#  Joint Range and Doppler Adaptive Processing for CBM based DFRC systems

Jifa Zhang Jifa Zhang is with the School of Electronic and Information
Engineering, Beihang University, Beijing 100191, China, E-mail:
<EMAIL_ADDRESS>
###### Abstract

Recently, dual-function radar communication (DFRC) systems have been proposed
to integrate radar and communication into one platform for spectrum sharing.
Various signalling strategies have been proposed to embed communication
information into the radar transmitted waveforms. Among these, complex
beampattern modulation (CBM) embeds communication information into the complex
transmit beampattens via changing the amplitude and phase of the beampatterns
towards the communication receiver. The embedding of random communication
information causes the clutter modulation and high range-Doppler sidelobe.
What’s more, transmitting different waveforms on a pulse to pulse basis
degrades the radar target detection capacity when traditional sequential pulse
compression (SPC) and moving-target detection (MTD) is utilized. In this
paper, a minimum mean square error (MMSE) based filter, denoted as joint range
and Doppler adaptive processing (JRDAP) is proposed. The proposed method
estimates the targets’ impulse response coefficients at each range-Doppler
cell adaptively to suppress high range-Doppler sidelobe and clutter
modulation. The performance of proposed method is very close to the full-
dimension adaptive multiple pulses compression (AMPC) while reducing
computational complexity greatly.

###### Index Terms:

Complex beampattern modulation, Dual-function radar communication, Joint range
and Doppler adaptive processing, Minimum mean square error.

##  I Introduction

The competition between radar and communication for scarce spectrum resources
has become more and more fierce recently. Spectrum sharing has been proposed
to mitigate the competition of radar and communications for spectrum[1, 2, 3].
Spectrum sharing is roughly divided into co-existence and co-design. The focus
of co-existence is to mitigate the interference between radar and
communication[4, 5]. And the focus of co-design is the shared waveform design
in order to embed communication information into radar emission or embed
sensing into communication signal[6, 7]. Dual-function radar communication
(DFRC) systems, where the radar and communication are integrated into one
platform, is a classical example of co-design[8]. The DFRC systems have the
advantages of low cost, compact size, improved performance and safety. There
are many applications for DFRC systems, such as defence, internet of
vehicles[9, 10, 11] and autonomous vehicles[12], which is the main focus of
this paper.

As for the DFRC systems, where the radar is regarded as the primary function
and communication is regarded as the secondary function, many signal embedding
strategies, such as complex beampattern modulation (CBM)[13, 14], index
modulation (IM)[15, 16], spatial modulation (SM)[17], fast time
modulation[18], have been proposed. The essence of CBM is to steer the
mainlobe towards the targets of interest while changing the complex
beampattern towards the communication receiver according to the random
communication symbols. And the communication symbols are represented by
specific complex beampatterns towards the communication direction, such as
sidelobe levels (SLLs), phases or both. In [19], a phase modulation (PM) based
DFRC system was proposed where the communication symbols are represented by
specific phases towards communication direction. A quadrature amplitude
modulation (QAM) based DFRC system was proposed[20]. As for the IM, the
communication symbols are embedded into specific antenna indexes, frequency
indexes or phase indexes. The communication symbols are represented by
specific radar waveforms for fast time modulation, such as waveform diversity.

As for the waveform diversity based DFRC systems, the transmitted waveforms
change on a pulse to pulse basis to deliver random communication information,
which causes range sidelobe modulation (RSM) and degrades the target detection
capacity of radar when traditional sequential pulse compression (SPC) and
moving-target detection (MTD) is utilized[21]. Lots of methods have been
proposed to solve the RSM caused by waveform diversity[22, 23, 24, 25, 26,
27]. In [22], the authors proposed an adaptive pulse compression (APC) method
where the targets’ impulse response coefficients are estimated adaptively via
minimum mean square error (MMSE) estimation. In [23], a MMSE based adaptive
multiple pulses compression (AMPC) was proposed to suppress both range and
Doppler sidelobes. In [24], a non-identical multiple pulses compression was
proposed to mitigate the clutter. The authors proposed a joint least squares
(JLS) mismatch filter to achieve nearly identical range sidelobe responses
[27]. In [25], the JLS mismatch filter was adopted to moving-target indication
(MTI) radar. However, all these methods are restricted to pure radar systems
and not applicable to DFRC systems.

Like the waveform diversity, the transmitted waveforms of the CBM based DFRC
systems change on a pulse to pulse basis to synthesize specific beampatterns
causing the range-Doppler sidelobe modulation. What’s more, the clutter
located in different azimuth bins are modulated by different complex scale
causing clutter modulation[28]. In order to analyse the effect of clutter,
knowledge-aided radar systems[29, 30, 31, 32], where the radar systems have
access to some prior information of a dynamic environmental database, such as
a geographical information system (GIS), tracking files, meteorological
information and interference models, have been proposed recently.

In this paper, we assume the clutter impulse responses obey the Gaussian
distribution and the clutter is assumed as stationary clutter, i.e., the
normalized Doppler frequency of the clutter is zero. We first analyse the
range-Doppler sidelobe modulation and clutter modulation of DFRC systems due
to information embedding. Then, we review the full-dimension AMPC for
mitigating range-Doppler sidelobe modulation. Finally, a novel method based on
MMSE framework is proposed to mitigate the range-Doppler sidelobe modulation
and clutter modulation for CBM based DFRC systems. The proposed method, named
by joint range and Doppler adaptive processing (JRDAP), is adopted to replace
traditional SPC and MTD (SPC & MTD). In JRDAP, a joint range and Doppler
adaptive filter based on MMSE framework is designed to estimate the targets’
refection coefficients in each range-Doppler cell under clutter and noise
situation. Since the computational complexity of full-dimension AMPC is very
high[33], the proposed JRDAP is a dimensionality reduction algorithm, which
computational complexity is lower than the full-dimension AMPC.

The contributions of this paper are listed as below:

  * •

Firstly, we analyse the range-Doppler sidelobe modulation and clutter
modulation of DFRC systems due to information embedding.

  * •

Secondly, we derive the full-dimension AMPC for DFRC systems based on MMSE
framework to mitigate the range-Doppler sidelobe modulation and clutter
modulation.

  * •

Finally, we propose a joint range and Doppler adaptive processing, which
computational complexity is lower than that of full-dimension AMPC and
performance is very close to the full-dimension AMPC.

The rest of this paper is organized as below. The CBM based signaling strategy
is reviewed in Section II. The signal model is described in section III. The
adaptive multiple pulses compression is presented in section IV. And the
proposed JRDAP is described in section V. Simulation results are presented in
Section VI and conclusions are drown in Section VII.

Notations: In this paper, the matrices and vectors are denoted by bold
uppercase letters (i.e. $\mathbf{A}$) and lowercase letters (i.e.
$\mathbf{a}$), respectively. The transpose and the conjugate operation are
$(\cdot)^{T}$ and $(\cdot)^{H}$, respectively. $\otimes$ denotes the Kronecker
product. $\mathbb{C}^{N}$ denotes the set of $N$-dimensional vectors of
complex numbers. $\mathbf{I}$ denotes the identity matrix, whose dimension is
determined by the context. $\mathbb{E}\\{\cdot\\}$ denotes the mathematical
expectation, $|\cdot|$ denotes the module of a complex number and $a^{*}$
denotes the conjugate of $a$. $\|\mathbf{x}\|$ denotes the Euclidean norm of
the vector $\mathbf{x}$. $\mathbf{B}^{-1}$ denotes the inverse matrix of the
invertible matrix $\mathbf{B}$. $j$ denotes the imaginary unit. $\max(a,b)$
returns the larger of $a$ and $b$. $\text{tr}(\cdot)$ denotes the trace of the
matrix argument.

##  II Complex Beampattern Modulation Based Signaling Strategy

The CBM signalling strategy, which modulates the transmitted complex
beampatterns towards the communication direction, is widely utilized in DFRC
systems. In this section, we give a brief review of the CBM signaling
strategy.

Suppose a DFRC platform is equipped with a $M$-element uniform linear array
(ULA) with inter-element spacing of half wavelength for transmit and an
omnidirectional antenna for receive. We aim to minimize the peak sidelobe
level (PSL) and keep the specific complex beampatterns towards the targets of
interest and the single communication receiver. The $k$th beamforming weight
vector for the CBM-based signalling strategy can be obtained by solving the
following optimization problem:

$\mathcal{P}_{1}\left\\{\begin{array}[]{ll}\min_{\tilde{\mathbf{w}}_{k},\epsilon}&\
\epsilon^{2}\\\ \text{subject
to}&|\tilde{\mathbf{w}}_{k}^{H}\mathbf{a}(\theta)|^{2}\leq\epsilon^{2}\quad\theta\in\bar{\bm{\Theta}}\\\
&\ \tilde{\mathbf{w}}_{k}^{H}\mathbf{a}(\theta_{t})=1\\\ &\
\tilde{\mathbf{w}}_{k}^{H}\mathbf{a}(\theta_{c})=\Delta_{k}e^{j\phi_{k}},\quad
1\leq k\leq K\end{array}\right.$ (1)

where $\bar{\bm{\Theta}}$ denotes the radar sidelobe region,
$\mathbf{a}(\theta_{t})=[1,e^{-j2\pi\frac{d}{\lambda}\sin\theta_{t}},e^{-j2\pi\frac{2d}{\lambda}\sin\theta_{t}},\cdots,e^{-j2\pi(M-1)\frac{d}{\lambda}\sin\theta_{t}}]^{T}$
and
$\mathbf{a}(\theta_{c})=[1,e^{-j2\pi\frac{d}{\lambda}\sin\theta_{c}},\cdots,e^{-j2\pi(M-1)\frac{d}{\lambda}\sin\theta_{c}}]^{T}$
denotes the steering vector of the target and communication receiver,
$\theta_{t}$ and $\theta_{c}$ denotes the direction of the target and
communication receiver, and $\tilde{\mathbf{w}}_{k}$ is the $k$th beamforming
weight vector to form the beampattern whose phase and sidelobe level (SLL)
towards the communication receiver are $\phi_{k}$ and $\Delta_{k}$,
respectively. Each SLL $\Delta_{k}$ and phase $\phi_{k}$ are chosen from the
predefined dictionary $\Omega$ contained $L_{1}$ allowable SLLs and $Q_{1}$
phases. Therefore, the total number of communication symbols which can be
embedded through the CBM is $K=L_{1}Q_{1}$. The optimization problem
$\mathcal{P}_{1}$ is convex [34], and can be solved by off-the-shelf
optimization solvers, such as CVX.

##  III Signal Model

The range processing window length is $L$ and $Q$ discrete Doppler cells are
considered in this paper. Suppose $N_{c}$ discrete clutter patches are evenly
distributed in the angle interval $[-60^{\circ},60^{\circ}]$ and the clutter
is assumed to be stationary clutter, i.e., the Doppler frequency of clutter
patches is zero. Consider the DFRC system transmits a burst of $P$ slow-time
pulses in a coherent processing interval (CPI). The transmitted weight matrix
$\mathbf{W}=[\mathbf{w}_{1},\mathbf{w}_{2},\cdots,\mathbf{w}_{P}]\in\mathbb{C}^{M\times
P}$ modulates the radar pulses in a CPI. Each column vector of $\mathbf{W}$ is
chosen from the $K$ vectors obtained by solving $\mathcal{P}_{1}$.

The Swerling-I target model is adopted in this paper. What’s more, we assume
that the reflection coefficients of targets are stationary during the $P$
consecutive pulse transmissions. Therefore, the $\ell$th sample during the
$p$th pulse can be expressed as

$\displaystyle y_{p}(\ell)$
$\displaystyle=\sum_{q=0}^{Q-1}\mathbf{x}^{T}(\ell,q)\mathbf{s}e^{j2\pi(p-1)\psi_{q}}$
(2)
$\displaystyle+\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{p}\mathbf{a}(\theta_{i})\mathbf{c}^{T}_{i}(\ell)\mathbf{s}+n_{p}(\ell),$

where $\mathbf{s}=[s(0),s(1),\cdots,s(N-1)]^{T}$ denotes the samples of
transmitted waveform, $\ell=1,\cdots,L$, $p=1,2\cdots,P$,
$\mathbf{x}(\ell,q)=[x(\ell,q),x(\ell-1,q),\cdots,x(\ell-N+1,q)]^{T}$ are $N$
contiguous samples of the range profile impulse response corresponding to the
normalized Doppler frequency $\psi_{q}=-0.5+\frac{q-1}{Q}\
(\psi_{q}\in[-0.5,0.5])$ for $q=1,\cdots,Q$.
$\mathbf{c}_{i}(\ell)=[c_{i}(\ell),\cdots,c_{i}(\ell-N+1)]^{T}$ denotes $N$
contiguous samples of the clutter impulse response of stationary clutter patch
located in $i$th azimuth bin and $n_{p}(\ell)$ denotes the noise sample.

Let $\mathbf{y}_{p}(\ell)=[y_{p}(\ell),\cdots,y_{p}(\ell+N-1)]^{T}$ denotes
the $N$ contiguous samples of $y_{p}(\ell)$ and the matched filter output of
$\mathbf{y}_{p}(\ell)$ can be formulated as

$\displaystyle y_{mf,p}(\ell)$
$\displaystyle=\mathbf{s}^{H}\left(\sum_{q=0}^{Q-1}\mathbf{A}^{T}(\ell,q)e^{j2\pi(p-1)\psi_{q}}\right)\mathbf{s}$
(3)
$\displaystyle+\mathbf{s}^{H}\left(\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{p}\mathbf{a}(\theta_{i})\mathbf{C}^{T}_{i}(\ell)\right)\mathbf{s}+\tilde{n}_{p}(\ell),$

for $p=1,2,\cdots,P$, and

$\mathbf{A}(\ell,q)=\begin{bmatrix}x(\ell,q)&\cdots&x(\ell+N-1,q)\\\
x(\ell-1,q)&\cdots&x(\ell+N-2,q)\\\ \vdots&\ddots&\vdots\\\
x(\ell-N+1,q)&\cdots&x(\ell,q)\end{bmatrix},$ (4)

is a set of $N$ length-$N$ sample-shifted snapshots of the impulse response
and

$\mathbf{C}_{i}(\ell)=\begin{bmatrix}c_{i}(\ell)&\cdots&c_{i}(\ell+N-1)\\\
c_{i}(\ell-1)&\cdots&c_{i}(\ell+N-2)\\\ \vdots&\ddots&\vdots\\\
c_{i}(\ell-N+1)&\cdots&c_{i}(\ell)\end{bmatrix},$ (5)

is a set of $N$ length-$N$ sample-shifted snapshots of the impulse response of
stationary clutter patch located $i$th azimuth bin.

Besides, $\mathbf{n}_{p}(\ell)=[n_{p}(\ell),\cdots,n_{p}(\ell+N-1)]^{T}$ is
the $N$ noise samples and
$\tilde{n}_{p}(\ell)=\mathbf{s}^{H}\mathbf{n}_{p}(\ell)$.

When non complex beampattern modulation (NCBM) is utilized, $y_{mf,p}(\ell)$
can be reformulated as

$\displaystyle y_{mf,p}(\ell)$
$\displaystyle=\mathbf{s}^{H}\left(\sum_{q=0}^{Q-1}\mathbf{A}^{T}(\ell,q)e^{j2\pi(p-1)\psi_{q}}\right)\mathbf{s}$
(6)
$\displaystyle+\mathbf{s}^{H}\left(\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{1}\mathbf{a}(\theta_{i})\mathbf{C}^{T}_{i}(\ell)\right)\mathbf{s}+\tilde{n}_{p}(\ell).$

Comparing Eq. (3) with Eq. (6), we can observe that the communication
embedding modulates clutter through the term
$\mathbf{w}^{H}_{p}\mathbf{a}(\theta_{i})$ cross different pulses in a CPI
resulting in the undesired spreading of the clutter spectrum and clutter
modulation. Due to the waveform diversity, the traditional SPC & MTD has high
range-Doppler sidelobe causing the small targets (low SINR) masked by the
large nearby targets (high SINR).

##  IV Adaptive multiple pulses compression

In order to compress the range-Doppler sidelobe, full-dimension AMPC is
proposed to replace the traditional SPC & MTD [23]. In AMPC, a MMSE filter is
utilized to estimate the targets’ impulse response coefficients at each range-
Doppler cell. The output of MMSE filter at $\ell$th range cell and $q$th
Doppler cell is given by

$\hat{x}(\ell,q)=\mathbf{h}^{H}(\ell,q)\mathbf{y}(\ell),$ (7)

where $\hat{x}(\ell,q)$ denotes the MMSE filter estimate of the $\ell$th
delayed sample of a length-$L$ section of the range profile response
corresponding to $q$th Doppler cell, $\mathbf{h}(\ell,q)=[h_{11}(\ell,q)$,
$\cdots$, $h_{1N}(\ell,q)$, $h_{21}(\ell,q)$, $\cdots$,
$h_{PN}(\ell,q)]^{T}\in\mathbb{C}^{NP\times 1}$ is the wight vector of MMSE
filter corresponding to $\ell$th range cell and $q$th Doppler cell, and
$\mathbf{y}(\ell)$ is given by

$\displaystyle\mathbf{y}(\ell)$
$\displaystyle=\left[\mathbf{y}^{T}_{1}(\ell),\mathbf{y}^{T}_{2}(\ell),\cdots,\mathbf{y}^{T}_{P}(\ell)\right]^{T}$
(8) $\displaystyle=\begin{bmatrix}\sum_{q=0}^{Q-1}\mathbf{A}^{T}(\ell,q)\\\
\vdots\\\
\sum_{q=0}^{Q-1}\mathbf{A}^{T}(\ell,q)e^{j2\pi(P-1)\psi_{q}}\end{bmatrix}\mathbf{s}$
$\displaystyle+\sum_{i=1}^{N_{c}}\begin{bmatrix}\mathbf{w}^{H}_{1}\mathbf{a}(\theta_{i})\mathbf{C}^{T}_{i}(\ell)\\\
\vdots\\\
\mathbf{w}^{H}_{P}\mathbf{a}(\theta_{i})\mathbf{C}^{T}_{i}(\ell)\end{bmatrix}\mathbf{s}+\begin{bmatrix}\mathbf{n}_{1}\\\
\vdots\\\ \mathbf{n}_{P}\end{bmatrix},$

where $\mathbf{y}_{i}(\ell)$ denotes the vector of $N$ contiguous samples of
the received signal during $i$th pulse for $i=1,\cdots,P$,
$\mathbf{n}_{i}=[n_{i}(\ell),\cdots,n_{i}(\ell+N-1)]$ denotes the noise vector
for $i=1,2,\cdots,P$. The cost function for MMSE filter at $\ell$th range cell
and $q$th Doppler cell is given by

$\mathcal{J}(\ell,q,\mathbf{h}(\ell,q))=\mathbb{E}\left[|x(\ell,q)-\mathbf{h}^{H}(\ell,q)\mathbf{y}(\ell)|^{2}\right].$
(9)

The MMSE cost function $\mathcal{J}(\ell,q,\mathbf{h}(\ell,q))$ is minimized
by differentiating with respect to $\mathbf{h}^{*}(\ell,q)$ and setting the
result is equal to zero. Therefore, the optimal $\tilde{\mathbf{h}}(\ell,q)$
takes the following form

$\tilde{\mathbf{h}}(\ell,q)=\left(\mathbb{E}\\{\mathbf{y}(\ell)\mathbf{y}^{H}(\ell)\\}\right)^{-1}\mathbb{E}\\{\mathbf{y}(\ell)x^{*}(\ell,q)\\}.$
(10)

We assume the neighboring target impulse response terms are uncorrelated, and
the target impulse response terms are uncorrelated with clutter impulse
response terms and noise terms too [23]. Therefore, the cross-correction
vector can be formulated as

$\mathbb{E}\left\\{\mathbf{y}(\ell)x^{*}(\ell,q)\right\\}=\rho(\ell,q)[\mathbf{s}^{T},\mathbf{s}^{T}e^{j2\pi\psi_{q}},\cdots,\mathbf{s}^{T}e^{j2\pi(P-1)\psi_{q}}]^{T},$
(11)

where $\rho(\ell,q)=\mathbf{E}\\{|x(\ell,q)|^{2}\\}$ denotes the power of
$x(\ell,q)$. However, it’s impossible to known the impulse response
coefficient at the $(\ell,q)$th range-Doppler cell. Therefore, the traditional
SPC & MTD is utilized to obtain the initial estimation of $\rho(\ell,q)$
firstly. Followed by that, the MMSE filter is performed on the received signal
to obtain the enhance estimation taking the initial estimation of
$\rho(\ell,q)$ as a prior information.

What’s more, we assume that the clutter impulse response terms obey Gaussian
distribution and identically distributed (IID), i.e.,
$\mathbb{E}\\{|c(\ell,q)|^{2}\\}=\sigma^{2}_{c},\forall\ell,q$, where
$\sigma^{2}_{c}$ denotes the power of the clutter. Therefore,
$\mathbb{E}\\{\mathbf{y}(\ell)\mathbf{y}^{H}(\ell)\\}\in\mathbb{C}^{NP\times
NP}$ is given by

$\mathbb{E}\left\\{\mathbf{y}(\ell)\mathbf{y}^{H}(\ell)\right\\}=\mathbf{R}_{t}(\ell)+\mathbf{R}_{c}+\mathbf{R}_{n},$
(12)

where the signal covariance matrix
$\mathbf{R}_{t}(\ell)\in\mathbb{C}^{NP\times NP}$ is given by

$\mathbf{R}_{t}(\ell)=\begin{bmatrix}\mathbf{R}_{t,1,1}(\ell)&\cdots&\mathbf{R}_{t,1,P}(\ell)\\\
\vdots&\ddots&\vdots\\\
\mathbf{R}_{t,P,1}(\ell)&\cdots&\mathbf{R}_{t,P,P}(\ell)\end{bmatrix}.$ (13)

###### Lemma IV.1

The $(l_{1},l_{2})$ element of $\mathbf{R}_{t}(\ell)$ is formulated as

$\mathbf{R}_{t,l_{1},l_{2}}(\ell)=\sum_{q=0}^{Q-1}\mathbf{\Phi}(\ell,q)e^{j2\pi(l_{1}-l_{2})\psi_{q}},$
(14)

where
$\mathbf{\Phi}(\ell,q)=\sum_{n=-N+1}^{N-1}\rho(n+\ell,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}$,
$\mathbf{s}_{n}$ denotes the transmitted waveform shifted by $n$ samples and
the remainder is filled with zero, e.g.,
$\mathbf{s}_{2}=[0,0,s(0),s(1),\cdots,s(N-3)]^{T}$ when $n=2$, and
$\mathbf{s}_{-2}=[s(2),\cdots,s(N-1),0,0]^{T}$ when $n=-2$.

###### Proof IV.1

See Appendix A.

In Eq. (12), $\mathbf{R}_{c}$ denotes the clutter covariance matrix and is
given by

$\mathbf{R}_{c}=\begin{bmatrix}\mathbf{R}_{c,1,1}&\cdots&\mathbf{R}_{c,1,P}\\\
\vdots&\ddots&\vdots\\\
\mathbf{R}_{c,P,1}&\cdots&\mathbf{R}_{c,P,P}\end{bmatrix}.$ (15)

###### Lemma IV.2

The $(l_{1},l_{2})$ element of $\mathbf{R}_{c}$ is formulated as

$\mathbf{R}_{c,l_{1},l_{2}}=\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{l_{1}}\mathbf{a}(\theta_{i})(\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i}))^{H}\mathbf{\Upsilon},$
(16)

where
$\mathbf{\Upsilon}=\sum_{n=-N+1}^{N-1}\sigma^{2}_{c}\mathbf{s}_{n}\mathbf{s}^{H}_{n}$.

###### Proof IV.2

See Appendix B.

In Eq. (12), $\mathbf{R}_{n}$ denotes the noise covariance matrix and the
$(l_{1},l_{2})$ element of $\mathbf{R}_{n}$ is formulated as

$\mathbf{R}_{n,l_{1},l_{2}}=\mathbb{E}\\{\mathbf{n}_{l_{1}}\mathbf{n}^{H}_{l_{2}}\\}=\begin{cases}\sigma^{2}_{n}\mathbf{I},&\mbox{if
}l_{1}=l_{2}\\\ \mathbf{0},&\mbox{otherwise}.\end{cases}$ (17)

Therefore, $\mathbf{R}_{n}=\sigma^{2}_{n}\mathbf{I}$, where $\sigma^{2}_{n}$
denotes the noise power.

However, the computational complexity of the full-dimension AMPC is very high
because the computational complexity of $NP\times NP$ matrix inversion in Eq.
(10) is very high. Therefore, the use of AMPC is limited for real-time
systems. In next section, we propose a dimensionality reduction method,
denoted as JRDAP, which can decease the computational complexity greatly.

##  V Joint Range and Doppler Adaptive Processing

###  V-A Proposed joint Range and Doppler Adaptive Processing

The standard joint range-Doppler matched filter (JRDMF) can be formulated as
[24]

$\hat{\mathbf{h}}(\ell,q)=\mathbf{d}(\psi_{q})\otimes\mathbf{s},$ (18)

where
$\mathbf{d}(\psi_{q})=[1,e^{j2\pi\psi_{q}},\cdots,e^{j2\pi(P-1)\psi_{q}}]^{T}\in\mathbb{C}^{P}$
denotes the temporal steering vector corresponding to normalized Doppler
frequency $\psi_{q}$. Inspired by the structure of Eq. (18), we reformulate
the $\mathbf{h}(\ell,q)$ in Eq. (9) as

$\mathbf{h}(\ell,q)=\mathbf{v}_{q}^{*}\otimes\mathbf{u}_{\ell},$ (19)

where $\mathbf{v}_{q}\in\mathbb{C}^{P}$ denotes the weight vector of Doppler
adaptive filter corresponding to $q$th Doppler cell and
$\mathbf{u}_{\ell}\in\mathbb{C}^{N}$ denotes the weight vector of range
adaptive filer corresponding to $\ell$th range cell. Therefore, the cost
function of MMSE based filter can be formulated equivalently as[35]

$\tilde{\mathcal{J}}(\ell,q,\mathbf{u}_{\ell},\mathbf{v}_{q})=\mathbb{E}\left\\{|x(\ell,q)-\mathbf{u}_{\ell}^{H}\mathbf{Y}(\ell)\mathbf{v}_{q}|^{2}\right\\},$
(20)

where
$\mathbf{Y}(\ell)=[\mathbf{y}_{1}(\ell),\mathbf{y}_{2}(\ell),\cdots,\mathbf{y}_{P}(\ell)]\in\mathbb{C}^{N\times
P}$. It is easy to verify that
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u}_{\ell},\mathbf{v}_{q})=\tilde{\mathcal{J}}(\ell,q,\gamma\mathbf{u}_{\ell},\gamma^{-1}\mathbf{v}_{q})$
for nonzero constant number $\gamma$. As a result, there is scale uncertainty
between $\mathbf{u}_{\ell}$ and $\mathbf{v}_{q}$. In order to eliminate the
scale uncertainty, we make the Euclidean norm of $\mathbf{u}_{\ell}$ equal to
$1$ in each iteration. In order to decrease computational complexity, we
optimize $N$-dimensional vector $\mathbf{u}_{\ell}$ and $P$-dimensional vector
$\mathbf{v}_{q}$ to minimize the cost function rather than optimize $N\times
P$-dimensional vector $\mathbf{h}(\ell,q)$. We propose an effective method
which optimizes one of $\mathbf{u}_{\ell}$ and $\mathbf{v}_{q}$ when the other
is fixed iteratively until the stopping criterion is met. The $N$-dimensional
vector $\mathbf{u}_{\ell}$ and $P$-dimensional vector $\mathbf{v}_{q}$ are
utilized to obtain the $N\times P$-dimensional weight vector by Eq. (19). In
the following, we omit the subscript of $\mathbf{u}_{\ell}$ and
$\mathbf{v}_{q}$ for ease of presentation, i.e., $\mathbf{u}_{\ell}$ is
replaced by $\mathbf{u}$ and $\mathbf{v}_{q}$ is replaced by $\mathbf{v}$.

When $\mathbf{u}$ is fixed, the MMSE cost function
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})$ is a quadratic function
with respect to $\mathbf{v}$ and is convex. Therefore,
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})$ is minimized by
differentiating with respect to $\mathbf{v}^{*}$ and then setting the result
equal to zero, i.e.,

$\displaystyle\frac{\partial\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})}{\partial\mathbf{v}^{*}}$
$\displaystyle=\mathbb{E}\left\\{-x(\ell,q)\mathbf{Y}^{H}(\ell)\mathbf{u}\right\\}$
(21)
$\displaystyle+\mathbb{E}\left\\{\mathbf{Y}^{H}(\ell)\mathbf{u}\mathbf{u}^{H}\mathbf{Y}(\ell)\mathbf{v}\right\\}$
$\displaystyle=0,$

thus, the optimal Doppler adaptive filter $\tilde{\mathbf{v}}$ takes the form

$\tilde{\mathbf{v}}=\left(\mathbb{E}\\{\mathbf{Y}^{H}(\ell)\mathbf{u}\mathbf{u}^{H}\mathbf{Y}(\ell)\\}\right)^{-1}\mathbb{E}\left\\{x(\ell,q)\mathbf{Y}^{H}(\ell)\mathbf{u}\right\\}.$
(22)

Since the impulse response of target and clutter are uncorrelated,
$\mathbb{E}\\{x(\ell,q)\mathbf{Y}^{H}(\ell)\mathbf{u}\\}$ is given by

$\mathbb{E}\\{x(\ell,q)\mathbf{Y}^{H}(\ell)\mathbf{u}\\}=\rho(\ell,q)\begin{bmatrix}\mathbf{s}^{H}\mathbf{u}\\\
\vdots\\\ \mathbf{s}^{H}\mathbf{u}e^{-j2\pi(P-1)\psi_{q}}\end{bmatrix}.$ (23)

$\mathbb{E}\\{\mathbf{Y}^{H}(\ell)\mathbf{u}\mathbf{u}^{H}\mathbf{Y}(\ell)\\}\in\mathbb{C}^{P\times
P}$ is formulated as

$\mathbb{E}\\{\mathbf{Y}^{H}(\ell)\mathbf{u}\mathbf{u}^{H}\mathbf{Y}(\ell)\\}=\tilde{\mathbf{R}}_{t}(\ell)+\tilde{\mathbf{R}}_{c}+\tilde{\mathbf{R}}_{n}.$
(24)

###### Lemma V.1

The $(l_{1},l_{2})$th element of $\tilde{\mathbf{R}}_{t}(\ell)$ can be
formulated as

$\displaystyle\tilde{\mathbf{R}}_{t,l_{1},l_{2}}(\ell)$ (25) $\displaystyle=$
$\displaystyle\mathbf{u}^{H}\left(\sum_{q=0}^{Q-1}\mathbf{\Phi}(\ell,q)e^{-j2\pi(l_{1}-l_{2})\psi_{q}}\right)\mathbf{u}.$

###### Proof V.1

See Appendix C.

###### Lemma V.2

The $(l_{1},l_{2})$th element of $\tilde{\mathbf{R}}_{c}$ can be formulated as

$\tilde{\mathbf{R}}_{c,l_{1},l_{2}}=\sum_{i=1}^{N_{c}}\mathbf{a}^{H}(\theta_{i})\mathbf{w}_{l_{1}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i})\mathbf{u}^{H}\mathbf{\Upsilon}\mathbf{u}.$
(26)

###### Proof V.2

See Appendix D.

The $(l_{1},l_{2})$th element of $\tilde{\mathbf{R}}_{n}$ can be formulated as

$\displaystyle\tilde{\mathbf{R}}_{n,l_{1},l_{2}}$
$\displaystyle=\mathbb{E}\\{\mathbf{n}^{H}_{l_{1}}\mathbf{u}\mathbf{u}^{H}\mathbf{n}_{l_{2}}\\}$
(27)
$\displaystyle=\mathbb{E}\\{\text{tr}(\mathbf{n}^{H}_{l_{1}}\mathbf{u}\mathbf{u}^{H}\mathbf{n}_{l_{2}})\\}$
$\displaystyle=\text{tr}\left(\mathbf{u}\mathbf{u}^{H}\mathbb{E}\left\\{\mathbf{n}_{l_{2}}\mathbf{n}^{H}_{l_{1}}\right\\}\right),$

since

$\mathbb{E}\left\\{\mathbf{n}_{l_{2}}\mathbf{n}^{H}_{l_{1}}\right\\}=\begin{cases}\sigma^{2}_{n}\mathbf{I},&\mbox{if
}l_{1}=l_{2}\\\ \mathbf{0},&\mbox{otherwise}.\end{cases}$ (28)

Therefore, $\tilde{\mathbf{R}}_{n,l_{1},l_{2}}$ is given by

$\tilde{\mathbf{R}}_{n,l_{1},l_{2}}=\begin{cases}\sigma^{2}_{n}\|\mathbf{u}\|^{2},&\mbox{if
}l_{1}=l_{2}\\\ 0,&\mbox{otherwise}.\end{cases}$ (29)

thus, $\tilde{\mathbf{R}}_{n}=\sigma^{2}_{n}\|\mathbf{u}\|^{2}\mathbf{I}$.

When $\mathbf{v}$ is fixed, the MMSE cost function
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})$ is a quadratic function
with respect to $\mathbf{u}$ and is convex. Therefore,
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})$ is minimized by
differentiating with respect to $\mathbf{u}^{*}$, then setting the result
equal to zero, i.e.,

$\displaystyle\frac{\partial\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})}{\partial\mathbf{u}^{*}}$
$\displaystyle=\mathbb{E}\left\\{-x^{*}(\ell,q)\mathbf{Y}(\ell)\mathbf{v}\right\\}$
(30)
$\displaystyle+\mathbb{E}\left\\{\mathbf{Y}(\ell)\mathbf{v}\mathbf{v}^{H}\mathbf{Y}^{H}(\ell)\mathbf{u}\right\\}$
$\displaystyle=0,$

thus, the optimal range adaptive filter $\tilde{\mathbf{u}}$ takes the form

$\tilde{\mathbf{u}}=\left(\mathbb{E}\left\\{\mathbf{Y}(\ell)\mathbf{v}\mathbf{v}^{H}\mathbf{Y}^{H}(\ell)\right\\}\right)^{-1}\mathbb{E}\left\\{x^{*}(\ell,q)\mathbf{Y}(\ell)\mathbf{v}\right\\}.$
(31)

Upon assuming neighboring impulse response terms are uncorrelated,
$\mathbb{E}\\{x^{*}(\ell,q)\mathbf{Y}(\ell)\mathbf{v}\\}$ can be formulated as

$\mathbb{E}\\{x^{*}(\ell,q)\mathbf{Y}(\ell)\mathbf{v}\\}=\rho(\ell,q)[\mathbf{s},\cdots,\mathbf{s}e^{j2\pi(P-1)\psi_{q}}]\mathbf{v},$
(32)

and
$\mathbb{E}\\{\mathbf{Y}(\ell)\mathbf{v}\mathbf{v}^{H}\mathbf{Y}^{H}(\ell)\\}\in\mathbb{C}^{N\times
N}$ is formulated as

$\mathbb{E}\left\\{\mathbf{Y}(\ell)\mathbf{v}\mathbf{v}^{H}\mathbf{Y}^{H}(\ell)\right\\}=\bar{\mathbf{R}}_{t}(\ell)+\bar{\mathbf{R}}_{c}+\bar{\mathbf{R}}_{n}.$
(33)

###### Lemma V.3

$\bar{\mathbf{R}}_{t}(\ell)$ can be written as

$\bar{\mathbf{R}}_{t}(\ell)=\sum_{q=0}^{Q-1}|\beta_{q}|^{2}\mathbf{\Phi}(\ell,q),$
(34)

where
$\beta_{q}=\mathbf{v}(1)+\mathbf{v}(2)e^{j2\pi\psi_{q}}+\cdots+\mathbf{v}(P)e^{j2\pi(P-1)\psi_{q}}$.

###### Proof V.3

See Appendix E.

###### Lemma V.4

$\bar{\mathbf{R}}_{c}$ can be written as

$\bar{\mathbf{R}}_{c}=\sum_{i=1}^{N_{c}}|\beta_{i}|^{2}\mathbf{\Upsilon},$
(35)

where
$\beta_{i}=\mathbf{v}(1)\mathbf{w}^{H}_{1}\mathbf{a}(\theta_{i})+\cdots+\mathbf{v}(P)\mathbf{w}^{H}_{P}\mathbf{a}(\theta_{i})$.

###### Proof V.4

See Appendix F.

$\bar{\mathbf{R}}_{n}$ can be written as

$\displaystyle\bar{\mathbf{R}}_{n}$
$\displaystyle=\mathbb{E}\left\\{([\mathbf{n}_{1},\cdots,\mathbf{n}_{P}]\mathbf{v})([\mathbf{n}_{1},\cdots,\mathbf{n}_{P}]\mathbf{v})^{H}\right\\}$
(36)
$\displaystyle=\mathbb{E}\left\\{\left(\sum_{i=1}^{P}\mathbf{v}(i)\mathbf{n}_{i}\right)\left(\sum_{m=1}^{P}\mathbf{v}(m)\mathbf{n}_{m}\right)^{H}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\sum_{i=1}^{P}|\mathbf{v}(i)|^{2}\mathbf{n}_{i}\mathbf{n}^{H}_{i}\right\\}$
$\displaystyle=\sum_{i=1}^{P}|\mathbf{v}(i)|^{2}\mathbb{E}\left\\{\mathbf{n}_{i}\mathbf{n}^{H}_{i}\right\\}$
$\displaystyle=\sigma^{2}_{n}\|\mathbf{v}\|^{2}\mathbf{I}.$

In order to eliminate scale uncertainty, we normalize the vector
$\tilde{\mathbf{u}}$ in every iteration. Let $\mathbf{v}^{i}$ denotes the
value of $\mathbf{v}$ at $i$th iteration and $\mathbf{u}^{i}$ denotes the
value of $\mathbf{u}$ at $i$th iteration. The iteration rule of proposed
method is formulated as below:

$\mathbf{v}^{i}=(\mathbb{E}\\{\mathbf{Y}^{H}(\ell)\mathbf{u}^{i-1}\mathbf{u}^{(i-1)H}\mathbf{Y}(\ell)\\})^{-1}\mathbb{E}\\{x(\ell,q)\mathbf{Y}^{H}(\ell)\mathbf{u}^{i-1}\\},$
(37)
$\mathbf{u}^{i}=\frac{\left(\mathbb{E}\left\\{\mathbf{Y}(\ell)\mathbf{v}^{i}\mathbf{v}^{(i)H}\mathbf{Y}^{H}(\ell)\right\\}\right)^{-1}\mathbb{E}\left\\{x^{*}(\ell,q)\mathbf{Y}(\ell)\mathbf{v}^{i}\right\\}}{\|\left(\mathbb{E}\left\\{\mathbf{Y}(\ell)\mathbf{v}^{i}\mathbf{v}^{(i)H}\mathbf{Y}^{H}(\ell)\right\\}\right)^{-1}\mathbb{E}\left\\{x^{*}(\ell,q)\mathbf{Y}(\ell)\mathbf{v}^{i}\right\\}\|},$
(38)

for $i=1,2,\cdots$. The detailed description of proposed JRDAP is shown in
Algorithm 1.

Algorithm 1 Proposed joint range and Doppler adaptive processing method

Input: nonzero normalized vector $\mathbf{u}^{0}$, $\eta$, $\rho(\ell,q)$,
$G$, $x(\ell,q)$, $\mathbf{Y}(\ell)$, $\mathbf{s}$, $P$, $i=1$;
Output: $\mathbf{u}^{(i)}$, $\mathbf{v}^{(i)}$;

1: repeat

2: Substitute $\mathbf{u}^{i-1}$ into Eq. (37) to obtain $\mathbf{v}^{i}$;

3: Substitute $\mathbf{v}^{i}$ into Eq. (38) to obtain $\mathbf{u}^{i}$;

4: $i=i+1$;

5: until $\|\mathbf{u}^{i}-\mathbf{u}^{i-1}\|\leq\eta$ or reaches the maximum
iteration number $G$;

The output of joint range and Doppler adaptive filter corresponding to
$\ell$th range cell and $q$th Doppler cell is formulated as

$\hat{x}(\ell,q)=(\mathbf{v}_{q}^{*}\otimes\mathbf{u}_{\ell})^{H}\mathbf{y}(\ell).$
(39)

The JRDAP is utilized to estimate the targets’ impulse coefficient in all
range-Doppler cells. As can be seen in Eq. (25), Eq. (26), Eq. (34), Eq. (35)
some matrices such as $\mathbf{\Upsilon}$,
$\sum_{q=0}^{Q-1}\mathbf{\Phi}(\ell,q)e^{j2\pi(l_{1}-l_{2})\psi_{q}}$
$\forall$$l_{1}$, $l_{2}$$\in\\{1,\cdots,P\\}$ are often utilized, we can
calculate these terms before using them to reduce the amount of calculation.
The block diagram of JRDAP is shown in Algorithm 2.

Algorithm 2 Block diagram of JRDAP

Input: $\rho(\ell,q)$, $\mathbf{W}$, $\mathbf{y}(\ell)$, $L$, $Q$, $N_{c}$,
$\sigma^{2}_{c}$, $\sigma^{2}_{n}$, $\mathbf{s}$, $P$, $G$, $N$;
Output: $\hat{x}(\ell,q),\ \forall\ell,q$;

1: Calculate matrix $\mathbf{\Upsilon}$;

2: for $\ell=1$; $\ell<=L$; $\ell++$ do

3: Calculate matrices
$\sum_{q=0}^{Q-1}\mathbf{\Phi}(\ell,q)e^{j2\pi(l_{1}-l_{2})\psi_{q}}$,
$\forall$$l_{1}$, $l_{2}$$\in\\{1,\cdots,P\\}$;

4: for $q=0$; $q<=Q-1$; $q++$ do

5: Calculate $\mathbf{u}_{\ell}$ and $\mathbf{v}_{q}$ by utilizing Algorithm
1;

6:
$\hat{x}(\ell,q)=(\mathbf{v}_{q}^{*}\otimes\mathbf{u}_{\ell})^{H}\mathbf{y}(\ell)$;

7: end for

8: end for

###  V-B Computational complexity

The computational complexity of AMPC is $\mathcal{O}(N^{3}P^{3})$ and proposed
adaptive range filter and Doppler filter is
$\mathcal{O}[\hat{L}\max(QP^{2}N^{2},N_{c}P^{2}N^{2})]$, where $\hat{L}$
denotes the number of iterations. In practice, $Q$ and $N_{c}$ are far less
than $N\times P$. Besides, as shown in Fig. 1, the proposed method converges
in two iterations, where the estimation error is defined by

$\hat{e}^{i}(\ell,q)=|\bar{x}(\ell,q)-\hat{x}^{i}(\ell,q)|,$ (40)

where $\bar{x}(\ell,q)$ denotes the target impulse response at $(\ell,q)$th
range-Doppler cell estimated by SPC & MTD and
$\hat{x}^{i}(\ell,q)=(\mathbf{v}^{(i)*}_{q}\otimes\mathbf{u}^{i}_{\ell})^{H}\mathbf{y}(\ell)$
denotes the target impulse response at $(\ell,q)$th range-Doppler cell
estimated by proposed JRDAP at $i$th iteration. Therefore, the proposed method
is more effective than AMPC.

Figure 1: Convergence of proposed method

###  V-C Convergence

The cost function $\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})$ given in
Eq. (20) is continuous and differentiable. What’s more,
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v})\geq 0$, i.e., the cost
function has a lower bound.

It’s evident that Eq. (37) and Eq. (38) are equivalently to solve
$\min_{\mathbf{v}}\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i-1},\mathbf{v})$ for
a fixed $\mathbf{u}^{i-1}$ and solve
$\min_{\mathbf{u}}\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v}^{i})$ for a
fixed $\mathbf{v}^{i}$, respectively. Therefore, we can obtain

$\displaystyle\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i-1},\mathbf{v}^{i-1})\geq\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i-1},\mathbf{v}^{i})=$
(41)
$\displaystyle\min_{\mathbf{v}}\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i-1},\mathbf{v})\geq\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i},\mathbf{v}^{i})=\min_{\mathbf{u}}\tilde{\mathcal{J}}(\ell,q,\mathbf{u},\mathbf{v}^{i}),$

for $i=1,2,\cdots$. Eq. (41) implies that
$\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i},\mathbf{v}^{i})\leq\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{0},\mathbf{v}^{0})$
for $i=1,2,\cdots$ and the sequence
$\\{\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i},\mathbf{v}^{i})\\}$ is
monotonically decreasing. According to Dedekind theorem [36], the sequence
$\\{\tilde{\mathcal{J}}(\ell,q,\mathbf{u}^{i},\mathbf{v}^{i})\\}$ is
convergent.

##  VI Simulation Results

In this section, simulation results are presented to illustrate the benefits
of the proposed method. The simulation parameters are set as follows unless
specified otherwise. The DFRC system is equipped with a $10$-element ULA with
an inter-element spacing of half-wavelength for transmit and an
omnidirectional antenna for receive. In the simulation, the transmit waveform
is linear frequency modulation (LFM) waveform with length $N=32$, pulse width
$\tau=4\mu$s, bandwidth $B=4$MHz and initial frequency is $1$GHz. We assume
that $P=30$ pulses are transmitted during a CPI and the transmitted weight
vectors for each pulse is randomly selected from the specially designed
dictionary $\Omega$. The length of processing window is set as $L=80$ and the
number of Doppler cell is set as $Q=64$. It’s assumed that there are three
moving targets under test. The angle, range cell, Doppler cell and SNR of
targets are listed in Table I. The noise power is set as $\sigma^{2}_{n}=0$dB.
The number of clutter patches is $N_{c}=100$ and the clutter patches are
evenly distributed in the angle interval $[-60^{\circ},60^{\circ}]$. The
clutter to noise ratio (CNR) is set as $28$dB. A single communication receiver
is located at $\theta_{c}=-50^{\circ}$. The sidelobe region of radar is set as
$\bar{\bm{\Theta}}=[-90^{\circ},-5^{\circ}]\cup[5^{\circ},90^{\circ}]$. The
maximum iteration number is set as $G=10^{3}$.

TABLE I: Parameters of moving targets Angle(∘) | Range cell | Doppler cell | SNR(dB)
---|---|---|---
0 | 35 | 52 | 10
0 | 50 | 47 | 5
0 | 40 | 52 | -5

We assume that $2$ bits of communication information are delivered in each
radar pulse, i.e., $K=4$. For the CBM based signalling strategy mentioned
above, $L_{1}=2$ communication SLLs of $-25$dB and $-30$dB are used, and each
level is associated with $Q_{1}=2$ phases of $0$ and $\pi$ radian. Fig. 2
shows the four transmit patterns with different phases and communication SLLs.
The four beamforming weight vectors are utilized in the following simulations.

Figure 2: Transmit beampattern corresponding to the dictionary $\Omega$ of
weighting vectors $\tilde{\mathbf{w}}_{k}$

We performance $51200$ Monte-carlo simulations using MATLAB softwave on a PC
(3.0GHz Intel(R) Core(TM) i7-9700 CPU, 8GB RAM) to compare the average run
time per range-Doppler cell of AMPC and JRDAP. The results are listed in
Table. II. As shown in Table. II, the average run time of proposed JRDAD is
much less than that of AMPC both in CBM and NCBM situation.

As for the clutter free situation, i.e., $N_{c}=0$, the range-Doppler map of
SPC & MTD, JRDMF, AMPC, and JRDAP are shown in Fig. 3. As shown in the figure,
both SPC & MTD and JRDMF suffer high range-Doppler sidelobe, the small target
is masked by the nearby large target. For example, the target located the
$(40,52)$th range-Doppler cell is masked by the target located the $(35,52)$th
range-Doppler cell. What’s more, both AMPC and JRDAP can compress range-
Doppler sidelobe effectively and all three targets are visible in the range-
Doppler map.

The range-Doppler maps of SPC & MTD, JRDMF, AMPC, and JRDAP under clutter and
NCBM situation are shown in Fig. 4. The results of SPC & MTD and JRDMF show
that all three targets are masked by the clutter background. On the contrary,
two targets with $10$dB SNR and $5$dB SNR are visible in the range-Doppler
maps produced by AMPC and JRDAP while the target with $-5$dB SNR is masked by
the clutter background. What’s more, the clutter spectrum becomes narrower and
clutter power becomes weaker. As a consequence, the performance of AMPC and
JRDAP is much better than SPC & MTD and JRDMF. What’s more, the performance of
JRDAP is very close to that of AMPC.

The range-Doppler maps of SPC & MTD, JRDMF, AMPC, and JRDAP under clutter and
CBM situation are shown in Fig. 5. The results of SPC & MTD and JRDMF show
that none of targets are visible due to spreading clutter. However, the two
targets with $10$dB SNR and $5$dB SNR are uncovered from the spreading clutter
by AMPC and JRDAP. What’s more, the clutter spectrum becomes narrower and
clutter power becomes weaker. Therefore, both AMPC and JRDAP can effectively
suppress the spreading clutter. What’s more, the performance of JRDAP is very
close to that of AMPC.

TABLE II: Comparison of the average run time (in seconds). Algorithm | AMPC & CBM | JRDAP & CBM | AMPC & NCBM | JRDAP & NCBM
---|---|---|---|---
Average computational time [s] | 0.0682 | 0.0534 | 0.0482 | 0.0449

(a) SPC & MTD

(b) JRDMF

(c) AMPC

(d) JRDAP

Figure 3: Range-Doppler map in dB of (a) SPC & MTD, (b) JRDMF, (c) AMPC and
(d) JRDAP in clutter free; the targets are indicated by the white circles

(a) SPC & MTD

(b) JRDMF

(c) AMPC

(d) JRDAP

Figure 4: Range-Doppler map in dB of (a) SPC & MTD, (b) JRDMF, (c) AMPC and
(d) JRDAP in clutter and NCBM; the targets are indicated by the white circles

(a) SPC & MTD

(b) JRDMF

(c) AMPC

(d) JRDAP

Figure 5: Range-Doppler map in dB of (a) SPC & MTD, (b) JRDMF, (c) AMPC and
(d) JRDAP in clutter and CBM; the targets are indicated by the white circles

##  VII Conclusion

In this paper, we analyzed the effect of CBM on clutter modulation of DFRC
systems. The CBM based DFRC systems transmits different waveforms on a pulse
to pulse basis causing the clutter modulation and range-Doppler sidelobe
modulation. We presented the received signal model of shared waveforms and
proposed an effective MMSE based filter to estimate the targets’ impulse
response coefficients adaptively. Simulation results validated that the
proposed MMSE-based filter, named by JRDAP, can suppress the range-Doppler
sidelobe and spreading clutter effectively. What’s more, performance of
proposed JRADP is close to full-dimension adaptive multiple pulses compression
but the computational complexity of proposed JRADP is much lower than that of
full-dimension adaptive multiple pulses compression.

##  VIII Appendix

##  Appendix A PROOF OF LEMMA IV.1

$\mathbf{R}_{t,l_{1},l_{2}}(\ell)$ is formulated as

$\displaystyle\mathbf{R}_{t,l_{1},l_{2}}(\ell)$
$\displaystyle=\mathbb{E}\Bigg{\\{}\left(\sum_{q=0}^{Q-1}\mathbf{A}^{T}(\ell,q)\mathbf{s}e^{j2\pi(l_{1}-1)\psi_{q}}\right)\times$
(42)
$\displaystyle\left(\sum_{m=0}^{Q-1}\mathbf{A}^{T}(\ell,m)\mathbf{s}e^{j2\pi(l_{2}-1)\psi_{m}}\right)^{H}\Bigg{\\}}.$

Since the impulse response of targets is uncorrelated with each other, Eq.
(42) is formulated as

$\displaystyle\mathbf{R}_{t,l_{1},l_{2}}(\ell)$
$\displaystyle=\sum_{q=0}^{Q-1}\mathbb{E}\left\\{\mathbf{A}^{T}(\ell,q)\mathbf{s}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\right\\}e^{j2\pi(l_{1}-l_{2})\psi_{q}},$
(43)

Since
$\mathbb{E}\left\\{\mathbf{A}^{T}(\ell,q)\mathbf{s}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\right\\}=\sum_{n=-N+1}^{N-1}\rho(n+\ell,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}$
[22], Eq. (43) can be formulated as

$\displaystyle\mathbf{R}_{t,l_{1},l_{2}}(\ell)=$
$\displaystyle\sum_{q=0}^{Q-1}\sum_{n=-N+1}^{N-1}\rho(n+\ell,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}e^{j2\pi(l_{1}-l_{2})\psi_{q}}$
(44) $\displaystyle=$
$\displaystyle\sum_{q=0}^{Q-1}\mathbf{\Phi}(\ell,q)e^{j2\pi(l_{1}-l_{2})\psi_{q}},$

where
$\mathbf{\Phi}(\ell,q)=\sum_{n=-N+1}^{N-1}\rho(n+\ell,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}$.

##  Appendix B PROOF OF LEMMA 16

$\mathbf{R}_{c,l_{1},l_{2}}$ is formulated as

$\displaystyle\mathbf{R}_{c,l_{1},l_{2}}$ (45)
$\displaystyle=\mathbb{E}\left\\{\left(\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{l_{1}}\mathbf{a}(\theta_{i})\mathbf{C}^{T}_{i}(\ell)\mathbf{s}\right)\left(\sum_{m=1}^{N_{c}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{m})\mathbf{C}^{T}_{m}(\ell)\mathbf{s}\right)^{H}\right\\},$

since the impulse response of clutter is uncorrelated with each other, Eq.
(45) is formulated as

$\displaystyle\mathbf{R}_{c,l_{1},l_{2}}$
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{l_{1}}\mathbf{a}(\theta_{i})(\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i}))^{H}\mathbb{E}\left\\{\mathbf{C}^{T}_{i}(\ell)\mathbf{s}\mathbf{s}^{H}\mathbf{C}^{*}_{i}(\ell)\right\\}$
(46)
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{l_{1}}\mathbf{a}(\theta_{i})(\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i}))^{H}\sum_{n=-N+1}^{N-1}\sigma^{2}_{c}\mathbf{s}_{n}\mathbf{s}^{H}_{n}$
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{w}^{H}_{l_{1}}\mathbf{a}(\theta_{i})(\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i}))^{H}\mathbf{\Upsilon},$

where
$\mathbf{\Upsilon}=\sum_{n=-N+1}^{N-1}\sigma^{2}_{c}\mathbf{s}_{n}\mathbf{s}^{H}_{n}$.

##  Appendix C PROOF OF LEMMA V.1

$\tilde{\mathbf{R}}_{t,l_{1},l_{2}}(\ell)$ is formulated as

$\displaystyle\tilde{\mathbf{R}}_{t,l_{1},l_{2}}(\ell)=$
$\displaystyle\mathbb{E}\Bigg{\\{}\left(\sum_{q=0}^{Q-1}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\mathbf{u}e^{-j2\pi(l_{1}-1)\psi_{q}}\right)$
(47) $\displaystyle\times$
$\displaystyle\left(\sum_{m=0}^{Q-1}\mathbf{u}^{H}\mathbf{A}^{T}(\ell,m)\mathbf{s}e^{j2\pi(l_{2}-1)\psi_{m}}\right)\Bigg{\\}},$

since the impulse response of targets is uncorrelated with each other, Eq.
(47) is formulated as

$\displaystyle\tilde{\mathbf{R}}_{t,l_{1},l_{2}}(\ell)$ (48)
$\displaystyle=\sum_{q=0}^{Q-1}\mathbb{E}\left\\{\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\mathbf{u}\mathbf{u}^{H}\mathbf{A}^{T}(\ell,q)\mathbf{s}\right\\}e^{-j2\pi(l_{1}-l_{2})\psi_{q}}$
$\displaystyle=\mathbf{u}^{H}\left(\sum_{q=0}^{Q-1}\mathbb{E}\left\\{\mathbf{A}^{T}(\ell,q)\mathbf{s}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\right\\}e^{-j2\pi(l_{1}-l_{2})\psi_{q}}\right)\mathbf{u}$
$\displaystyle=\mathbf{u}^{H}\left(\sum_{q=0}^{Q-1}\sum_{n=-N+1}^{N-1}\rho(n+\ell,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}e^{-j2\pi(l_{1}-l_{2})\psi_{q}}\right)\mathbf{u}$
$\displaystyle=\mathbf{u}^{H}\left(\sum_{q=0}^{Q-1}\mathbf{\Phi}(\ell,q)e^{-j2\pi(l_{1}-l_{2})\psi_{q}}\right)\mathbf{u}.$

##  Appendix D PROOF OF LEMMA 26

$\tilde{\mathbf{R}}_{c,l_{1},l_{2}}$ is formulated as

$\displaystyle\tilde{\mathbf{R}}_{c,l_{1},l_{2}}$
$\displaystyle=\mathbb{E}\Bigg{\\{}\left(\sum_{i=1}^{N_{c}}(\mathbf{w}^{H}_{l_{1}}\mathbf{a}(\theta_{i}))^{H}\mathbf{s}^{H}\mathbf{C}^{*}_{i}(\ell)\mathbf{u}\right)$
(49)
$\displaystyle\times\left(\sum_{m=1}^{N_{c}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{m})\mathbf{u}^{H}\mathbf{C}^{T}_{m}(\ell)\mathbf{s}\right)\Bigg{\\}}.$

Since the impulse response of clutter is uncorrelated with each other, Eq.
(49) is formulated as

$\displaystyle\tilde{\mathbf{R}}_{c,l_{1},l_{2}}$
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{a}^{H}(\theta_{i})\mathbf{w}_{l_{1}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i})\mathbb{E}\left\\{\mathbf{s}^{H}\mathbf{C}^{*}_{i}(\ell)\mathbf{u}\mathbf{u}^{H}\mathbf{C}^{T}_{i}(\ell)\mathbf{s}\right\\}$
(50)
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{a}^{H}(\theta_{i})\mathbf{w}_{l_{1}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i})\mathbb{E}\left\\{\mathbf{u}^{H}\mathbf{C}^{T}_{i}(\ell)\mathbf{s}\mathbf{s}^{H}\mathbf{C}^{*}_{i}(\ell)\mathbf{u}\right\\}$
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{a}^{H}(\theta_{i})\mathbf{w}_{l_{1}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i})\mathbf{u}^{H}\left(\sum_{n=-N+1}^{N-1}\sigma^{2}_{c}\mathbf{s}_{n}\mathbf{s}^{H}_{n}\right)\mathbf{u}$
$\displaystyle=\sum_{i=1}^{N_{c}}\mathbf{a}^{H}(\theta_{i})\mathbf{w}_{l_{1}}\mathbf{w}^{H}_{l_{2}}\mathbf{a}(\theta_{i})\mathbf{u}^{H}\mathbf{\Upsilon}\mathbf{u}.$

##  Appendix E PROOF OF LEMMA V.3

$\bar{\mathbf{R}}_{t}(\ell)$ is formulated as

$\displaystyle\bar{\mathbf{R}}_{t}(\ell)$ (51) $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{\\{}\left(\sum_{q=0}^{Q-1}(\mathbf{v}(1)+\cdots+\mathbf{v}(P)e^{j2\pi(P-1)\psi_{q}})\mathbf{A}^{T}(\ell,q)\mathbf{s}\right)$
$\displaystyle\times$
$\displaystyle\left(\sum_{m=0}^{Q-1}(\mathbf{v}(1)+\cdots+\mathbf{v}(P)e^{j2\pi(P-1)\psi_{m}})\mathbf{A}^{T}(\ell,m)\mathbf{s}\right)^{H}\Bigg{\\}}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\sum_{q=0}^{Q-1}|\beta_{q}|^{2}\mathbf{A}^{T}(\ell,q)\mathbf{s}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\right\\}$
$\displaystyle=$
$\displaystyle\sum_{q=0}^{Q-1}|\beta_{q}|^{2}\mathbb{E}\\{\mathbf{A}^{T}(\ell,q)\mathbf{s}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\\},$

where $\beta_{q}=\mathbf{v}(1)+\cdots+\mathbf{v}(P)e^{j2\pi(P-1)\psi_{q}}$.
Since
$\mathbb{E}\\{\mathbf{A}^{T}(\ell,q)\mathbf{s}\mathbf{s}^{H}\mathbf{A}^{*}(\ell,q)\\}=\sum_{n=-N+1}^{N-1}\rho(\ell+n,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}$,
Eq. (51) can be formulated as

$\displaystyle\bar{\mathbf{R}}_{t}(\ell)$
$\displaystyle=\sum_{q=0}^{Q-1}|\beta_{q}|^{2}\sum_{n=-N+1}^{N-1}\rho(\ell+n,q)\mathbf{s}_{n}\mathbf{s}^{H}_{n}$
(52) $\displaystyle=\sum_{q=0}^{Q-1}|\beta_{q}|^{2}\mathbf{\Phi}(\ell,q).$

##  Appendix F PROOF OF LEMMA V.4

$\bar{\mathbf{R}}_{c}$ is formulated as

$\displaystyle\bar{\mathbf{R}}_{c}$ (53) $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{\\{}\sum_{i=1}^{N_{c}}\left((\mathbf{v}(1)(\mathbf{w}^{H}_{1}\mathbf{a}(\theta_{i}))+\cdots+\mathbf{v}(P)(\mathbf{w}^{H}_{P}\mathbf{a}(\theta_{i})))\mathbf{C}_{i}^{T}(\ell)\mathbf{s}\right)$
$\displaystyle\times$
$\displaystyle\sum_{i=1}^{N_{c}}\left((\mathbf{v}(1)(\mathbf{w}^{H}_{1}\mathbf{a}(\theta_{i}))+\cdots+\mathbf{v}(P)(\mathbf{w}^{H}_{P}\mathbf{a}(\theta_{i})))\mathbf{C}_{i}^{T}(\ell)\mathbf{s}\right)^{H}\Bigg{\\}},$

since the impulse response of clutter is uncorrelated with each other, Eq.
(53) is formulated as

$\displaystyle\bar{\mathbf{R}}_{c}$
$\displaystyle=\sum_{i=1}^{N_{c}}|\beta_{i}|^{2}\mathbb{E}\\{\mathbf{C}^{T}_{i}(\ell)\mathbf{s}\mathbf{s}^{H}\mathbf{C}^{*}_{i}(\ell)\\}$
(54)
$\displaystyle=\sum_{i=1}^{N_{c}}|\beta_{i}|^{2}\sum_{n=-N+1}^{N-1}\sigma^{2}_{c}\mathbf{s}_{n}\mathbf{s}^{H}_{n}$
$\displaystyle=\sum_{i=1}^{N_{c}}|\beta_{i}|^{2}\mathbf{\Upsilon},$

where
$\beta_{i}=\mathbf{v}(1)\mathbf{w}^{H}_{1}\mathbf{a}(\theta_{i})+\cdots+\mathbf{v}(P)\mathbf{w}^{H}_{P}\mathbf{a}(\theta_{i})$.

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