# Coexistence of Communications and Cognitive MIMO Radar: Waveform Design and
Prototype

Mohammad Alaee-Kerahroodi, Ehsan Raei, Sumit Kumar, , Bhavani Shankar Mysore
Rama Rao

###### Abstract

New generation of radar systems will need to coexist with other radio
frequency (RF) systems, anticipating their behavior and reacting appropriately
to avoid interference. In light of this requirement, this paper designs,
implements, and evaluates the performance of phase-only sequences (with
constant power) for intelligent spectrum utilization using the custom built
cognitive Multiple Input Multiple Output (MIMO) radar prototype. The proposed
transmit waveforms avoid the frequency bands occupied by narrowband
interferers or communication links, while simultaneously have a small cross-
correlation among each other to enable their separability at the MIMO radar
receiver. The performance of the optimized set of sequences obtained through
solving a non-convex bi-objective optimization problem, is compared with the
state-of-the-art counterparts, and its applicability is illustrated by the
developed prototype. A realistic Long Term Evolution (LTE) downlink is used
for the communications, and the real-time system implementation is validated
and evaluated through the throughput calculations for communications and the
detection performance measurement for the radar system.

###### Index Terms:

Coexistence, Communications, Prototype, Radar, SDR, Spectral Shaping, USRP,
Waveform Design.

##  I Introduction

Spectrum congestion has become an imminent problem with multitude of radio
services like wireless communications, active Radio-Frequency (RF) sensing and
radio astronomy vying for the scarce usable spectrum. Within this conundrum of
spectrum congestion, radars need to cope with simultaneous transmissions from
other RF systems. Spectrum sharing with communications being the highly
plausible scenario given the need for high bandwidth in both systems [1, 2,
3]. While elaborate allocation policies are in place to regulate the spectral
usage, the rigid allocations result in inefficient spectrum utilization when
the subscription is sparse. In this context, smart spectrum utilization offers
a flexible and a fairly promising solution for improved system performance in
the emerging smart sensing systems [4].

Two paradigms, Cognition and Multiple Input Multiple Output (MIMO) have been
central to the prevalence of smart sensing systems. Herein, the former concept
offers ability to choose intelligent transmission strategies based on
prevailing environmental conditions and a prediction of the behavior of the
emitters in the scene, in addition to the now ubiquitous receiver adaptation
[5, 6, 7, 8]. The second concept, offers a canvas of transmission strategies
to the cognition manager to select from; these strategies exploit waveform
diversity and the available degrees of freedom [9, 10]. Smart sensing opens up
the possibility of coexistence of radar systems with incumbent communication
systems in the earlier mentioned spectrum sharing instance. A representative
coexistence scenario is illustrated in Figure 1, where an understanding of the
environment is essential for seamless operation of radar systems while
opportunistically using the spectrum allocated to communication [11, 1, 2,
12].

Figure 1: An illustration of coexistence between radar and communications. The
radar aims to detect the airplane, without creating interference to the
communication links, and similarly avoiding interference from the
communication links.

In this paper, we design a cognitive MIMO radar system towards fostering
coexistence with communications; it involves spectrum sensing and transmission
strategies adapted to the sensed spectrum while accomplishing the radar tasks
and without degrading the performance of communications. Particularly, a set
of transmit sequences is designed to focus the emitted energy in the bands
suggested by the spectrum sensing module while limiting the out-of-band
interference. The waveforms, along with the receive processing, are designed
to enhance the radar detection performance. The designed system is then
demonstrated for the representative scenario of Figure 1 using a custom built
Software-Defined Radio (SDR) based prototype developed on Universal Software
Radio Peripheral (USRP) s111USRPs are inexpensive programmable radio platforms
used in wireless communications and sensing prototyping, teaching and
research. from National Instruments (NI) [13, 14]. These USRP s operate at
sub-$6$ GHz frequencies with a maximum instantaneous bandwidth of $160$ MHz.

###  I-A Background and Related Works

##### Coexistence of Communication Systems

Spectral coexistence has been found to be a promising method to alleviate the
spectrum scarcity problem. For example, in the ISM band, several wireless
standards operate on overlapping frequency bands such as the IEEE 802.11
(popularly known as Wi-Fi) and the IEEE 802.15.4 (popularly known as ZigBee)
family of standards in the $2.4$ GHz ISM band. As the ISM band does not
require a license to transmit, the participating standards transmit at will
(frequency and power) leading to interference and a loss of throughput[15, 16,
17]. Similarly, in the 5 GHz ISM band, it has been speculated that there will
be interference between IEEE 802.11n/ac and Long Term Evolution (LTE) [18].
For the two aforementioned interference scenarios, several spectral
coexistence methods like successive Interference Cancellation (SIC)[19, 20]
for single antenna and Technology Independent Multi-Output (TIMO) [21, 22, 23]
for multi-antennas have been proposed.

##### Radar-Communication Coexistence

The interfered bands, including those occupied by communications, are not
useful for the radar system, and traditional radars aim to mitigate these
frequencies at their receivers. To avoid energy wasted due to transmissions on
these bands while pursuing coexistence applications, research into the
transmit strategy of spectrally shaped radar waveforms has been driving
coexistence studies since the last decade [24, 25, 26, 27, 28, 29, 30, 31,
32]. In fact, it is possible to radiate the radar waveform in a smart way by
using two key elements of the cognition; spectrum sensing and spectrum sharing
[33, 7]. Further, it is possible to increase the total radar bandwidth, and
consequently improve the range resolution by combining several clear bands
together [24].

##### Radar Waveform Optimization for Coexistence

To enable waveform design in coexistence scenarios, several optimization
methods including alternative optimization, Majorization-Minimization (MM),
Coordinate Descent (CD), Alternating Direction Method of Multipliers (ADMM),
power like iterations, etc., have been developed recently for designing
waveforms exhibiting deep notches on their spectrum. Herein, we provide a
brief summary of the recent works and refer the interested readers to [24, 34,
25, 35, 26, 27, 36, 37, 38, 39, 30, 40, 32, 41, 42] and references therein for
an elaborate literature on spectral shaping optimization.

Spectral/ Template Matching: In [25, 26, 27, 37, 38, 41, 42] spectral/template
matching222Spectral matching is when the spectrum of the designed sequence is
forced to be similar to a given template in the least-squares sense. approach
is pursued to shape the spectral behaviour of the transmit sequences. The
approach of template matching is also used in [27] for designing sequences
with a specific auto-correlation function, by considering a weighted objective
of the both spectral and auto-correlation functions. However, matching to a
template in the vicinity of the spectrum edges may decrease the possibility of
synthesizing deep notches in the undesired frequency bands.

Signal to Interference plus Noise Ratio (SINR) optimization: In [30, 32], SINR
enhancement is considered to design the transmit sequences, while the spectral
behaviour is considered as a constraint. Particularly, in [30] mutual
information (MI) is maximized based on MM approach while, in [32], a CD based
technique is utilized to maximize the SINR; both consider a local control on
the interference energy radiated in each frequency band. Both the SINR and
mutual information metrics require a knowledge of the possible location of the
targets, which cannot be obtained easily in practice. Further, in case of MIMO
radar systems, maximizing MI/ SINR as the only design metric has the drawback
of leading to a set of waveforms that are fully correlated, i. e., converting
the MIMO radar to a conventional phased-array radar [43].

Stopband/ Passband levels: In [39, 40], the ratio of the maximum stopband
level to the minimum passband level is considered as the objective function to
shape the spectrum of the transmit waveforms with deeper notches compared to
its counterparts. This metric is particularly useful in spectrally dense
environments, where only the stopbands and passbands of the interfering
signals can be estimated by energy detection, matched filtering or feature
detection[6]. However, [39, 40], does not consider the correlation behaviour
of the designed sequences towards providing the required orthogonality in MIMO
radars.

Orthogonality: Traditionally Integrated Sidelobe Level (ISL) has been used as
a metric to design set of approximately orthogonal sequences in a Code
Division Multiplexing (CDM)-MIMO radar system, i.e., having small auto- and
cross-correlation sidelobes [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. It
has been shown that if a sequence set of size $M$, has atleast one $N$-length
sequence, then the ISL lower bound ${\cal{B}}_{\text{ISL}}=N^{2}M(M-1)$, [55,
56, 57, 51], can nearly be met by a set of unimodular sequences generated by
the algorithms proposed in [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54].

##### Summary of the Shortcomings in Prior-Art

The well-known notch shaping or SINR enhancement methods do not offer
satisfactory performance in terms of simultaneously providing sets of
uncorrelated sequences and deep notches in the spectrum. In Figure 2, we show
that the Integrated Sidelobe Level Ratio (ISLR) values of different sets of
random-phase333Random-phase refers to the family of unimodular sequences
defined by  $x_{m,n}=e^{j\phi_{m,n}},~{}~{}~{}n=1,\ldots,N$ (1) where
$\phi_{m,n}$, $n=1,\ldots,N$, are independent and identically distributed
random variables with a uniform probability density function over $[0,2\pi)$.
sequences nearly approach the lower bound on aperiodic ISLR. This fact shows
that the traditional ISL may not be an ideal choice for the objective function
if the goal is to design set of sequences which provide orthogonality in a
MIMO radar system. Furthermore, there are no hardware (HW) prototypes to
validate the waveform design in a coexistence context. Our work aims to
address these shortcomings.

###  I-B Contribution

In this paper, following a radar-centric approach, we demonstrate for the
first time, a coexistence set-up in incumbent communication bands by
optimizing, implementing and validating a complete cognitive MIMO radar
prototype.

Figure 2: Aperiodic/periodic ISLR values of a set of $M=4$ random-phase
sequences averaged over $100$ independent trials: comparison for different
sequence lengths.

The key contributions of this paper can be summarized as:

  1. 1.

Building a SDR based cognitive MIMO radar system capable of performing real-
time spectrum sensing and smart spectrum utilization for coexistence in
addition to the classical radar tasks like matched filtering and Doppler
processing. The testbed implements two-channels achieving a transmitting
bandwidth close to $80$ MHz with the possibility of changing the parameters of
the processing units during the operation.

  2. 2.

Setting up of the coexistence system using the aforementioned cognitive MIMO
radar and an incumbent LTE link which serves as the communication counterpart.

  3. 3.

Designing a set of optimized sequences with deep notches in their spectrum
that can also be well-separated at the MIMO radar receiver. In this context,
we propose an objective function based on the weighted sum Spectral Integrated
Level Ratio (SILR) and Integrated Cross Correlation Level (ICCL). These two
objectives have been proposed in this paper to encourage the design of
waveforms having required spectral as well as cross-correlation properties
simultaneously. The optimization is undertaken using the low complexity
iterative CD methodology which offers an attractive framework in the context
of dynamic spectral shaping. In fact, the SILR renders the CD approach
attractive through a simplified single variable optimization.

  4. 4.

Evaluating and validating the performance of the designed sequences
experimentally, through a real-time visualization of the detection performance
of the MIMO radar system and a measurement of the LTE throughput. This ensures
the development, implementation and validation of one of the first prototypes
for co-existing radar and communications with waveform optimization.

###  I-C Paper Overview and Notation

The rest of the paper is organized as follows. Section II describes the
prototype architecture and different applications developed for the
demonstration of the coexistence paradigm. Section III describes the proposed
optimization of the spectral and cross-correlation of the MIMO radar by
defining an novel objective function. In Section IV, the performance of the
proposed method is compared with prior-art and the numerical experiments of
the USRP implementation of the prototype are given in section V and section VI
summarizes the paper. This paper uses lower-case and upper-case boldface for
vectors ($a$) and matrices ($A$) respectively. The conjugate, transpose and
the conjugate transpose operators are denoted by $(.)^{*}$, $(.)^{T}$ and
$(.)^{H}$ respectively. The Frobenius norm, $l_{2}$ norm, absolute value and
round operator are denoted by $\mathinner{\\!\left\lVert.\right\rVert}_{F}$,
$\mathinner{\\!\left\lVert.\right\rVert}_{2}$, $|.|$ and $\lfloor.\rceil$
respectively. For a complex number $a$, $\Re(a)$ and $\Im(a)$ denote the real
and imaginary part respectively and $j=\sqrt{-1}$. The letter $(i)$ is use as
step of a procedure. Finally $\odot$ denotes the Hadamard product.

##  II The Proposed Prototype Architecture

The prototype consists of three application frameworks as depicted in Figure
3; a) LTE Application Framework, b) Spectrum sensing application, and c)
Cognitive MIMO radar application. Various building blocks of the proposed
coexistence prototype are shown in Figure 4(b). The HW consists of three main
modules: 1) USRP 2974 for LTE communications 2) USRP B210 for spectrum
sensing, and 3) USRP 2944R for cognitive MIMO radar with specifications given
in TABLE I. The design details of each module are presented in the sequel.
USRP s are used for the transmission and reception of the wireless RF signals
and the Rohde and Schwarz spectrum analyzer is used for the validation of the
transmission444 A video of the Over-The-Air (OTA) operation of the proposed
coexistence prototype can be found in radarmimo.com. .

(a) LTE Application

(b) Spectrum Sensing

(c) MIMO Radar

Figure 3: Application frameworks forming the proposed prototype: LTE
application developed by NI, spectrum sensing and cognitive MIMO radar
applications developed in this paper.

(a)

(b)

Figure 4:  The photograph (a) and connection diagram (b) and of the proposed
coexistence prototype. The photo shows communication BS and user, spectrum
sensing, and cognitive MIMO radar systems.

###  II-A LTE Application Framework

The LabVIEW LTE Application Framework (Figure 3(a)) is an add-on software that
provides a real-time physical layer LTE implementation in the form of an open
and modifiable source-code [58]. The framework complies with a selected subset
of the 3GPP LTE which includes a closed-loop OTA operation with channel state
and ACK/ NACK feedback, $20$ MHz bandwidth, Physical Downlink Shared Channel
(PDSCH) and Physical Downlink Control Channel (PDCCH), up to $75$ Mbps data
throughput, FDD and TDD configuration $5$-frame structure, QPSK, $16$-QAM, and
$64$-QAM modulation, channel estimation and zero-forcing channel equalization.
The framework also has a basic MAC implementation to enable packet-based data
transmission along with a MAC adaptation framework for rate adaptation. Since
the NI-USRP 2974 has two independent RF chains and the Application Framework
supports single antenna links, we emulated both the BS and communications user
on different RF chains of the same USRP.

TABLE I: Hardware characteristics of the proposed prototype Parameters | 2974/2944R | B210
---|---|---
Frequency range |  $10$ MHz $-6$ GHz |  $70$ MHz $-6$ GHz
Max. output power |  $20$ dBm |  $10$ dBm
Max. input power |  $+10$ dBm |  $-15$ dBm
Noise figure |  $5-7$ dB |  $8$ dB
Bandwidth |  $160$ MHz |  $56$ MHz
DACs |  $200$ MS/s, $16$ bits |  $61.44$ MS/s, $12$ bits
ADCs |  $200$ MS/s, $14$ bits |  $61.44$ MS/s, $12$ bits
TABLE II: Characteristics of the developed applications Parameters | Spectrum sensing | Cognitive MIMO radar
---|---|---
Bandwidth |  $50$ MHz / $100$ MHz |  $1-80$ MHz
Window type | Rectangle, Hamming, Hanning, Blackman, etc. | Rectangle, Hamming, Hanning, Blackman, etc.
Averaging mode | RMS averaging, vector averaging, peak hold | Coherent integration (Fast Fourier Transform (FFT))
Processing units |  | RSSI calculation in $1$ MHz bins
---
based on spectrum waterfall average/peak
Matched filtering, range-Doppler processing
Transmitting waveforms | – |  | Random-polyphase, Frank, Golomb,
---
Random-Binary, Barker, m-Sequence, Gold, Kasami,
Up-LFM, Down-LFM,
and the Optimized sequences based on proposed algorithm

###  II-B Spectrum Sensing Application

To perform the cognition and continuously sensing the environment, we
developed an application based on LabView NXG 3.1 that connects to Ettus USRP
B2xx (Figure 3(b)). The developed application is flexible in terms of changing
many parameters on the fly, e.g., averaging modes, window type, energy
detection threshold, and the USRP configurations (gain, channel, start
frequency, etc.). The features and flexibilities of the developed application
are reported in TABLE II. Figure 5 depicts a snapshot of the developed
application when the WiFi ISM band $2.4$ GHz with $50$ MHz bandwidth was
analyzed. The center frequency can be adjusted to any arbitrary value in the
interval $70$ MHz to $6$ GHz, and the span bandwidth can be selected from the
two values of $50$ MHz, and $100$ MHz555Note that USRP B2xx provides $56$ MHz
of real-time bandwidth by using AD9361 RFIC direct-conversion transceiver.
However, the developed application can analyze larger bandwidths by sweeping
the spectrum with efficient implementation.. The obtained frequency chart in
Figure 5-(b) is being transferred through a network connection (LAN/Wi-Fi) to
the cognitive MIMO radar application.

Figure 5: A snapshot of the developed two-channel Spectrum Sensing
Application. (a) Power spectrum. (b) The energy detector.

###  II-C Cognitive MIMO Radar

The cognitive MIMO radar application was developed based on LabView NXG 3.1,
and was connected to the HW platform NI-USRP 2944R. This USRP consists of a
$2\times 2$ MIMO RF transceiver with a programmable Kintex-7 field
programmable gate array (FPGA). The developed application is quite flexible in
terms of changing the transmit waveform on the fly, such that it can adapt
with the environment. TABLE II details the features and flexibilities of the
developed application. The center frequency can be adjusted to any arbitrary
value in the interval $70$ MHz to $6$ GHz, and the radar bandwidth can be
adjusted to any arbitrary value in the interval $1$ MHz to $80$ MHz.

Further, up to two targets can be also emulated in this application. Passing
the transmitting waveforms through the $30$ dB attenuators as indicated in
Figure 4(b), a reflection will be generated; this will be used to generate the
targets, contaminated with the communications interference. The received
signal in this way will be further shifted in time, frequency and spatial
direction to create the simulated targets. These targets will be detected
after calculating the absolute values of the range-Doppler maps.

The block diagram of the developed cognitive MIMO radar is depicted in Figure
6. Note that the application is connected through a network (LAN/Wi-Fi) to the
spectrum sensing application to receive the information about the occupied
frequency bands. The radar adapts its transmit waveform and the matched filter
coefficients based on this information. Then, using the two transmit channels
of NI-USRP 2944R, the optimized MIMO waveforms will be transmitted; the design
algorithm for the optimization is described in Section III.

Figure 7 depicts a snapshot of the developed cognitive MIMO radar application
framework, when the licensed band $3.78$ GHz with $40$ MHz bandwidth was used
for transmission666SnT has experimental licence to use $3.75$ \- $3.8$ GHz for
5G research in Luxembourg.. All the parameters related to the radar waveform,
processing units, and targets can be changed and adjusted during the operation
of the radar system.

(a) Transmitter

(b) Receiver

Figure 6: Block diagram of the developed cognitive MIMO radar application. In
the transmit side (a), a list of occupied frequency bands will be received
through the network connection between the spectrum sensing application and
the radar application. Based on this information, the proposed design
algorithm optimizes the transmitting waveforms. At the receive side (b), the
optimized waveforms will be used for appropriate matched filtering in the
fast-time dimension. Consequently, the modulus of the range-Doppler plots will
be calculated after taking FFT in the slow-time dimension.  Figure 7: A
snapshot of the developed cognitive MIMO radar application. (a) Settings for
device, radar, and processing parameters. (b) I and Q signals of two receive
channels. (c) Spectrum of the received signals in two receive channels. (d)
Matched filters to two transmitting waveforms at the first receive channel.
(e) Matched filters to two transmitting waveforms at the second receive
channel. (f) Received information from the energy detector of the spectrum
sensing application.

##  III Waveform Design

We consider a colocated narrow-band MIMO radar system, with $M$ transmit
antennas, each transmitting a sequence of length $N$ in the fast-time domain.
Let the matrix $\mbox{\boldmath{$X$}}\in\mathbb{C}^{M\times
N}\triangleq[\mathbf{x}_{1}^{T},\dots,\mathbf{x}_{M}^{T}]^{T}$ denotes the
transmitted set of sequences in baseband, where the vector
$\mathbf{x}_{m}\triangleq[x_{m,1},\ldots,x_{m,N}]^{T}\in\mathbb{C}^{N}$
($m=\\{1,\dots,M\\}$) indicates the $N$ samples of $m^{th}$ transmitter. We
aim to design a transmit set of sequences which have small cross-correlation
among each others, while each of the sequences having a desired spectral
behaviour. To this end, in the following, we introduce the SILR and ICCL
metrics and subsequently the optimization problem to handle them.

Let
$\mbox{\boldmath{$F$}}\triangleq[\mathbf{f}_{0},\dots,\mathbf{f}_{N-1}]\in\mathbb{C}^{N\times
N}$ be the Discrete Fourier Transform (DFT) matrix, where,
$\mathbf{f}_{k}\triangleq[1,e^{j\frac{2\pi k}{N}},\dots,e^{j\frac{2\pi
k(N-1)}{N}}]^{T}\in\mathbb{C}^{N},\ k=\\{0,\dots,N-1\\}$. Let $\mathcal{V}$
and $\mathcal{U}$ be the desired and undesired discrete frequency bands for
MIMO radar, respectively. These two sets satisfy
$\mathcal{V}\cup\mathcal{U}=\\{0,\dots,N-1\\}$ and
$\mathcal{V}\cap\mathcal{U}=\emptyset$. We define SILR as,

$g_{s}(\mbox{\boldmath{$X$}})\triangleq\frac{\sum_{m=1}^{M}\mathinner{\\!\left\lVert\mathbf{f}_{k}^{\dagger}\mathbf{x}_{m}\right\rVert}^{2}|k\in\mathcal{U}}{\sum_{m=1}^{M}\mathinner{\\!\left\lVert\mathbf{f}_{k}^{\dagger}\mathbf{x}_{m}\right\rVert}^{2}|k\in\mathcal{V}}$
(2)

which is the energy of the radar waveform interfering with other incumbent
services (like communications) relative to the energy of transmission in the
desired bands. Optimizing the above objective function may shape the spectral-
power of the transmitting sequence and satisfy a desired mask in the spectrum.
However, in a MIMO radar it is necessary to separate the transmitting
waveforms in the receiver to investigate the waveform diversity, which ideally
requires orthogonality between the transmitting sequences. To make this
orthogonality feasible by CDM, we need to transmit a set of sequences which
have small cross-correlations among each other. The aperiodic cross-
correlation777In this paper, we provide the solution to the design of
sequences with good aperiodic correlation functions. However, following the
same steps as indicated in [53], the design procedure can be extended to
obtain sequences with good periodic correlation properties. of
$\mathbf{x}_{m}$ and $\mathbf{x}_{l}$ is defined as,
$r_{m,m^{\prime}}(l)=\sum_{n=1}^{N-l}x_{m,n}x_{m^{\prime},n+l}^{*}$, where
$m\neq m^{\prime}\in\\{1,\dots,M\\}$ are indices of the transmit antennas and
$l\in\\{-N+1,\dots,N-1\\}$ denotes the cross-correlation lag. We define ICCL
as,

$\widetilde{g}_{c}(\mbox{\boldmath{$X$}})\triangleq\sum_{m=1}^{M}\sum_{\begin{subarray}{c}{m^{\prime}=1}\\\
{m^{\prime}\neq
m}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}|r_{m,m^{\prime}}(l)|^{2},$ (3)

which can be used to promote the orthogonality between the transmitting
sequence.

###  III-A Problem Formulation

We aim to design sets of sequences that simultaneously have good properties in
terms of SILR and ICCL, under constant modulus and discrete phase constraints.
The optimization problem can be represented as,

$\begin{dcases}\min_{\mbox{\boldmath{$X$}}}&g_{s}(\mbox{\boldmath{$X$}}),g_{c}(\mbox{\boldmath{$X$}})\\\
s.t&C_{1}\ \text{or}\ C_{2}\end{dcases}$ (4)

where
$g_{c}(\mbox{\boldmath{$X$}})=\frac{1}{(2MN)^{2}}\widetilde{g}_{c}(\mbox{\boldmath{$X$}})$
is the scaled version of the ICCL as defined in (3). Further,
$C_{1}\triangleq\\{\mbox{\boldmath{$X$}}\mid
x_{m,n}=e^{j\phi_{m,n}},\phi_{m,n}\in\Omega_{\infty}\\},\Omega_{\infty}=[0,2\pi)$
is the constant modulus constraint and
$C_{2}\triangleq\\{\mbox{\boldmath{$X$}}\mid
x_{m,n}=e^{j\phi_{m,n}},\phi_{m,n}\in\Omega_{L}\\},\Omega_{L}=\left\\{0,\frac{2\pi}{L},\dots,\frac{2\pi(L-1)}{L}\right\\}$
is discrete phase constraint. In fact, $\Omega_{L}$ indicates the $M$-ary
Phase Shift Keying (MPSK) with alphabet size $L$.

Problem (4) is a bi-objective optimization problem in which a feasible
solution that minimizes the both the objective functions may not exist [52,
59]. Scalarization is a well known technique that converts the bi-objective
optimization problem to a single objective problem by replacing a weighted sum
of the objective functions. Using this technique, the following Pareto-
optimization problem will be obtained,

$\mathcal{P}\begin{dcases}\min_{\mbox{\boldmath{$X$}}}&g(\mbox{\boldmath{$X$}})\triangleq\theta
g_{s}(\mbox{\boldmath{$X$}})+(1-\theta)g_{c}(\mbox{\boldmath{$X$}})\\\
s.t&C_{1}\ \text{or}\ C_{2},\end{dcases}$ (5)

The coefficient $\theta\in\left[0,1\right]$ is a weight factor that effects
trade-off between SILR and ICCL. In (5), $g_{s}(\mbox{\boldmath{$X$}})$ is a
fractional quadratic function while $g_{c}(\mbox{\boldmath{$X$}})$ is quartic
function, both with multiple variables. Further, both $C_{1}$ and $C_{2}$
constraints are not an affine set, besides $C_{2}$ is non-continuous and non-
differentiate set. Therefore, we encounter a non-convex, multi-variable and
NP-hard optimization problem [52, 60].

###  III-B The Proposed Method

To solve (5) directly, we utilize the CD framework wherein the multi variable
problem is solved as a sequence of single variable problems. The methodologies
based on CD, generally start with a feasible matrix
$\mbox{\boldmath{$X$}}=\mbox{\boldmath{$X$}}^{(0)}$ as the initial waveform
set. Then, in each iteration, the waveform set is updated entry by entry
several times [61]. In particular, an entry of $X$ is considered as the only
variable while others are held fixed and then the objective function is
optimized with respect to this identified variable. Let us assume that
$x_{t,d}$ ($t\in\\{1,\dots,M\\}$ and $d\in\\{1,\dots,N\\}$) is the only
variable and the fixed entries are stored in the matrix
$\mbox{\boldmath{$X$}}_{-(t,d)}^{(i)}$. The resulting single-variable
objective function at $(i)^{th}$ iteration can be written as (see Appendix A),

$\displaystyle g(x_{t,d},\mbox{\boldmath{$X$}}_{-(t,d)}^{(i)})=$
$\displaystyle\theta\frac{a_{0}x_{t,d}+a_{1}+a_{2}x_{t,d}^{*}}{b_{0}x_{t,d}+b_{1}+b_{2}x_{t,d}^{*}}$
(6) $\displaystyle+$
$\displaystyle(1-\theta)\left(c_{0}x_{t,d}+c_{1}+c_{2}x_{t,d}^{*}\right)$

where, the coefficients $a_{i}$, $b_{i}$ and $c_{i}$ depend on
$\mbox{\boldmath{$X$}}^{(i)}_{-(t,d)}$ and are derived in Appendix A. By
substituting $x_{t,d}=e^{j\phi}$, $\mathcal{P}$ in (5) the optimization
problem at the $i^{th}$ iteration can be recast as a function of $\phi$888For
the convenience we use $\phi$ instead of $\phi_{t,d}$ in the rest of the
paper. as follows,

$\begin{dcases}\min_{\phi}&\theta\frac{a_{0}e^{j\phi}+a_{1}+a_{2}e^{-j\phi}}{b_{0}e^{j\phi}+b_{1}+b_{2}e^{-j\phi}}+(1-\theta)\left(c_{0}e^{j\phi}+c_{1}+c_{2}e^{-j\phi}\right)\\\
s.t&C_{1}\ \text{or}\ C_{2},\end{dcases}$ (7)

At the $i^{th}$ iteration, for $t=1,\ldots,M$, and $d=1,\ldots,N$, the
$(t,d)^{th}$ entry of $X$ will be updated by solving (7). After updating all
the entries, a new iteration will be started, provided that the stopping
criteria is not met. This procedure will continue until the objective function
converges to an optimal value. A summary of the proposed method is reported in
Algorithm 1.

Input: Initial set of feasible sequences, $\mbox{\boldmath{$X$}}^{(0)}$.

Initialization: $i\mathrel{\mathop{\ordinarycolon}}=0$.

Optimization:

  1. 1.

while the stopping criteria is not met, do

  2. 2.

$i\mathrel{\mathop{\ordinarycolon}}=i+1$;

  3. 3.

for $t=1,\dots,M$ do

  4. 4.

for $d=1,\dots,N$ do

  5. 5.

Optimize $x_{t,d}^{(i-1)}$ and obtain $x^{\star}_{t,d}$;

  6. 6.

Update $x^{(i)}_{t,d}=e^{j\phi^{\star}}$;

  7. 7.

$\mbox{\boldmath{$X$}}^{(i)}=\mbox{\boldmath{$X$}}^{(i)}_{-(t,d)}|_{x_{t,d}=x^{(i)}_{t,d}}$;

  8. 8.

end for

  9. 9.

end for

  10. 10.

end while

Output: $\mbox{\boldmath{$X$}}^{\star}=\mbox{\boldmath{$X$}}^{(i)}$.

Algorithm 1 The proposed method for designing set of sequences that avoid
reserved frequency bands, and in a same time have small cross-correlation
among each other.

Let $\phi^{\star}$ be the optimized solution of Problem (7). Algorithm 1
considers a feasible set of sequences as the initial waveforms. Then, at each
single variable update, it selects $x_{t,d}^{(i-1)}$ as the variable and
updates it with the optimized ${s_{t,d}^{(i)}}$, denoted by
${s_{t,d}^{\star}}$. This procedure is repeated for other entries and is
undertaken until all the entries are optimized at least once. After optimizing
the $MN^{th}$ entry, the algorithm examines that the waveform converges to a
stationary point. If the stopping criteria is not met the algorithm repeats
the aforementioned steps. The solution to the single variable optimization
problem $\mathcal{P}^{(i)}$ can be obtained by following similar steps of the
proposed method in [54], where the the key steps are detailed in the
following.

####  III-B1 Continuous phase constraint

The solution under continuous phase will be obtained by finding the critical
points of the objective function and selecting the one that minimizes the
objective. This is omitted for brevity.

####  III-B2 Discrete phase constraint

In this case, the feasible set is limited to a set of $L$ phases. Thus, the
objective function with respect to the indices of $\Omega_{L}$ can be written
as,

$g(l)=\theta\frac{\sum_{n=0}^{2}a_{n}e^{-j\frac{2\pi
nl}{L}}}{\sum_{n=0}^{2}b_{n}e^{-j\frac{2\pi nl}{L}}}+(1-\theta)e^{\frac{j2\pi
l}{L}}\sum_{n=0}^{2}c_{n}e^{-j\frac{2\pi nl}{L}}$ (8)

where, $l=\\{0,\dots,L-1\\}$. The summation term in the numerator and
denominator in Equation (8) is exactly the definition of $L-$point DFT of
sequences $[a_{0},a_{1},a_{2}]$ , $[b_{0},b_{1},b_{2}]$ and
$[b_{0},b_{1},b_{2}]$ respectively. Therefore, $g(l)$ can be written as,

$g(l)=\theta\frac{\mathcal{F}_{L}\\{a_{0},a_{1},a_{2}\\}}{\mathcal{F}_{L}\\{b_{0},b_{1},b_{2}\\}}+(1-\theta)\mbox{\boldmath{$h$}}\odot\mathcal{F}_{L}\\{c_{0},c_{1},c_{2}\\}.$
(9)

where,
$\mbox{\boldmath{$h$}}=[1,e^{-j\frac{2\pi}{L}},\ldots,e^{-j\frac{2\pi(L-1)}{L}}]^{T}\in\mathbb{C}^{L}$
and $\mathcal{F}_{L}$ is $L-$point DFT operator. The current function is valid
only for $L>2$. According to periodic property of DFT, for binary $g(l)$ can
be written as,

$g(l)=\theta\frac{\mathcal{F}_{L}\\{a_{0}+a_{2},a_{1}\\}}{\mathcal{F}_{L}\\{b_{0}+b_{2},b_{1}\\}}+(1-\theta)\mbox{\boldmath{$h$}}\odot\mathcal{F}_{L}\\{c_{0}+c_{2},c_{1}\\}.$
(10)

Finally
${l^{\star}}=\arg\displaystyle{\min_{l=1,\dots,L}}\left\\{g(l)\right\\}$, and
${\phi_{d}^{\star}}=\frac{2\pi({l^{\star}}-1)}{L}.$

##  IV Performance Analysis

This section deals with a simulation framework to compare the performance of
the proposed method in terms of SILR and ICCL with the counterparts. To
terminate Algorithm 1, we consider
$\Delta\mbox{\boldmath{$X$}}^{(i)}\triangleq\mathinner{\\!\left\lVert\mbox{\boldmath{$X$}}^{(i)}-\mbox{\boldmath{$X$}}^{(i-1)}\right\rVert}_{F}\leq\zeta$
as the stopping criterion, and set $\zeta=10^{-5}$ for all the ensuing
examples. Let $\mathcal{S}=\cup_{k=1}^{K_{s}}(s_{k,1},s_{k,2})$ be the $K_{s}$
number of normalized frequency stop bands, where $0\leq s_{k,1}<s_{k,2}\leq 1$
and $\cap_{k=1}^{K_{s}}(s_{k,1},s_{k,2})=\emptyset$. Thus, the undesired
discrete frequency band is given by, $\mathcal{U}=\cup_{k=1}^{K_{s}}(\lfloor
Ns_{k,1}\rceil,\lfloor Ns_{k,2}\rceil)$.

Figure 8(a) shows the convergence behavior of the objective function of the
proposed method under $C_{1}$ and $C_{2}$ constraints for different values of
$\theta$. Algorithm 1 was initialized using random-phase sequences and
identical initial sequences was used for all the counterparts in case of any
comparison. As can be seen, the objective function decreases monotonically and
converges to a certain value under both $C_{1}$ and $C_{2}$ constraints. In
case of the $C_{1}$ constraint, a better performance can be observed due to
the higher degrees of freedom in comparison to the $C_{2}$ constraint.
Further, Figure 8(b) numerically shows the convergence of the argument to the
stationary points ($\Delta\mbox{\boldmath{$X$}}^{(i)}$) for different values
of $\theta$.

(a) Objective function.

(b) $\Delta\mbox{\boldmath{$X$}}^{(i)}$.

Figure 8: Convergence behavior of the proposed method under continuous and
discrete phase constraints for different $\theta$ values ($M=4$, $N=64$ and
$L=16$).

In Figure 9, we set $\theta=1$ to design a set of sequences optimized based on
SILR minimization. We set $M=3$, and denote the spectral behaviour of the
optimized sequence of $m^{th}$ transmitter as $\mbox{\boldmath{$x$}}_{m}$ in
Figure 9. The figure shows that the spectral behaviour of the optimized
waveforms obtained by the proposed method provide deeper notches under $C_{1}$
constraint, when compared with SHAPE [26] under a similar constraint. Under
the $C_{2}$ constraint, the proposed algorithm outperforms the Spectrum
Discrete Phase Modulation (SDPM) [42], indicating the effectiveness of the
proposed method over the counterparts in shaping the spectrum of the
waveforms.

Figure 9: Comparing the performance of the proposed method ($\theta=1$) with
SHAPE [26] and SDPM [42] in case of spectral shaping ($M=3$, $m=\\{1,2,3\\}$,
$N=512$, $L=64$ and $\mathcal{S}=[0.05,0.1]$ $\cup[0.2,0.25]$ $\cup[0.4,0.5]$
$\cup[0.7,0.85]$ Hz). All the methods are initialized with the same set of
random-phase sequences with $L=64$.

Next, we set $\theta=0$ and evaluate the performance of the proposed method in
terms of ICCL minimization. In this case, we consider Multi-CAN [45] and
Binary Sequences seTs (BiST) [52] (preforming ISL minimization) as the
benchmarks to compare the performance under $C_{1}$ and $C_{2}$ constraints.
Figure 10(a) compares the cross-correlation of the first and second optimized
waveforms obtained by the proposed method, Multi-CAN and BiST algorithms, for
$N=64$ and $M=3$. Intuitively, it can be seen that the proposed method offers
set of sequences with smaller cross-correlation levels. In Figure 10(b) and
Figure 10(c), we fix $M=3$ and evaluate the ICCL of the waveforms obtained by
the proposed method by changing the sequence length, and vice versa, fixing
$N=64$ and changing number of transmitting waveforms. In both cases, we
compare the performance of the proposed method with Multi-CAN and BiST. It can
be observed from this figure that the proposed method obtains smaller ICCL in
compare with the counterparts.

(a) $M=3$, and $N=64$.

(b) $M=3$.

(c) $N=64$.

Figure 10:  Comparing the performance of the proposed method ($\theta=0$) with
Multi-CAN [45] and BiST [52] in terms of ICCL. All the methods are initialized
with the same set of random-phase sequences.

In Figure 11, we evaluate the trade-off between a good spectrum shaping and
good orthogonality by choosing $\theta$. Figure 11(a) shows the impact of
$\theta$ on the spectral behaviour. As can be seen, by choosing $\theta=0$,
the optimized waveforms are not able to put notches on the undesired
frequencies. By increasing $\theta$, the notches will appear gradually and in
case of $\theta=1$, we obtain the deepest notches. However, when $\theta=1$
the cross-correlation is at the highest level which decreases with $\theta$.
In $\theta=0$, we obtain the best orthogonality. This fact is shown in Figure
11(b). Therefore, by choosing an appropriate value of $\theta$, one can make a
good trade-off between these two metrics. For instance, choosing $\theta=0.75$
is able to put a null level around $50$ dB (see Figure 11(a)), while having
good cross-correlation level (see Figure 11(b)).

(a) Spectrum of the proposed method for different $\theta$ values
($\mathcal{S}=[0.05,0.1]$ $\cup[0.4,0.45]$ $\cup[0.7,0.8]$ Hz).

(b) Cross-Correlation of the proposed method for different $\theta$ values.

Figure 11: The impact of $\theta$ value on trade-off between (a) spectral
shaping and (b) cross-correlation levels in comparision with SHAPE [26] ($M=2$
and $N=512$).

##  V Experiments and Results

In this section, we present present experiments conducted using the developed
prototype and analyze the HW results. The prototype setup in Figure 4(a)
depicts the cognitive MIMO radar, the spectrum sensing application and the LTE
application framework as well as RF cable enclosure, USRP s, and the spectrum
analyzer. For the practical applicability of our methods and verification of
the simulation, we established all the connections shown in Figure 4(b) using
RF cables and splitters/ combiners, and measured the performance in a
controlled environment. Nevertheless, a video of the OTA operation of the
proposed prototype can be found in radarmimo.com.

The developed cognitive MIMO radar system in the proposed coexistence
prototype consists of a $2\times 2$ transmit/receive unit. Two targets with
adjustable attenuation paths, Doppler and spatial angles can be also augmented
in the receive side as described in Section II. The transmitting waveforms can
be selected based on the options in TABLE II or obtained based on Algorithm 1.
When executing the application, input parameters to optimize the waveforms
pass from the Graphical User Interface (GUI) to MATLAB, and the optimized set
of sequences are passed to the application through the GUI. The other
processing blocks of the radar system including matched filtering, Doppler
processing, and scene generation are developed in the LabView G dataflow
application. TABLE III and TABLE IV summarize the parameters used for radar
and targets in this experiment.

TABLE III: Radar experiment parameters Parameters | Value
---|---
Center frequency |  $2$ GHz
Real-time bandwidth |  $40$MHz
Transmit and receive channels | $2\times 2$
Transmit power |  $10$ dBm
Duty cycle | $50\%$
Transmit code length | $400$
Pulse repetition interval |  $20\mu$s
TABLE IV: Target experiment parameters Parameters | Target 1 | Target 2
---|---|---
Range delay |  $2\mu$s |  $2.6\mu$s
Normalized Doppler |  $0.2$ Hz |  $-0.25$ Hz
Angle | $25\deg$ | $15\deg$
Attenuation |  $30$ dB |  $35$ dB

For the LTE communications, we established the downlink between a BS and one
user. Nonetheless, the experiments can be also be performed with uplink LTE as
well as bi-directional LTE link. LabVIEW LTE framework offers the possibility
to vary the Modulation and Coding Schemes (MCS) of PDSCH from $0$ to $28$
where the constellation size goes from QPSK to $64$QAM [62]. LTE uses PDSCH
for the transport of data between the BS and the user. TABLE V indicates the
experimental parameters used in our test set-up for the communications.

TABLE V: Communications experiment parameters Parameters | Value
---|---
Communication MCS |  | MCS0 (QPSK $0.12$)
---
MCS10 (16QAM $0.33$)
MCS17 (64QAM $0.43$)
Center frequency (Tx and Rx) |  $2$ GHz
Bandwidth |  $20$ MHz

###  V-A Impact of SILR Minimization

In this experiment, we show the impact of SILR enhancement on the coexistence.
We use the experiment parameters reported in TABLE III and TABLE V for radar
and communications, respectively.

According to these tables, we utilize the radar with a $50\%$ duty cycle. By
transmitting a set of $M=2$ waveforms with length $N=400$, radar transmissions
will occupy a bandwidth of $40$ MHz. On the other side, the LTE communications
framework utilizes $20$ MHz bandwidth for transmission. Further, we select the
allocation $1111111111110000000111111$ for the LTE resource block ($4$
physical resource blocks/bit), where the entry “1” indicates the use of the
corresponding time-bandwidth resources in the LTE application framework. The
spectrum of this LTE downlink is measured with the developed spectrum sensing
application as depicted in Figure 12(a). By setting $\theta=1$ in (5), the
radar optimizes its transmit waveform to avoid interference with the bands
occupied by the LTE resource blocks, while exploiting the holes in the $20$
MHz of the LTE spectrum. A screen capture of the resulting spectrum occupied
by radar and communications is depicted in Figure 12. This figure serves two
purposes, (i) focusing on the LTE downlink spectrum, it validates the spectrum
analyzer application with a commercial product, and (ii) it clearly indicates
that the desired objective of spectrum shaping is met. The impact of this
matching on performance of radar and communications are presented next.

(a) LTE at developed application.

(b) LTE at R&H spectrum analyzer.

(c) Radar and LTE at developed application.

Figure 12: Screen captures of the resulting spectrum occupied by the LTE
communications and radar signals at the developed two-channel spectrum sensing
application and R&H spectrum analyzer. The spectrum of the LTE downlink in (a)
is validated by a commercial product in (b), and (c) indicates the the
resulting spectrum of both communications (blue) and radar (red) at the
developed two-channel spectrum sensing application.

When the radar transmits random-phase sequences, it utilizes the entire
bandwidth and the two system mutually interfere. In fact, the operations of
both radar and communications are disrupted as depicted in Figure 13 (a and
c), thereby creating difficulties for their coexistence. In this case, by
utilizing the SILR optimized waveforms obtained by setting $\theta=1$ in
Algorithm 1, the performance of both systems are enhanced as indicated
pictorially in Figure 13 (b and d). A possible short-coming is the potentially
higher cross-correlation levels of the two transmit waveforms leading to false
targets impacting the radar performance.

(a) Random-phase sequences.

(b) Optimized sequences ($\theta=1$).

(c) Random-phase sequences.

(d) Optimized sequences ($\theta=1$).

Figure 13: LTE application framework in the presence of radar signal. In case
of transmitting random-phase sequences in radar at the same frequency band of
communications, the throughput of communications decreases drastically which
is depicted in (a). In this case, radar also cannot detect targets as depicted
in (c). In case of transmitting the optimized waveforms, the performance of
both radar and communications enhances (b and c).

###  V-B ICCL Minimization

In the absence of the LTE downlink, the radar can optimize its waveform based
on ICCL minimization by setting $\theta=0$ in Algorithm 1. Figure 14 indicates
the performance of the radar system in this case in comparison with a set of
random-phase sequences. The lower cross-correlation of the optimized sequences
can be visually observed from the lack of sidelobes in this figure. Since the
SILR is not considered, this optimized sequence is observed to occupy the
entire band causing degradation to communication link.

(a) Random-phase sequences.

(b) Optimized waveforms ($\theta=0$).

Figure 14: Range-Doppler plots for random-phase and optimized sequences
obtained by Algorithm 1 with $\theta=0$.

###  V-C Trade-off between SILR and ICCL

This section evaluates the performance of the optimized waveforms when
$\theta=0.75$, a value shown in Section IV to offer an adequate trade-off
between spectrum shaping and low cross-correlation. We first evaluate the
performance of radar and communication experiencing mutual interference when
the radar does not undertake the waveform optimization. Subsequently, we let
the radar optimize its waveform by setting $\theta=0.75$. Figure 15 depicts
range-Doppler plots of this case.

Figure 15: The performance of the radar in case of transmitting optimized
waveforms with $\theta=0.75$.

To evaluate the corresponding performances, we calculate the SINR of the two
targets for radar while on the communication side, report the PDSCH throughput
calculated by the LTE application framework. We perform our experiments in
following steps:

  * •

Step-1: In the absence of radar transmission, we collect the LTE PDSCH
throughput for MCS0, MCS10 and MCS 17. For each MCS, we use LTE transmit power
of $5$ dBm, $10$ dBm, $15$ dBm and $20$ dBm.

  * •

Step-2: In the absence of LTE transmission, we obtain the received Signal to
Noise Ratio (SNR) for the two targets. The SNR is calculated as the ratio of
the peak power of the detected targets to the average power of the cells close
to the target location in the range-Doppler map.

  * •

Step-3: We transmit a set of random-phase sequences as the radar waveform. At
the same time, we transmit the LTE waveform and let the two waveforms
interfere with each other. We log the PDSCH throughput as well as the SINR of
Target-$1$ and Target-$2$. We perform this experiment for MCS0, MCS10 and
MCS17 and for each MCS, we increase the LTE transmit power from $5$ dBm to
$20$ dBm in steps of $5$ dBm. Throughout the experiment we keep the radar
transmit power fixed. For each LTE MCS and LTE transmit power combination, we
average over $5$ experiments before logging the PDSCH throughput and target
SINR s.

  * •

Step-4: We repeat step-3, but using the optimized waveforms with $\theta=0.75$
at the radar transmitter. This optimization is carried out in real-time as
mentioned earlier using the spectrum sensing application and the Algorithm 1
implementation.

(a) MCS 0 (QPSK $0.12$)

(b) MCS 10 (16QAM $0.33$)

(c) MCS 17 (64QAM $0.43$)

Figure 16: PDSCH throughput of LTE under radar interference. We observe that
with radar interference reduces the PDSCH throughput but with cognitive
spectrum sensing followed by spectral shaping of the radar waveform PDSCH
throughput improves for all the LTE MCS.

(a) MCS 0 (QPSK $0.12$)

(b) MCS 10 (16QAM $0.33$)

(c) MCS 17 (64QAM $0.43$)

Figure 17: SINR of targets under interference from donwlink LTE link. We
observe that by optimizing the transmitting waveforms, the SNR of both the
targets improves. Note that in this experiment the SNR upper-bound for the
first and the second target in the absence of communications interference was
$22$ dB, and $17$ dB, respectively.

The performance of the communication link in terms of PDSCH throughput is
shown in Figure 16. We first observe that, the throughput of the link degrades
in the presence of radar interference. The degradation becomes prominent at
higher MCS since the SINR requirement to obtain a clean constellation for
larger modulations is also high. Subsequently, when the radar optimizes its
waveform as per Algorithm 1, the LTE throughput improves. Again we see that
the improvement is prominent in the higher MCS. This is due to the fact that
after a certain SINR, the lower MCS do not show any symbol error as the
constellation points are already well separated. But as the distance between
the constellation points decrease, even a small improvement of SINR leads to
improved Error Vector Magnitude (EVM) which leads to improved decoding and
hence a prominent increase in throughput.

We set $\theta=0.75$, such that the optimized waveforms avoid the frequency
bands occupied by communications, and simultaneously have an acceptable cross-
correlation. In this case, the performance of the radar in terms of target SNR
is shown in Figure 17. In the presence of LTE interference, we observe that
the SINR s of Target-1 and Target-2 degrade. These quantities improve when the
radar optimizes the transmitting waveforms. Interestingly, when the LTE
transmission power is high ($15$ dBm, and $20$ dBm), higher improvement
results from the avoidance of the used LTE bands. Precisely, when the
communication system is transmitting with a power of $20$ dBm, use of the
optimized waveforms enhances the SINR of Target-1, and Target-2 in excess of
$7$ dB in all the MCS values. Note that, due to the different attenuation
paths that is considered for the two targets (see TABLE III ), the measured
SINR s for these targets are different. Also, in the absence of the LTE
interference, the achieved SINR of Target-1, and Target-2 is $22$ dB and $17$
dB, respectively, which is the upper bound for the achievable SINR through the
optimized waveforms in presence of the communications interference.

##  VI Conclusion

In this paper, a radar-centric approach has been pursued towards a coexistence
with communication systems. Particularly, the paper developed a SDR based
cognitive MIMO radar prototype using USRP devices that coexist with LTE links.
To enable seamless operation of incumbent LTE links and opportunistic radar
sensing, the paper relied on cognition achieved through the implementation of
a spectrum sensing followed by the development of a MIMO waveform design
process. An algorithm based on CD approach is considered to design a set of
sequences, where the optimization is based on real-time feedback received from
the environment through the spectrum sensing application. The developed
prototype is tested both in controlled environment and OTA to validate its
functionalities and the experimental results indicating adherence to system
requirements and performance enhancement are noted. A further development of
this first coexistence prototype can be spatial beamforming in addition to the
spectral shaping that can be considered in advanced cognitive MIMO radar
systems.

## Appendix A

#####  SILR coefficients

Let $g_{a}(\mbox{\boldmath{$X$}})$ and $g_{b}(\mbox{\boldmath{$X$}})$ be the
numerator and denominator of $g_{s}(\mbox{\boldmath{$X$}})$ respectively.
Therefore $g_{a}(\mbox{\boldmath{$X$}})$ can be written as,

$\displaystyle g_{a}(\mbox{\boldmath{$X$}})\triangleq$
$\displaystyle\textstyle\sum_{m=1}^{M}\mathinner{\\!\left\lVert\mathbf{f}_{k}^{\dagger}\mathbf{x}_{m}\right\rVert}^{2}|k\in\mathcal{U}=\sum_{m=1}^{M}\sum_{k\in\mathcal{U}}\mathbf{x}_{m}^{\dagger}\mathbf{f}_{k}\mathbf{f}_{k}^{\dagger}\mathbf{x}_{m}$
(11) $\displaystyle=$
$\displaystyle\textstyle\sum_{m=1}^{M}\mathbf{x}_{m}^{\dagger}\mbox{\boldmath{$F$}}_{\mathcal{U}}\mathbf{x}_{m}=\sum_{m=1}^{M}\sum_{n=1}^{N}\sum_{l=1}^{N}x_{m,n}^{*}u_{n,l}x_{m,l}$

where,
$\mbox{\boldmath{$F$}}_{\mathcal{U}}\triangleq\sum_{k\in\mathcal{U}}\mathbf{f}_{k}\mathbf{f}_{k}^{\dagger}\in\mathbb{C}^{N\times
N}$ and $u_{n,l}$ are the elements of matrix
$\mbox{\boldmath{$F$}}_{\mathcal{U}}$. By some mathematical manipulation it
can be shown that,
$g_{a}(\mbox{\boldmath{$X$}})=a_{0}x_{t,d}+a_{1}+a_{2}x_{t,d}^{*}$, where,

$\displaystyle a_{0}$
$\displaystyle\triangleq\textstyle\sum_{\begin{subarray}{c}{n=1}\\\ {n\neq
d}\end{subarray}}^{N}x^{*}_{t,n}u_{n,d},\ a_{2}\triangleq a_{0}^{*},$ (12)
$\displaystyle a_{1}$
$\displaystyle\triangleq\textstyle\sum_{\begin{subarray}{c}{n=1}\\\ {n\neq
d}\end{subarray}}^{N}\sum_{{n,l=1}}^{N}x^{*}_{m,n}u_{n,l}x_{m,l}+\sum_{\begin{subarray}{c}{n,l=1}\\\
{n,l\neq d}\end{subarray}}^{N}x^{*}_{t,n}u_{n,l}x_{t,l}+u_{d,d}.$

Let us assume that,
$\mbox{\boldmath{$F$}}_{\mathcal{V}}\triangleq\sum_{k\in\mathcal{V}}\mathbf{f}_{k}\mathbf{f}_{k}^{\dagger}\in\mathbb{C}^{N\times
N}$. Likewise,
$g_{b}(\mbox{\boldmath{$X$}})=b_{0}x_{t,d}+b_{1}+b_{2}x_{t,d}^{*}$, where
$b_{i}$’s are obtained similar to $a_{i}$ with $v_{n,d}$, $v_{n,l}$ and
$v_{d,d}$ replacing $u_{n,d}$, $u_{n,l}$ and $u_{d,d}$ respectively, and
$v_{n,l}$ are the elements of matrix $\mbox{\boldmath{$F$}}_{\mathcal{V}}$.

#####  ICCL coefficients

The ICCL can be written as,
$\tilde{g}_{c}(\mbox{\boldmath{$X$}})=\sum_{\begin{subarray}{c}{m=1}\\\ {m\neq
t}\end{subarray}}^{M}\sum_{\begin{subarray}{c}{m^{\prime}=1}\\\
{m^{\prime}\neq
m,t}\end{subarray}}^{M}\sum_{k=-N+1}^{N-1}|r_{m,m^{\prime}}(l)|^{2}+\sum_{\begin{subarray}{c}{m^{\prime}=1}\\\
{m^{\prime}\neq
t}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}|r_{t,m^{\prime}}(l)|^{2}+\sum_{\begin{subarray}{c}{m=1}\\\
{m\neq t}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}|r_{m,t}(l)|^{2}$. Since
$\mathinner{\\!\left\lVert
r_{t,m}(l)\right\rVert}_{2}^{2}=\mathinner{\\!\left\lVert
r_{m,t}(l)\right\rVert}_{2}^{2}$, hence,
$\tilde{g}_{c}(\mbox{\boldmath{$X$}})$ becomes,

$\tilde{g}_{c}(\mbox{\boldmath{$X$}})=\gamma_{t}+2\textstyle\sum_{\begin{subarray}{c}{m=1}\\\
{m\neq t}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}|r_{m,t}(l)|^{2}$ (13)

where, $\gamma_{t}\triangleq\sum_{\begin{subarray}{c}{m=1}\\\ {m\neq
t}\end{subarray}}^{M}\sum_{\begin{subarray}{c}{m^{\prime}=1}\\\
{m^{\prime}\neq
m,t}\end{subarray}}^{M}\sum_{k=-N+1}^{N-1}|r_{m,m^{\prime}}(l)|^{2}$. Then
$r_{m,t}(l)$ can be written as,
$r_{m,t}(l)=\alpha_{mtdl}x_{t,d}+\gamma_{mtdl}$, where [52],

$\gamma_{mtdl}\triangleq\textstyle\sum_{\begin{subarray}{c}{n=1}\\\ {n\neq
d-k}\end{subarray}}^{N-k}x_{m,n}s_{t,n+l}^{*},\alpha_{mtdl}\triangleq
x_{m,d-l}I_{A}(d-l)$ (14)

where, $I_{A}(p)$ is the indicator function of set
$A=\left\\{1,\dots,N\right\\}$. By some mathematical manipulation the
$g_{c}(\mbox{\boldmath{$X$}})$ can be expressed as,
$g_{c}(\mbox{\boldmath{$X$}})=c_{0}x_{t,d}+c_{1}+c_{2}x_{t,d}^{*}$, where,

$\displaystyle c_{0}$
$\displaystyle\triangleq\textstyle\frac{2}{(2MN)^{2}}\sum_{\begin{subarray}{c}{m=1}\\\
{m\neq
t}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}\alpha_{mtdl}\gamma_{mtdl}^{*},\quad
c_{2}\triangleq c_{0}^{*}$ (15) $\displaystyle c_{1}$
$\displaystyle\triangleq\textstyle\frac{1}{(2MN)^{2}}(\gamma_{t}+2\sum_{\begin{subarray}{c}{m=1}\\\
{m\neq t}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}|\alpha_{mtdl}|^{2}$
$\displaystyle+\textstyle 2\sum_{\begin{subarray}{c}{m=1}\\\ {m\neq
t}\end{subarray}}^{M}\sum_{l=-N+1}^{N-1}|\gamma_{mtdl}|^{2}).$

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