Signal Processing - September 2017 - 21

of the carrier with equal power when s 4 is in the quadraturephase (Q) portion of the carrier, with a times power of s 1. The
modulation constellations of the direct superposition signal
s DS in the case a = 3 are illustrated as open markers in Figure 4(a). It can be seen that s DS does not have a constant envelope.
Four representative design methods-MV+QPSK, Interplex,
POCET, and CEMIC-are employed, and their power efficiencies in this scenario with respect of a are shown in Figure 4(b)
for comparison.
As a WDP-based method, CEMIC can output the explicit
expression of the composite signal: when a = 3, s MUX ^ t h =
s 1 + s 2 + s 3 + 3 j $ s 4 - 0.51 $ s 1 s 2 s 3 - 0.28j $ (s 1 s 2 s 4 + s 1 s 3 s 4
+ s 2 s 3 s 4) . The modulation constellations of this signal are
illustrated as solid markers in Figure 4(a). The role the IM term
plays in the I component is moving the phase points ! 0.51
horizontally (indicated by red arrows), while the combined
action of the three IM terms in the Q component makes the
phase points move ! 0.28 or ! 0.28 # 3 vertically (indicated
by green arrows). The overall effect of the auxiliary components is to bring all of the phase points to a circle (indicated by
gray arrows).
In Figure 4(b), we can see that the structured and quasistructured approaches have relatively low power efficiency
compared to optimization-based design approaches. Note
that different values of N and r may lead to different power
efficiencies. For general optimization-based CEM design
approaches, since there are multiple degrees of freedom, no
obvious general relationship exists between power allocation and multiplexing efficiency. However, in either PDP or
WDP, the design process based on optimization ensures the
maximum efficiency of the result under the given conditions.

ods are emerging. Through the comparison of (4) and (5),
one can see that PDP optimizes the phase of a signal, so the
solution meets the envelope constancy constraint inherently,
while the transparency constraint is used to limit the feasible region. However, WDP is just the opposite: it optimizes
the value of the auxiliary component so the transparency
constraint can be met inherently as long as m ! W =, but
the envelope constancy constraint has to be given specifically. Compared with (5), it is easier to solve (4). Therefore,
the early CEM design methods are mostly based on PDP.
However, WDP is more flexible and extendable. Since WDP
directly designs the weight of each IM term in the auxiliary
component, additional constraints can be added during the
optimization process to flexibly control the proportion of
every IM term in the composite signal. As discussed in [17],
this enables the adjustment of the spectrum distribution of
the multiplexed signal, imposing the cancelation of unwanted IM terms, or relaxing the constant envelope constraint
to realize a quasi-constant envelope that may have a higher
power efficiency.

Representation and implementation of CEM
In spite of the significant difference between the starting
points of these two design patterns, PDP and WDP have mutual consistency. Early research [24] has shown that some
specific CEM techniques have more than one representation, such as coherent adaptive subcarrier modulation [20]
and Interplex [21]. By using the results of [15] and [17], the
relationship and mutual transformation of different CEM
representation can be generalized. Especially true for bipolar
signals, every CEM technique has at least three different expressive forms, i.e., the
■ phase-mapping-based (PMB) form
■ phase-synthesis-based (PSB) form
■ waveform-synthesis-based (WSB) form

Summary
Both PDP and WDP design patterns have advantages and
disadvantages and, in both of them, new CEM design meth-

1
3

sDS
sMUX

0.9
Power Efficiency

Q Component

2
1
0
-1

0.7
MV-QPSK
Interplex

0.6

-2
-3

0.8

-3

-2

-1
0
1
I Component

POCET
CEMIC

0.5
0.5

1.5
2
Power Ratio α
(b)

(a)

2.5

FIGURE 4. (a) The modulation constellations of a direct superposition signal and CEM signal constructed by CEMIC technique with a = 3 . (b) Power
efficiencies of four representative design methods with respect to a .
IEEE SIGNAL PROCESSING MAGAZINE

September 2017

Table of Contents for the Digital Edition of Signal Processing - September 2017