1

Towards Dual-functional Radar-Communication
Systems: Optimal Waveform Design

arXiv:1711.05220v1 [] 14 Nov 2017

Fan Liu, Student Member, IEEE, Longfei Zhou, Student Member, IEEE, Christos Masouros, Senior Member, IEEE,
Ang Li, Student Member, IEEE, Wu Luo, Member, IEEE and Athina Petropulu, Fellow, IEEE

Abstract—We focus on a dual-functional multi-input-multioutput (MIMO) radar-communication (RadCom) system, where
a single transmitter communicates with downlink cellular users
and detects radar targets simultaneously. Several design criteria
are considered for minimizing the downlink multi-user interference. First, we consider both the omnidirectional and directional
beampattern design problems, where the closed-form globally
optimal solutions are obtained. Based on these waveforms, we
further consider a weighted optimization to enable a flexible
trade-off between radar and communications performance and
introduce a low-complexity algorithm. The computational costs
of the above three designs are shown to be similar to the
conventional zero-forcing (ZF) precoding. Moreover, to address
the more practical constant modulus waveform design problem,
we propose a branch-and-bound algorithm that obtains a globally
optimal solution and derive its worst-case complexity as a
function of the maximum iteration number. Finally, we assess
the effectiveness of the proposed waveform design approaches
by numerical results.
Index Terms—Spectrum sharing, radar-communication, multiuser interference, non-convex optimization, global minimizer.

I. I NTRODUCTION

I

T has been reported that by 2020, the number of connected devices will jump to more than 20 million, which
brings forward impending needs for extra frequency spectrum
resources. Realizing the scarcity of the spectrum, network
providers and policy regulators are exploring the feasibility
to use the spectrum that is currently occupied by other
applications [1]–[4], such as airborne radars and navigation
systems close to the 3.4GHz band [5] and shipborne and Vessel
Traffic Service (VTS) radar at 5.6GHz [6], which may be
shared with LTE and Wi-Fi systems in the near future. As an
emerging research topic, the communications-radar spectrum
sharing (CRSS) not only presents the advantage for enabling
the efficient usage of the spectrum, but also provides a new
F. Liu is with the School of Information and Electronics, Beijing Institute
of Technology, Beijing, 100081, China, and is also with the Department of
Electronic and Electrical Engineering, University College London, London,
WC1E 7JE, UK (e-mail: liufan92@bit.edu.cn).
L. Zhou and W. Luo are with the State Key Laboratory of Advanced Optical
Communication Systems and Networks, Peking University, Beijing 100871,
China (e-mail: zhoulongfei@pku.edu.cn, luow@pku.edu.cn).
C. Masouros and A. Li are with the Department of Electronic and Electrical
Engineering, University College London, London, WC1E 7JE, UK (e-mail:
chris.masouros@ieee.org, ang.li.14@ucl.ac.uk).
A. P. Petropulu is with the Department of Electrical and Computer
Engineering, Rutgers, the State University of New Jersey, 94 Brett Road,
Piscataway, NJ 08854, United States (e-mail: athinap@rutgers.edu).
This work has been submitted to the IEEE for possible publication.
Copyright may be transferred without notice, after which this version may
no longer be accessible.

way for designing novel systems that can benefit from the
cooperation of radar and communications.
As a straightforward way to achieve the spectral coexistence
for communication and radar, opportunistic spectrum sharing
[7] provides a naive approach, where the communication
system transmits when the space and frequency spectra are
not occupied by radar. Nevertheless, it does not allow the two
systems to work simultaneously. In view of this, the pioneering
work [8] proposes a null-space projection (NSP) method,
which has been widely applied to different spectral coexistence
scenarios between MIMO radar and communication systems
[9], [10]. In such schemes, a radar beamformer is designed
to project the signals onto the null-space of the interference channel between the radar and base station (BS)/user
equipment (UE), such that the interference from the radar
to the communication link is zero. This, however, results in
performance loss for the radar, since the beamforming is no
longer optimal for target detection and estimation. Trade-offs
between the performance of radar and communications can
be achieved by relaxing the zero-forcing precoder to impose
controllable interference levels on the communication systems
[11], which offers a more realistic coexistence.
More recent contributions have exploited optimization techniques to realize CRSS. In [12], the radar beamformer and
communication covariance matrix are jointly designed to maximize the Signal-to-Interference-plus-Noise-Ratio (SINR) of
the radar subject to capacity and power constraints at the
communication’s side. Similar work has been done for the coexistence between the MIMO-matrix completion (MIMO-MC)
radar and point-to-point (P2P) MIMO communications [13],
[14], where the radar sub-sampling matrix is further introduced
as an optimization variable. To address the more practical
coexistence issue between MIMO radar and multi-user MIMO
(MU-MIMO) communications, recent work in [15] considers
the robust beamforming designs with imperfect channel state
information (CSI) at the communication’s side, where the
detection probability of the radar is maximized subject to
SINR constraints of the downlink users and the power budget
of the BS. As a further development of the technique, a novel
beamforming design has been proposed in [16] that exploits
the interference as a useful power source, which demonstrates
orders-of-magnitude power-savings. While the above coexistence approaches are well-designed, a critical shortfall is that
radar and communication devices are required to exchange
side-information for achieving a beneficial cooperation, such
as the CSI, radar probing waveforms and communication
modulation formats. Typically, these exchanges are realized

2

by an all-in-one control center that is connected to both
systems via either a wireless link or a backhaul channel
[14], which conducts the coordination of the cooperation.
In practical scenarios, however, such a control center brings
forward considerable extra complexity to the system, and is
therefore difficult to implement.
In contrast to the above coexistence schemes, a more favorable approach for CRSS is to design a novel dual-functional
system that carries out both radar and communications, where
the above problem does not exist. Note that such methods
are distinctly different from the classic cognitive radio based
techniques, as they require the use of specific radar constraints
and designs. Recent information theoretical work has shown
great potential [17], [18], but it remains to be seen what
benefits can be implemented in practice. As an enabling
solution, dual-functional waveform design can support target
detection while carrying information at the same time. Such
possibilities have been explored for single-antenna systems,
where several integrated waveforms that combine the radar
and communication signals have been proposed [19]–[21].
Nevertheless, all of these schemes lead to performance loss for
either radar or communication, e.g., high peak-average-powerratio (PAPR) and limited dynamic range [20]. As a step further,
recent works consider dual-functional waveform design for
MIMO systems. In [22], a transmit beampattern for MIMO
radar is designed to embed the information bits in sidelobe
levels. Related works consider waveform shuffling across the
antennas or Phase Shift Keying (PSK) by different beamformer
weighting factors as the communication modulation schemes
[23], [24]. It should be noted that in the above approaches,
one communication symbol is represented by one or several
radar pulses, which leads to a low date rate in the order of
the radar pulse repetition frequency (PRF). To support multiuser transmission for the cellular downlink, the previous work
[25] develops a series of beamforming approaches for dualfunctional RadCom systems, which will not affect the original
modulation scheme and the data rate of the communication
system. Nevertheless, the beamforming approaches only focus
on the average power constraints, and do not address the
design of the constant modulus signals.
As an important requirement for both radar and communication applications, the utilization of constant modulus
waveforms can avoid signal distortion when the low-cost nonlinear power amplifiers are used [26], which leads to an
energy-efficient transmission. Such topics have been widely
studied for massive MIMO communication scenarios [27]–
[30] as well as the MIMO radar waveform designs [31]–
[34], where optimization problems with non-convex constant
modulus constraint (CMC) are formulated. Due to the NPhardness of these problems, only suboptimal solutions can be
obtained via either convex relaxation methods or local algorithms, such as Semidefinite Relaxation (SDR) [31], [32] and
Riemannian manifold methods [28], [30]. Recent MIMO radar
work proposes to approach the constant modulus solution by
a successive Quadratic Constrained Quadractic Programming
(QCQP) Refinement (SQR) procedure [33]. Nevertheless, this
technique still only guarantees the local optimality of the
obtained solution. To the best of our knowledge, the efficient

global algorithm for constant modulus waveform design is
widely unexplored in the existing literature.
In this paper, we propose several optimization-based waveform designs for the dual-functional RadCom systems. It
is worth highlighting that all the proposed methods yield
probably globally optimal waveforms, which can be used for
both target detection and downlink communications. Throughout the paper, we aim to minimize the downlink multi-user
interference (MUI) under radar-specific constraints. First, we
consider an orthogonal waveform design, which is often used
for the initial omnidirectional probing by MIMO radar. Based
on this waveform, we extend to the design of a directional
radar beampattern that points to the targets of interest. The
aforementioned two optimization problems are non-convex,
but the optimal solutions can be readily obtained in closedforms. Still, the obtained performance for the communication
system is limited. To allow a trade-off between radar and communication performance, we consider a weighted optimization
under non-convex power budget constraint, and obtain its
global optimum via a well-designed low-complexity algorithm.
Given that both radar and communication systems require
constant modulus signals for power-efficient transmission, we
finally consider a more practical optimization by enforcing
constant modulus constraints and similarity constraints (SC)
on the waveform design. In contrast to the existing approaches
in both radar and communication works that obtain the local
minimizers of problems with CMC [27]–[34], we propose a
branch-and-bound method that can efficiently yield a globally
optimal solution for the problem. Our numerical results show
that the proposed branch-and-bound algorithm considerably
outperforms the conventional SQR method [33]. For clarity,
we summarize our contributions as follows:
•

•

•

•

We propose dual-functional waveform design approaches
for both omnidirectional and directional radar beampatterns, and derive the closed-form solutions.
We propose a weighted optimization that achieves a flexible trade-off between the radar and communication performance, and solve the problem with a low-complexity
global algorithm.
We consider the waveform design with CMC and SC
constraints, and develop a branch-and-bound algorithm to
obtain the globally optimal solutions, which outperforms
the conventional SQR algorithm.
We derive the computational complexity for the proposed
algorithms analytically.

The remainder of this paper is organized as follows, Section
II introduces the system model, Section III proposes the
closed-form waveform optimizations for radar beampattern
design, Section IV considers the trade-off design between
radar and communications, Section V solves the problem
with CMC and SC constraints, Section VI provides numerical
results, and finally Section VII concludes the paper.
Notations: Unless otherwise specified, matrices are denoted
by bold uppercase letters (i.e., H), vectors are represented by
bold lowercase letters (i.e., x), and scalars are denoted by
normal font (i.e., ρ). Subscripts indicate the location of the
entry in the matrices or vectors (i.e., si,j and ln are the (i, j)-th

3

For each user, the entry of S is assumed to be drawn from
the same constellation. The second term in (2) represents the
MUI signals. The total MUI energy can be measured as
PMUI = kHX − Sk2F .

(3)

It has been proven in [27] that the MUI energy in (3) directly
links to the achievable sum-rate of the downlink users. For the
i-th user, the SINR per frame is given as [27]


2
E |si,j |

γi = 
,
(4)
2
+N0
E hTi xj − si,j
{z
}
|
MUI energy

Fig. 1. Dual-functional Radar-Communication System.

and the n-th element in S and l, respectively). tr (·), (·)T , (·)H
∗
and (·) stand for trace, transpose, Hermitian transpose and
complex conjugate, respectively. Re (·) and Im (·) denote the
real and imaginary part of the argument, k·k, k·k∞ and k·kF
denote the l2 norm, l∞ and the Frobenius norm respectively.
II. S YSTEM M ODEL
We consider a dual-functional MIMO RadCom system,
which simultaneously transmits radar probing waveforms to
the targets and communication symbols to the downlink users.
The joint system is equipped with a uniform linear array
(ULA) with N antennas, serving K single-antenna users while
detecting radar targets at the same time.
A. Communication Model
The received symbol matrix at the downlink users can be
given as
Y = HX + W,
(1)
where H = [h1 , h2 , ..., hK ]T ∈ CK×N is the channel matrix,
X = [x1 , x2 , ..., xL ] ∈ CN ×L is the transmitted signal
matrix, with L being the length of the communication frame,
W = [w1 , w2 , ..., wL ] ∈ CK×L is the noise matrix, with
wj ∼ CN (0, N0 IN ) , ∀j.
Following [25], we rely on the following assumptions: 1)
The transmitted signal matrix X is used as dual-functional
waveform for both radar and communication operations. In
this case, each communication symbol is also a snapshot of
a radar pulse; 2) The downlink channel H is flat Rayleigh
fading, and remains unchanged during one communication
frame/radar pulse; 3) The channel H is assumed to be perfectly
estimated by pilot symbols.
Given the desired constellation symbol matrix S ∈ CK×L
for the downlink users, the received signals can be rewritten
as
(2)
Y = S + (HX − S) +W,
| {z }
MUI

where si,j is the (i, j)-th entry of S, E denotes the ensemble
average with respect to the time index. It follows that the
achievable sum-rate of the users can be given as
R=

K
X

log2 (1 + γi ).

(5)

i=1

For a given constellation
 with fixed energy, the power of the

2
useful signal E |si,j | is also fixed. Hence, the sum-rate can
be maximized by minimizing the MUI energy.
B. Radar Model
It is widely known that by employing uncorrelated waveforms, MIMO radar achieves higher Degrees of Freedom
(DoFs) than the traditional phased-array radar [35], [36]. The
existing literature indicates that the design of such a beampattern is equivalent to designing the covariance matrix of the
probing signals, where convex optimization can be employed.
We refer readers to [36]–[38] for more details on this topic.
Here we focus on designing the dual-functional waveform
matrix X, which has the following spatial covariance matrix
1
XXH .
(6)
L
To ensure that RX is positive-definite, we assume L ≥ N
without loss of generality. Further, the transmit beampattern
for the RadCom system can be given as
RX =

Pd (θ) = aH (θ) RX a (θ) ,

(7)

where θ denotes the detection angle, a (θ)
=
 j2π∆ sin(θ)
T
1, e
, ..., ej2π(N −1)∆ sin(θ)
∈ CN ×1 is the
steering vector of the transmit antenna array with ∆ being
the spacing between adjacent antennas normalized by the
wavelength.
In the following, we formulate optimization problems that
minimize PMUI under MIMO radar-specific constraints.
III. C LOSED - FORM WAVEFORM D ESIGN
R ADAR B EAMPATTERNS

FOR

G IVEN

In this section, we first consider the omnidirectional beampattern design, which is usually used in MIMO radar for initial
probing. After that, we consider a directional beampattern
design that points to the directions of interest.

4

A. Omnidirectional Beampattern Design

C. Complexity Analysis

For an omnidirectional beampattern, the transmitted waveform matrix X has to be orthogonal, i.e., the corresponding
covariance matrix must be the identity matrix. To minimize
the MUI energy, the optimization problem is formulated as

The omnidirectional beampattern design includes two matrix multiplications and one SVD, which needs a total of
O N KL + 2N L2 complex floating-point-operations (flops),
where one complex flop is defined as one complex addition or multiplication. The directional beampattern design, which needs one Cholesky decomposition, four matrix multiplications and one SVD, has the total
complexity

of O 2N L2 + N 2 L + N KL + N 3 + N 2 K . For the conventional communication-only zero-forcing (ZF) precoding,
which involves one pseudoinverse for H, and one matrix
multiplication between the precoder and the transmitted
sym
bol matrix, the complexity is O N KL + N 2 K . It is worth
noting that the computational costs of the proposed closedform approaches share the same order of magnitude with that
of the zero-forcing precoder.

min kHX − Sk2F
X

(8)
1
PT
XXH =
IN ,
L
N
where PT is the total transmit power, IN denotes the N × N
identity matrix. Problem (8) is obviously non-convex due to
the equality constraint, which indicates that X is a point on the
Stiefel manifold. Fortunately, it has been proven that (8) can
be classified as the so-called Orthogonal Procrustes problem
(OPP), which has a simple closed-form global solution based
on the Singular Value Decomposition (SVD), and is given as
[39]
r
LPT
X=
UIN ×L VH ,
(9)
N
s.t.

where UΣVH = HH S is the SVD of HH S with U ∈ CN ×N
and V ∈ CL×L being the unitary matrices, IN ×L is an N × L
rectangular matrix composed by an N ×N identity matrix and
an N × (L − N ) zero matrix.
B. Directional Beampattern Design
Given a covariance matrix Rd that corresponds to a welldesigned MIMO radar beampattern, the MUI minimization
problem is given as

IV. T RADE - OFF B ETWEEN R ADAR AND C OMMUNICATION
P ERFORMANCE
It should be highlighted that both problem (8) and (10)
enforce a strict equality constraint, in which case the radar
performance is guaranteed to be optimal while the communication counterpart may suffer from serious performance loss.
This is particularly pronounced in the cases that the covariance
matrices of the communication channel are ill conditioned,
where the resulting MUI minimum is still high. We therefore
consider a trade-off design by allowing a tolerable mismatch
between the designed and the desired radar beampatterns.

2

min kHX − SkF
X

(10)
1
s.t. XXH = Rd ,
L
where Rd is the desired Hermitian positive semidefinite covariance matrix, and tr (Rd ) = PT . We consider its Cholesky
decomposition, which is
Rd = FFH ,

(11)

where F ∈ CN ×N is a lower triangular matrix. Without loss of
generality, we assume Rd is positive-definite to ensure that F
is invertible. Hence, the constraint in (10) can be equivalently
written as
1 −1
F XXH F−H = IN .
(12)
L
q
Denoting X̃ = L1 F−1 X, problem (10) can be reformulated
as
2
√
LHFX̃ − S
min
F
(13)
X̃
H
s.t. X̃X̃ = IN ,
which is again an OPP problem, and its globally optimal
solution is given by
X̃ = ŨIN ×L ṼH ,
H

H

H

(14)
H

H

where ŨΣ̃Ṽ = F H S is the SVD of F H S. It follows
that the solution of the original problem (10) is given as
√
(15)
X = LFŨIN ×L ṼH .

A. Problem Formulation
Let us first denote the optimal solution obtained from (8)
or (10) as X0 . Given X0 , the trade-off problem can be then
formulated as
2

2

min ρ kHX − SkF + (1 − ρ) kX − X0 kF
X

(16)
1
2
kXkF = PT ,
L
where 0 ≤ ρ ≤ 1 is a weighting factor that determines
the weights for radar and communication performance in the
dual-functional system. For coherence between (16) and the
previous two problems, we enforce an equality constraint for
the power budget, as the radar is often required to transmit at
its maximum available power in practice.
Note that the two Frobenius norms in the objective function
can be combined in the form
s.t.

ρ kHX − Sk2F + (1 − ρ) kX − X0 k2F
iT
iT
h√
h√
p
p
ρHT , 1 − ρIN X −
ρST , 1 − ρXT0
=

2

.
F

(17)
√ T √
T
Let us denote A =
ρH , 1 − ρIN ∈ C(K+N )×N , B =
T
√ T √
∈ C(K+N )×L , problem (16) can be
ρS , 1 − ρXT0
written compactly as
2

min kAX − BkF
X

2

s.t. kXkF = LPT ,

(18)

5

which is a non-convex QCQP, and can be readily transformed
into a Semidefinite Programming (SDP) using SDR technique.
Since it has only one quadratic constraint, according to [40],
[41], the SDR is tight, i.e., the solution of the SDR is
rank-1, which yields the globally optimal solution of (18).
Nevertheless, due to the large number of variables in the
problem, SDR is not computationally efficient in general.
Hence, we propose a low-complexity algorithm that achieves
the global optimum in the following.
B. Low-complexity Algorithm
We further expand the objective function of (18) as


2
H
kAX − BkF = tr (AX − B) (AX − B)


= tr XH AH AX − tr XH AH B


− tr BH AX + tr BH B .

(19)

Defining Q = AH A, G = AH B, problem (18) can be
rewritten as


min tr XH QX − 2 Re tr XH G
X
(20)
2
s.t. kXkF = LPT .
Since Q is a Hermitian matrix, problem (20) can be viewed
as the matrix version of the trust-region subproblem (TRS),
for which the strong duality holds [42], i.e., the duality gap is
zero. Let us formulate the Lagrangian multiplier as


L (X, λ) = tr XH QX − 2 Re tr XH G


(21)
2
+λ kXkF − LPT ,

where λ is the dual variable associated with the equality
constraint. Let Xopt and λopt be the primal and dual optimal
points with zero duality gap, the optimality conditions for the
above TRS can be given as [43]

∇L (Xopt , λopt ) = 2 (Q + λopt IN ) Xopt − 2G = 0, (22a)
2

kXopt kF = LPT , (22b)
Q + λopt IN  0, (22c)

where (22b) and (22c) guarantee the primal and the dual
feasibility respectively. It follows from (22a) that
†

Xopt = (Q + λopt IN ) G,

(23)

†

where (·) denotes the Moore-Penrose pseudoinverse of the
matrix. Based on (22b) and (22c) we have
(Q + λopt IN )† G

2
F

= V(Λ + λopt IN )

−1

VH G

2
F

= LPT ,

that there exists an unique solution for the equations (24). Let
us define
−1

P (λ) = V(Λ + λIN ) VH G

 2
N X
L
VH G i,j
X
,
=
2
(λ + λi )
i=1 j=1

2
F

(25)

where λi is the i-th eigenvalue of Q. It can be seen that P (λ)
is strictly decreasing and convex on the interval λ ≥ −λmin ,
which suggests that λopt can be obtained by simple line search
methods, e.g., Golden-section search [44]. Thanks to the
eigenvalue decomposition, in each iteration we only need to
calculate the inversion of a diagonal matrix. Once the optimal
λ is obtained, the optimal solution to (16) can be computed
by (23). For clarity, we summarize the above approach in
Algorithm 1.
Algorithm 1 Low-complexity Algorithm for Solving (16)
Input: H, S, x0 , weighting factor 0 ≤ ρ ≤ 1, PT ,
Output: Global minimizer Xopt
1. Compute A, B, Q and G.
2. Compute the eigenvalue decomposition of Q, set the
searching interval as [−λmin , b], where b ≥ 0 is a searching
upper-bound.
3. Find the optimal solution λopt to (24) using Goldensection search.
†
4. Xopt = (Q + λopt IN ) G.

C. Complexity Analysis
We end this section by analyzing the complexity of Algorithm 1. The Golden-section search method is known to
have linear convergence rate, which finds an ε0 -solution within
O (log (1/ε0 )) iterations. In each iteration we calculate the
value of a 1-dimensional function, which suggests that the
complexity of the Golden-section search can be omitted in
general. Hence the complexity for Algorithm 1 is domainated
by the matrix multiplications, the pseudoinverse and the
eigenvalue decomposition. Both of the latter
 two operations
involve the computational costs of O N 3 complex flops,
and the matrix multiplications  involve the complexity of
O N 2 L + N KL + N 3 + N 2 K . Therefore, the total com
plexity for Algorithm 1 is O N 2 L + N KL + 3N 3 + N 2 K ,
which again shares the same order of magnitude with the
communication-only ZF precoding. For the sake of clarity,
we summarize the computational costs of the proposed three
waveform design approaches in TABLE I.

(24)

λopt ≥ −λmin .
where Q = VΛVH is the eigenvalue decomposition of Q
with V and Λ being the orthogonal and diagonal matrices
that contain the eigenvectors and eigenvalues respectively, and
λmin is the minimum eigenvalue of Q. One can further show

V. C ONSTANT M ODULUS WAVEFORM D ESIGN
In the previous sections, the dual-functional RadCom waveform is designed under total power constraints, which is
not guaranteed to generate constant modulus signals. In this
section, we consider the RadCom waveform design that minimizes the communication MUI energy given the CMC.

6

Im

TABLE I
C OMPUTATIONAL C OMPLEXITY FOR THE P ROPOSED A PPROACHES

ar
c(

l, u

)

exp ( jl )

q

)

q

(

Omnidirectional Design
Directional Design
Trade-off Design
Zero-forcing (benchmark)

Complex Flops

O N KL + 2N L2

O 2N L2 + N 2 L + N KL + N 3 + N 2 K

O N 2 L + N KL + N 3 + N 2 K

O N KL + N 2 K

=

Method

O

Re

PR 2

exp ( ju )

A. Problem Formulation

PR1

x

Following the same notations in the previous section, our
optimization problem can be formulated as
2

min kHX − SkF

(26a)

s.t kvec (X − X0 )k∞ ≤ η,
r
PT
, ∀i, j,
|xi,j | =
N

(26b)

X

(26c)

where X0 ∈ CN ×L is a known benchmark radar signal matrix
that has constant-modulus entries, e.g., chirp signals, vec (·)
denotes the vectorization of a matrix, and xi,j is the (i, j)-th
entry of X. The constraint (26b) is called similarity constraint
(SC) in the radar literature [32], which controls the difference
between the designed waveform and the benchmark with η
being the tolerable difference.
It is trivial to see that the objective function of (26) is
separable, since
2

kHX − SkF =

L
X
i=1

2

kHxi − si k .

x

r

PT
Hx − s
N

q

2

(28)

N
PT

, x ∈ CN ×1 , x0 ∈ CN ×1 are the columns of
q
X and X0 normalized by PNT , s ∈ CK×1 is the column of
S, and x (n) denotes the n-th entry of x. Since problem (26)
can be solved by solving the problem (28) for each column
of X concurrently, we will focus on (28) in the following
discussion. For notational convenience, we omit the column
index.
Note that 0 ≤ ε ≤ 2 since both x and x0 have unit modulus.
According to [32], the similarity constraint can be rewritten
as
arg x (n) ∈ [ln , un ] , ∀n,
where

 
ln = arg x0 (n) − arccos 1 − ε2 2 ,
 
un = arg x0 (n) + arccos 1 − ε2 2 ,

min f (x) = H̃x − s

2

x

s.t
q

arg x (n) ∈ [ln , un ] , ∀n,

(31)

|x (n)| = 1, ∀n,

PT
where H̃ =
N H. For each x (n), the feasible region is
an arc on the unit circle as shown in Fig. 2, which makes the
problem non-convex, and NP-hard in general. In the following,
we consider a global optimization algorithm for solving (28),
which is based on the general framework of the branch-andbound (BnB) methodology [45].

B. The Branch-and-bound Framework

s.t kx − x0 k∞ ≤ ε,
|x (n)| = 1, ∀n,
where ε = η

which leads to the following equivalent formulation of the
problem

(27)

Hence, it can be further simplified using the normalized vector
variable, which is
min

Fig. 2. Feasible region and convex hull of problem (28).

(29)

(30)

A typical BnB algorithm requires to partition the feasible
region into several subregions, where we formulate corresponding subproblems. For each subproblem, we obtain a
sequence of asymptotic lower-bounds and upper-bounds by
well-designed bounding functions. In each iteration, we update
the bounds and the set of the subproblems following the BnB
rules until convergence, i.e., the difference between the upperbound and lower-bound goes to zero.
It is well-known that the worst-case complexity for the BnB
algorithm is of the exponential order with respect to N , i.e.,
to search all the branches of the subproblems exhaustively
[45]. Nevertheless, by carefully choosing the tightest bounds,
it is possible to efficiently identify and prune the unqualified
branches, which accelerates the algorithm significantly.
Let us denote the feasible region, i.e., the arc shown in Fig.
2, as θn = arc (ln , un ). Problem (31) can be compactly written
as
P (Θ0 ) : min f (x)
x
(32)
s.t. x ∈ Θ0 .
where Θ0 = θ1 ×θ2 ×...×θN , and f (x) is defined in (31). By
the above notations, a subproblem can be denoted as P (Θ),
where Θ ⊆ Θ0 is the corresponding subregion. We then find
a lower-bound of P (Θ) by a lower-bounding function
fL (Θ) = f (xl ) ,

(33)

7

where xl is a relaxed solution that achieves the bound. In order
to compute the upper-bound, we find a feasible solution xu
for P (Θ). The upper-bounding function is thus given by
fU (Θ) = f (xu ) .

(34)

The above bounding functions (33) and (34) will be specified
in the next subsection. Here we only use fL and fU to
introduce the BnB framework for notational convenience. In
the BnB algorithm, we store all the subproblems in a problem
set S, which will be updated together with the global bounds
in each iteration by the following rules [45]
1) Branching: Pick a problem P (Θ) ∈ S that yields
the smallest lower-bound. Equally divide Θ into two
subregions following some subdivision rules detailed
in the following, and generate two subproblems. Then
delete P (Θ) in the problem set.
2) Pruning (optional): Evaluate the qualification of the
two subproblems, if their lower-bounds are less than the
current global upper-bound, add them into S.
3) Bounding: Choose the smallest lower-bound and upperbound from S as the bounds for the next iteration.
Note that the pruning step is only for saving the memory of
storing S, and will not affect the effectiveness of the BnB
procedure. This is because by choosing the smallest bounds
in S we can always avoid the unqualified branches. For clarity,
we summarize our BnB algorithm in Algorithm 2.
Algorithm 2 Branch-and-Bound Method for Solving (28)
Input: H̃, S, x0 , 0 ≤ ε ≤ 2, tolerance threshold δ > 0, bound
functions fL and fU .
Initialization: Let Θ0 be the initial feasible region of
problem (27), S = {P (Θ0 ) , fU (Θ0 ) , fL (Θ0 )} be the
initialized subproblem set. Set U B = fU (Θ0 ), LB =
fL (Θ0 ).
while U B − LB > δ do
Branching
a) Pick P (Θ) ∈ S, such that fL (Θ) = LB. Update
S = S\P (Θ).
b) Divide Θ into ΘA and ΘB following the chosen
subdivision rule.
Bounding
a) Compute fU (Θi ) and fL (Θi ) for P (Θi ) , i = A, B,
and add them to S.
b) Update U B and LB as the smallest upper-bound and
lower-bound in S, respectively.
end while
Output: xopt = the feasible solution that achieves U B.
To ensure that Algorithm 2 converges in a finite number of
iterations, the chosen subproblem for branching, the subdivision rule and the bounding functions fL and fU should satisfy
the following conditions [45]
1) The branching is bounding-improving, i.e., in each iteration we choose the problem that yields the smallest
lower-bound as the branching node.
2) The subdivision is exhaustive, i.e., the maximum length
of the subregions converges to zero as the iteration
number goes to infinity.

3) The bounding is consistent with branching, i.e., U B −
fopt converges to zero as the maximum length of the
subregions goes to zero, where fopt is the optimal value
of the original problem.
Our Algorithm 2 satisfies condition 1) automatically. We then
choose the subdivision rules to obtain the subproblems from
the branching node. For a given node P (Θ), we consider the
following two rules:
• Basic rectangular subdivision (BRS): Equally divide
Θ along arc (ln , un ) and keep arc (li , ui ) , ∀i 6= n
unchanged, where
n = arg max {φn |φn = un − ln } .
n

•

(35)

Adaptive rectangular subdivision (ARS): Equally divide Θ along arc (ln , un ) and keep arc (li , ui ) , ∀i 6= n
unchanged, where
n = arg max {dn |dn = |xu (n) − xl (n)| } .
n

(36)

In (35) xu and xl are the solutions associated with fU (Θ)
and fL (Θ), respectively.
According to [45, Theorem 6.3 and 6.4], both the above two
rules satisfy condition 2). In practical simulations, we observe
that ARS has a faster convergence rate than BRS.
C. Upper-bound and Lower-bound Acquisition
It remains to develop approaches to acquire the lower
and upper bounds, which are key to accelerating the BnB
procedure. Following the approach in [46], we compute the
lower-bound by the convex relaxation of (32). As shown in
Fig. 2, the convex hull for each entry x (n), denoted as Q (θn ),
is a circular segment, and can be given as
Q (θn ) : {x |arg (x) ∈ θn , |x| ≤ 1 } .

(37)

By simple analytic geometry, the angle constraint can be
equivalently written as



  ju
u−l
e + ejl
∗
≥ cos
,
(38)
Re x
2
2
which is nothing but a linear constraint. It follows that the
constraint for the vector variable is

 ju



e + ejl
u−l
Re x∗ ◦
≥ cos
,
(39)
2
2
T

T

where u = [u1 , u2 , ..., uN ] ∈ RN ×1 , l = [l1 , l2 , ..., lN ] ∈
RN ×1 , and ◦ denotes the Hadamard product. Hence, the convex relaxation can be given as the following QCQP problem
2

QP-LB : min kHx − sk
(40a)
x




 ju
jl
u−l
e +e
∗
≥ cos
,
s.t. Re x ◦
2
2
(40b)
|x (n)|2 ≤ 1, ∀n.

(40c)

Problem (40) can be efficiently solved via numerical solvers,
e.g., the CVX toolbox. By doing so, we can readily obtain the

8

lower-bound for each subproblem.
A natural way to compute the upper-bound is to project each
entry of the obtained solution xl of (40) on the corresponding
arc to get a feasible solution. Such a projector can be given
in an element-wise manner as follows


 x/|x|, arg x ∈ [l, u] ,

exp (jl) , arg x ∈ [(l + u)/2 + π, l + 2π] ,
exp (ju) , arg x ∈ [u, (l + u)/2 + π] ,
(41)
where we omit the subscripts for convenience.
The upper-bound obtained by the projector (41) is still loose
in general. To get a tighter bound, one can use PR1 (xl ) as
the initial point, and solve the following non-convex QCQP
PR1 (x) =




definitions of φn and dn in (35) and (36). By denoting
φmax = max {φn } , dmax = max {dn }, we have the following
Lemma 1.
Lemma 1. As φmax or dmax goes to zero, the difference
between U B and LB uniformly converges to zero, i.e.,
∀δ > 0, ∃η1 , η2 ≥ 0 s.t.
φmax ≤ η1 or dmax ≤ η2 ⇒ U B − LB ≤ δ.

Proof. Let us first denote the points that generate U B and LB
as xu and xl , i.e., U B = f (xu ) , LB = f (xl ). Following the
Lagrange Mean-value Theorem we have
U B − LB = f (xu ) − f (xl )
= ∇f H (z) (xu − xl )

2

QP-UB : min kHx − sk
(42a)
x

 ju



e + ejl
u−l
s.t. Re x∗ ◦
≥ cos
,
2
2
(42b)
2

|x (n)| = 1, ∀n.

(42c)

which can be locally solved via the fmincon solver in MATLAB. Since the solver employs descent methods, the obtained
local minimizer is guaranteed to yield a smaller value than
f (PR1 (xl )).
To further accelerate the speed for solving QP-LB and obtaining the bounds, we consider accelerated gradient projection
(GP) methods [47] in addition to the QCQP solvers. Given
xn ∈ C, the projector PR2 projects xn to the nearest point in
the corresponding convex hull Q (θn ). The details for deriving
PR2 are provided in the Appendix. Here we briefly introduce
our iterative scheme as

k − 1  (k)
(43)
x − x(k−1) ,
v = x(k) +
k+2



(44)
x(k+1) = PR2 v − 2sH̃H H̃v − s ,

where we start from x(0) and x(1) = x(0) . For the least-squares
objective function, we choose the stepsize as s = 1/λ̃max ,
where λ̃max is the maximum eigenvalue of H̃H H̃, i.e., the
Lipschitz constant.
Note that the above iteration scheme can only be used for
convex feasible regions due to the interpolation operation (43).
For the non-convex QP-UB problem (42), we use x(k) instead
of the interpolated point v, and replace the projector PR2 with
PR1 , which projects the point onto the arc, i.e., the feasible
region. Similar to (40), we use PR1 (xl ) as the initial point.
Based on [48], the complexity for using interior-point
method to solve the QCQP problems is O N 3 per iteration.
For both gradient-based methods, the costs are O (N K) in
each iteration, which are far more efficient in terms of a fixed
iteration number.
D. Convergence Analysis and Worst-case Complexity
We end this section by analyzing the convergence behavior
and the worst-case complexity for the proposed Algorithm
2. The convergence proof is to show that our bounding
functions fL and fU satisfy the condition 3). Recall the

(45)

(46)

≤ k∇f (z)k k(xu − xl )k ,

where
z ∈ {w |w = txu + (1 − t) xl , t ∈ [0, 1] } .
The upper-bound of the gradient is given as


k∇f (z)k = 2 H̃H H̃z − s
≤ 2 H̃H H̃z + 2 H̃H s
√
≤ 2 N λ̃max + 2 H̃H s ,

(47)

(48)

where the second line of (48) is based on the triangle inequality, the third line is based on the definition of the matrix l2
norm.
For the convex hull of each arc (ln , un ), the longest line
segment is the chord shown in Fig. 2 (φn ≤ π) or the diameter
(φn ≥ π). By simple geometric relations we have
v
uN


uX
√
min (φn , π)
sin2
.
kxu − xl k ≤ N dmax ≤ 2t
2
n=1
(49)
For φn ≤ π, ∀n, it follows that


√
√
φmax
kxu − xl k ≤ N dmax ≤ 2 N sin
.
(50)
2
By using (46), (48) and (50) we obtain




√
φmax
H
, (51)
U B − LB ≤ 4 N λ̃max + N H̃ s sin
2


√
U B − LB ≤ 2 N λ̃max + N H̃H s dmax .
(52)

Given any δ > 0, let




η1 = min π, 2 arcsin  
4 N λ̃max +
η2 =

δ
√

N H̃H s

δ
,

√
2 N λ̃max + N H̃H s

we have U B − LB ≤ δ if φmax ≤ η1 or dmax ≤ η2 .



  ,

(53)
(54)

9

Proof. Algorithm 2 satisfies both conditions 1) and 2). Furthermore, according to the definition of U B and LB we have
0 ≤ U B − fopt ≤ U B − LB.

(55)

According to Lemma 1, the bounding is consistent with
branching for the proposed two subdivision rules. Therefore, Algorithm 2 satisfies condition 3), which completes the
proof.
The following Theorem 2 specifies the worst-case complexity of Algorithm 2 for using BRS.
Theorem 2. When the BRS is used, Algorithm 2 converges to
a δ-optimal solution for at most
&
'
2N +1 arccosN 1 − ε2 /2
(56)
T =
η1

18

Average Achievable Sum Rate (bps/Hz)

Theorem 1. Algorithm 2 converges in a finite number of
iterations to a value arbitrary close to fopt .

16
14

16×4
Omni-Strict
Directional-Strict
Omni-Tradeoff, ρ = 0.1
Directional-Tradeoff, ρ = 0.1
ZF
Zero MUI

12
10
8
6
4
2
-2

0

2

4

N
Proof. Define vol (Θ0 ) = 2 arccos 1 − ε2 /2
as the volume of the initialized feasible region. Assume that Algorithm
2 terminates at the T-th iteration. According to [49], we have

It follows that
T ≥

2
2 vol (Θ0 )
=
η1

N +1

N

arccos
η1


1 − ε /2
2

,

(58)

which completes the proof.
Although the worst-case costs of both BnB-ARS and BnBBRS are at the exponential order with N , our simulations show
that in most cases, the algorithm terminates at a small iteration
number thanks to the tight bounds.
VI. N UMERICAL R ESULTS
In this section, we present numerical results to validate the
effectiveness of the proposed waveform design approaches.
For convenience, we set PT = 1, and each entry of the
channel matrix H subject to standard Complex Gaussian
distribution, i.e., hi,j ∼ CN (0, 1). In all the simulations, we
set N = 16 and employ a ULA with half-wavelength spacing
between the adjacent antennas. The constellation chosen for
the communication users is the unit-power QPSK alphabet,
i.e., the power of each entry in the symbol matrix S is 1.
A. Dual-functional Waveform Design with Given Radar
Beampatterns
We first show the communication sum-rate obtained by
different approaches as well as the associated radar beampatterns in Fig. 3 and Fig. 4, respectively, where we define
SNR = PT /N0 , and use ‘Omni’, ‘Directional’ and ‘ZF’
to represent omnidirectional beampattern design, directional
beampattern design and zero-forcing precoding based on the
problems (8) and (10). Further, we denote the waveform design

10

12

16×4

8
6
4

Beampattern (dBi)

(57)

8

Fig. 3. Sum-rate comparison for different approaches, N = 16, K = 4.

iterations, where η1 is given by (53).

vol (Θ0 )
η1
φmax
≤
≤ .
2
T
2

6

Transmit SNR (dB)

2
0
-2
-4
-6

Omni-Strict
Directional-Strict
Omni-Tradeoff, ρ = 0.1
Directional-Tradeoff, ρ = 0.1

-8
-10
-90

-60

-30

0

30

60

90

θ (deg)
Fig. 4. Radar beampatterns obtained by different approaches, N = 16, K =
4.

with strict equality constraints (8) and (10) and the trade-off
design (16) as ‘Strict’ and ‘Tradeoff’ respectively. The length
of the communication frame/radar pulse is set as L = 20. For
directional beampattern design, we consider three targets of
interest with angles of −π/3, 0 and π/3, and exploit the classic
Least-Squares techniques [38] to obtain the desired covariance
matrix Rd as defined in (10). It can be observed in Fig.3
that, the proposed two strict waveform designs outperform
the communication-only zero-forcing precoding significantly,
despite that their computational costs remain at the same
level as we have discussed in Section III. The resultant radar
beampatterns are shown in Fig. 4 for ‘Strict’, which are
exactly the same with the desired beampatterns. Moreover,
by introducing a small weighting factor ρ = 0.1 to the communication side, the sum-rates for trade-off designs increase
significantly by approaching to that of the zero MUI case, i.e.,
the AWGN capacity. Meanwhile in Fig. 4, the corresponding

10

Omnidirectional Design
Objective Value

1

0.95

10-2
Lower-bound
Upper-bound

10-4

0

5

10

PD

0.9

BnB - ARS

100

Objective Value

0.85
K=4
K=6
K=8

0.8

0.75
2.6

2.8

3

3.2

3.4

3.6

Average Achievable Rate (bps/Hz/user)
Fig. 5. Trade-off between the achievable rate per user and the radar detection
probability for omnidirectional beampattern design, N = 16, radar SNR =
−6dB, PF A = 10−7 .

Average MSE (dB)

5

K=4
K=6
K=8

0

25

30

35

100

10-2
Lower-bound
Upper-bound

10-4

0

200

400

600

800

1000

Iteration Number
Fig. 7. Convergence Behavior of BnB Algorithm for N = 16, K = 4, ε = 1.

B. Dual-functional Constant Modulus Waveform Design

-5
-10
-15
-20
1.5

20

can be further minimized by increasing the DoFs. The same
trends appear in Fig. 6 for the directional beampattern, where
we employ the mean squared error (MSE) between the desired
and obtained directional beampatterns as the radar metric. Both
figures prove that our approach can achieve a favorable tradeoff between radar and communications.

Directional Design

10

15

Iteration Number
BnB - BRS

2

2.5

3

3.5

Average Achievable Rate (bps/Hz/user)
Fig. 6. Trade-off of the achievable rate per user and the average MSE between
the designed and desired directional beampattern, N = 16.

radar beampatterns only experience slight performance-loss.
In Fig. 5 and 6, we aim to explicitly show the tradeoffs between the communication and radar performance. For
omnidirectional beampattern design, the detection probability
PD is used as the metric, where we consider the constant
false-alarm rate (CFAR) detection for a point-like target in
the far field, located at the angle of π/5. The receive SNR
is fixed at -6dB. The false-alarm probability for radar is
PF A = 10−7 . We calculate the detection probability based on
[9, eq. (69)]. It can be seen that there exists a trade-off between
the communication rate and the radar detection performance.
For a fixed PD , the achievable rate increases with the decrease
of the number of users, which suggests that the MUI energy

We show the results for solving the waveform optimization
problem with CMC and SC in Fig. 7-9. Following the simulation configurations in [33], we employ the orthogonal chirp
waveform matrix as the reference signal. The convergence
behavior of the proposed BnB algorithm for solving (28) is
shown in Fig. 7, with N = 16, K = 4, ε = 1, where we
compare the performance of the two different subdivision
rules, i.e., ARS and BRS. Both methods converge in a finite
number of iterations with a nearly constant upper-bound,
which suggests that we can reach the optimal value of problem
(28) by iteratively using the local algorithms for several times,
e.g., QCQP solver or the proposed gradient projection method.
Nevertheless, due to the non-convexity of the problem, we
need BnB algorithm to confirm that this is indeed a global
optimum. It can be also observed that the BnB-ARS has a
faster convergence rate than BnB-BRS, which is consistent
with the analysis in [45].
In Fig. 8 and 9, we show the trade-off between communication sum-rate and radar waveform similarity for the constant
modulus designs, where we employ the SQR-Binary Search
(SQR-BS) algorithm proposed by [33] as our benchmark
technique. Fig. 8 demonstrates the communication sum-rate
with increasing ε for N = 16, K = 4, SNR = 10dB. As
expected, the proposed BnB algorithm outperforms the SQRBS significantly, since the result obtained by BnB is the
global optimum, while SQR-BS can only yield local minimum
solutions. It is worth highlighting that the performance of
BnB is very close to the convex relaxation bound, which is

11

16 × 4, SNR = 10dB
Convex Relaxation Bound
BnB
SQR-BS [33]
Zero MUI
ZF

12

10

0

Pulse Comperession Gain (dB)

Average achievable sum rate (bps/Hz)

14

8

6

4

BnB, ǫ = 0.05
SQR-BS, ǫ = 0.05
Original Chirp

-10
-20
-30
-40
-50
-60
-70

2
0

0.5

1

1.5

2

-80
0

ǫ

40

80

120

160

IFFT Index
Fig. 8. Trade-off between the communication sum-rate and radar waveform
similarity, N = 16, K = 4, SNR = 10dB.

(a)

obtained by solving QP-LB. When the similarity tolerance ε is
big enough, our BnB algorithm can approximate the AWGN
capacity, i.e., the MUI can be fully eliminated.
Fig. 9 shows the results of radar pulse compression with
different similarity tolerance ε, where we use the waveform
transmitted by the first antenna, and employ the classic FFTIFFT pulse compression method [50] with a Taylor window
to reduce the power of sidelobes. From Fig. 9 we see that
there exists a trade-off between the communication sum-rate
and radar pulse compression performance. Moreover, the pulse
compression results of BnB and SQR-BS are nearly the same,
as their performance is guaranteed by the same waveform
similarity constraint, which again proves the superiority of the
proposed BnB Algorithm.

Pulse Comperession Gain (dB)

0
BnB, ǫ = 0.4
SQR-BS, ǫ = 0.4
Original Chirp

-10
-20
-30
-40
-50
-60
-70
-80
0

40

80

120

160

IFFT Index
(b)

VII. C ONCLUSION
0

Pulse Comperession Gain (dB)

In this paper, we discuss the waveform design for dualfunctional radar-communication system, which can be used
for both target detection and downlink communications. First
of all, two design approaches are proposed to minimize the
multi-user interference while formulating an appropriate radar
beampattern, which have been further extended as a weighted
optimization to achieve a flexible trade-off between radar and
communications. It has been proven that the computational
costs for the above three approaches are all at the same level
with the communication-only ZF precoding. Numerical results
show that all the proposed methods outperform the ZF precoding, while guaranteeing both the radar and communication
performance. Moreover, our trade-off design can significantly
improve the communication performance by allowing a slight
performance loss at radar. Finally, we consider the non-convex
constant modulus waveform design with similarity constraints,
where an efficient global optimization algorithm based on the
branch-and-bound framework has been developed. Gradient
projection algorithms are used to efficiently obtain the upper
and lower bounds. Simulations show that the proposed BnB

BnB, ǫ = 1
SQR-BS, ǫ = 1
Original Chirp

-10
-20
-30
-40
-50
-60
-70
-80
0

40

80

120

160

IFFT Index
(c)
Fig. 9. Radar pulse compression for different similarity tolerance, N =
16, K = 4. (a) ε = 0.05; (b) ε = 0.4; (c) ε = 1.

12

X ∈ M4 , we project it onto the line AB, and the projection
4
S
Mi , we
is the foot of perpendicular. For ∀X ∈ M5 = C\

Im

M3

M5

B

i=1

use the normalization as its projection. By basic plane analytic
geometry, we define the following lines

M1

Line AB : f1 (X) = Re (T ∗ (X − T )) = 0,

T

A

O

∗

The projector is then given as

X, f1 (X) ≥ 0, |X| ≤ 1 (X ∈ M1 ) ,






 A, f2 (X) ≤ 0 ≤ f4 (X) (X ∈ M2 ) ,
PR2 (X) = B, f3 (X) ≤ 0 ≤ f4 (X) (X ∈ M3 ) ,


 XT , f1 (X) , f4 (X) , f5 (X) ≤ 0 (X ∈ M4 ) ,




X/ |X| , else,
(61)
where XT is the foot of perpendicular on AB, i.e.,
XXT ⊥AB, XT ∈ AB. This is given by
∗  T
.
(62)
XT = X − Re (X − T ) T
|T |

Q
(a)

Q

Im

M4

M2

P

A

T
O

For the case of φ ≥ π the projector is the same. The only
difference is that f1 (X) should be defined as

M 3 Re

f1 (X) = − Re (T ∗ (X − T )) .

B

(63)

R EFERENCES

M1
M5

(b)
Fig. 10. Projector for GP. (a) φ ≤ π; (b) φ ≥ π.

algorithm for constant modulus waveform design with similarity constraints considerably outperforms the conventional
SQR-BS algorithm by obtaining the global optimum of the
problem.

D ERIVATION

(60)


Line AQ : f4 (X) = Re (A − B) (X − A) = 0,

∗
Line BP : f5 (X) = Re (B − A) (X − B) = 0.

M2

M4

P

Re

Line OA : f2 (X) = − Re (jA∗ X) = 0,
Line OB : f3 (X) = Re (jB ∗ X) = 0,

A PPENDIX
P ROJECTOR PR2

OF THE

The projector are derived for two cases respectively, i.e., the
open angle of the circular segment is (a) less than π or (b)
greater than π. We start from the first case. As shown in Fig.
10 (a), the whole complex plane C has been divided into five
parts. The lower and the upper bounds for the angle are l and
u respectively. Let us define
A = exp (jl) , B = exp (ju) , T = (A + B) /2,

(59)

where T is the midpoint of AB. Given X ∈ C, we aim to
find a nearest point PR2 (X) ∈ M1 as the projection. Note
that ∀X ∈ M1 , the projection is itself. For X ∈ M2 and
X ∈ M3 , the nearest points are A and B respectively. For

[1] FCC. (2010) Connecting America: The national broadband plan. [Online]. Available: https://www.fcc.gov/general/national-broadband-plan
[2] NSF. (2013) Enhancing access to the radio spectrum (EARS). [Online].
Available: https://www.nsf.gov/pubs/2013/nsf13539/nsf13539.htm
[3] DARPA.
(2016)
Shared
spectrum
access
for
radar
and
communications
(SSPARC).
[Online].
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