1

An Efficient Manifold Algorithm for Constructive
Interference based Constant Envelope Precoding

arXiv:1706.02900v1 [] 9 Jun 2017

Fan Liu, Student Member, IEEE, Christos Masouros, Senior Member, IEEE, Pierluigi Vito
Amadori, Student Member, IEEE, and Huafei Sun

Abstract—In this letter, we propose a novel manifold-based
algorithm to solve the constant envelope (CE) precoding problem
with interference exploitation. For a given power budget, we
design the precoded symbols subject to the CE constraints,
such that the constructive effect of the multi-user interference
(MUI) is maximized. While the objective for the original problem
is non-differentiable on the complex plane, we consider the
smooth approximation of its real representation, and map it onto
a Riemannian manifold. By using the Riemmanian conjugate
gradient (RCG) algorithm, a local minimizer can be efficiently
found for the problem. The complexity of the algorithm is
analytically derived in terms of floating-points operations (flops)
per iteration. Numerical results show that the proposed algorithm
outperforms the conventional methods on both symbol error rate
and computational complexity.
Index Terms—Constant envelope, MU-MISO downlink, massive MIMO, manifold optimization.

I. I NTRODUCTION
S one of the most promising approaches in 5G technology, massive multi-input-multi-output (mMIMO) communication systems are expected to provide significant benefits
over conventional MIMO systems by employing much larger
antenna arrays [1], [2]. Nevertheless, such systems face numerous challenges brought by the increasing number of antennas,
e.g., higher hardware costs and power consumption, which
may delay its deployment in future 5G systems. Hence, cheap
and efficient RF power amplifiers (PA) are required for making
the technology realizable in practical scenarios.
It is important to note that most of power-efficient PAs are
made by non-linear components, therefore waveforms with
low peak-to-average-power-ratio (PAPR) are needed to avoid
signal distortions when the PA is operated at the saturation
region [3]. Pioneered by [4], [5], the constant envelope precoding (CEP) has been proposed as an enabling solution,
where the MUI is minimized subject to the CE constraints.
The optimization in [5] is a non-convex non-linear least
square (NLS) problem, and is solved by sequential gradient
descent (GD) method, which converges to a local minimum. To

A

Manuscript received ***. This work was supported by the Engineering and
Physical Sciences Research Council (EPSRC) project EP/M014150/1 and the
China Scholarship Council (CSC).
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).
C. Masouros and P. V. Amadori are with the Department of Electronic and
Electrical Engineering, University College London, London, WC1E 7JE, UK
(e-mail: chris.masouros@ieee.org, uceepva@ucl.ac.uk).
H. Sun is with the School of Mathematics and Statistics, Beijing Institute
of Technology, Beijing, 100081, China (email: huafeisun@bit.edu.cn).

further improve the performance, a cross-entropy optimization
(CEO) solver is introduced in [6]. More recently, by using
the fact that the feasible region of the CE problem can be
geometrically viewed as a complex circle manifold, a RCG
algorithm is proposed by [7], where the NLS problem is
solved with much lower complexity than both GD and CEO.
While the interference reduction (IR) methods in above works
are relatively straightforward, their performance is strongly
dependent with the constellation energy [5], which is difficult
to optimally set in advance. In addition, IR approaches ignore
that MUI is known to the base station (BS) in general, and thus
can be utilized as a source of useful power. Realizing these
facts, the previous work [8] considers a novel CEP approach
with the concept of constructive interference (CI) [9], which
can overcome the above drawbacks. Due to the CE constraints,
the CI-CEP problem is non-convex, but can be solved using
CEO solver as well. Moreover, by relaxing the constraints,
the CI-CEP problem becomes convex, thus can be solved by
standard numerical tools. However, both of the above methods
demand large amount of computations inevitably.
Based on the previous works on manifold optimizations
[10], [11], we consider a manifold-based algorithm to solve
the CI-CEP problem in this letter. Since the objective is not
complex differentiable, we first equivalently transform the
problem into its real representation, and use a smooth upperbound to obtain a differentiable approximation. By viewing
the feasible region as an oblique manifold, a RCG algorithm
is employed to find a local minimizer of the problem. Unlike
the relaxed convex problem in [8], the proposed algorithm is
guaranteed to yield precoded symbols with exactly constant
envelopes, and has better performance than the methods of
[8] in terms of both symbol error rate (SER) and complexity.
II. S YSTEM M ODEL
We consider a multi-user multi-input-single-output (MUMISO) downlink scenario where a N-antenna BS transmits
signals to M single-antenna users. The received signal vector
is given as
y = HT x + w,
(1)
where y = [y1 , y2 , ..., yM ]T ∈ CM×1 with ym being the received symbol for the m-th user, x =
T
[x1 , x2 , ..., xN ] ∈ CN ×1 represents the transmitted symbols,
T
w = [w1 , w2 , ..., wM ] ∈ CM×1 ∼ CN (0, N0 I) is the
Gaussian noise, and H = [h1 , h2 , ..., hM ] ∈ CN ×M is the
channel matrix, with hm being the channel vector for the m-th
user. Without loss of generality, the channel is assumed to be

2

Rayleigh fading, i.e., each entry of H subjects to i.i.d Complex
Gaussian distribution with zero-mean, and is perfectly known
to the BS. The transmitted signal is expected to have constant
envelope, which is
p
xn = PT /N ejθn , ∀n,
(2)

where PT is the total transmit power, θn is the phase of the
n-th transmitted symbol.
that the desired symbol for the m-th user is sm =
√ Assume
Em ejφm , where Em and φm denote the power and the phase
of the symbol respectively. The received symbol for the m-th
user can be written as

ym = sm + hTm x − sm +wm ,
(3)
|
{z
}
MUI

where the second term represents the interfering signal for the
user. The total MUI power is then given by
PMUI =

M
X

m=1

hTm x − sm
T

2

= HT x − s

2

,

(4)

where s = [s1 , s2 , ..., sM ] is the desired symbol vector.

a convex approximation problem can be efficiently solved
by numerical solvers, e.g., CVX toolbox. The results are
then normalized to obtain transmitted symbols with constant
envelopes [8]. Nevertheless, using CEO or CVX to solve (6)
requires significant computation resources. In the next section,
we propose a manifold based optimization technique to solve
(6), which has much lower complexity.
IV. P ROPOSED A LGORITHM BASED
M ANIFOLD

ON

O BLIQUE

Since Re (·) and Im (·) are not complex differentiable, we
formulate the real representation of (6). First we rewrite tm as

(7)
tm = hTm x − sm e−jφm = h̃Tm x − u,
where h̃m = hm e−jφm . We then separate the real and
imaginary parts of complex notations as follows

H̃ = H̃R + j H̃I , h̃m = h̃Rm + j h̃Im , x = xR + jxI . (8)
i
h
where H̃ = h̃1 , h̃2 , ..., h̃M . It follows that

III. P ROBLEMS F ORMULATION

Re (tm ) = h̃TRm xR − h̃TIm xI − u,

Aiming at minimizing the MUI power, the conventional
CEP approaches are designed to solve the following optimization problem [5]

By using the fact that |a| = max (a, −a), and denoting β =
tan ψ we have

2

HT x − s
p
s.t. |xn | = PT /N , ∀n.

min

Im (tm ) = h̃TIm xR + h̃TRm xI .

x

(5)

|Im (tm )| − Re (tm ) tan ψ = max (g2m−1 , g2m ) − uβ, (10)
where

Problem (5) is a NLS problem, which is obviously non-convex,
and has multiple local minima. Fortunately, it has been proven
that most of the local minima yield small values [5], and can
be obtained by a variety of approaches [5]–[7]. However, it
should be highlighted that by treating all the interference as
harmful, these techniques ignore the fact that MUI can be
employed as a green signal power source to benefit the symbol
demodulation. This has been first proposed by [12], where
the MUI is classified as constructive and destructive parts.
CI based beamformers aim at minimizing destructive and
exploiting constructive interference, which enable a relaxed
feasible region for the optimization [9]. Based on this, previous
work [8] focuses on maximizing the constructive effect of the
MUI to achieve CE precoding, where the PSK modulations
are employed. We refer the reader to the above literature for
detailed discussions. Here we recapture the CI-CEP problem
in [8] as follows
min max |Im (tm )| − Re (tm ) tan ψ
x
m
p
s.t. |xn | = PT /N , ∀n,

tm = hTm x − sm e−jφm , ∀m,

(9)

(6)

where sm = uejφm , ψ = π/L, u is the amplitude for the PSK
symbols, L is the PSK modulation order. The above problem
can be solved by CEO suboptimally, and has been further
relaxed as a convex problem by replacingpthe equality constraints on xn as inequalities, i.e., |xn | ≤ PT /N , ∀n. Such

T

T

g2m−1 = h̃Im − β h̃Rm xR + h̃Rm + β h̃Im xI ,
T
T


g2m = β h̃Im − h̃Rm xI − h̃Im + β h̃Rm xR .
(11)
q
T
Denoting X̃ = PNT [xR , xI ] , the real representaion of the
problem can be written compactly as follows
min max gi
i
X̃


s.t. X̃T X̃

(12)
nn

= 1, n = 1, 2, ..., N,

where i = 1, 2, ..., 2M . It is clear that the feasible region of
(12) can be given as
n
o
M = X̃ ∈ R2×N : (X̃T X̃)nn = 1, ∀n .
(13)

We say that M forms a manifold, and X̃ is a point on M. To
be more specific, M is a 2N -dimensional oblique manifold
[13]. In Riemannain geometry, a manifold is defined as a set
of points that endowed with a locally Euclidean structure near
each point. Given a point p on M, a tangent vector at p is
defined as the vector that is tangent to any smooth curves
on M through p. The set of all such vectors at p forms the
tangent space, denoted by Tp M, which is an Euclidean space.
Specially, the tangent space at X̃ is given as
n
o
TX̃ M = U ∈ R2×N : (X̃T U)nn = 0, ∀n .
(14)

3

Xk
mk

Xk

where an,m , bn,m , cn,m and dn,m denote the (n, m)-th enrty
of the following matrices

-ÑXk f

TXk

A = H̃I − β H̃R , B = −H̃I − β H̃R ,

Xk

( )

C = H̃R + β H̃I , D = H̃R − β H̃I ,

- grad f Xk

( Π k -1 )

dk Πk
Xk

X k +1

Fig. 1. Riemannian conjugate gradient algorithm.

If the tangent spaces of a manifold are equipped with a
smoothly varying inner product, the manifold is called Riemannian manifold [14]. Accordingly, the family of inner products is called Riemannian metric, which allows the existence
of rich geometric structure on the manifold. Here we use
the usual Euclidean inner product as the metric, which is
hU, ViX̃ = tr UT V , where U, V ∈ TX̃ M. The algorithm
that we employ is the so-called Riemannian conjugate gradient
(RCG) algorithm [15], which needs to first compute the
gradient of the objective. Since the objective in (12) is still
not differentiable, we consider
  the well-known smooth logsum-exp upper-bound f X̃ for the max function, which is
gmax

!
 
X
≤ f X̃ = ε log
exp (gi /ε)

(19)

In the RCG algorithm, (16) is called Euclidean gradient,
and can be used to compute the Riemannian gradient, which
is defined as the tangent vector belongs
 to TX̃ M that indicates
the steepest ascent direction of f X̃ . It can be viewed as
the orthogonal projection of the Euclidean gradient onto the
tangent space [16], which is given as
 

grad f X̃ = PX̃ ∇X̃ f = ∇X̃ f − X̃ diag(X̃T ∇X̃ f ),
(20)
where PX̃ (·) denotes the projector, diag (·) sets all offdiagonal entries of a matrix to zero. At the k-th iteration, the
descent direction Πk is obtained as
 
(21)
Πk = − grad f X̃k + µk PX̃k (Πk−1 ) .
Here the projector is used as vector transport, which maps the
vector from one tangent space to another. µk is given by the
Riemannian version of the Polak-Ribière formula, which is
µk
=

D

 
 


E
grad f X̃k , grad f X̃k − PX̃k grad f X̃k−1
X̃k
D



E
.
grad f X̃k−1 , grad f X̃k−1
X̃k−1

(22)
The k+1-th update is thus given by
X̃k+1 = RX̃k (δk Πk ) ,

(23)

(15)

where RX̃k (·) is called retraction, which maps a point on
TX̃k M to M with a local rigidity condition that preserves
gradients at X̃k [16], and is given as

where
 ε > 0 is some small positive number. The gradient of
f X̃ is thus given as

RX̃k (δk Πk )
 


 
X̃k + δk Πk
X̃k + δk Πk
(24)
1 , ..., 
N  ,
= 
X̃k + δk Πk
X̃k + δk Πk
1
N


where X̃k + δk Πk
is the n-th column of the matrix
n


X̃k + δk Πk , and the stepsize δk is obtained by backtracking line search algorithms, e.g., Armijo rule. Fig. 1 shows a
single iteration of the RCG algorithm on M, which has also
been summarized in Algorithm 1.
Note that the complexity of Algorithm 1 mainly comes from
line 4 and line 5, where 16N 2 + 14M N + 18M + 16N and
4N 2 + 6N flops are required
respectively, leading to a total

complexity of O N 2 for each iteration. By
 contrast, the
complexity of GD and RCG-IR are O M N 2 and O (M N )
per iteration [5], [7], respectively. For CEO, the complexity
is O (KM N ) in each iteration [8], where K stands for the
number of random samples, which may be quite larger than M
and N . While the RCG-IR requires less computations compared to the proposed algorithm, the latter brings significant
performance gain as we will show in the next section, and
therefore offers a favourable performance-complexity tradeoff.

i

≤ gmax + ε log (2M ) ,

∇X̃ f =




∂f
∂f ∂f
,
, ...,
,
∂x̃1 ∂x̃2
∂x̃N

(16)

where x̃n is the n-thqcolumn of X̃. Noting that xR =
q
N
N
T
T
PT X̃ (:, 1) , xI =
PT X̃ (:, 2), which are the first and
second column of X̃T respectively, we have
r
r
∂xR
∂xI
N
N
=
[en , 0],
=
[0, en ],
∂x̃n
PT
∂x̃n
PT

(17)

where en ∈ RN ×1 have all-zero entries except that its nth entry equals to 1. Based on (17), the n-th column of the
gradient is given by

g
"
#
2m−1
r
M
N X  an,m , bn,m  exp
 g ε  
2m
PT m=1
cn,m , dn,m
exp
∂f
ε
=
,
g 
2M
P
∂x̃n
i
exp
ε
i=1
(18)

4

3

IR Methods

10-1

10-2
10-2

2.5
2

CI Methods

10-3
10-4

1.5

RCG-CI
CVX-CI
CEO-CI
RCG-IR
GD-IR
CEO-IR

1
10-5
0.5
0
16

18

SNR = 8dB

10-1

IR Methods

SER

Average Execution Time (s)

4
3.5

M = 20

100

RCG-CI
CVX-CI
CEO-CI
RCG-IR
GD-IR
CEO-IR

20

22

24

10-6

SER

4.5

0

2

10-3

10-4

4

6

8

10

12

10-5
12

RCG-CI
CVX-CI
CEO-CI
RCG-IR
GD-IR
CEO-IR

CI Methods

14

16

18

Users (M)

Transmit SNR (dB)

Users (M)

(a)

(b)

(c)

20

22

24

Fig. 2. Numerical results. (a) Average execution time vs. number of users for different algorithms; (b) SER vs. SNR for different algorithms; (c) SER vs.
User for different algorithms.

Algorithm 1 RCG for CI-based CEP
Input: s, H, ∆ > 0, kmax > 0.
Output: Local minimizer X̃∗ for (12).
1. Initialize randomly
 X̃0 ∈ M,
set Π0 = − grad f X̃0 , k = 0,
 
while k ≤ kmax & grad f X̃k
≥ ∆ do
F
2. k = k + 1,
3. Compute stepsize δk−1 by Armijo rule, and set X̃k
using the retraction defined in (23),
4. Compute µk by (22),
5. Compute Πk by (21).
end while

V. N UMERICAL R ESULTS
In this section, numerical results based on Monte Carlo
simulations have been provided to compare the performance of
different algorithms. We consider the following 6 algorithms:
• The proposed RCG algorithm for CI (RCG-CI);
• Convex relaxation for CI (CVX-CI) [8];
• Cross-entropy optimization for CI (CEO-CI) [8];
• RCG algorithm for IR (RCG-IR) [7];
• Gradient descent algorithm for IR (GD-IR) [5];
• Cross-entropy optimization for IR (CEO-IR) [6].
Without loss of generality, we use QPSK modulation for all the
approaches. We set u = 1, ∀m, which is a common assumption
in related literature for the reason that the optimal u is
difficult to determine for IR methods [5], [6] while arbitrary
u can be accepted by CI methods [8]. We also assume that
PT = 1, N = 64 for all the algorithms, and each entry of the
channel H subjects to standard complex Gaussian distribution,
i.e., hn,m ∼ CN (0, 1) , ∀n, ∀m. For CEO methods, we use
the same parameter configuration with [8], which is T = 1000
(the number of iterations), K = 500 (the number of initialized
random samples), ρ = 0.05 (quantile), α = 0.08 (the smooth
parameter). For GD-IR, the number of iterations is set as 50.
While the analytic complexity per iteration of the most
algorithms has already been given, we compare the overall
complexity in terms of average execution time in Fig. 2 (a)
since it is difficult to specify the complexity of the CVX-CI
approach. The simulation is performed on an Intel Core i7-

4790 CPU 32GB RAM computer with 3.6GHz. As expected,
the RCG methods require least execution time to solve the
problem while other methods need much more. Although the
proposed RCG-CI algorithm is more complex than RCG-IR
by each iteration, the total time needed is still comparable
with the latter. More importantly, RCG-CI is robust to the
increasing users because its complexity is mainly determined
by the antenna number of the BS.
In Fig. 2 (b), we show the error performance of all 6
approaches in terms of SER with increased transmit signal-tonoise-ratio (SNR), where M = 20, SNR = PT /N0 . Note that
all the IR methods show negligible difference under the given
parameter configuration, and all the CI methods outperform
the IR methods thanks to the utilization of the MUI power.
It is worth noting that the proposed RCG-CI has the best
performance among all the 6 approaches with 2dB gain over
IR methods, and 1dB gain against the CVX-CI algorithm.
We further consider the error performance with increased
number of users in Fig. 2 (c), where the SNR is fixed at 8dB
with the number of users ranging from 12 to 24. It can be
observed that the SER becomes worse with the growth of the
users due to the reduction of the Degrees of Freedom (DoFs).
Once again, we see that the proposed RCG-CI achieves the
lowest SER among all the approaches, and the CI methods
achieve far better performance than IR methods, while the
latter maintains an SER of 10−2 for all the users numbers.

VI. C ONCLUSION
A low-complexity manifold optimization algorithm has been
introduced to solve the CEP problem with the exploitation
of the MUI power. By viewing the feasible region of the
optimization as an oblique manifold, the proposed method can
efficiently find a near-optimal solution using the Riemannnian
conjugate gradient algorithm. Numerical results show that the
proposed RCG-CI algorithm outperforms the existing 5 other
approaches in terms of error performance, with a comparable
complexity to the fastest RCG-IR algorithm. It is further
shown that when the DoFs of the system are limited, the
proposed RCG-CI still performs far better than other methods.

5

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