

where $1\leq\rho\leq L$, $n=1,\cdots,N_{t}$, and

$\textrm{PAPR}({\boldsymbol{s}}_{n})=\frac{\textrm{max}_{l}|s_{n}(l)|^{2}}{\frac{1}{L}\sum_{l=1}^{L}|s_{n}(l)|^{2}},\
l=1,\cdots,L.$

Particularly, if $\rho=1$, the PAPR constraint is reduced to the constant-
envelope constraint:

$|s_{n}(l)|=\sqrt{p_{s}},\ n=1,\cdots,N_{t},\ l=1,\cdots,L,$

where $p_{s}=e_{t}/({LN_{t}})$.

  3. 3.

Multi-Spectral Constraint
Owing to the massive increase in the number of radio devices and the limited
spectrum resources, radar systems may have to share the frequency band with
communication systems, which will cause mutual interference and deteriorate
the performance of both systems. To improve the spetral compatibility, one
possible way is to control the radar transmit waveforms to form notches in the
stopbands (i.e., minimize the energy spectral density (ESD) of radar transmit
waveforms in the working frequency bands of communication systems). In this
respect, assume that $K_{rad}$ licensed radiators are coexisting with the MIMO
radar system. Let $\Omega_{k}=[f_{1}^{k},f_{2}^{k}]$ denote the normalized
frequency band of the $k$th radiator, where $f_{1}^{k}$ and $f_{2}^{k}$
indicate the lower and the upper normalized frequencies, $k=1,\cdots,K_{rad}$.
Note that the ESD of the $n$th waveform is written as

$S_{n}(f)=|{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{a}}(f)|^{2},$ (15)

where ${\boldsymbol{a}}(f)=[1,e^{j2\pi f},\cdots,e^{j2\pi(L-1)f}]^{\top}$.
Therefore, the energy of ${\boldsymbol{s}}_{n}$ leaked on the $k$th stopband
can be expressed as

$\int_{f_{1}^{k}}^{f_{2}^{k}}S_{n}(f)df=\int_{f_{1}^{k}}^{f_{2}^{k}}|{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{a}}(f)|^{2}df={\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n},$

where the $(m,l)$th element of ${\boldsymbol{R}}_{I}^{k}$ is given by

${\boldsymbol{R}}_{I}^{k}(m,l)=\begin{cases}f_{2}^{k}-f_{1}^{k},&m=l,\\\
\frac{e^{j2\pi f_{2}^{k}(m-l)}-e^{j2\pi f_{1}^{k}(m-l)}}{j2\pi(m-l)},&m\neq
l.\end{cases}$

To enhance the spectral compatibility of the radar signals with the licensed
radiators, the following spectral constraint is enforced on the transmit
waveforms, which is given by

${\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},$ (16)

where $E_{I}^{k}$ denotes the maximum allowed interference energy of
${\boldsymbol{s}}_{n}$ on the $k$th frequency band
($n=1,\cdots,N_{t},k=1,\cdots,K_{rad}$). Note that when the constraint in (16)
is satisfied, we can precisely control the interference energy of each
waveform on every frequency band, meaning that it is possible to ensure the
quality of service for each licensed radiator. In the sequel, similar to [38,
28, 27, 30], we call the constraint in (16) a multi-spectral constraint 111We
point out that the multi-spectral constraint is enforced on multiple
waveforms, whereas the studies in [38, 28, 27, 30] enforce the multi-spectral
constraint on a single waveform..

###  2.4 Problem Formulation

By considering the constraints in (12), (14), and (16), we formulate the
following joint design problem to maximize the output SINR of MIMO radar in
spectrally crowded environments:

$\mathcal{P}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{w}},{\boldsymbol{s}}}\
&\textrm{SINR}({\boldsymbol{w}},{\boldsymbol{s}})\\\ \textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},\\\ &n=1,\cdots,N_{t},\ k=1,\cdots,K_{rad}.\end{aligned}\end{cases}$
(17)

Note that $\mathcal{P}$ is in general a non-convex problem, due to the PAPR
constraint. In the next section, we develop a cyclic method to provide high-
quality solutions to the above waveform design problem.

_Remark_ : In the formulation of (17), we have assumed that the prior
knowledge of the interference characteristics and the operating frequency
bands of the licensed radiators are available. Indeed, these knowledge can be
obtained via cognitive methods (see, e.g., [39, 40, 33, 31, 32, 41] for more
details for the application of cognitive methods in radar systems). We also
highlight that if the clutter is non-stationary (e.g., due to internal clutter
motion), we will assume that the normalized Doppler frequency of the $k$th
clutter patch in the $p$th range ring (i.e., $f_{c,p,k}$) is uniformly
distributed around the mean$\bar{f}_{c,p,k}$, that is,

$f_{c,p,k}\sim\mathcal{U}(\bar{f}_{c,p,k}-\delta_{c,p,k}/2,\bar{f}_{c,p,k}+\delta_{c,p,k}/2),$
(18)

where $\delta_{c,p,k}$ rules the uncertainty of clutter Doppler frequency. In
this case, the clutter covariance matrix
${\boldsymbol{R}}_{c}({\boldsymbol{s}})$ can calculated by the method in [40,
42].

##  3 Algorithm Design

In this section, we develop cyclic optimization methods to tackle the non-
convex problem in (17). For each cyclic optimization method, two sub-problems
are involved at the $(t+1)$th iteration: the optimization of receive filters
for fixed transmit waveforms (i.e., ${\boldsymbol{s}}^{(t)}$ is fixed) and the
optimization of transmit waveforms for fixed receive filters (i.e.,
${\boldsymbol{w}}^{(t+1)}$ is fixed). Next we present solutions to the two
subproblems. To lighten the notations, we omit the superscripts if doing so
does not have a risk of confusion.

If ${\boldsymbol{s}}^{(t)}$ is fixed, the receive filters can be optimized by
solving the following maximization problem:

$\max\limits_{{\boldsymbol{w}}}\
\frac{|{\boldsymbol{w}}^{\dagger}{\boldsymbol{v}}_{t}({\boldsymbol{s}})|^{2}}{{\boldsymbol{w}}^{\dagger}{\boldsymbol{R}}_{v}({\boldsymbol{s}}){\boldsymbol{w}}}.$
(19)

It can be seen that the minimum variance distortionless response (MVDR)
beamformer [43] maximizes the objective, i.e., the solution is given by

${\boldsymbol{w}}={\boldsymbol{R}}_{v}^{-1}({\boldsymbol{s}}){\boldsymbol{v}}_{t}({\boldsymbol{s}}).$
(20)

To optimize ${\boldsymbol{s}}_{n},n=1,\cdots,N_{t}$ (for fixed
${\boldsymbol{w}}^{(t+1)}$), we note that the SINR can be expressed as

$\displaystyle\textrm{SINR}({\boldsymbol{w}},{\boldsymbol{s}})$
$\displaystyle=\frac{|\alpha_{t}|^{2}|{\boldsymbol{w}}^{\dagger}{\boldsymbol{V}}(w_{t},\theta_{t}){\boldsymbol{s}}|^{2}}{{\boldsymbol{w}}^{\dagger}{\boldsymbol{R}}_{c}({\boldsymbol{s}}){\boldsymbol{w}}+{\boldsymbol{w}}^{\dagger}{\boldsymbol{R}}_{\textrm{u}}{\boldsymbol{w}}}.$
(21)

In addition,

${\boldsymbol{w}}^{\dagger}{\boldsymbol{R}}_{c}({\boldsymbol{s}}){\boldsymbol{w}}={\boldsymbol{s}}^{\dagger}{\boldsymbol{Q}}{\boldsymbol{s}},$
(22)

where

${\boldsymbol{Q}}=\sum_{p=-P}^{P}\sum_{k=1}^{N_{c}}\sigma_{c,p,k}^{2}{\boldsymbol{V}}_{c,p,k}^{\dagger}{\boldsymbol{w}}{\boldsymbol{w}}^{\dagger}{\boldsymbol{V}}_{c,p,k},$
(23)

and
${\boldsymbol{V}}_{c,p,k}\triangleq{\boldsymbol{V}}(w_{c,p,k},\theta_{c,p,k})$.

Let

${\boldsymbol{D}}={\boldsymbol{V}}^{\dagger}(w_{t},\theta_{t}){\boldsymbol{w}}{\boldsymbol{w}}^{\dagger}{\boldsymbol{V}}(w_{t},\theta_{t}).$
(24)

Then, SINR can be expressed as

$\textrm{SINR}({\boldsymbol{w}},{\boldsymbol{s}})=|\alpha_{t}|^{2}\frac{{\boldsymbol{s}}^{\dagger}{\boldsymbol{D}}{\boldsymbol{s}}}{{\boldsymbol{s}}^{\dagger}{\boldsymbol{Q}}{\boldsymbol{s}}+\beta({\boldsymbol{w}})},$
(25)

where
$\beta({\boldsymbol{w}})={\boldsymbol{w}}^{\dagger}{\boldsymbol{R}}_{\textrm{u}}{\boldsymbol{w}}$.

Therefore, the optimization of the multiple transmit waveforms (given
${\boldsymbol{w}}^{(t+1)}$) can be given by

$\mathcal{P}_{{\boldsymbol{s}}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}}\
&\frac{{\boldsymbol{s}}^{\dagger}{\boldsymbol{D}}{\boldsymbol{s}}}{{\boldsymbol{s}}^{\dagger}{\boldsymbol{Q}}{\boldsymbol{s}}+\beta({\boldsymbol{w}})}\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},\\\ &n=1,\cdots,N_{t},k=1,\cdots,K_{rad}.\end{aligned}\end{cases}$
(26)

Note that $\mathcal{P}_{{\boldsymbol{s}}}$ is a fractional programming
problem. Next we resort to the Dinkelbach’s transform [34] and MM to replace
the fractional objective with a quadratic surrogate, respectively. Then, with
the quadratic surrogate function, we propose an ADMM algorithm to tackle the
non-convex QCQP problem. The corresponding algorithms are referred to as DK-
ADMM and MM-ADMM, respectively.

###  3.1 DK-ADMM

Let ${\boldsymbol{s}}^{(t,l)}$ denote the waveform in the $(t,l)$th iteration
of the proposed algorithm, where the superscript $t$ denotes the outer
iteration for the cyclic optimization, and $l$ denotes the inner iteration for
Dinkelbach’s transform. Let $f^{(t,l)}$ denote the SINR associated with
${\boldsymbol{s}}^{(t,l)}$. By applying the Dinkelbach’s transform, we
formulate the following optimization problem at the $(t,l+1)$th iteration

$\hat{\mathcal{P}}_{{\boldsymbol{s}}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}}\
&{\boldsymbol{s}}^{\dagger}\hat{{\boldsymbol{T}}}{\boldsymbol{s}}\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},\\\ &n=1,\cdots,N_{t},k=1,\cdots,K_{rad}.\end{aligned}\end{cases}$
(27)

where $\hat{{\boldsymbol{T}}}={\boldsymbol{T}}+\eta{\boldsymbol{I}}$,

${\boldsymbol{T}}={\boldsymbol{D}}-f^{(t,l)}({\boldsymbol{Q}}+\beta({\boldsymbol{w}})/e_{t}\cdot{\boldsymbol{I}}),$
(28)

and $\eta$ is a constant to ensure
$\hat{{\boldsymbol{T}}}\succeq{\boldsymbol{0}}$.

Next we use the block coordinate descent (CD) method to deal with the
optimization problem $\hat{\mathcal{P}}_{{\boldsymbol{s}}}$ (We refer to [44]
for a comprehensive review of the CD method). To apply the CD method, we
define $\bar{{\boldsymbol{s}}}=\textrm{vec}({\boldsymbol{S}}^{\top})$. Note
that ${\boldsymbol{s}}={\boldsymbol{P}}\bar{{\boldsymbol{s}}}$ [16], where
${\boldsymbol{P}}$ is a commutation matrix. Therefore, the objective function
of $\hat{\mathcal{P}}_{{\boldsymbol{s}}}$ can be rewritten as

${\boldsymbol{s}}^{\dagger}\hat{{\boldsymbol{T}}}{\boldsymbol{s}}=\bar{{\boldsymbol{s}}}^{\dagger}\bar{{\boldsymbol{T}}}\bar{{\boldsymbol{s}}},$
(29)

where
$\bar{{\boldsymbol{T}}}={\boldsymbol{P}}^{\dagger}\hat{{\boldsymbol{T}}}{\boldsymbol{P}}$.
Next, let us partition $\bar{{\boldsymbol{T}}}$ into $N_{t}\times N_{t}$
blocks, each of which is an $L\times L$ matrix. Let
$\bar{{\boldsymbol{T}}}_{n,m}$ denote the $(n,m)$th block of
$\bar{{\boldsymbol{T}}}$. Then
$\bar{{\boldsymbol{s}}}^{\dagger}\bar{{\boldsymbol{T}}}\bar{{\boldsymbol{s}}}$
can be rewritten as

$\bar{{\boldsymbol{s}}}^{\dagger}\bar{{\boldsymbol{T}}}\bar{{\boldsymbol{s}}}={\boldsymbol{s}}_{n}^{\dagger}\bar{{\boldsymbol{T}}}_{n,n}{\boldsymbol{s}}_{n}+2\textrm{Re}({\boldsymbol{s}}_{n}^{\dagger}\sum_{\begin{subarray}{c}m=1\\\
m\neq
n\end{subarray}}^{N_{t}}\bar{{\boldsymbol{T}}}_{n,m}{\boldsymbol{s}}_{m})+const_{0},$
(30)

where

$const_{0}=\sum_{\begin{subarray}{c}m=1\\\ m\neq
n\end{subarray}}^{N_{t}}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
n\end{subarray}}^{N_{t}}{\boldsymbol{s}}_{m}^{\dagger}\bar{{\boldsymbol{T}}}_{m,m^{\prime}}{\boldsymbol{s}}_{m^{\prime}}.$
(31)

Based on the observation in (30), we formulate the following problem to
optimize ${\boldsymbol{s}}_{n}$:

$\mathcal{P}_{s_{n}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}_{n}}\
&{\boldsymbol{s}}_{n}^{\dagger}\bar{{\boldsymbol{T}}}_{n,n}{\boldsymbol{s}}_{n}+2\textrm{Re}({\boldsymbol{b}}_{n}^{\dagger}{\boldsymbol{s}}_{n})\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},\ k=1,\cdots,K_{rad},\end{aligned}\end{cases}$ (32)

where

${\boldsymbol{b}}_{n}=\sum_{\begin{subarray}{c}m=1\\\ m\neq
n\end{subarray}}^{N_{t}}\bar{{\boldsymbol{T}}}_{n,m}{\boldsymbol{s}}_{m}.$
(33)

Next we use the ADMM method to deal with the optimization problem
$\mathcal{P}_{s_{n}}$ (we refer to [45] for a tutorial review of the ADMM
method). To proceed, we reformulate $\mathcal{P}_{s_{n}}$ as

$\mathcal{P}_{{\boldsymbol{s}}_{n},t,{\boldsymbol{g}}_{k},{\boldsymbol{z}}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}_{n},t,{\boldsymbol{g}}_{k},{\boldsymbol{z}}}\
&t+2\textrm{Re}({\boldsymbol{b}}_{n}^{\dagger}{\boldsymbol{s}}_{n})\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{g}}_{k}={\boldsymbol{B}}_{k}^{1/2}{\boldsymbol{s}}_{n},\\\
&\|{\boldsymbol{g}}_{k}\|^{2}\leq 1,\ k=1,\cdots,K_{rad},\\\
&{\boldsymbol{z}}=\bar{{\boldsymbol{T}}}_{n,n}^{1/2}{\boldsymbol{s}}_{n},\|{\boldsymbol{z}}\|^{2}\geq
t,\\\ \end{aligned}\end{cases}$ (34)

where $t$, ${\boldsymbol{g}}_{k}$, and ${\boldsymbol{z}}$ are the introduced
auxiliary variables, and
${\boldsymbol{B}}_{k}={\boldsymbol{R}}_{I}^{k}/E_{I}^{k}$. The augmented
Lagrangian function corresponding to
$\mathcal{P}_{{\boldsymbol{s}}_{n},t,{\boldsymbol{g}}_{k},{\boldsymbol{z}}}$
can be expressed as

$\displaystyle
L_{\vartheta}({\boldsymbol{s}}_{n},{\boldsymbol{z}},t,{\boldsymbol{g}}_{k},{\boldsymbol{c}}_{k},{\boldsymbol{d}})$
(35) $\displaystyle=$
$\displaystyle-t-2\textrm{Re}({\boldsymbol{b}}_{n}^{\dagger}{\boldsymbol{s}}_{n})$
$\displaystyle+\frac{\vartheta}{2}\left\\{\sum_{k=1}^{K_{rad}}\left(||{\boldsymbol{g}}_{k}-{\boldsymbol{B}}_{k}^{1/2}{\boldsymbol{s}}_{n}+{\boldsymbol{c}}_{k}||^{2}-||{\boldsymbol{c}}_{k}||^{2}\right)\right\\}$
$\displaystyle+\frac{\vartheta}{2}\left\\{||{\boldsymbol{z}}-\bar{{\boldsymbol{T}}}_{n,n}^{1/2}{\boldsymbol{s}}_{n}+{\boldsymbol{d}}||^{2}-||{\boldsymbol{d}}||^{2}\right\\},$

where $\vartheta$ is the penalty parameter,
${\boldsymbol{c}}_{k}(k=1,2,\cdots,K_{rad})$ and ${\boldsymbol{d}}$ are the
Lagrange multiplier vectors. Then, during the $(m+1)$th iteration of the ADMM
method, we carry out the following steps in (36), shown at the bottom of this
page:

Next we present solutions to (36a), (36b), and (36c).

$\displaystyle{\boldsymbol{s}}_{n}^{(m+1)}$
$\displaystyle=\arg\min\limits_{{\boldsymbol{s}}_{n}}\
L_{\vartheta}({\boldsymbol{s}}_{n},{\boldsymbol{z}}^{(m)},t^{(m)},{\boldsymbol{g}}_{k}^{(m)},{\boldsymbol{c}}_{k}^{(m)},{\boldsymbol{d}}^{(m)}),$
(36a) $\displaystyle({\boldsymbol{z}}^{(m+1)},t^{(m+1)})$
$\displaystyle=\arg\min\limits_{{\boldsymbol{z}},t}\
L_{\vartheta}({\boldsymbol{s}}_{n}^{(m+1)},{\boldsymbol{z}},t,{\boldsymbol{g}}_{k}^{(m)},{\boldsymbol{c}}_{k}^{(m)},{\boldsymbol{d}}^{(m)}),$
(36b) $\displaystyle{\boldsymbol{g}}_{k}^{(m+1)}$
$\displaystyle=\arg\min_{{\boldsymbol{g}}_{k}}\
L_{\vartheta}({\boldsymbol{s}}_{n}^{(m+1)},{\boldsymbol{z}}^{(m+1)},t^{(m+1)},{\boldsymbol{g}}_{k},{\boldsymbol{c}}_{k}^{(m)},{\boldsymbol{d}}^{(m)}),$
(36c) $\displaystyle{\boldsymbol{c}}_{k}^{(m+1)}$
$\displaystyle={\boldsymbol{c}}_{k}^{(m)}+{\boldsymbol{g}}_{k}^{(m+1)}-{\boldsymbol{B}}_{k}^{1/2}{\boldsymbol{s}}_{n}^{(m+1)},$
(36d) $\displaystyle{\boldsymbol{d}}^{(m+1)}$
$\displaystyle={\boldsymbol{d}}^{(m)}+{\boldsymbol{z}}^{(m+1)}-\bar{{\boldsymbol{T}}}_{n,n}^{1/2}{\boldsymbol{s}}_{n}^{(m+1)},$
(36e)

1) Update of ${\boldsymbol{s}}_{n}^{(m+1)}$
Define

${\boldsymbol{Y}}_{n}=-\frac{\vartheta}{2}(\bar{{\boldsymbol{T}}}_{n,n}+\sum_{k=1}^{K_{rad}}{\boldsymbol{B}}_{k}),$
(37)

and

${\boldsymbol{h}}=\frac{\vartheta}{2}(\bar{{\boldsymbol{T}}}_{n,n}^{1/2}({\boldsymbol{z}}+{\boldsymbol{d}})+\sum_{k=1}^{K_{rad}}{\boldsymbol{B}}_{k}^{1/2}({\boldsymbol{g}}_{k}+{\boldsymbol{c}}_{k})).$
(38)

Let ${\boldsymbol{v}}={\boldsymbol{h}}+{\boldsymbol{b}}$. Then the update of
${\boldsymbol{s}}_{n}^{(m+1)}$ can be given by

$\mathcal{P}_{{\boldsymbol{s}}_{n}}^{(m+1)}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}_{n}}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{Y}}_{n}{\boldsymbol{s}}_{n}+2\textrm{Re}({\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{v}})\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho.\\\ \end{aligned}\end{cases}$
(39)

We can tackle the maximization problem
$\mathcal{P}_{{\boldsymbol{s}}_{n}}^{(m+1)}$ leveraging the MM method [46]. To
proceed, note that

$({\boldsymbol{s}}_{n}-{\boldsymbol{s}}_{n}^{(m,j)})^{\dagger}({\boldsymbol{Y}}_{n}-\lambda_{\min}({\boldsymbol{Y}}_{n}){\boldsymbol{I}})({\boldsymbol{s}}_{n}-{\boldsymbol{s}}_{n}^{(m,j)})\geq
0,$ (40)

where ${\boldsymbol{s}}_{n}^{(m,j)}$ is the waveform at the $(m,j)$th
iteration, and $\lambda_{\min}({\boldsymbol{Y}}_{n})$ is the smallest
eigenvalue of ${\boldsymbol{Y}}_{n}$. We can derive from (40) that

${\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{Y}}_{n}{\boldsymbol{s}}_{n}\geq
2\textrm{Re}({\boldsymbol{s}}_{n}^{\dagger}({\boldsymbol{Y}}_{n}-\lambda_{\min}({\boldsymbol{Y}}_{n}){\boldsymbol{I}}){\boldsymbol{s}}_{n}^{(m,j)})+const_{1},$
(41)

where
$const_{1}=-({\boldsymbol{s}}_{n}^{(m,j)})^{\dagger}{\boldsymbol{Y}}_{n}{\boldsymbol{s}}_{n}^{(m,j)}+2\lambda_{\min}({\boldsymbol{Y}}_{n})e_{t}/N_{t}$.
Let

${\boldsymbol{u}}^{(m,j)}=({\boldsymbol{Y}}_{n}-\lambda_{\min}({\boldsymbol{Y}}_{n}){\boldsymbol{I}}){\boldsymbol{s}}_{n}^{(m,j)}+{\boldsymbol{v}},$
(42)

then the minorized problem based on (41) at the $(m,j+1)$th iteration can be
formulated as

$\displaystyle\max\limits_{{\boldsymbol{s}}_{n}}$
$\displaystyle\textrm{Re}({\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{u}}^{(m,j)})$
(43) s.t.
$\displaystyle{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},$
$\displaystyle\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho.$

In [47], an algorithm is provided to solve the above problem. Particularly, if
$\rho=1$, this problem has a closed-form solution

$s_{n}^{(m,j+1)}(l)=\sqrt{p_{s}}\textrm{exp}(j\textrm{arg}(u^{(m,j)}(l))),$
(44)

where $s_{n}^{(m,j+1)}(l)$ and $u^{(m,j)}(l)$ denote the $l$th element of
${\boldsymbol{s}}_{n}^{(m,j+1)}$ and ${\boldsymbol{u}}^{(m,j)}$, respectively.

2) Update of ${\boldsymbol{z}}^{(m+1)}$ and $t^{(m+1)}$
Let
${\boldsymbol{q}}=\bar{{\boldsymbol{T}}}_{n,n}^{1/2}{\boldsymbol{s}}_{n}-{\boldsymbol{d}}$,
then the update of ${\boldsymbol{z}}^{(m+1)}$ and $t^{(m+1)}$ can be given by

$\mathcal{P}_{{\boldsymbol{z}},t}^{(m+1)}\begin{cases}\begin{aligned}
\min\limits_{{\boldsymbol{z}},t}\
&\frac{\vartheta}{2}||{\boldsymbol{z}}-{\boldsymbol{q}}||^{2}-t\\\
\textrm{s.t.}\ &||{\boldsymbol{z}}||^{2}\geq t.\end{aligned}\end{cases}$ (45)

It is evident that if $t=\|{\boldsymbol{z}}\|^{2}$, the objective function
achieves the smallest value. As a result, we can obtain the solution to
$\mathcal{P}_{{\boldsymbol{z}},t}^{(m+1)}$ through solving the following
unconstrained optimization:

$\displaystyle\min_{{\boldsymbol{z}}}\frac{\vartheta}{2}\|{\boldsymbol{z}}-{\boldsymbol{q}}\|^{2}-||{\boldsymbol{z}}||^{2}.$
(46)

Assume that $\vartheta>2$. Then the optimal solution to ${\boldsymbol{z}}$ is
shown as follows

${\boldsymbol{z}}=\frac{\vartheta{\boldsymbol{q}}}{\vartheta-2}.$ (47)

3) Update of ${\boldsymbol{g}}_{k}^{(m+1)}$
Let
${\boldsymbol{x}}_{k}={\boldsymbol{B}}_{k}^{1/2}{\boldsymbol{s}}_{n}-{\boldsymbol{c}}_{k}$.
Then the update of ${\boldsymbol{g}}_{k}$ can be given by

$\mathcal{P}_{{\boldsymbol{g}}_{k}}^{(m+1)}\begin{cases}\begin{aligned}
\min\limits_{{\boldsymbol{g}}_{k}}\
&||{\boldsymbol{g}}_{k}-{\boldsymbol{x}}_{k}||^{2},\\\ \textrm{s.t.}\
&||{\boldsymbol{g}}_{k}||^{2}\leq 1.\end{aligned}\end{cases}$ (48)

Obviously, the solution to $\mathcal{P}_{{\boldsymbol{g}}_{k}}^{(m+1)}$ is
given by

${\boldsymbol{g}}_{k}=\begin{cases}\begin{aligned} &{\boldsymbol{x}}_{k},\
&||{\boldsymbol{x}}_{k}||^{2}\leq 1,\\\
&{\boldsymbol{x}}_{k}/||{\boldsymbol{x}}_{k}||,\
&||{\boldsymbol{x}}_{k}||^{2}>1.\\\ \end{aligned}\end{cases}$ (49)

We sum up the proposed ADMM algorithm in Algorithm 1, where the algorithm
terminates if $\|{\boldsymbol{r}}^{(m+1)}\|<\xi$ or the algorithm reaches a
maximum number of iterations, $\xi>0$ is a small user-defined value, and

$\displaystyle{\boldsymbol{r}}^{(m+1)}=$
$\displaystyle{\boldsymbol{z}}^{(m+1)}-\bar{{\boldsymbol{T}}}_{n,n}^{(m+1)}{\boldsymbol{s}}_{n}^{(m+1)}$
$\displaystyle+\sum_{k=1}^{K_{rad}}({\boldsymbol{g}}_{k}^{(m+1)}-{\boldsymbol{B}}_{k}^{1/2}{\boldsymbol{s}}_{n}^{(m+1)}).$
(50)

Input: $e_{t}$, $N_{t}$, $\bar{{\boldsymbol{R}}}_{n,n}$, $\rho$,
${\boldsymbol{R}}_{I}^{k}$, $E_{I}^{k}$, and $\xi$.

Output: ${\boldsymbol{s}}_{n}^{(t,l+1)}$.

1 Initialize: $m=0$, ${\boldsymbol{s}}_{n}^{(m)}$, ${\boldsymbol{z}}$, $t$,
${\boldsymbol{g}}_{k}$, ${\boldsymbol{c}}_{k}$, ${\boldsymbol{d}}$ and
$\vartheta$.

2 repeat

3 // Update of ${\boldsymbol{s}}_{n}^{(m+1)}$

4 $j=0$, ${\boldsymbol{s}}_{n}^{(m,j)}={\boldsymbol{s}}_{n}^{(m)}$;

5 repeat

6 Compute ${\boldsymbol{u}}^{(m,j)}$ by (42);

7 Update ${\boldsymbol{s}}_{n}^{(m,j+1)}$ by solving (43);

8 $j=j+1$;

9

10 until  _convergence_ ;

11 ${\boldsymbol{s}}_{n}^{(m+1)}={\boldsymbol{s}}_{n}^{(m,j+1)}$;

12 // Update of ${\boldsymbol{z}}^{(m+1)}$

13
${\boldsymbol{q}}^{(m)}=\bar{{\boldsymbol{R}}}_{n,n}{\boldsymbol{s}}_{n}^{(m)}-{\boldsymbol{d}}^{(m)}$;

14
${\boldsymbol{z}}^{(m+1)}={\vartheta{\boldsymbol{q}}^{(m)}}/{(\vartheta-2)}$;

15 $t^{(m+1)}=\|{\boldsymbol{z}}^{(m+1)}\|_{2}^{2}$;

16 // Update of ${\boldsymbol{g}}_{k}^{(m+1)}$

17 Update ${\boldsymbol{g}}_{k}^{(m+1)}$ by (49);

18
${\boldsymbol{c}}_{k}^{(m+1)}={\boldsymbol{c}}_{k}^{(m)}+{\boldsymbol{g}}_{k}^{(m+1)}-{\boldsymbol{B}}_{k}^{1/2}{\boldsymbol{s}}_{n}^{(m+1)}$;

19
${\boldsymbol{d}}^{(m+1)}={\boldsymbol{d}}^{(m)}+{\boldsymbol{z}}^{(m+1)}-\bar{{\boldsymbol{R}}}_{n,n}^{1/2}{\boldsymbol{s}}_{n}^{(m+1)}$;

20 $m=m+1$

21until  _$\|{\boldsymbol{r}}^{(m)}\| <\xi$_;

${\boldsymbol{s}}_{n}^{(t,l+1)}={\boldsymbol{s}}_{n}^{(m+1)}$.

Algorithm 1 ADMM algorithm for $\mathcal{P}_{s_{n}}$.

###  3.2 MM-ADMM

Substituting (20) into (10), we rewrite SINR as

$\textrm{SINR}({\boldsymbol{s}})=|\alpha_{t}|^{2}{\boldsymbol{v}}_{t}^{\dagger}({\boldsymbol{s}}){\boldsymbol{R}}_{v}^{-1}({\boldsymbol{s}}){\boldsymbol{v}}_{t}({\boldsymbol{s}}),$
(51)

According to [48, Lemma 1], $\textrm{SINR}({\boldsymbol{s}})$ is minorized by:

$-{\boldsymbol{s}}^{\dagger}{\boldsymbol{R}}{\boldsymbol{s}}+2\textrm{Re}({\boldsymbol{c}}^{\dagger}{\boldsymbol{s}})+const_{2},$
(52)

where
${\boldsymbol{c}}={\boldsymbol{V}}^{\dagger}(w_{t},\theta_{t}){\boldsymbol{R}}_{v}^{-1}({\boldsymbol{s}}^{(k)}){\boldsymbol{V}}(w_{t},\theta_{t}){\boldsymbol{s}}^{(k)}$,
$const_{2}=-\textrm{tr}({\boldsymbol{B}}_{k}{\boldsymbol{R}}_{u})$,

$\displaystyle{\boldsymbol{R}}=$
$\displaystyle\sum_{p=-P}^{P}\sum_{k=1}^{N_{c}}\sigma_{c,p,k}^{2}{\boldsymbol{V}}_{c,p,k}{\boldsymbol{B}}_{k}{\boldsymbol{V}}_{c,p,k}^{\dagger},$

and ${\boldsymbol{B}}_{k}={\boldsymbol{u}}_{k}{\boldsymbol{u}}_{k}^{\dagger}$,
${\boldsymbol{u}}_{k}={\boldsymbol{R}}_{v}^{-1}({\boldsymbol{s}}^{(k)}){\boldsymbol{V}}(w_{t},\theta_{t}){\boldsymbol{s}}^{(k)}$,
the superscript “$k$” denotes the $k$th iteration in the MM-based algorithm.
Let $\hat{{\boldsymbol{R}}}=\mu{\boldsymbol{I}}-{\boldsymbol{R}}$, where $\mu$
is set to ensure $\hat{{\boldsymbol{R}}}>0$. By omitting the constant terms,
the optimization of ${\boldsymbol{s}}$ can be formulated as

$\overline{\mathcal{P}}_{{\boldsymbol{s}}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}}\
&{\boldsymbol{s}}^{\dagger}\hat{{\boldsymbol{R}}}{\boldsymbol{s}}+2\textrm{Re}({\boldsymbol{c}}^{\dagger}{\boldsymbol{s}})\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},\\\ &n=1,\cdots,N_{t},k=1,\cdots,K_{rad}.\end{aligned}\end{cases}$
(53)

Let ${\boldsymbol{c}}={\boldsymbol{P}}\bar{{\boldsymbol{c}}}$, and the
objective function of $\overline{\mathcal{P}}_{{\boldsymbol{s}}}$ can be
rewritten as

${\boldsymbol{s}}^{\dagger}\hat{{\boldsymbol{R}}}{\boldsymbol{s}}+2\textrm{Re}({\boldsymbol{c}}^{\dagger}{\boldsymbol{s}})=\bar{{\boldsymbol{s}}}^{\dagger}\bar{{\boldsymbol{R}}}\bar{{\boldsymbol{s}}}+2\textrm{Re}(\bar{{\boldsymbol{c}}}^{\dagger}\bar{{\boldsymbol{s}}}),$
(54)

where
$\bar{{\boldsymbol{R}}}={\boldsymbol{P}}^{\dagger}\hat{{\boldsymbol{R}}}{\boldsymbol{P}}$.
Next, let us partition $\bar{{\boldsymbol{R}}}$ and $\bar{{\boldsymbol{c}}}$
into $N_{t}\times N_{t}$ and $N_{t}\times 1$ blocks, each of which are an
$L\times L$ matrix and an $L\times 1$ vector, respectively. Let
$\bar{{\boldsymbol{R}}}_{n,m}$ and ${\boldsymbol{c}}_{n}$ denote the $(n,m)$th
and the $n$th block of $\bar{{\boldsymbol{R}}}$ and $\bar{{\boldsymbol{c}}}$.
Then
$\bar{{\boldsymbol{s}}}^{\dagger}\bar{{\boldsymbol{R}}}\bar{{\boldsymbol{s}}}+2\textrm{Re}(\bar{{\boldsymbol{c}}}^{\dagger}\bar{{\boldsymbol{s}}})$
can be rewritten as

$\bar{{\boldsymbol{s}}}^{\dagger}\bar{{\boldsymbol{R}}}\bar{{\boldsymbol{s}}}+2\textrm{Re}(\bar{{\boldsymbol{c}}}^{\dagger}\bar{{\boldsymbol{s}}})={\boldsymbol{s}}_{n}^{\dagger}\bar{{\boldsymbol{R}}}_{n,n}{\boldsymbol{s}}_{n}+2\textrm{Re}({\boldsymbol{f}}_{n}^{\dagger}{\boldsymbol{s}}_{n})+const_{3}$
(55)

where ${\boldsymbol{f}}_{n}=\frac{1}{2}\sum_{\begin{subarray}{c}m=1\\\ m\neq
n\end{subarray}}^{N_{t}}\bar{{\boldsymbol{R}}}_{n,m}{\boldsymbol{s}}_{m}+{\boldsymbol{c}}_{n}$,

$const_{3}=\sum_{\begin{subarray}{c}m=1\\\ m\neq
n\end{subarray}}^{N_{t}}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
n\end{subarray}}^{N_{t}}{\boldsymbol{s}}_{m}^{\dagger}\bar{{\boldsymbol{R}}}_{m,m^{\prime}}{\boldsymbol{s}}_{m^{\prime}}+\sum_{\begin{subarray}{c}m=1\\\
m\neq
n\end{subarray}}^{N_{t}}{\boldsymbol{c}}_{m}^{\dagger}{\boldsymbol{s}}_{m}.$

By using (55), we formulate the following problem to optimize
${\boldsymbol{s}}_{n}$:

$\overline{\mathcal{P}}_{s_{n}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}_{n}}\
&{\boldsymbol{s}}_{n}^{\dagger}\bar{{\boldsymbol{R}}}_{n,n}{\boldsymbol{s}}_{n}+2\textrm{Re}({\boldsymbol{f}}_{n}^{\dagger}{\boldsymbol{s}}_{n})\\\
\textrm{s.t.}\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{s}}_{n}=e_{t}/{N_{t}},\\\
&\textrm{PAPR}({\boldsymbol{s}}_{n})\leq\rho,\\\
&{\boldsymbol{s}}_{n}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{n}\leq
E_{I}^{k},\ k=1,\cdots,K_{rad}.\end{aligned}\end{cases}$ (56)

Note the similarity between (56) and (32). Thus, we can use Algorithm 1 to
tackle (56).

Table 2: Computational complexity analysis Algorithm 1 | Algorithm 2
---|---
Computation | Complexity | Computation | Complexity
${\boldsymbol{u}}^{(m,j)}$ | $O(L^{2})$ | ${\boldsymbol{R}}_{c}({\boldsymbol{s}})$ | $O((2P+1)(MN_{r})^{3}LN_{t}(N_{t}+L))$
${\boldsymbol{s}}_{n}^{(m,j+1)}$ | $O(L)$ | ${\boldsymbol{w}}$ | $O((LMN_{r})^{3})$
${\boldsymbol{q}}^{(m)}$ | $O(L^{2})$ | ${\boldsymbol{D}}$ | $O((LN_{t})^{2})$
${\boldsymbol{z}}^{(m+1)}$ | $O(L)$ | ${\boldsymbol{Q}}$ | $O((2P+1)(LN_{t}^{3}M^{2}N_{r}^{2}+L^{2}N_{t}^{3}MN_{r}))$
$t^{(m+1)}$ | $O(L)$ | $\beta({\boldsymbol{w}}^{(t+1)})$ | $O(LMN_{r})$
${\boldsymbol{g}}_{k}^{(m+1)}$ | \- or $O(L)$ | $f^{(t,l)}$ | $O((LN_{t})^{2})$
${\boldsymbol{c}}_{k}^{(m+1)}$ | $O(L^{2})$ |  $\bar{{\boldsymbol{T}}}_{n,n}^{(t,l)}$ or $\bar{{\boldsymbol{R}}}_{n,n}^{(t)}$ | $O((LN_{t})^{2})$
${\boldsymbol{d}}^{(m+1)}$ | $O(L^{2})$ |  ${\boldsymbol{b}}_{n}^{(t,l)}$ or ${\boldsymbol{f}}_{n}^{(t)}$ | $O((N_{t}-1)L^{2})$
- | - |  ${\boldsymbol{B}}_{k}$ or ${\boldsymbol{R}}$ | $O(L^{3}M^{2}N_{r}^{2}N_{t}+L^{2}M^{2}N_{r}^{2}N_{t})$
- | - | ${\boldsymbol{c}}$ | $O(L^{3}M^{2}N_{r}^{2}N_{t}+L^{2}N_{t}^{2})$

###  3.3 Algorithm Summary and Computational Complexity Analysis

We summarize the proposed multi-spectrally constrained waveform design
algorithm in Algorithm 2, where $\epsilon_{1},\epsilon_{2}>0$ are user-defined
small values. The computational complexity of the proposed algorithm at each
iteration is analyzed in Table 2. To alleviate the computational burden, we
calculate ${\boldsymbol{R}}_{c}({\boldsymbol{s}})$ and ${\boldsymbol{Q}}$ by
the method in [18, Appendix C]. We can see that if $P$ and $N_{t}$ is large,
the computational complexity of DK-ADMM is higher than that of MM-ADMM;
otherwise, if $L$ is large, the computational complexity of MM-ADMM is higher
than that of DK-ADMM.

Input: ${\boldsymbol{R}}_{\textrm{u}}$, ${\boldsymbol{V}}(w_{t},\theta_{t})$,
$\mu$, $\epsilon_{1}$, $\epsilon_{2}$,

Output: ${\boldsymbol{s}}_{\textrm{opt}}$ and
${\boldsymbol{w}}_{\textrm{opt}}$.

1 Initialize: $t=0$, ${\boldsymbol{s}}_{n}^{(t)},n=1,\cdots,N_{t}$.

2

3repeat

4 // Update of ${\boldsymbol{w}}^{(t+1)}$

5 Compute ${\boldsymbol{R}}_{c}({\boldsymbol{s}}^{(t)})$ by (8);

6
${\boldsymbol{R}}_{v}({\boldsymbol{s}}^{(t)})={\boldsymbol{R}}_{c}({\boldsymbol{s}}^{(t)})+{\boldsymbol{R}}_{\textrm{u}}$;

7
${\boldsymbol{w}}^{(t)}={\boldsymbol{R}}^{-1}_{v}({\boldsymbol{s}}^{(t)}){\boldsymbol{v}}_{t}({\boldsymbol{s}}^{(t)})$;

8 // Update of ${\boldsymbol{s}}^{(t+1)}$

9

10 for  _DK-ADMM Algorithm_ do

11 $l=0$, ${\boldsymbol{s}}^{(t,l)}={\boldsymbol{s}}^{(t)}$;

12 repeat

13 Compute ${\boldsymbol{D}}$, ${\boldsymbol{Q}}$, and
$\beta({\boldsymbol{w}}^{(t+1)})$;

14 Compute $f^{(t,l)}$;

15 Compute $\bar{{\boldsymbol{T}}}_{n,n}^{(t,l)}$ and
${\boldsymbol{b}}_{n}^{(t,l)}$;

16 for  _$n=1$ to $N_{t}$_ do

17 Update ${\boldsymbol{s}}_{n}^{(t,l+1)}$ using Algorithm 1;

18

19 end for

20 $l=l+1$;

21 until  _$|f^{(t,l+1)}-f^{(t,l)}|/f^{(t,l+1)} <\epsilon_{1}$_;

22 ${\boldsymbol{s}}_{n}^{(t+1)}={\boldsymbol{s}}_{n}^{(t,l)}$;

23 end for

24

25 for  _MM-ADMM Algorithm_ do

26 Compute ${\boldsymbol{B}}_{k}^{(t)}$, ${\boldsymbol{R}}^{(t)}$, and
${\boldsymbol{c}}^{(t)}$;

27 Compute $\bar{{\boldsymbol{R}}}_{n,n}^{(t)}$ and
${\boldsymbol{f}}_{n}^{(t)}$;

28 for  _$n=1$ to $N_{t}$_ do

29 Update ${\boldsymbol{s}}_{n}^{(t+1)}$ using Algorithm 1;

30

31 end for

32

33 end for

34

35 $t=t+1$;

36until  _$|\textrm{SINR}^{(t+1)}-\textrm{SINR}^{(t)}|/\textrm{SINR}^{(t+1)}
<\epsilon_{2}$_;

37${\boldsymbol{s}}_{\textrm{opt}}={\boldsymbol{s}}^{(t+1)}$;

${\boldsymbol{w}}_{\textrm{opt}}={\boldsymbol{w}}^{(t+1)}$.

Algorithm 2 Multi-spectrally constrained waveform design for MIMO STAP.

###  3.4 Extension to Multiple Space-Frequency Constraints

In some situations, the directions of the licensed radiators might be
approximately known. Assume that the direction of the $k$th licensed radiator
belongs to $\Theta_{k}=[\theta_{1}^{k},\theta_{2}^{k}]$, where
$\theta_{1}^{k}$ and $\theta_{2}^{k}$ are the lower and upper angles,
respectively, $k=1,\cdots,K_{rad}$. Therefore, the energy of
${\boldsymbol{s}}$ leaked on the $k$th space-frequency band can be expressed
as

$\int_{v_{1}^{k}}^{v_{2}^{k}}\int_{f_{1}^{k}}^{f_{2}^{k}}|{\boldsymbol{s}}_{\theta}^{\dagger}{\boldsymbol{a}}(f)|^{2}dfd\theta=\int_{v_{1}^{k}}^{v_{2}^{k}}{\boldsymbol{s}}_{\theta}^{\dagger}{\boldsymbol{R}}_{I}^{k}{\boldsymbol{s}}_{\theta}d\theta={\boldsymbol{s}}^{\dagger}{\boldsymbol{F}}_{I}^{k}{\boldsymbol{s}},$

where $v_{1}^{k}=\sin(\theta_{1}^{k})$, $v_{2}^{k}=\sin(\theta_{2}^{k})$,
${\boldsymbol{s}}_{\theta}=\textrm{vec}({\boldsymbol{a}}^{\top}(\theta){\boldsymbol{S}})=({\boldsymbol{I}}_{L}\otimes{\boldsymbol{a}}^{\top}(\theta)){\boldsymbol{s}}$,
${\boldsymbol{F}}_{I}^{k}={\boldsymbol{R}}_{I}^{k}\otimes{\boldsymbol{U}}$,
the $(p,q)$th entry of ${\boldsymbol{U}}\in{\mathbb{C}}^{N_{t}\times N_{t}}$
is given by

${\boldsymbol{U}}(p,q)=\begin{cases}v_{2}^{k}-v_{1}^{k},&p=q,\\\
\frac{e^{j2\pi v_{2}^{k}(q-p)d_{t}/\lambda}-e^{j2\pi
v_{1}^{k}(q-p)d_{t}/\lambda}}{j2\pi(q-p)d_{t}/\lambda},&p\neq q,\end{cases}$

and we have assumed that the transmit array is a uniform linear array (ULA)
with inter-element spacing denoted $d_{t}$. Then we can enforce a space-
frequency constraint to control the energy leaked on the space-frequency band.
When multiple space-frequency constraints and the PAPR constraint are imposed,
the optimization of ${\boldsymbol{s}}$ (at each iteration) can be formulated
by the following:

$\mathcal{P}_{{\boldsymbol{s}}}\begin{cases}\begin{aligned}
\max\limits_{{\boldsymbol{s}}}\
&\frac{{\boldsymbol{s}}^{\dagger}{\boldsymbol{D}}{\boldsymbol{s}}}{{\boldsymbol{s}}^{\dagger}{\boldsymbol{Q}}{\boldsymbol{s}}+\beta({\boldsymbol{w}})}\\\
\textrm{s.t.}\ &{\boldsymbol{s}}^{\dagger}{\boldsymbol{s}}=e_{t},\\\
&\textrm{PAPR}({\boldsymbol{s}})\leq\rho,\\\
&{\boldsymbol{s}}^{\dagger}{\boldsymbol{F}}_{I}^{k}{\boldsymbol{s}}\leq
E_{I}^{k},\\\ &k=1,\cdots,K_{rad}.\end{aligned}\end{cases}$ (57)

Similarly, we can use Algorithm 2 to tackle the above optimization problem.

##  4 Numerical Examples

In this section, numerical experiments are conducted to evaluate the
performance of the proposed algorithm. The considered MIMO radar system has
$N_{t}=4$ transmitters and $N_{r}=4$ receivers, where both transmit array and
receive array are assumed to be ULAs, with inter-element spacing
$d_{t}=2\lambda$ and $d_{r}=\lambda/2$, respectively ($\lambda$ is the
wavelength). The radar system is at an altitude of $h_{a}=9000$ m and moving
with a constant speed of $v_{a}=75$ m/s. The total transmit energy of the
waveforms is $e_{t}=1$. The waveform has a bandwidth of $800$ kHz and a
duration of $T=200\mu$s, sampled with a frequency of $f_{s}=800$ kHz (i.e.,
the code length is $L=160$). Additionally, we use a linear frequency modulated
(LFM) waveform with a chirp rate of $\gamma_{s}=3.5\times 10^{9}$
$\textrm{s}^{-2}$ as the initial waveform for all the transmit waveforms (Note
that such waveforms do not satisfy the multi-spectral constraint, meaning that
the initial waveforms are infeasible). The radar transmits $M=16$ pulses in a
CPI with a constant PRF of $f_{r}=1000$ Hz. The target of interest is at an
azimuth of $0^{\circ}$, and a range of $R_{t}=12728$ m. To establish the
clutter model, we assume that $P=3$ and $N_{c}=361$ clutter patches are
uniformly distributed in each iso-range ring. Additionally,
$\sigma_{c,p,k}^{2}=1,p=-P,\cdots,P,k=1,\cdots,N_{c}$. The noise power is
$\sigma^{2}=1$. $K_{rad}=3$ licensed radiators are coexisting with the AEW
radar system. The normalized frequency bands of the licensed radiators are
$\Omega_{1}=[0.2218,0.2773]$, $\Omega_{2}=[0.4609,0.6132]$, and
$\Omega_{3}=[0.7223,0.76328]$. The maximum allowed interfered energy of each
waveform on these bands are $E_{I}^{1}=-35$ dB, $E_{I}^{2}=-35$ dB, and
$E_{I}^{3}=-30$ dB, respectively. Regarding the ADMM algorithm, we set the
penalty parameter to $\vartheta=4$, and the maximum number of iterations to
$1000$. For the stopping criterion of the ADMM algorithm, the Dinkelbach’s
transform, and the cyclic optimization, we set $\xi=5\times 10^{-10}$,
$\epsilon_{1}=3\times 10^{-3}$, and $\epsilon_{2}=3\times 10^{-4}$,
respectively. Finally, the experiments are conducted on a standard PC with
Intel(R) Core(TM) i7-9750H CPU and 16GB RAM.

Figure 2: Convergence of SINR versus CPU time. $e_{t}=1$. $v_{t}=52.5$ m/s.
$E_{I}^{1}=E_{I}^{2}=-35$ dB, $E_{I}^{3}=-30$ dB.

(a) $\rho=1$, DK-ADMM (b) $\rho=2$, DK-ADMM (c) $\rho=3$, DK-ADMM (d)
$\rho=L$, DK-ADMM

(e) $\rho=1$, MM-ADMM (f) $\rho=2$, MM-ADMM (g) $\rho=3$, MM-ADMM (h)
$\rho=L$, MM-ADMM

Figure 3: ESDs of the designed waveforms. The blue and the yellow lines represent the ESDs of the initial waveforms and the optimized waveforms. $e_{t}=1$. $E_{I}^{1}=E_{I}^{2}=-35$ dB, $E_{I}^{3}=-30$ dB. (a) and (e): Constant-envelope waveforms. (b) and (f): $\rho=2$. (c) and (g): $\rho=3$. (d) and (h): Energy-constrained waveforms. Table 3: SINR at convergence SINR (dB) | $\rho=1$ | $\rho=2$ | $\rho=3$ | $\rho=L$
---|---|---|---|---
DK-ADMM | 23.8059 | 23.8101 | 23.8101 | 23.8102
MM-ADMM | 23.6952 | 23.7439 | 23.7794 | 23.7795
Table 4: CPU time needed to reach convergence CPU time (s) | $\rho=1$ | $\rho=2$ | $\rho=3$ | $\rho=L$
---|---|---|---|---
DK-ADMM | 380.843 | 177.167 | 166.941 | 141.049
MM-ADMM | 403.326 | 383.573 | 366.424 | 318.698

First, we analyze the convergence of the proposed algorithm. Figure 2 shows
the SINR curves of the proposed algorithm versus the CPU time, under the PAPR
constraints of $\rho=1$ (i.e., the constant-envelope constraint), $\rho=2$,
$\rho=3$, and $\rho=L$ (i.e., the energy constraint), respectively, where the
target velocity is $v_{t}=52.5$ m/s (i.e., $f_{t}=0.35$). Note that for both
DK-ADMM and MM-ADMM, the SINR monotonically increases as the iterations, which
confirms the convergence of the proposed algorithm. The SINR of the waveforms
synthesized by the DK-ADMM algorithm and the MM-ADMM algorithm at convergence
is shown in Table 3. We can see that a larger PAPR corresponds to a higher
SINR, because of the larger feasibility region. In addition, even the
stringent constant-envelope constraint is enforced on the waveforms, the SINR
of the synthesized low-PAPR waveforms is very close to that of energy-
constrained waveforms. Moreover, the SINR achieved by the DK-ADMM algorithm is
slightly higher than that of the MM-ADMM algorithm. Regarding the CPU time to
reach convergence, as shown in Table 4, the DK-ADMM algorithm is faster than
the MM-ADMM algorithm. Interestingly, the results therein also imply that a
larger PAPR results in a faster convergence.

Figure 3 presents the ESDs of the designed waveforms. The three stopbands are
shaded in gray with red dash-dot lines. The blue lines indicate the ESDs of
the initial waveforms, and the yellow lines denote the ESDs of the designed
waveforms. From Figure 3, we can observe that all the transmit waveforms form
deep nulls in the stopbands and satisfy the spectral constraints. In other
words, the designed waveform can precisely control the energy leaked on the
stopbands, which enhance the coexistence between the radar system and other
radio frequency systems. Moreover, we can observe that the ESDs of the
waveforms synthesized by the DK-ADMM algorithm is smoother than by the MM-ADMM
algorithm. Considering that the DK-ADMM algorithm achieves a larger SINR in a
shorter time and the associated ESDs of the synthesized waveforms are
smoother, we use the DK-ADMM algorithm to synthesize the multi-spectrally
constrained waveforms in the sequel.

Next we analyze the space-time cross-ambiguity (STCA) function of the devised
waveforms under different constraints, where the STCA function is defined as
[16]

$P_{{\boldsymbol{w}},{\boldsymbol{s}}}(\theta,f)=|{\boldsymbol{w}}^{\dagger}{\boldsymbol{V}}(\theta,f){\boldsymbol{s}}|^{2},$
(58)

where
${\boldsymbol{V}}(\theta,f)={\boldsymbol{d}}(f)\otimes{\boldsymbol{I}}_{L}\otimes{\boldsymbol{A}}(\theta)$,
and ${\boldsymbol{d}}(f)=[1,\cdots,e^{j2\pi(M-1)f}]^{\top}$. Figure 4 shows
the STCA function of the constant-envelope waveforms and the energy-
constrained waveforms. We can observe the mainlobes of all the STCA functions
at zero spatial frequency (which corresponds to an azimuth of $0^{\circ}$) and
a normalized Doppler frequency of $0.35$. Additionally, these functions form
deep nulls along the clutter ridges. Therefore, the devised waveforms and
filters can successfully suppress the clutter and improve the SINR
performance.

(a) $\rho=1$ (b) $\rho=1$

(c) $\rho=L$ (d) $\rho=L$

Figure 4: STCA of the designed waveforms. $e_{t}=1$, $v_{t}=52.5$ m/s.
$E_{I}^{1}=E_{I}^{2}=-35$ dB, $E_{I}^{3}=-30$ dB. (a) 3D STCA function of the
constant-envelope waveforms. (b) Top view associated with (a). (c) 3D STCA
function of the energy-constrained waveforms. (d) Top view associated with
(c).

To assess the impact of initial points on the performance of the designed
algorithms, various randomly generated waveforms are set to be the initial
points, where the random waveforms are constant-envelope waveforms with
modulus of $\sqrt{p_{s}}$ and phases following a zero-mean Gaussian
distribution. The SINRs at convergence and the associated CPU time for
different initial points are shown in Figure 5, where 50 Monte Carlo trials
are conducted. Table 5 and Table 6 show the maximum, the average, and the
minimum value of the SINR at convergence and the CPU time needed to reach
convergence for the different PAPR-constrained waveforms (i.e.,
$\rho=1,2,3,L$). From Figure 5 and the results in Table 5 and 6, we find that
the SINR of the designed algorithm is insensitive to the initial points, but
the convergence speed is affected by the initial points. To show that the
synthesized waveforms satisfy the spectral constraint, we randomly select a
set of the results and plot the ESDs of the designed waveforms in Figure 6.
The results indicate that compared with the initial waveforms, the waveforms
devised via the proposed algorithm achieve better spectral compatibility.
Interestingly, the spectrum of the waveforms initialized by randomly generated
waveforms is not as smooth as that in Figure 3.

Figure 5:  The impact of initial waveforms on SINR and CPU time. Random waveforms are used as the initial point. $e_{t}=1$, $v_{t}=52.5$ m/s. $E_{I}^{1}=E_{I}^{2}=-35$ dB, $E_{I}^{3}=-30$ dB. (a) SINR. (b) CPU time. Table 5: SINR at convergence SINR (dB) | Maximum | Average | Minimum
---|---|---|---
Constant envelope | 23.736 | 23.696 | 23.419
$\rho=2$ | 23.777 | 23.750 | 23.569
$\rho=3$ | 23.778 | 23.751 | 23.553
Energy constraint | 23.782 | 23.755 | 23.706
Table 6: CPU time needed to reach convergence CPU time (s) | Maximum | Average | Minimum
---|---|---|---
Constant envelope | 484.717 | 329.859 | 253.514
$\rho=2$ | 384.215 | 288.868 | 229.881
$\rho=3$ | 372.878 | 255.552 | 199.061
Energy constraint | 298.581 | 227.069 | 151.206

Figure 6: ESDs of the designed waveforms. The blue and yellow lines represent
the ESDs of the initial random waveforms and the optimized waveforms.
$e_{t}=1$. $E_{I}^{1}=E_{I}^{2}=-35$ dB, $E_{I}^{3}=-30$ dB. (a) Constant-
envelope waveforms. (b) $\rho=2$. (c) $\rho=3$. (d) Energy-constrained
waveforms.

In the following, we analyze the impact of spectral notch depths on the
achieved SINR. Figure 7(a) shows the achieved SINR with respect to different
spectral notch depths and compares with that of the waveforms devised via the
algorithm in [30] 222It should be noted that the algorithm in [30] focuses on
designing constant-envelope waveforms for a SISO radar system. Herein, we
extend this algorithm to deal with the MIMO case. However, it is difficult for
the algorithm in [30] to secure a feasible initial point to satisfy both the
equality constraint ${\boldsymbol{s}}^{\dagger}{\boldsymbol{s}}=e_{t}$ and the
multi-spectral constraint. Therefore, when using the algorithm in [30] to
design the constant-envelope waveforms, we replace this equality constraint
with the inequality constraint ${\boldsymbol{s}}^{\dagger}{\boldsymbol{s}}\leq
e_{t}$., which is initialized by a heuristic initialization via alternating
optimization with MM (HIVAM) or a heuristic initialization via alternating
optimization with CD (HIVAC) method. For simplicity we assume that
$E_{I}^{1}=E_{I}^{2}=E_{I}^{3}=E_{I}$. It can be seen that as the notch depth
goes deeper, the waveforms devised via the proposed algorithm attain higher
SINR than those synthesized by the algorithm in [30]. This is because that the
algorithm in [30] needs to scale the energy of the waveforms to satisfy the
multi-spectral constraint. To see this, Figure 7(b) draws the energy of the
waveforms. It can be observed that as the notch depth goes deeper, the energy
of the waveforms devised via the algorithm in [30] drops to a low level (to
satisfy the stringent multi-spectral constraint), while the energy of the
waveforms devised via the proposed algorithm always reaches the highest
possible level. Since the SINR performance improves with the waveform energy,
the performance of our waveforms is superior to that of the waveforms devised
via the algorithm in [30].

Figure 7: (a) The impact of $E_{I}^{k}$ on SINR. (b) The energy of the
waveforms at convergence. $e_{t}=1$, $v_{t}=52.5$m/s.
$E_{I}^{1}=E_{I}^{2}=E_{I}^{3}$.

Figure 8 compares the SINR of the waveforms devised via the proposed algorithm
versus the normalized target Doppler frequencies with the algorithm in [30],
where the performance of the energy-constrained waveforms is also included as
a benchmark. From Figure 8, we can see that the waveforms devised via the
proposed algorithm achieve better detection performance than those devised via
the algorithm in [30], especially at the low Doppler frequency area.

Figure 8: (a) SINR versus normalized target Doppler frequency.
$f_{t}\in[-0.5,0.5]$. (b) SINR at low Doppler frequencies.
$f_{t}\in[-0.1,0.1]$. $E_{I}^{1}=E_{I}^{2}=-35$ dB, $E_{I}^{3}=-30$ dB.

Next, we assess the robustness of the proposed algorithm with respect to the
Doppler uncertainty of the clutter patches. Figure 9 shows the SINR of the
constant-envelope waveform versus the normalized Doppler frequency under
different clutter uncertainty. Note that the Doppler uncertainty degrades the
target detection performance, especially in the low Doppler frequency region
(about $2\sim 3$ dB loss in this area). However, the proposed algorithm still
achieves better radar detection performance than the competing algorithms.

Figure 9: (a) SINR versus normalized target Doppler frequency. (b) SINR at low
Doppler frequencies. $f_{t}\in[-0.5,0.5]$, $E_{I}^{1}=E_{I}^{2}=-35$ dB,
$E_{I}^{3}=-30$ dB.  Figure 10: Convergence of SINR with respect to CPU time.
$d_{t}=\lambda/2$, $d_{r}=2\lambda$. $M=24$. $\phi=10^{\circ}$.
$E_{I}^{1}=E_{I}^{2}=E_{I}^{3}=-35$ dB.
$\Theta_{1}=[-60^{\circ},-25^{\circ}]$, $\Theta_{2}=[20^{\circ},60^{\circ}]$,
$\Theta_{1}=[25^{\circ},70^{\circ}]$

Figure 11: The spectral distributions on the spatial-frequency domain.
$d_{t}=\lambda/2$, $d_{r}=2\lambda$. $M=24$. $\phi=10^{\circ}$.
$E_{I}^{1}=E_{I}^{2}=E_{I}^{3}=-35$ dB.
$\Theta_{1}=[-60^{\circ},-25^{\circ}]$, $\Theta_{2}=[20^{\circ},60^{\circ}]$,
$\Theta_{1}=[25^{\circ},70^{\circ}]$. (a) Constant-envelope waveforms (b)
Energy-constrained waveforms.

Finally, we extend the proposed ADMM algorithm to deal with the multiple
space-frequency constraints. Figure 10 analyzes the SINR of the proposed
algorithm versus the CPU time, where the spatial regions associated with the
three radiators are $\Theta_{1}=[-60^{\circ},-25^{\circ}]$,
$\Theta_{2}=[20^{\circ},60^{\circ}]$, and
$\Theta_{1}=[25^{\circ},70^{\circ}]$, respectively,
$E_{I}^{1}=E_{I}^{2}=E_{I}^{3}=-35$ dB. The inter-element spacing is set to be
$d_{t}=\lambda/2$ and $d_{r}=2\lambda$. The elevation of the target of
interest is set to be $\phi=10^{\circ}$. The MIMO radar transmits $M=24$
pulses in a CPI. The results show the monotonically increasing SINR of the
waveforms synthesized by the proposed algorithm. Figure 11 shows the spectral
distribution of the synthesized waveforms over the spatial-frequency domain.
We can see that the synthesized waveforms can precisely control the energy
leaked on the spatial-frequency domains corresponding to the radiators,
further improving the spectral coexistence of the MIMO radar system and the
nearby radiators.

##  5 Conclusions

We derived efficient algorithms to design low-PAPR waveforms for airborne MIMO
radar in spectrally crowded environments. The purpose was to maximize the
output SINR by jointly optimizing the transmit waveforms and receive filters.
To tackle the multi-spectrally constrained waveform optimization problem, we
developed two iterative algorithms. which were based on cyclic optimization,
Dinkelbach’s transform, MM, and ADMM. Results showed that the waveforms
devised via the proposed algorithm not only improved the detection performance
of airborne MIMO radar, but also attained better spectral compatibility.

Possible future work includes the design of filter banks to account for
unknown target Doppler (see, e.g., [49] for a discussion on this topic), the
investigation of the correlation properties of the designed waveforms, and the
performance analysis of the waveforms on hardware. It’s also crucial to
develop computationally efficient algorithms to design the waveforms in real
time. Finally, the theoretical analysis for the convergence of the proposed
ADMM algorithm will be left as a future topic.

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##  6 Biography Section

[]Da Li was born in Luoyang, Henan Province, China, in 1995. He received the
B.S. degree from Lanzhou University, Lanzhou, China, in 2018, and the M.S.
degree in 2020 from National University of Defense Technology, Hefei, China,
where he is currently working toward the Ph.D. degree in information and
communication engineering at the Department of College of Electronic
Engineering. His research interests mainly include signal processing and radar
waveform design.

[]Bo Tang was born in Linchuan, Jiangxi Province, China, in 1985. He received
the B.S. and Ph.D. degrees in electrical engineering from Tsinghua University,
Beijing, China, in 2006 and 2011, respectively. From July 2011 to June 2017,
he was with Electronic Engineering Institute, as a Lecturer. Since July 2017,
he has been with the College of Electronic Engineering, National University of
Defense Technology, Hefei, China, where he is currently a Professor. His
research interests mainly include adaptive radar signal processing and radar
waveform design. He was selected as the “Young Elite Scientists Sponsorship
Program” by China Association for Science and Technology and sponsored by the
Anhui Provincial Natural Science Foundation for Distinguished Young Scholars.
He is currently an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL
PROCESSING.

[]Lei Xue was born in Shouxian, Anhui, China, in 1963. He is currently a
Professor of Information and Communication Engineering from the College of
Electronic Engineering, National University of Defense Technology.