To appear on IEEE Transactions on Wireless communications
1
Symbol-Level Multiuser MISO Precoding for
Multi-level Adaptive Modulation
Maha Alodeh, Member, IEEE, Symeon Chatzinotas, Senior Member, IEEE, Bj
¨
orn Ottersten, Fellow Member, IEEE
Abstract—Symbol-level precoding is a new paradigm for mul-
tiuser multiple-antenna downlink systems which aims at creating
constructive interference among the transmitted data streams.
This can be enabled by designing the precoded signal of the
multiantenna transmitter on a symbol level, taking into account
both channel state information and data symbols. Previous
literature has studied this paradigm for Mary phase shift
keying (MPSK) modulations by addressing various performance
metrics, such as power minimization and maximization of the
minimum rate. In this paper, we extend this to generic multi-
level modulations i.e. Mary quadrature amplitude modulation
(MQAM) by establishing connection to PHY layer multicasting
with phase constraints. Furthermore, we address adaptive mod-
ulation schemes which are crucial in enabling the throughput
scaling of symbol-level precoded systems. In this direction, we
design signal processing algorithms for minimizing the required
power under per-user signal to interference noise ratio (SINR)
or goodput constraints. Extensive numerical results show that
the proposed algorithm provides considerable power and energy
efficiency gains, while adapting the employed modulation scheme
to match the requested data rate.
Index Terms—Symbol-level precoding, Constructive interfer-
ence, Multiuser MISO Channel, MQAM, Multi-level modulation.
I. INTRODUCTION
In a generic framework, precoding can be loosely defined
as the design of the transmitted signal to efficiently deliver
the desired information to multiple users exploiting the mul-
tiantenna space. Focusing on multiuser downlink systems, the
precoding techniques can be classified as:
1) Group-level precoding in which multiple codewords are
transmitted simultaneously but each codeword (i.e. a se-
quence of symbols) is addressed to a group of users. This
case is also known as multigroup multicast precoding [1]-
[4] and the precoder design is dependent on the channels
in each user group.
2) User-level precoding in which multiple codewords are
transmitted simultaneously but each codeword (i.e. a
Maha Alodeh, Symeon Chatzinotas and Bj
¨
orn Ottersten are with In-
terdisciplinary Centre for Security Reliability and Trust (SnT) at the
University of Luxembourg, Luxembourg. E-mails:{ maha.alodeh@uni.lu,
symeon.chatzinotas @uni.lu, and bjorn.ottersten@uni.lu}.
This work is supported by Fond National de la Recherche Luxembourg (FNR)
projects, Smart Resource Allocation for Satellite Cognitive Radio (SRAT-
SCR), Spectrum Management and Interference Mitigation in Cognitive Radio
Satellite Networks (SeMiGod), and SATellite SEnsor NeTworks for spectrum
monitoring (SATSENT). Part of this work is published in Globecom 2015
[28]. This work is protected under the filed patent, System and Method for
Symbol-level Precoding in Multiuser Broadcast Channels EP No. 15186548.2.
sequence of symbols) is addressed to a single user. This
case is also known as multiantenna broadcast channel
precoding [5]- [17] and the precoder design is dependent
on the channels of the individual users.
3) Symbol-level precoding in which multiple symbols are
transmitted simultaneously and each symbol is addressed
to a single user [18]- [27]. This is also known as a con-
structive interference precoding and the precoder design
is dependent on both the channels and the symbols of the
users.
It has been shown in various literature that symbol-level
precoding shows considerable gains in comparison to the
conventional group- or user-level precoding schemes [18]-
[30]. The main reason is that in symbol-level precoding
the vector of the aggregate multiuser interference can be
manipulated, so that it contributes in a constructive manner
from the perspective of each individual user. This approach
cannot be exploited in conventional precoding schemes, since
each codeword includes a sequence of symbols and the phase
component of each symbol rotates the interference vector in a
different direction. As a result, conventional schemes focus
on controlling solely the power of the aggregate multiuser
interference, neglecting the vector phase in the signal domain.
However, it should be highlighted here that the anticipated
symbol-level gains come at the expense of additional com-
plexity at the system design level. More specifically, the
precoded signal has to be recalculated on a symbol- instead of
a codeword-basis. Therefore, faster precoder calculation and
switching is requisite for symbol-level precoding, which can
be translated to more complex algorithms at the transmitter
side.
Before highlighting the contributions of this paper, the
following paragraphs present a detailed overview of related
work. The paradigm of symbol-level precoding was firstly
proposed in the context of directional modulation [40]- [41].
The idea of exploiting this paradigm for multiuser multiple
input single output (MISO) downlink to exploit the interfer-
ence was proposed in [18], but it was strictly limited to PSK
modulations. The main concept relies on the fact that the
multiuser interference can be pre-designed at the transmitter,
so that it steers the PSK symbol deeper into the correct
detection region. Based on a minimum mean square error
(MMSE) objective, two techniques were proposed based on
partial zero-forcing [18] and correlation rotation [19]. These
techniques were based on decorrelating the user channels
2
before designing the constructive interference. However, this
step leads to suboptimal performance, as channel correlation
can be beneficial while aiming for constructive interference.
Based on this observation, a maximum ratio transmission
based solution was proposed in [21]- [22] to perform interfer-
ence rotation without channel inversion, which outperformed
previous techniques.
All aforementioned techniques have a commonality, namely
they were based on the conventional approach of applying
a precoding matrix to the user symbol vector for designing
the transmitted signal. Interestingly, authors in [21] [22] have
shown that in symbol-level precoding more efficient solutions
can be found while designing the transmitted signal directly.
Following this intuition, a novel multicast-based symbol-level
precoding technique was initially proposed in [21] and later
elaborated in [22] for MPSK modulations. In more detail,
the transmitted signal can be designed directly by solving
an equivalent PHY-layer multicasting problem with additional
phase constraints on the received user signal. Subsequently,
the calculated complex coefficients can be utilized to modulate
directly the output of each antenna instead of multiplying the
desired user symbol vector with a precoding matrix. Based
on this novel approach, authors in [25] have extended the
multicast-based symbol-level precoding for imperfect chan-
nel state information (CSI) by proposing a robust precoding
scheme.
Going one step further, the above techniques were general-
ized in [26]- [27] taking into account that the desired MPSK
symbol does not have to be constrained by a strict phase
constraint for the received signal, as long as it remains in
the correct detection region. The flexible phase constraints
can obviously introduce a higher symbol error rate (SER)
if not properly designed. In this direction, the work in [27]
studies the optimal operating point in terms of flexible phase
constraints that maximizes the system energy efficiency.
In the context of the above related work, the main contri-
butions of this paper are:
• The extension of symbol-level precoding from single-
level to any generic multi-level modulations, such as
MQAM.
• The definition of a system architecture for a symbol-level
precoding transmitter.
• The extension of the connections between symbol-level
precoding and phase-constrained PHY multicasting for
generic multi-level modulations.
• The derivation of the probability density function (PDF)
for the equivalent channel power and amplitude.
• The derivation of a symbol-level precoding algorithm for
the power minimization with SINR or goodput constraints
under an adaptive modulation scheme.
The remainder of this paper is organized as follows: the
system model is described in section (II). A multicast char-
acterization of symbol-level precoding is explained in section
(IV). In section V, we propose symbol-level precoding for any
generic modulation. In section (VII), we propose a goodput-
based optimization algorithm. Finally, the numerical results
are displayed in section (VIII).
Notation: We use boldface upper and lower case letters for
matrices and column vectors, respectively. (·)
H
, (·)
∗
stand for
Hermitian transpose and conjugate of (·). E(·) and k·k denote
the statistical expectation and the Euclidean norm, and A 0
is used to indicate the positive semi-definite matrix. ∠(·), | ·|
are the angle and magnitude of (·) respectively. Finally, I(·),
Q(·) denote the in phase and the quadrature components of
(·).
II. SYSTEM AND SIGNAL MODELS
Let us consider a single-cell multiple-antenna downlink sce-
nario, where a single base station (BS)
1
is equipped with N
t
transmit antennas that serves K user terminals simultaneously,
each one of them is equipped with a single receive antenna.
As depicted in Fig. 1, the transmission scheme is based on K
frames (one per user) which include a common preamble for
the pilot symbols and signaling information, followed by N
useful symbols for each user (data payload). It should be noted
that the preamble is not precoded, while the useful symbols
are precoded on a symbol-level.
Similar to conventional multiuser precoding schemes, the
pilots are exploited by each user in order to estimate its
channel through standard CSI estimation methods and feed
it back to the BS, so that it can be used in the design of the
precoded signal. In this context, we assume a quasi static block
fading channel h
j
∈ C
1×N
t
between the BS antennas and the
j
th
user
2
. This is assumed to be known at the BS based on
the CSI feedback and fixed for each frame, i.e. N symbols.
Remark 1. Channel information estimation in conventional
precoding comprises of two steps: CSI estimation step to
design the precoding matrix and SINR estimaton step to select
the appropriate modulation and its corresponding detection
region at the receivers [42]. However, it can be conjectured
that the SINR estimation step cannot be performed easily in the
systems that adopt symbol-level precoding. In SINR estimation
step, a precoded sequence is transmitted to estimate the SINR
at each receiver. In the user-level precoding (conventional
linear beamforming), this sequence is designed based on
the acquired CSI in the first step. However in symbol-level
precoding, the output of the precoded pilot depends both
on the symbols and channel. The difficulty of SINR stems
from the fact the precoded pilot should be designed taking
into consideration different vector combinations to provide a
reliable averaging process for the SNR estimation. It should be
noted that the number of symbol vector combinations increases
with the constellation size. In this section, we propose a simple
modulation allocation based on the user’s goodput demands.
1
The described system can be straightforwardly extended for a multicell
system where the signal design takes place in a centralized manner, e.g.
Coordinated MultiPoint (CoMP), Cloud Radio Access Network (RAN) etc.
2
The proposed algorithms can be applied to Very High Speed Digital
Subscriber Line (VDSL) [12] and satellite communications [13], where the
channel remains constant for a long period
3
User Feedback &
Scheduling
CSI
Goodput
Targets
Symbol-level
precoding
P/S
1
N symbols
N symbols
N symbols
2
K
S/P
K symbols
Physical
Layer
Framing
Preamble
Signalling
Pilots
RF chains
Frame Level
K symbols
Symbol Level
K precoded
symbols
K precoded
symbols
N symbols
N symbols
N symbols
N symbols
Pr
Pr
N symbolsPr
N symbols
Pr
x
d
H
r
Fig. 1. Transmitter block diagram for symbol-level precoding. The block operations are classified into frame-level and symbol-level.
Regarding the useful symbols, the BS can serve each user
with a different modulation to support different user rates.
This is enabled through an adaptive modulation scheme. In
more detail, the modulation for each user is selected from the
set M = {1, . . . , M} based on the user’s requested rate and
the minimum and maximum SINR thresholds. The supported
SINR range is ζ ∈ [ζ
0
, ζ
max
] and thus, signal to interference
noise ratio (SINR) lower than ζ
0
leads to unavailability (i.e.
zero goodput), while SINR larger than ζ
max
do not provide a
further goodput increase.
It should be noted that although the precoding changes on
a symbol-basis, the modulation types are allocated to users
on a frame-basis. This is necessary because the user expects
to receive the same modulation type for all useful symbols
in a frame in order to properly adjust the detection regions.
The users are notified about their corresponding modulations
through the signaling preamble of the frame
3
.
For a single symbol period n = 1 . . . N, the received signal
at j
th
user can be written as
y
j
[n] = h
j
x[n] + z
j
[n]. (1)
x[n] ∈ C
N
t
×1
is the transmitted symbol sampled signal
vector at the n th symbol period from the multiple antennas
transmitter and z
j
denotes the noise at jth receiver, which
is assumed as an i.i.d complex Gaussian distributed variable
CN(0, σ
2
z
). A compact formulation of the received signal at
all users’ receivers can be written as
y[n] = Hx[n] + z[n]. (2)
Assuming linear precoding, let x[n] be written as x[n] =
P
K
j=1
w
j
[n]d
j
[n], where w
j
is the C
N
t
×1
precoding vector
for user j. The received signal at j
th
user y
j
in n
th
symbol
period is given by
y
j
[n] = h
j
w
j
[n]d
j
[n] +
X
k6=j
h
j
w
k
[n]d
k
[n] + z
j
[n]. (3)
3
Changing the modulation on a symbol-basis is unfeasible, as the user
would have to be notified about the used modulation on a symbol-basis and
this would lead to unacceptable overhead.
A more detailed compact system formulation is obtained by
stacking the received signals and the noise components for the
set of K selected users as
y[n] = HW[n]d[n] + z[n] (4)
with H = [h
T
1
, . . . , h
T
K
]
T
∈ C
K×N
t
, W[n] =
[w
1
[n], . . . , w
K
[n]] ∈ C
N
t
×K
as the compact channel and
precoding matrices. Notice that the transmitted symbol vector
d ∈ C
K×1
includes the uncorrelated data symbols d
k
for all
users with E
n
[|d
k
|
2
] = 1. From now on, we drop the symbol
period index for the sake of notation.
A. Power constraints for user-level and symbol-level precod-
ings
In the conventional user-level precoding (linear beam-
forming), the transmitter needs to precode every τ
c
which
means that the power constraint has to be satisfied along
the coherence time E
τ
c
{kxk
2
} ≤ P . Taking the expectation
of E
τ
c
{kxk
2
} = E
τ
c
{tr(Wdd
H
W
H
)}, and since W is
fixed along τ
c
, the previous expression can be reformulated
as tr(WE
τ
c
{dd
H
}W
H
) = tr(WW
H
) =
P
K
j=1
kw
j
k
2
,
where E
τ
c
{dd
H
} = I due to uncorrelated symbols over
τ
c
. However, in symbol level precoding the power constraint
should be guaranteed for each symbol vector transmission
namely for each τ
s
. In this case the power constraint equals
to kxk
2
= Wdd
H
W
H
= k
P
K
j=1
w
j
d
j
k
2
.
III. CONSTRUCTIVE INTERFERENCE DEFINITION
Interference can deviate the desired signal in any random
direction. The power of the interference can be used as an
additional source of power to be utilized in wireless systems.
In conventional user-level precoding, multiuser interference
treated as harmful factor that should be mitigated, without
paying attention to the fact the interference in some scenario
can push the received signal deeper in the detection region. As
consequence, an additional parameter that can be optimized. In
the literature, the multiuser interference has been be classified
4
into constructive or destructive based on whether it facilitates
or deteriorates the correct detection of the received symbol.
For MPSK scenarios, a detailed classification of interference
is discussed thoroughly in [18], [22]. In this situation, the
interference is tackled at each set of users’ symbol which
manages to find the optimal precoding strategy that can utilize
the interference in a constructive fashion rather than just
mitigating it. Therefore, the symbol-level precoding tailors
the multiuser MISO transmission strategy to suit the adopted
modulation by exploiting its detection regions.
Furthermore, it is worth mentioning that symbol-level pre-
coding is different from the interference alignment techniques
[23]- [24]. It should be noted that symbol-level precoding does
not attempt to project interference in a certain subspace of
the degrees of freedom so that it can be removed easily. On
the contrary, it uses all the degrees of the freedom for all
users by operating on a symbol-level. This allows to mitigate
interference in the signal domain rather than in the power
domain, as done in conventional user-level precoding.
In multi-level modulations, each constellation can consist
of inner, outer, and outermost constellation points. The in-
terference can be utilized to push the received signal deeper
in the detection region for outer and outermost constellation
points. However, for inner constellation points, the interference
can have limited constructive contribution to the target signals.
In the remainder of paper, a detailed symbol-level precoding
technique that exploits the interference in multiuser MISO for
any multi-level modulation is proposed.
IV. THE RELATION BETWEEN SYMBOL-LEVEL PRECODING
AND PHYSICAL-LAYER MULTICASTING
A. PHY-layer Multicasting Preliminaries
The PHY-layer multicasting aims at sending a single mes-
sage to multiple users simultaneously through multiple trans-
mit antennas [35]- [39]. In this context, the power min problem
for PHY-layer multicasting can be written as:
x(H, ζ) = arg min
x
kxk
2
s.t |h
j
x|
2
≥ ζ
j
σ
2
z
, ∀j ∈ K (5)
where ζ
j
is the SINR target for the j
th
user that should be
granted by the BS, and ζ = [ζ
1
, . . . , ζ
K
] is the vector that
contains all the SINR targets. This problem has been efficiently
solved using semidefinite relaxation [34] in [35].
In-phase
Quadrature
Fig. 2. The first quadrant of a generic modulation constellation.
B. Symbol-level Precoding Through Multicasting
Let us define a generic constellation represented by the
symbol set D, where d
j
∈ D represent symbols (see Fig.
(2)). Each symbol can have two equivalent representations:
1) Magnitude |d
j
|
2
and phase ∠(d
j
)
2) In-phase Re{d
j
} and quadrature Im{d
j
} components.
Let us also denote the received signal at the antenna of the
jth user (ignoring the receiver noise) as s
j
= h
j
P
K
k=1
w
k
d
k
.
In this context, a generic formulation for power minimization
in a single symbol period under symbol-level precoding and
SINR constraints
4
can be written using the I-Q representation:
w
k
(d, H, ζ) = arg min
w
k
k
K
X
k=1
w
k
d
k
k
2
s.t C
1
: I{h
j
K
X
k=1
w
k
d
k
} E
p
ζ
j
σ
z
I{d
j
}, ∀j ∈ K
C
2
: Q{h
j
K
X
k=1
w
k
d
k
} E
p
ζ
j
σ
z
Q{d
j
}, ∀j ∈ K,(6)
where E denotes the correct detection region. The desired
amplitude for each user depends on two factors: a long and
a short-term one. The long-term factor refers to the target
SINR ζ which determines the SER and remains constant across
all the symbol vectors of a frame. Assuming that the entire
symbols set D has unit average power i.e. E
D
[|d
j
|
2
] = 1.
Using the magnitude-phase representation, an equivalent way
of formulating the problem can be expressed as:
w
k
(d, H, ζ) = arg mink
K
X
k=1
w
k
d
k
k
2
s.t C
1
: kh
j
K
X
k=1
w
k
d
k
k
2
E κ
2
j
ζ
j
σ
2
z
, ∀j ∈ K
C
2
: ∠(h
j
K
X
k=1
w
k
d
k
) = ∠(d
j
), ∀j ∈ K(7)
where κ
j
= |d
j
|/
p
E
D
[|d
j
|
2
] denotes short-term factor
changes on a symbol-basis and adjusts the long-term SINR
based on the amplitude of the desired symbol. The set of
constraints C
1
, C
2
guarantees that each user receives its corre-
sponding data symbol d
j
with a correct amplitude and phase
5
.
Theorem 1. In symbol-level precoding, the power minimiza-
tion problem under SINR constraints (7) is equivalent to a
PHY-layer multicasting problem with an effective channel
ˆ
H
and phase constraints (10).
Proof. Before starting the proof, it should be noted that the
variable amplitude of each target symbol has been already
incorporated in the SINR constraints of C
1
. In other words,
4
The complete algorithm including goodput constraints is elaborated in
section VII-A.
5
C
1
and C
2
depend on the type of modulation and the constellation point
as elaborated in section V.
5
the multi-level amplitudes for each user have been expressed
as weighting factors for the frame-level SINRs ζ. Building on
this, the proof is based on two steps: a) defining an effective
channel, where each symbol phase is absorbed in the user’s
channel vector, b) observing that the transmitted signal vector
x can be designed directly and not as a linear product of the
precoding matrix with the symbol vector i.e. Wd.
By denoting the contribution of each user’s precoded sym-
bol to the transmit signal as x
k
= w
k
d
k
, and assuming a
unit-norm symbol d with a reference phase, let us define the
effective channel
ˆ
H = AH, where A is a diagonal K × K
matrix expressed as:
[A]
j,j
=
exp(∠(d − d
j
)i)
κ
j
. (8)
Using the above notations, an equivalent optimization prob-
lem can be formulated below:
x
k
(
ˆ
H, ζ) = arg min k
K
X
k=1
x
k
k
2
s.t C
1
: k
ˆ
h
j
K
X
k=1
x
k
k
2
≥ ζ
j
σ
2
z
, ∀j ∈ K
C
2
: ∠(
ˆ
h
j
K
X
k=1
x
k
) = ∠(d), ∀j ∈ K. (9)
It should be noted that the original user symbols do not
appear in the optimization problem anymore, as they have
been incorporated in the weighted SINR constraints and the
effective channel. Based on this observation, we can design
directly the transmit signal x, by dropping its dependency on
the individual user’s symbols. Replacing x =
P
K
j=1
x
j
yields:
x(
ˆ
H, ζ) = arg min
x
kxk
2
s.t C
1
: k
ˆ
h
j
xk
2
= ζ
j
σ
2
z
, ∀j ∈ K
C
2
: ∠(
ˆ
h
j
x) = ∠(d), ∀j ∈ K. (10)
which is equivalent to a PHY-layer multicasting problem (5)
for the effective channel
ˆ
H with additional phase constraints
on the received user signals C
2
.
Remark 2. In the equivalent problem, the effect of the input
symbols have been absorbed in the channel. As a result, the
equivalent channel is no longer fixed and it combines the
effects of the fixed channel and the current input symbols.
Treating this ergodically, we can model it as a random fast
fading channel which changes with the symbol index n.
In section VI, we derive the probability function of the
equivalent channel power, magnitude, and phase.
Corollary 1. An equivalent formulation of the optimization
problem (7) can be expressed by rewriting the magnitude
and phase constraints in the form of in-phase and quadrature
constraints:
x(
ˆ
H, ζ) = arg min
x
kxk
2
C
1
: I
j
E
p
ζ
j
σ
z
I{d}, ∀j ∈ K
C
2
: Q
j
E
p
ζ
j
σ
z
Q{d}, ∀j ∈ K, (11)
where I
j
, Q
j
are in-phase and out-of-phase components for
the detected signal at j
th
terminal and can be reformulated
as:
I
j
=
ˆ
h
j
x + (
ˆ
h
j
x)
∗
2
Q
j
=
ˆ
h
j
x − (
ˆ
h
j
x)
∗
2i
.
Remark 3. The PHY-layer multicasting problem in (5) is
based on constraints in the power domain (amplitude only),
while the symbol-level precoding problems in (10) and (12)
are based on constraints in the signal domain (both amplitude
and phase). This lower-level optimization is enabled by the fact
that the all components (both symbols and channel) that affect
the user received signal are taken into account in symbol-level
precoding.
C. Constructive Interference Power Minimization (CIPM) for
Multi-level Modulation
The power minimization with SINR constraints can be
expressed as:
x = arg min
x
kxk
2
s.t.
(
C
1
: I
j
E
p
ζ
j
σ
z
I{d
j
}, ∀j ∈ K
C
2
: Q
j
E
p
ζ
j
σ
z
Q{d
j
}, ∀j ∈ K.
(12)
For any practical modulation scheme, the above problem can
be solved by constructing appropriate C
1
, C
2
constraints as
explained in sec. V. Subsequently, an equivalent channel can
be constructed and x can be straightforwardly calculated using
Theorem 1.
Theorem 2. The symbol-level precoding can be solved by
finding the Lagrange function of (12) which can be expressed
as:
L(x) = x
H
x +
X
j
λ
j
(I
j
(x) −
p
ζ
j
σ
z
I{d
j
})
+
X
j
µ
j
(Q
j
(x) −
p
ζ
j
σ
z
Q{d
j
}). (13)
The derivative of L(x) with respect to x
∗
, λ
j
, and µ
j
can be
expressed:
∂L(x)
∂x
∗
= x +
X
j
λ
j
dI
j
(x)
dx
∗
+
X
j
µ
j
dQ
j
(x)
dx
∗
, (14)
∂L(x)
∂λ
j
= I
j
(x) −
p
ζ
j
σ
z
I{d
j
}, (15)
∂L(x)
∂µ
j
= Q
j
(x) −
p
ζ
j
σ
z
Q{d
j
}. (16)
