# Generalized Fully Coherent Closed-form Receiver Design for Joint Radar and
Communication System

Muhammad Zubair, Sajid Ahmed, Senior Member, IEEE, and Mohamed-Slim Alouini,
Fellow, IEEE
King Abdullah University of Science and Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In conventional radar, the transmission of the same waveform is repeated after
a predefined interval of time called pulse-repetition-interval (PRI). This
technique helps to estimate the range and Doppler shift of targets and
suppress clutter. In dual-function radar communication (DFRC), different
waveforms are transmitted after each PRI. Thus, each waveform yields different
range-side-lobe (RSL) levels at the receiver’s output. As a consequence,
Doppler shift estimation and clutter suppression become challenging tasks. A
state-of-the-art (SOTA) method claims that if the number of waveforms is more
than two, it is impossible to achieve fully coherent RSL levels with both
waveforms. Therefore, this algorithm uses iterative methods to achieve as much
as possible coherency and minimize the RSL levels. In contrast to that SOTA
method, we proposed two novel closed-form receivers for the DFRC that yield a
fully coherent response for several waveforms and suppress the RSL levels.
Experimental results demonstrate that the proposed receivers achieve full
coherency and the RSL levels are significantly lower than the conventional
method.

###### Index Terms:

Joint radar communication, constrained optimization, Closed-form receive
filters.

##  I Introduction

In wireless communication, the radio spectrum is the most precious and
expensive resource. Radio spectrum extending from 1MHz to 300GHz has favorable
propagation characteristics, therefore, this spectrum is being used for
different purposes. The use of this band for different applications is
summarized in Table I. There are currently 7.9 billion mobile subscriptions
and it is predicted that the total mobile subscriptions by the end of 2026
will reach 8.8 billion. Similarly, the overall Internet-of-Things (IoT)
connections have reached 12.6 billion and in 2026 they are expected to reach
26.9 billion. The fifth-generation (5G) standard was introduced in 2020 to
accommodate the high demand for subscriptions within the available spectrum.
The 5G will be much faster, reliable, more efficient, and will support data
rates up to 35.46 Gbps [1]. It will use frequency range 1 (FR1), also known as
a sub-6GHz band (2-7.125GHz), and frequency range 2 (FR2), known as mm-Wave
radio band (24.25-52.6GHz), as shown in Table. I. Future applications, such as
holographic calls and flying networks, will require even more high data rates.
In this perspective quest for 6G and 7G has been already started [1, 2, 3, 4,
5]. In the 6G and 7G, the integration of 5G, satellite network, and space
roaming will be carried out. Spacecraft, drones, and radars will be considered
a simple IoT devices [6, 7, 8].

As cellular communication systems have evolved from 1G to 5G (please see Table
I) and on the way to 6G, radar systems are also evolving at a much faster
pace. Until the second world war, radar systems had a significant size, were
bulky, power-hungry, and the cost was enormous. Moreover, the use of radar was
restricted to the military only. Now, the size and cost of radar have
significantly reduced, and it is entering the consumer market. For example,
now the radars are being used in the automotive industry, medical devices for
non-invasive testing, gesture control smart devices, and wearable devices [9,
10, 11, 12].

Until now, radar and cellular communication systems have been developed and
studied separately. Both systems use their frequency spectrum . In Table I, it
is summarized how radar bands for different applications underutilized the
limited radio spectrum. Now the evolution of applications requiring high data
rate communication has completely occupied the limited radio spectrum and
started to overlap with existing radar bands as shown in Table I. The scarcity
of radio spectrum urged researchers to develop new solutions to overcome this
problem. One of the possible solutions is the integration of both systems. As
both technologies rely on the same electromagnetic radiation phenomena, the
same equipment, spectrum, and signals can be used for both systems. For
example, suppose that the vicinity objects reflect the transmitted signal to
convey information from point A to B. These reflected signals from objects can
be received at point A and processed to estimate the position and velocity of
various objects. Therefore, to efficiently use the available spectrum for both
communication and radar, extensive research is being done to devise new
techniques. The spectrum sharing and the use of identical waveforms for
communications and radar will likely be the main feature of future
communications systems in 5G and beyond.

Band |  Frequencies |  Wavelengths |  Applications |  Evolution; 1G-5G and beyond
---|---|---|---|---
HF |  3 - 30MHz |  100 - 10m |  Over the horizon surveillance radar |
VHF |  30 - 300MHz |  10 - 1m |  Long range surveillance, Counter stealth |  $\left.{\begin{gathered}\hfill\rotatebox{90.0}{4G, 3G, 2G, 1G }\end{gathered}}\right\downarrow{}$ $\left.{\begin{gathered}\hfill\\\ \rotatebox{90.0}{5G(7.125GHz )}\\\ \end{gathered}}\right\downarrow{\rotatebox{90.0}{FR1}}$
UHF |  300MHz - 1GHz |  1 - 30cm |  Long range surveillance, Counter stealth
L |  1 - 2GHz |  30 - 15cm |  Air traffic control
S |  2 - 4GHz |  15 - 7.5cm |  Weather observation
C |  4 - 8GHz |  7.5 - 3.75cm |  Short range, Missile guidance |
X |  8 - 12GHz |  3.75 - 2.5cm |  Unmanned aerial vehicle, Mobile/Fixed service on primary basis |  FR3: More challenges due to attenuation, currently under study by ITU.
Ku |  12 - 18GHz |  2.5 - 1.67cm |  Police radar, Mobile/Fixed service on primary basis |
K |  18 - 27Ghz |  1.67 - 1.11cm |  Airport surveillance |  $\qquad\quad\left.{\begin{gathered}\rotatebox{90.0}{5G(mm Wave)}\\\ \end{gathered}}\right\downarrow{\rotatebox{90.0}{FR2}}$
Ka |  27 - 40 Ghz |  1.11 - 7.5mm |  Scientific remote sensing
mm |  40 - 52.6GHz |  7.5 - 5.7mm |  Automobile cruise control, NR-U, NR-Light
FR2+4) |  57 - 71GHz |  5.3 - 4.2mm |  Automobile cruise control, NR-Light
FR4 |  52.6 - 114.25GHz |  5.7 - 2.6mm |  Automobile cruise control |
FR5 (D band) |  110 - 170 GHz |  2.7 - 1.8mm |  Automobile cruise control, Fundamental 6G research |  $\qquad\quad\left.{\begin{gathered}\rotatebox{90.0}{6G(sub-THz)}\\\ \end{gathered}}\right\downarrow{\rotatebox{90.0}{FR5}}$
FR5 (G band) |  140 - 220 GHz |  2.1 - 1.4mm |  Automobile cruise control, Fundamental 6G research
FR5 |  220 - 275GHz |  2.1 - 1.1mm |  Automobile cruise control
|  252 - 325GHz |  1.2 - 0.923mm |  Automobile cruise control,  802.15.3d-2017 (P2P communication with 100 Gbps) |  $\qquad\quad\left.{\begin{gathered}\rotatebox{90.0}{6G (THz) }\hfill\\\ \end{gathered}}\right\downarrow{}$
|  0.3 - 3THz |  1mm - 100um |  Automobile cruise control
Table I: Frequency spectrum overlapping by radar and communication. In
application column bold and normal fonts respectively represent use of
frequency band for communication and radar [13, 14]. FR2+4) represents the
addition of FR4 frequency band from 52.6GHz to 71GHz to FR2 for the evolution
of 5G applications named as New radio Light(NR-Light) or reduced capability
(RedCap) devices.

In the literature, there are mainly four categories of joint radar and
communication research, which are as follows:

  1. 1.

Spectrum Sharing: In this technique, both communication and radar systems
share the same spectrum when it is free from the other [15, 16, 17, 18, 19,
20].

  2. 2.

Communication Centric Joint Communication Radar (JCR): In JCR, radar
parameters are estimated from the transmission of the communication system
[21, 22, 23, 24].

  3. 3.

Radar Centric Joint Radar Communication (JRC): In JRC technique, communication
is realized on a primary radar system. The pioneering work in this respect is
presented in [25, 26], where OFDM communication is employed on MIMO radar.

  4. 4.

dual function radar communication (DFRC): In DFRC, radar and communication
functions are implemented on a single hardware platform. Here, a single
waveform serve the purpose of both data communication and radar parameter
estimation. Based on the system model, DFRC can be further classified into two
categories: i) Communication Centric DFRC [27, 28, 29, 30] , and ii) Radar
Centric DFRC [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. Both of these
technologies rely either on the design of receive filter or waveform design.

Although, all of the above techniques, i.e., spectrum sharing, JCR, and JRC
have their significance but DFRC due to to its single hardware for dual
functionality has more applications compared to the other methods. Therefore,
the focus of this paper will be on DFRC, and its unaddressed problem will be
solved.

For example, the biggest problem that arises in the DFRC system is the
detection and estimation of radar parameters. Because, due to vast radar-
cross-section (RCS) and spread of clutter (the electromagnetic return from
mountains, ground, and buildings ), it shows high peak values in the range-
side-lobes(RSLs) that can mask weak targets. Moreover, due to different
waveforms in CPI, RSLs are not coherent that can degrade Doppler shift
estimation. To address this problem, state-of-the-art (SOTA) research designs
unique waveforms or receive filters for the suppression of the RSLs, and
coherent response for different waveforms. For instance, to design waveforms
for DFRC, one closed-form and two iterative algorithms are proposed in [27].
The proposed algorithms satisfy constant modulus and similarity constraints.
However, none of them addresses the suppression of RSLs. Therefore, in [28] to
reduce RSLs, integrated-side-lobes constraint is incorporated in the objective
function. Similarly, further developments on DFRC waveforms are presented in
[36, 37, 38, 39, 40, 41, 27, 28, 29, 30, 31, 32, 33, 34, 42]. For DFRC,
iterative receive filters are proposed in [43, 44, 45]. The proposed iterative
receive filters suppress the RSLs, but the RSLs of receive filters
corresponding to different waveforms are non-coherent . Due to this problem,
they fail to perform well.

Contributions: Two main contribution of this paper are:

  1. 1.

In this work, we transformed the problem into a constrained optimization
problem that guarantees a fully coherent receiver for any number of waveforms.
While the existing research claims that the design of the fully coherent
receiver is not possible for more than two waveforms.

  2. 2.

Two generalized closed-form receivers are designed based on linear and
circular convolution model for multiple waveforms. The output response of both
receivers is fully coherent and the RSL levels are also lower than the
existing iterative method.

Paper Organization:
The rest of the paper is organized as follows: The problem formulation is
discussed in Sec. II, proposed closed-form receivers design is described in
Sec. III. The receive signal model for radar and communication is discussed in
Sec. IV. Experimental results are discussed in Sec. VI, and conclusions are
drawn in Sec. VI.
Notations: Bold upper case letters, ${\bf X}$ represents matrices and lower
case letters, ${\bf x}$, denote vectors. Transpose and conjugate transposition
of a matrix are denoted by $(\cdot)^{T}$ and $(\cdot)^{H}$, respectively. The
conjugate of a scalar is denoted by $(\cdot)^{*}$ . Convolution operator is
denoted by $*$ and the close interval $\\{x:a\leq x\leq b\\}$ is denoted as
$[a,b]$.

##  II Problem Formulation

Consider a joint radar-communication scenario shown in Fig. 1, where a base
station (BS) intends to communicate with multiple mobile-users (MU) and
localize various targets using the transmitted waveforms for communication.
Depending on the antenna’s beampattern, the transmitted signal propagates in
different directions. It is received by multiple objects, such as mobile user,
vehicle, aeroplane, ship, ground, and buildings, present at different range
bins. In the figure 1, blue and yellow waveforms respectively represent the
transmitted signal parts for radar and communication, while the red waveform
represents the transmitted signal received by the stationary objects. If the
RCS of an object is sufficiently large, part of the transmitted signal energy
will be reflected back and it can be used to localize that object. A
significant portion of the reflected signal energy is from the ground and
buildings due to their large RCS. However, we are not interested in localizing
them. Therefore, the received signal from such targets is called clutter. Due
to the vast RCS of clutter compared to the objects we are interested in, it
becomes challenging to estimate these object’s parameters. However, the
exploitation of Doppler shift of moving objects can help to localize them.

Figure 1: Joint Radar-Communication scenario. Transmitted signal from the
base-station is used for radar (blue waveform) and communication (yellow
waveform) functions. The reflected signal from the stationary objects is
considered as a clutter signal.
Authors are thankful for the illustration created by Heno Hwang. Scientific
Illustrator at Research Communication and Publication Services. Office of the
Vice President for Research - King Abdullah University of Science and
Technology.

In conventional radar systems, a coherent-processing-interval (CPI) technique
is used to estimate the target’s Doppler shift. In this technique, $N$
identical waveforms are transmitted in each CPI. Receiver’s output identical
index samples from $N$ waveforms are called slow time samples. To estimate the
Doppler shift of a target fast-Fourier-transform (FFT) is applied on slow time
samples in the CPI [46]. However, for the joint radar-communication system, a
non-coherent processing interval (NCPI) technique is required that can allow
$N$ different waveforms in each processing interval to transmit various
communication bits in each waveform. For the transmission of $M$ symbols, the
NCPI technique is illustrated in Fig. 2. For a particular symbol, the system
can select a corresponding waveform to transmit from $K$ waveforms. Each
waveform/symbol carries $\log_{2}(K)$ bits. In the NCPI technique, since
waveforms are different, it becomes challenging to estimate Doppler shift by
applying FFT on slow time samples. The second issue arises due to the range
side-lobes. After matched-filtering, clutter due to its high RCS can produce
high range side-lobe values that can mask small targets. Moreover, different
waveforms can have different range side-lobe values for the same clutter,
affecting the receiver’s sensitivity, especially for small targets. Therefore,
for each waveform, range side-lobe values should be coherent and as-small-as
possible. To address this problem, in the sequel, we design receive filters
for $K$ waveforms and constrain them to yield coherent output responses.

If $\Psi_{k}(t)$ is the transmitted waveform, $h_{k}(t)$ is the corresponding
receive filter to be designed, and $e_{k}(t)$ is the desired output response,
mathematically, the receive filter design problem can be expressed as

$\displaystyle\Psi_{k}(t)*h_{k}(t)=e_{k}(t),\quad k=1,2,\ldots,K\hskip
50.00008pt$ subject to constraint
$\displaystyle\Psi_{1}(t)*h_{1}(t)=\Psi_{2}(t)*h_{2}(t)=\cdots=\Psi_{K}(t)*h_{K}(t).$
(1)

The solution of (1) yields optimal receive filters that can produce coherent
response for non-coherent waveforms.

Figure 2: Embedding data bits in radar pulses. Here, $\Psi_{k}$ is the $k$th
waveform, $T$ is the pulse duration, $T_{\textrm{PRI}}$ is pulse repetition
interval, and NCPI is the non-coherent processing interval.

The performance of the filter is assessed for differential phase shift keying
(DPSK) and minimum-phase-shift keying (MSK) symbols. Due to continuous phase
change, DPSK and MSK waveform gives narrow bandwidth and good auto-correlation
compared to the other phase modulation schemes. DPSK waveforms can be designed
in two steps. In the first step, a digital constant envelope waveform using
binary random sequence is generated as

$\displaystyle
s_{d}(t)=\sum_{n=0}^{N-1}e^{j\pi\left(\frac{x(n)+1}{2}\right)}\left[u\left(t-n\tau_{c}\right)-u\left(t-(n+1)\tau_{c}\right)\right],$

where $x(n)\in\pm 1$ generated from Gaussian distribution, $u(t)$ is the unit
step function, and $\tau_{c}$ is chip interval. While in the second step,
pulse shaping is applied across bit boundaries to convert discontinuous phase
transition to continuous. Thus final DPSK waveform for baseband frequency
$f_{b}=\frac{1}{2\tau_{c}}$ can be expressed as

$\displaystyle\Psi_{\textrm{DPSK}}(t)$
$\displaystyle=s_{d}\left(t-\frac{\tau_{c}}{2}\right)\left|\cos\left(2\pi
f_{b}t\right)\right|+js_{d}(t)\left|\sin\left(2\pi f_{b}t\right)\right|$
$\displaystyle=s_{c}\left(t-\frac{\tau_{c}}{2}\right)+js_{s}(t)\quad,0\leq
t\leq T,$ (2)

where $s_{c}(t)=s_{d}\left(t-\frac{\tau_{c}}{2}\right)\left|\cos(2\pi
f_{b}t)\right|$ and $s_{s}(t)=s_{d}(t)\left|\sin(2\pi f_{b}t)\right|$. If
$\tau_{c}$ is a single chip interval, for $N$ chips, $T=N\tau_{c}$. The time
domain passband version of complex envelope or baseband signal defined in (2)
is as follows,

$\displaystyle\Psi_{p}(t)$
$\displaystyle=s_{\mbox{\tiny{c}}}\left(t-\frac{\tau_{c}}{2}\right)\cos\left(2\pi
f_{c}t\right)-{s}_{\mbox{\tiny{s}}}(t)\sin(2\pi f_{c}t).$ (3)

Now the relationship between passband and baseband in exponential form can be
expressed as,

$\displaystyle\Psi_{p}(t)$
$\displaystyle=\mbox{Re}\left(\Psi_{\mbox{\tiny{DPSK}}}(t)e^{j2\pi
f_{c}t}\right)$ (4)

Envelope and Phase of Passband Signal:
The baseband signal $\Psi_{p}(t)$ can be expressed in polar form. Define the
envelope and phase $\theta(t)$ as

$\displaystyle\left|\Psi_{\mbox{\tiny{DPSK}}}(t)\right|$
$\displaystyle=\sqrt{{s}_{\mbox{\tiny{c}}}^{2}\left(t-\frac{\tau_{c}}{2}\right)+{s}_{\mbox{\tiny{s}}}^{2}(t))},$
$\displaystyle\theta$
$\displaystyle=\tan^{-1}\left(\frac{{s}_{\mbox{\tiny{s}}}(t)}{{s}_{\mbox{\tiny{c}}}(t-\frac{\tau_{c}}{2})}\right).$

Substituting
$\Psi_{\mbox{\tiny{DPSK}}}(t)=\left|\Psi_{\mbox{\tiny{DPSK}}}(t)\right|e^{j\theta(t)}$
into (4), we get another form of passband signal

$\displaystyle\Psi_{\mbox{\tiny{P}}}(t)=\left|\Psi_{\mbox{\tiny{DPSK}}}(t)\right|\cos(2\pi
f_{c}t+\theta(t)).$ (5)

Similar to DPSK, minimum-shift-keying (MSK) can also be used. MSK uses binary
codes and can be expressed as [35]

$\Psi_{\textrm{MSK}}(t)=e^{-j(\theta_{o}+s_{d}(t)\pi f_{b}t)},$ (6)

where $\theta_{o}$ is the final phase value from previous chip interval. In
the next section, DPSK and MSK waveforms will be used to solve the constrained
optimization problem in (1).

##  III Proposed Closed-form Receive Filter Design

To solve the constrained optimization problem defined in (1), an iterative
algorithm is proposed in [35]. The proposed algorithm reduces the RSLs but
does not achieve good coherency. Therefore, the algorithm is good to suppress
range side-lobe levels but not good for Doppler shift measurement. However,
the proposed solution in [35] can achieve fully coherent RSLs for only two
waveforms. This means, one can only transmit waveforms having only two bits of
information. Similarly, in [47] the same group of [35], for fixed system
parameters using a continuous phase modulation (CPM) framework [43], and based
on match and mismatch filters proposed a cascaded filter design. Though, using
these cascaded filters, one can enhance the coherency to reduce the RSMs but
at the expense of fixed system parameters and higher sidelobe levels (that can
reduce the sensing capability of the radar system). The motivation of our
proposed work is to provide a generalized closed-form solution applicable for
multiple waveforms, thus achieving the capacity to communicate any number of
information bits and without sacrificing the performance of a DFRC system.

To design our proposed receive filters, suppose there are $K$ waveforms as
expressed in (2) and (6). Considering $x_{k}(p)$ and $h_{k}(l)$ are $p$th and
$l$th sampled values of $\Psi_{k}(t)$ and $h_{k}(t)$, respectively. Using
sample values, the vector-form of convolution in (1) can be written as

$\displaystyle{\bf\Psi}_{k}{\bf h}_{k}={\bf e}_{k},$ (7)

where

$\displaystyle{\bf\Psi}_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}x_{k}(0)&0&\cdots&0\\\
x_{k}(1)&x_{k}(0)&\ddots&\vdots\\\ \vdots&x_{k}(1)&\ddots&0\\\
x_{k}(L-1)&\vdots&\ddots&x_{k}(0)\\\ 0&x_{k}(L-1)&&x_{k}(1)\\\
\vdots&\ddots&\ddots&\vdots\\\ 0&\cdots&0&x_{k}(L-1)\end{bmatrix}$
$\displaystyle{\bf h}_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}h_{k}(0)&h_{k}(1)&\cdots&h_{k}(L_{f}-1)\end{bmatrix}^{T},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}$ $\displaystyle{\bf
e}_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0&0&\cdots&1&\cdots&0&0\end{bmatrix}^{T},$ (8)

where ${\Psi}_{k}\in{\cal C}^{(L+L_{f}-1)\times L_{f}}$. Similarly, the
constraints in (1) can also be written as

$\displaystyle{\bf\Psi}_{1}\mathbf{h}_{1}={\bf\Psi}_{2}\mathbf{h}_{2}=\cdots={\bf\Psi}_{K}\mathbf{h}_{K}.$
(9)

Using (8) and (9), to design receive filters for $K$ waveforms, the objective
function in (1) can be defined as

$\displaystyle{\bf Xh=e}$ (10) $\displaystyle\mbox{and }\tilde{\bf X}{\bf
h=0},$ (11)

where

$\displaystyle\mathbf{X}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}{\bf\Psi}_{1}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&{\bf\Psi}_{2}&\cdots&\mathbf{0}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}&\mathbf{0}&\cdots&{\bf\Psi}_{K}\end{bmatrix},$
$\displaystyle\tilde{\bf X}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}{\bf\Psi}_{1}&-{\bf\Psi}_{2}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&{\bf\Psi}_{2}&-{\bf\Psi}_{3}&\cdots&\mathbf{0}&\mathbf{0}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&{\bf\Psi}_{K-1}&-{\bf\Psi}_{K}\end{bmatrix},$
$\displaystyle\mathbf{h}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbf{h}_{1}^{T}&\mathbf{h}_{2}^{T}&\cdots\mathbf{h}_{K}^{T}\end{bmatrix}^{T},$
$\displaystyle\mathbf{e}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbf{e}_{1}^{T}&\mathbf{e}_{2}^{T}&\cdots\mathbf{e}_{K}^{T}\end{bmatrix}^{T},\
$

where ${\bf X}\in{\cal C}^{(K(L+L_{f}-1))\times KL_{f}}$ and $\tilde{\bf
X}\in{\cal C}^{(K-1)(L+L_{f}-1)\times KL_{f}}$.

Using (10) and (11), the constrained optimization problem to estimate ${\bf
h}$ can be redefined as

$\displaystyle\underset{{\bf h}}{\text{minimize}}\left|\left|{{\bf X}{\bf
h}-{\bf e}}\right|\right|^{2}$ subject to
$\displaystyle\qquad\qquad{\mathbf{\tilde{X}}{\bf h}={\bf 0}}.$ (12)

The above constrained optimization problem can be solved using the Lagrangian
multiplier, and the cost-function to be minimised can be written as

$\displaystyle J=$
$\displaystyle\left|\left|{\mathbf{Xh-e}}\right|\right|^{2}+{\bf
h}^{H}\tilde{\bf X}^{H}{\bm{\lambda}},$ (13)

where vector ${\bm{\lambda}}$ is called Lagrangian multiplier. The
minimization of (13) with respect to ${\bf h}$ yields

${\bf h}=\left({\bf X}^{H}{\bf X}\right)^{-1}\left[{\bf X}^{H}{\bf
e}-\tilde{\bf X}^{H}{\bm{\lambda}}\right]$ (14)

Substituting (14) back into (13) yields

$\displaystyle{\bm{\lambda}}$ $\displaystyle={\bf D}^{-1}\tilde{\bf
X}\left({\bf X}^{H}{\bf X}\right)^{-1}{\bf X}^{H}{\bf e},$ (15)

where ${\bf D}=\left(\tilde{\bf X}\left({\bf X}^{H}{\bf
X}\right)^{-1}\tilde{\bf X}^{H}\right)$. Now substituting (15) into (14) we
get

${\bf h}=\left({\bf X}^{H}{\bf X}\right)^{-1}{\bf X}^{H}{\bf e}-\left({\bf
X}^{H}{\bf X}\right)^{-1}{\tilde{\bf X}^{H}}{\bf D}^{-1}{\tilde{\bf
X}\left({\bf X}^{H}{\bf X}\right)^{-1}}{\bf X}^{H}{\bf e}.$ (16)

Remark: The derived estimation vector ${\bf h}$ in (16) contains two inverse
matrices, i.e. $({\bf X}^{H}{\bf X})^{-1}$ and ${\bf D}^{-1}=\tilde{\bf
X}\left({\bf X}^{H}{\bf X}\right)^{-1}\tilde{\bf X}^{H}$. To make the matrix
$({\bf X}^{H}{\bf X})$ non-singular, the dimensions of ${\bf X}$ i.e, ${\cal
C}^{(K(L+L_{f}-1))\times KL_{f}}$ must fulfill the bound $KL_{f}\leq
K(L+L_{f}-1)$. Similarly, for the matrix ${\bf D}$ to be full rank, the
dimensions of $\tilde{\bf X}\in{\cal C}^{(K-1)(L+L_{f}-1)\times KL_{f}}$ must
fulfill the bound $(K-1)(L+L_{f}-1)\leq KL_{f}$ and this simply impose
$L\leq\frac{L_{f}}{K-1}+1$. Another way to express it for ill condition
avoidance is $(K-1)(L+L_{f}-1)\leq KL_{f}\leq K(L+L_{f}-1)$.

In (12), instead of using tall linear convolution matrix, ${\bf\Psi}_{k}$,
using square circular convolution matrix, ${\bf\Psi}_{ck}$, can be defined as

$\displaystyle{\bf\Psi}_{ck}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}x_{k}(0)&0&\cdots&\cdots&x_{k}(1)\\\
\vdots&\ddots&\ddots&\vdots&\vdots\\\
x_{k}(L-1)&&x_{k}(0)&\ddots&x_{k}(L-1)\\\ \vdots&\ddots&\vdots&\ddots&{\bf
0}\\\ {\bf 0}&\cdots&x_{k}(L-1)&\cdots&x_{k}(0)\end{bmatrix},$

ill matrix condition can be removed. The corresponding desired output response
of the designed receive filter, ${\bf h}_{ck}$, can be written as

$\displaystyle{\bf\Psi}_{ck}{\bf h}_{ck}={\bf e}_{ck}.$ (17)

In (17), ${\bf\Psi}_{ck}\in{\cal C}^{(L+L_{f}-1)\times(L+L_{f}-1)}$. Following
similar strategy used to generate ${\bf X}$ and $\tilde{\bf X}$ with
${\bf\Psi}_{k}$s, we can generate ${\bf X}_{c}$ and $\tilde{\bf X}_{c}$ with
${\bf\Psi}_{ck}$ and can find corresponding ${\bf h}_{c}$ by solving the
following constrained optimization problem,

$\displaystyle\underset{\mathbf{h}_{c}}{\text{minimize}}\left|\left|{\mathbf{X_{\textit{c}}h_{\textit{c}}-e_{\textit{c}}}}\right|\right|^{2}$
subject to
$\displaystyle\qquad\qquad{\mathbf{\tilde{X}_{\textit{c}}h_{\textit{c}}}={\bf
0}},$ (18)

where the dimension of ${\bf X}_{\textit{c}}\in{\cal
C}^{(K(L+L_{f}-1))\times(K(L+L_{f}-1))}$ and $\tilde{\bf
X}_{\textit{c}}\in{\cal C}^{(K-1)(L+L_{f}-1)\times K(L+L_{f}-1)}$, while the
dimensions of $\bf{h_{\textit{c}}}$ and $\bf{e_{\textit{c}}}$ are adjusted
accordingly. Solving (18) yields,

$\displaystyle{\bf h_{\textit{c}}}$ $\displaystyle=$ $\displaystyle\left({\bf
X}^{H}_{\textit{c}}{\bf X}_{\textit{c}}\right)^{-1}{\bf
X}^{H}_{\textit{c}}{\bf e}_{\textit{c}}-$ (19) $\displaystyle\left({\bf
X}^{H}_{\textit{c}}{\bf X}_{\textit{c}}\right)^{-1}{\tilde{\bf
X}}^{H}_{\textit{c}}{\bf D}^{-1}_{\textit{c}}{\tilde{\bf
X}_{\textit{c}}\left({\bf X}^{H}_{\textit{c}}{\bf
X}_{\textit{c}}\right)^{-1}}{\bf X}^{H}_{\textit{c}}{\bf e}_{\textit{c}},$

where ${\bf D}_{c}=\left({\tilde{\bf X}}_{c}\left({\bf X}^{H}_{c}{\bf
X}_{c}\right)^{-1}\tilde{\bf X}^{H}_{c}\right)$.

Figure 3: (a) Iterative receiver proposed in [35], RSL are not coherent, (b)
first proposed receiver using linear convolution modelling, RSL are fully
coherent and RSL levels are lover than [35], (c) second proposed receiver, RSL
are fully coherent and RSL levels are lover than [35].

##  IV Receive Signal Model

Receive signal can be model in two perspective, i.e., radar and communication.
For communication, generally different frequencies are transmitted for up- and
down-link channels. Therefore, for communication, an independent receiver can
be modeled at the base-station. The design of receiver for radar is a
challenging task, therefore, in the following discussion more focus will be on
it.

###  IV-A Receive Signal Model for Radar

Suppose a DFRC system, as shown in Fig.1, transmitting $M$ waveforms separated
by a PRI of $T_{\textit{PRI}}$. Consider, we have $P$ scaterers located at
different ranges and angles. Some of the scatterers are stationery, some are
slow moving, while some of them are fast moving. We are interested in finding
the fast moving ones. The stationary scatterers can be termed as clutter,
while the slow moving and fast moving can be categorized as the targets.
Consider a scatterer $i$ at the range $R_{i}$ from the transmitter. Since
transmitter and receiver are co-located, the echo from the target will be
received after time $\tau_{i}=\frac{2R_{i}}{c}$. When $m$th pulse is sent the
range of the target will be $R_{i}\pm mv_{i}T_{\textit{PRI}}$ depending the
target is moving toward or away from the base-station. Note that, it is
assumed that the target movement is negligible during NCPI but there will be a
phase shift after each symbol, it is valid assumption because EM wave moves
with the speed of light [48]. For such scenarios, the received signal due to
the $m$th transmitted signal, defined in (3), by ignoring noise can be written
as

$\displaystyle
r_{m}(t)=\sum_{i=1}^{P}\beta_{i}\Psi_{p}\left(t-\frac{2(R_{i}-m{v_{i}}T_{\textrm{PRI}})}{c}\right),$
(20)

where $m=0,\ldots,M-1$ and $\beta_{i}$ is the reflection coefficient of $i$th
scaterer. Using (3), (20) can be written as

$\displaystyle r_{m}(t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{P}\beta_{i}\bigg{[}\tilde{s}_{c}\left(t-\tau_{i}-\frac{\tau_{c}}{2}\right)\cos\left(2\pi
f_{c}t+\phi(t,R_{i})\right)$ (21) $\displaystyle\quad\quad\leavevmode\nobreak\
-\tilde{s}_{s}\left(t-\tau_{i}\right)\sin\left(2\pi
f_{c}t+\phi(t,R_{i})\right)\bigg{]},$

where $\phi(t,R_{i})=2\pi f_{di}mT_{\textrm{PRI}}-\frac{4\pi R_{i}}{\lambda}$
and ${\tau_{i}\leq t\leq\tau_{i}+\tau}$ while
$\tau_{i}=\frac{2(R_{i}-m{v_{i}}T_{PRI})}{c}$. Demodulating (21), in-phase,
$I(t)$, and quadrature-phase, $Q(t)$, components of the baseband received
signal due to the clutter can be written as

$\displaystyle I(t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{P}\beta_{i}\bigg{[}s_{\mbox{\tiny{c}}}\left(t-\tau_{i}-\frac{\tau_{c}}{2}\right)\sin\left(2\pi
f_{d}mT_{\textit{PRI}}-\frac{4\pi R_{i}}{\lambda}\right)$
$\displaystyle+s_{\mbox{\tiny{s}}}\left(t+\tau_{i}\right)\cos\left(2\pi
f_{d}mT_{\textit{PRI}}-\frac{4\pi R_{i}}{\lambda}\right)\bigg{]}.$
$\displaystyle Q(t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{P}\beta_{i}\bigg{[}s_{\mbox{\tiny{c}}}\left(t-\tau_{i}-\frac{\tau_{c}}{2}\right)\cos\left(2\pi
f_{d}mT_{\textit{PRI}}-\frac{4\pi R_{i}}{\lambda}\right)$ $\displaystyle-
s_{\mbox{\tiny{s}}}\left(t-\tau_{i}\right)\sin\left(2\pi
f_{d}mT_{\textit{PRI}}-\frac{4\pi R_{i}}{\lambda}\right)\bigg{]}.$

Now combining $I(t)$ and $Q(t)$, the baseband received signal due to the
clutter can be written as

$\displaystyle y_{m}(t)=I(t)+jQ(t).$ (22)

Further solving (22), the baseband received echo can be expressed as

$y_{m}(t)=\sum_{i=1}^{P}\beta_{i}\Psi_{\textrm{DPSK}}\left(t-\tau_{i}\right)e^{j(2\pi
f_{di}mT_{\textit{PRI}}-\frac{4\pi R_{i}}{\lambda})}.$ (23)

In discrete vector form, $y_{m}(t)$ can be written as

$\displaystyle{\bf
y}_{m}={\Psi}_{m}\left({\bm{\beta}}\odot{\mathbf{s}}(\phi)\odot{\mathbf{a}}(f_{d})\right),$
(24)

where,

$\displaystyle{\bf y}_{m}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}y_{m}(0)&y_{m}(1)&\cdots&y_{m}(L+L_{f}-1)\end{bmatrix}^{T},$
$\displaystyle{\bf\Psi}_{m}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}x_{m}(0)&0&\cdots&0\\\
x_{m}(1)&x_{m}(0)&\ddots&\vdots\\\ \vdots&x_{m}(1)&\ddots&\\\
x_{m}(L-1)&\vdots&\ddots&x_{m}(0)\\\ 0&x_{m}(L-1)&&x_{m}(1)\\\
\vdots&\ddots&\ddots&\vdots\\\ 0&\cdots&\cdots&x_{m}(L-1)\end{bmatrix},$
$\displaystyle{\bm{\beta}}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\beta_{1}&\beta_{1}&\cdots&\beta_{P}\end{bmatrix}^{T},$
$\displaystyle{\bf s}(\phi)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}e^{-\frac{j4\pi R_{1}}{\lambda}}&e^{-\frac{j4\pi
R_{2}}{\lambda}}&\cdots&e^{-\frac{j4\pi R_{P}}{\lambda}}\end{bmatrix}^{T},$
$\displaystyle{\bf a}(f_{d})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}e^{j2\pi f_{d1}mT_{\textit{PRI}}}&\cdots&e^{j2\pi
f_{dP}mT_{\textit{PRI}}}\end{bmatrix}^{T}.$

In (24), ${\bf\Psi}_{m}$ is the convolution matrix of $m$th waveform,
${\bm{\beta}}$ is the vector of reflection coefficients of $P$ scatterers,
${\bf s}(\phi)$ is the phase delay vector, and ${\bf a}(f_{d})$ is the Doppler
vector contains the Doppler shift of different scatterers.

###  IV-B Receive Signal Model for Communication

The receive signal model for communication system at the base-station or user
end is very straight forward. The baseband receive signal for the $m$th
transmitted waveform can be written as

$\displaystyle c_{m}(t)=\sum_{l=0}^{L-1}g_{l}e^{j2\pi f_{d}t}x_{m}(t)+v(t),$
(25)

where $L$ is the number of paths, $g_{l}$ is the path gain of $l$th path, and
$f_{d}$ is the Doppler shift due to the relative motion between the
transmitter and receiver. The transmitted symbols and channel can be easily
found using the conventional techniques [49].

(a)

(b)

(c)

Figure 4: Detection of six targets at CNR = $50$dB and SNR $10$dB. (a)
Iterative receiver proposed in [35], closely spaced targets are difficult to
distinguish, (b) 1st proposed receiver, all targets can be easily identified,
(c) 2nd proposed receiver, all targets can be easily identified.

(a)  (b)  (c)

Figure 5: Detection of six targets at CNR = $70$dB and SNR $10$dB. (a)
Iterative receiver proposed in [35], targets can not be identified, (b) 1st
proposed receiver, all targets can be easily identified, (c) 2nd proposed
receiver, all targets can be easily identified.

##  V Simulation Results

In this section, to assess the performance of our designed closed-form receive
filters, a number of simulations are performed. For all simulation the number
of waveforms $K=4$ and waveforms are DPSK.

In the first simulation, we compare the RSL levels of our proposed receive
filter with the iterative joint least-square (JLS) receive filter proposed in
[35]. Here, we consider the waveforms are DPSK and the number of waveforms is
four. The simulation results are shown in Fig. 3. It can be observed from Fig.
3a, the responses of receive filters corresponding to different waveforms are
not fully coherent and the computational complexity is high due to its
iterative nature. Therefore, this receiver can mask week targets in the
presence of a clutter, and the estimation of Doppler can be a challenging
task. In Fig. 3b, the output of our designed receive filters using linear
convolution is shown. As can be seen in the figure that the output of all
filters is fully coherent and the RSL levels are comparatively lower than the
iterative JLS receiver filter. Similarly, the output of our designed filters
using circulant convolution technique is shown in Fig. 3c. It can be seen from
3c that the output of all receivers is fully coherent and RSLs are remarkably
lower then both of the above techniques.

In the second simulation, we compare the target detection performance of the
proposed receivers with the iterative JLS receive filter proposed in [35]. For
this simulation, $M=50$ waveforms are transmitted in each NCPI. Each waveform
is randomly selected from 4 DPSK alphabet waveforms. The number of chips in
each waveform is $N=30$, and each chip is of duration $\tau_{c}=1$ms. The
total duration of each DPSK waveform is $T=30$ms. The pulse shaping sines and
cosines in DPSK waveform have the baseband frequency $f_{b}=500$Hz, and their
sampling rate is $3$kHz. There are 6 targets, 4 targets are moving towards the
DFRC system while the remaining 2 targets are moving away. The normalised
Doppler frequency of first 3 targets is $0.3$Hz and fourth target is $0.25$Hz.
The overall range of our DFRC system is divided into $450$ range gates, and
fast moving scatterers (targets) are placed around the central range gates.
For example, 3 targets with Doppler shift of $0.3$Hz are located in range
cells111The smallest range increment the radar is capable of detecting.225,
228 and 221. The other three targets with Doppler frequencies of $-0.3$Hz,
$-0.25$Hz and $0.2$Hz are located in range cells of 235, 215, and 245,
respectively. Putting received samples corresponding to different waveforms in
different columns can form a matrix. Samples in a particular row of this
matrix are called slow time samples and they represent echoes from a target in
the same range-bin. Applying fast-Fourier-transform (FFT) on the matrix in the
row dimension can find the Doppler shift of moving targets in different range-
bins. The corresponding FFT results are shown in Fig. 4 and 5. Note that only
$100$ central range gates are shown in the simulation results for better
visualization. Echos from slowly moving and stationary objects are considered
as clutter returns. In simulations, the clutter is modeled as a white Gaussian
noise process and have the normalized Doppler range of $-0.1$Hz to $0.1$Hz for
stationary and slowly moving scatterers. For simulation two scenarios are
considered, in the first scenario the signal-to-noise ratio (SNR) is $10$dB
and clutter-to-noise ratio (CNR) is $50$dB, while in the second scenario SNR
$=10$dB and CNR $=70$ dB. The corresponding performances are shown in Fig. 4
and Fig. 5. For the first scenario, Fig. 4a shows that with JLS iterative
receive filter, the three closely located targets in rang-gates 225, 228, and
221 each of Doppler shift $0.3$Hz are poorly resolvable. The other three
widely separated targets are resolvable, but due to the non-coherence response
of the receive filter, targets are heavily masked by nearby clutters. In
contrast, Fig. 4b shows the detection performance of our proposed linear-
convolution-based receiver. As can be seen in the figure, the closely located
targets are resolved efficiently due to the fully coherent response of our
proposed receiver. Similarly, the performance of our proposed circular-
convolution-based receiver is shown in Fig. 5, which is even better than the
linear-convolution-based receiver (please compare the amplitudes of peaks).

Similarly, Fig.5 shows the detection performance of all receivers for the
second scenario. It can be seen in Fig.5a, the JLS iterative receiver
completely fails to detect the targets in the presence of nearby strong
clutters. On the other hand, Fig. 5b shows the linear-convolution-based
receivers have successfully resolved and unmasked the targets buried in heavy
clutter. Similarly, Fig. 5c shows that the circular-convolution-based
receivers, as in the first scenario, have significantly better performance
compared to JLS and linear-convolution-based receivers.

In the last simulation of radar, the probability of detection of proposed
receivers is compared with the JLS iterative receiver. Fig. 6 shows that the
performance of proposed linear- and circular-convolution-based receivers is
much better than the JLS iterative receiver.

Finally, we perform a simulation to find the symbol-error-rate (SER) of the
communication. For simplicity, we consider a line-of-sight communication
(single-path channel). However, simulations can be easily extended for the
multipath channel. Here, the received waveform (symbol) is passed through all
receive filters. For detection of the symbol, the receiver that yields maximum
output amplitude, its corresponding waveform is considered as the detected
symbol. Since we have used only four waveforms, each can carry only two bits.
Fig. 7 compares the SER rate, where it can be seen that the proposed liner
convolution based receiver performs the best for SNR lower than 5dB. While for
higher SNR i.e., $SNR\geq 5$dB, circular-convolution based receiver performs
the best. This performance behavior of our proposed schemes is because at
higher SNR they provide better cross-correlation among received waveform and
receivers. At a typical SER of $10^{-2}$, both proposed liner-convolution and
circular-convolution based receivers yield a 2dB gain in SNR compared to the
iterative JLS receiver [35]. This gain is because, during optimization, JLS
iterative cost-function does not have to fulfill any constraint as in our
proposed schemes. As a consequence, compared to our proposed design, the
target detection and SER of the JLS iterative scheme are poor.

Figure 6: Probability of detection comparison of proposed schemes with the JLS
iterative receiver proposed in [35]. Figure 7: SER comparison of our proposed
schemes with the JLS iterative receiver proposed in [35].

##  VI Conclusions

In this paper, to reduce range side-lobes and achieve full coherency, two
novel closed-form algorithms based on linear- and circular-convolution are
proposed. The computationally complexity of the proposed algorithms is lower
than the proposed algorithms available in the literature and achieves full
coherency for any number of waveforms. As far as we know no one has achieved
full coherence for more than two waveforms. Simulation results demonstrate the
superiority of our proposed algorithms over the others for target detection
and symbol-error-rate (SER). In the future work, the focus will be to improve
the cross-correlation among different received waveform and their
corresponding receivers.

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