Phase Synchronization for Coherent MIMO Radar: Algorithms and Their Analysis

Abstract

Multiple-input multiple-output (MIMO) radar can achieve improved localization performance by employing a coherent processing approach with proper antenna positioning. Coherent processing, however, entails the challenge of ensuring phase coherence of the carrier signals from different distributed radar elements. In this work, we aim to address such a challenge by providing a systematic treatment of the phase synchronization problem in coherent MIMO radar systems. We propose and study three different approaches for reaching a common notion of phase in coherent MIMO radar, namely, the master-slave closed-loop algorithm, the round-trip algorithm and the broadcast consensus based algorithm. These algorithms range from centralized to distributed types, and include both noniterative and iterative approaches. They do not require a priori establishment of the time synchronization, and thus are all time asynchronous in nature. Under a similar analytical framework, we mathematically characterize each of these algorithms, and further derive and study the statistical properties of a few relevant figures of merit including the resulting phase synchronization error. Simulation results are presented to validate our theoretical analysis.

Full text

Page 1

1
Phase Synchronization for Coherent MIMO Radar:
Algorithms and Analysis
Yang Yang, Member, IEEE, Rick S. Blum, Fellow, IEEE
Abstract
Multiple-input multiple-output (MIMO) radar can achieve improved localization performance by employing a
coherent processing approach with proper antenna positioning. Coherent processing, however, entails the challenge
of ensuring phase coherence of the carrier signals from different distributed radar elements. In this work, we aim
to address such a challenge by providing a systematic treatment of the phase synchronization problem in coherent
MIMO radar systems. We propose and study three different approaches for reaching a common notion of phase in
coherent MIMO radar, namely, the master-slave closed-loop algorithm, the round-trip algorithm and the broadcast
consensus based algorithm. These algorithms range from centralized to distributed types, and include both non-
iterative and iterative approaches. They do not require a priori establishment of the time synchronization, and
thus are all time asynchronous in nature. Under a similar analytical framework, we mathematically characterize
each of these algorithms, and further derive and study the statistical properties of a few relevant figures of merit
including the resulting phase synchronization error. Simulation results are presented to validate our theoretical
analysis.
Index Terms
Multiple-input multiple-output (MIMO) radar, coherent processing, phase synchronization.
I. INTRODUCTION
The last few years have seen a great surge of interest and progress in understanding the concept of multiple-
input multiple-output (MIMO) radar and developing paradigms to assess its potential and actual effectiveness.
This material is based on research supported by the US Army Space and Missile Defense Command under contract number W9113M-
08-C-0221 to ANDRO Computational Solutions, LLC, who in turn, sponsored a subcontract to Syracuse University. The U.S. Government
is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.
The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official
polices or endorsements, either expressed or implied, of the US Army Space and Missile Defense Command, ANDRO Computational
Solutions, LLC, Syracuse University or the U.S. Government.
Y. Yang and R. S. Blum are with the Department of Electrical and Computer Engineering, Lehigh University, 19 Memorial Drive West,
Bethlehem, PA 18015, USA (email: yay204@lehigh.edu; rblum@ece.lehigh.edu).

Page 2

2
Currently there exist two types of MIMO radar, collocated MIMO radar and more widely separated MIMO radar.
In this work, we are more concerned with the latter. It is also known that MIMO radar with more widely separated
antennas can operate in two modes: noncoherent and coherent [1]. Regardless of which mode MIMO radar is tuned
in to, all the elements in MIMO radar, before transmitting waveform signals to detect, estimate or track single
or multiple targets, need to synchronize firstly in terms of, for example, time, frequency, or phase. As a matter
of fact, phase synchronization embodies a major difference between the operations of noncoherent MIMO radar
and coherent MIMO radar, as will be mathematically demonstrated later in this work. To be more specific, the
coherent approach can provide improved target localization performance but it requires phase synchronization
between the separated antenna elements1. Unfortunately, in reality, those radar elements are usually operated
with physically different local oscillators, and each of them suffers from statistically independent phase offset,
indicating that the phase of the carrier signal transmitted gets rotated by an unknown amount. As a consequence,
practical implementation of coherent MIMO radar entails the challenge of ensuring phase coherence of the carrier
signals from different distributed radar elements.
The imperfection in phase synchronization that would inevitably occur in coherent MIMO radar, on the
other hand, has recently attracted the attention of some researchers. For example, some theoretical analysis
conducted on coherent MIMO radar has taken into account the phase synchronization error or mismatch, such
as [2]-[5]. However, the technical feasibility of phase coherence in coherent MIMO radar as well as approaches
for achieving such phase coherence has received very little attention. Moreover, although the synchronization
problem has been intensely studied in several closely related contexts, for example, in traditional radar systems
(see, e.g., [6]-[10]) and in wireless networks including sensor networks (see e.g., [11]-[18]), we have not seen
much work that is specifically targeted at attaining phase coherence, or phase synchronization in coherent MIMO
radar. Without doubt, this line of work, which addresses the challenge that is inherent to coherent MIMO radar,
is of both practical and theoretical importance, and merits interest on its own terms.
As a result, a systemic treatment of the phase synchronization problem in coherent MIMO radar becomes
the primary objective of our study, which, is also a major contribution of our work. As far as the novelty
of our work goes, the very few efforts to date to this problem has made ours the first of its kind, according
to our best knowledge. More specifically, in our work, we propose and discuss three different yet efficient
approaches to achieve phase coherence in coherent MIMO radar, namely, the master-slave closed-loop algorithm,
the round-trip algorithm and the broadcast consensus based algorithm. These phase synchronization algorithms
range from centralized to distributed types, and include both non-iterative and iterative methods. They are versatile
and practically implementable, and should cater to different needs arising as a result of different number of
1In this paper, we use the terms sensor, node, and element interchangely.

Page 3

3
MIMO radar elements, different sensor placements, as well as different operational preferences. They are all
time asynchronous algorithms, and do not require the establishment of time synchronization in advance. In
particular, the master-slave closed-loop algorithm and the round-trip algorithm also require no a priori frequency
synchronization. Under a similar analytical framework, for each of these algorithms, we derive and also provide
an analysis of the statistical properties of some interesting figures of merit, such as the aggregate phase error
after synchronization and the phase deviation from consensus. Our analysis clearly reveals the impact of the
phase and frequency estimation errors on the phase synchronization accuracy.
Of particular note is that while each of these phase synchronization algorithms may resemble some al-
gorithm(s) in existing works that are designed for or applied to situations other than phase synchronization in
coherent MIMO radar, they do contain distinct features and contributions when in comparison. For example,
a similar approach to our master-slave closed-loop algorithm has been used in [14] to synchronize the carrier
phase of a group of nodes, but [14] implicitly assumes the a priori establishment of time synchronization. In
contrast, our work does not assume either time nor frequency synchronization, and features an asynchronous time
model and timing analysis. The round-trip type method proposed in [15]-[18] for global clock synchronization
in sensor networks or carrier synchronization for distributed beamforming share the same essential features as
our round-trip algorithm, but in general they cannot be applied to our problem in a straightforward manner.
Besides, in our work, we provide a thorough analysis of the resulting phase synchronization error at each radar
element, which is not available in these works. Finally, the application of broadcast-based consensus algorithm
to phase synchronization and the analysis of its convergence behavior in the presence of non-zero-mean phase
perturbations, is an extension of our work in [33], and is new on its own.
Therefore, each algorithm proposed in our work represents a unique addition to their respective category of
methods. Overall, our work, providing a practical orientation along with a theoretical depth, is a first attempt to
explicitly address practical issues related to phase synchronization in coherent MIMO radar. It is also worth noting
that in our work, we only consider space as the transmission medium through which phase synchronization is
implemented, but we note that these synchronization algorithms can remain effective when phase synchronization
is to be achieved across different radar elements through wired connections, such as coaxial cable or fiber
optic links. For example, the broadcast consensus based phase synchronization algorithm given in this paper is
applicable to other types of media than the wireless environment, as long as broadcast is enabled. Additionally, we
assume sensors operate in the half-duplex mode, which means that they cannot receive while they are transmitting.
The remainder of this paper is organized as follows. In Section II, we give a brief introduction to the
coherent MIMO radar model, demonstrate the difference between coherent and non-coherent processing for
target localization, and most importantly, elucidate the motivation of our work. In Section III, we describe a
simple master-slave closed-loop phase synchronization approach, and examine the statistical properties of the

Page 4

4
cumulative phase error. In Section IV, we propose a round-trip phase synchronization algorithm, and analyze
the phase synchronization error. In Section V, we propose a broadcast consensus based phase synchronization
algorithm, and study some convergence properties of this algorithm. In Section VI, we summarize and compare
these three phase synchronization algorithms. In Section VII, we provide some simulation results on the broadcast
consensus based algorithm. In Section VIII, we end this paper with some concluding remarks.
II. COHERENT MIMO RADAR MODEL
In this section, we provide a concise description of the coherent MIMO radar model, and illustrate the
difference between coherent and non-coherent processing in the context of target localization. Note that to make
the discussion very clear, we consider here some simplified assumptions. Let us consider a MIMO radar that is
equipped with M transmitters and N receivers. Thus, the total number of radar elements is M + N, and for
notational convenience we denote K = M + N. The positions of the k-th, k = 1,··· ,M transmitter and the
l-th, l = 1,··· ,N receiver are (xt
k,yt
k) and (xr
l,yr
l), respectively, in a two-dimensional Cartesian coordinate
system. The lowpass equivalent of the signal transmitted from the k-th transmitter is
?E/M sk(t), where E
denotes the total transmitted energy, and the waveform is normalized such that
?∞
−∞
|sk(t)|2dt = 1.
Assume the target’s location and velocity are (x,y) and (vx,vy), respectively. For notational convenience, we
define a parameter vector as follows:
θ = [x,y,vx,vy]T.
In the coherent mode, the antennas are assumed to be all within the effective beamwidth of the target, if
the target is modeled as an antenna [1]. Thus, the reflection coefficient is assumed to be the same for each
(l,k)-th path, and is denoted as ζ. We assume ζ is unknown but deterministic2. Then, the signal emanated from
transmitter k and captured by receiver l can be expressed as
rlk(t) =
?
E
Mζsk(t − τlk)ej2πflkt+ zlk(t),
(1)
where τlkand flkrepresent the time delay and Doppler shift, respectively, corresponding to the (l,k)-th path.
2Considering random ζ leads to very similar results but the deterministic case simplifies the analysis.

Page 5

5
In particular, the time delay τlkis a function of the unknown target location (x,y), i.e.,
τlk
=
?
dt
(xt
k− x)2+ (yt
k− y)2+?(xr
l− x)2+ (yr
l− y)2
c
=
k+ dr
c
l
,
where c denotes the speed of light, dt
kthe distance between the target and the k-th transmitter, and dr
lthe distance
between the target and the l-th receiver. The Doppler shift flkis a function of the unknown target location (x,y)
and velocity (vx,vy), i.e.,
flk=vx(xt
k− x) + vy(yt
λdt
k− y)
k
+vx(xr
l− x) + vy(yr
λdr
l− y)
l
,
where λ denotes the wavelength of the carrier. In (1), zlk(t) denotes the noise-plus-clutter for the (l,k)-th path.
For simplicity of discussion, we assume zlk(t) is a temporally white, zero-mean complex Gaussian random
process. More specifically, we have
E{zlk(t)z∗
lk(u)} = σ2
zδ(t − u),
where σ2
zis a constant and δ(t) is a unit impulse function. Additionally, to explain the concept of coherent
processing in a simple way, we assume elements of the noise-plus-clutter are spatially white, i.e.,
E{zlk(t)z∗
l′k′(u)} = 0,∀l ?= l′,or k ?= k′.
For simplicity, we set σ2
z= 1.
Let r(t) be an MN × 1 vector which contains all the signals received at the receive elements, i.e.,
r(t) = [r11(t),r12(t),··· ,rNM(t)]T.
The scatter reflectivity ζ can be estimated using the maximum likelihood (ML) method, and we provide this
result directly as follows
˜ζ =
1
MN
M
?
k=1
N
?
l=1
?∞
−∞
rk,l(t)s∗
k(t − τk,l)dt
(2)
As the noise components are i.i.d. Gaussian, the log-likelihood function of the received signal vector, after
simplification, can be expressed as
Λ(r(t)|θ) = c′
?????
M
?
k=1
N
?
l=1
ej2πfcτlk
?∞
−∞
˜ rlk(t)s∗
k(t − τlk)e−j2πflktdt
?????
2
+ c′′
(3)
where c′and c′′are both constants. Thus, the ML estimate of the deterministic but unknown parameter vector