Signal Processing - September 2017 - 19

each the appearing probability of + 1 and - 1 is the same, they
have 2 N value combinations with equal probability. At a time
when the value combination corresponds to the k th row of
this LUT, the composite signal takes the value s M, k . Let m ki
correspond to the value of s i for the k th combination and let
m i = 6m 1i, m 2i, f, m 2N i@T and s MUX = 6s M,1, s M,2, f, s M,2N@T correspond to the ith column and the last column of the LUT in
Figure 3, respectively.
For the receiving channel of s i, the aligned correlator output is R MUX, i = T -1 # s MUX (t) s )i (t) dt. Statistically, when T
T
is long enough, this integral can be viewed as a time-domain
implementation of E " s MUX (t) s *i (t) , = m Ti s )MUX /2 N . The
transparency constraint requires that R MUX,i must equal to the
design input value Pi e jzi, for all i = 1 + N, which equals to
r = M T s )MUX 2 -N ,

(2)

where M = 6m 1, m 2, f, m N@, as highlighted in Figure 3.
Thus, from a mathematical point of view, the design of a CEM
scheme is equivalent to finding a solution s MUX of (2) under the
transparency constraint r and the complex envelope constraint
s M,1 = g = s M,2N .
Since (2) is underdetermined, it has either no solution, if r is
not in the span of the columns of M, or infinitely many solutions.
The complex envelope constraint further increases the difficulty in finding feasible solution. So far, most research in existing
CEM design approaches are essentially looking for this solution.

Design methodologies of CEM
Over the last 15 years, many novel and advanced CEM techniques
have been proposed. Although the various methodologies differ,
the essential distinction of these assorted methods is on the mapping objects and the solving approaches. There exist two general
design patterns: phase domain processing (PDP) and waveform
domain processing (WDP), both of which we will explore.

PDP
Since the constant envelope composite signal s MUX ^ t h inherently has a phase-modulated form A exp ^ ji ^ t hh, the main
concept of PDP is specifying a mapping rule i ^s 1, f, s N h
from the value combination of the component signals to the
phase of the composite signal. At a time when the value combination of s 1, f, s N corresponds to the kth row of the LUT
as in Figure 3, the composite signal takes the phase angle i k,
thus the value s M, k = Ae jik. Therefore, in the PDP view, the
general constraint (2) is customized to
r = AM T sl )MUX ^ih 2 -N ,

analytical way (e.g., Interplex [21]) or by numerical optimization (e.g., the phase-optimized constant-envelope transmission
(POCET) method [14]). In some methods, the phase-mapping
rule may be an explicit expression, while in others it can only be
presented in a more general LUT form. Furthermore, the mapping relation from component values to phase angle can be oneto-one (e.g., POCET) or many-to-one (e.g., majority voting (MV)
multiplexing [22]).

Structured and quasi-structured PDP approaches

For unique cases of N and r, some CEM techniques can directly construct particular solutions of (3). This kind of methods are
collectively referred to as structured PDP design methods. For
example, in MV multiplexing, the composite signal takes the
value of the majority of the component signals, which is equivalent to i = r {1 - sgn (R iN= 1 s i)} 2, mapping multiple combination values to one of two phase angles: 0 and r. Despite
its simplicity, MV multiplexing has poor applicability to the
number, power distribution, and phase relationships of component signals because of its overrigid mapping form. In [22], an
interlacing operation is employed that improves the flexibility
of MV multiplexing on power distribution to some extent.
Some quasi-structured PDP design methods, while presupposing the concrete form of the phase-mapping rules, can
obtain a degree of flexibility in the power ratio and phase relationship of components by tuning undetermined coefficients.
For example, in the Interplex technique [21], the phase angle is
preset to the form i = a 1 s 1 + R iN= 2 a i s 1 s n in which adjusting
indices a i can change the power ratio of the components in the
composite signal. However, as with other structured and quasistructured PDP design methods, the imposed prior restraint on
the combination form artificially reduces the feasible domain
of (3), which not only limits the flexibility of the phase relationship between components, but also affects power optimization efficiency in many cases (see the "Case Study: CEM of
Four Bipolar Signals" section for an example).

Optimization-based PDP approaches
There have been some optimization-based PDP design methods proposed in recent years that further extend the flexibility
and applicability of the solution while guaranteeing the optimality of the solution. In (1), for a given design constraint r,
the optimal solution for maximizing power efficiency is the
one with the smallest A. Conforming to the PDP design pattern, the CEM design problem is equivalent to solving the following nonlinear programming problem:
min A ^ih, s.t. A ^ih M T sl )MUX ^ih 2 -N = r.

(3)

T
where i = 6i 1, f, i 2 N@T and slMUX ^ih = 6e ji1, f, e ji2 @ , and
the design of a CEM scheme is equivalent to finding a solution
" A, i , of (3) for a given design constraint r.
Different PDP-based methods, such as [7], [8], [14], and [20]-
[22], obtain the solution of this underdetermined and nonlinear
complex equation (3) differently, and the ways to present i may
also be different. The phase-mapping rule can be obtained in an
N

i

(4)

By introducing the penalty function, Dafesh et al. [14]
convert the aforeconstrained optimization into an approximately equivalent but unconstrained search of the objective
2
function, minimizing F ^ih = A 2 + n a R i ! V ^ R MUX, i - Pi h
)
-j^z i - z jh
+ n b R i, j ! U Im " e
R MUX, i R MUX, j ,, where the penalty
n
n
factors a and b are positive, and V and U are the sets
of component indices that have power and phase constraints,

IEEE SIGNAL PROCESSING MAGAZINE

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September 2017

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19



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

Signal Processing - September 2017 - Cover1
Signal Processing - September 2017 - Cover2
Signal Processing - September 2017 - 1
Signal Processing - September 2017 - 2
Signal Processing - September 2017 - 3
Signal Processing - September 2017 - 4
Signal Processing - September 2017 - 5
Signal Processing - September 2017 - 6
Signal Processing - September 2017 - 7
Signal Processing - September 2017 - 8
Signal Processing - September 2017 - 9
Signal Processing - September 2017 - 10
Signal Processing - September 2017 - 11
Signal Processing - September 2017 - 12
Signal Processing - September 2017 - 13
Signal Processing - September 2017 - 14
Signal Processing - September 2017 - 15
Signal Processing - September 2017 - 16
Signal Processing - September 2017 - 17
Signal Processing - September 2017 - 18
Signal Processing - September 2017 - 19
Signal Processing - September 2017 - 20
Signal Processing - September 2017 - 21
Signal Processing - September 2017 - 22
Signal Processing - September 2017 - 23
Signal Processing - September 2017 - 24
Signal Processing - September 2017 - 25
Signal Processing - September 2017 - 26
Signal Processing - September 2017 - 27
Signal Processing - September 2017 - 28
Signal Processing - September 2017 - 29
Signal Processing - September 2017 - 30
Signal Processing - September 2017 - 31
Signal Processing - September 2017 - 32
Signal Processing - September 2017 - 33
Signal Processing - September 2017 - 34
Signal Processing - September 2017 - 35
Signal Processing - September 2017 - 36
Signal Processing - September 2017 - 37
Signal Processing - September 2017 - 38
Signal Processing - September 2017 - 39
Signal Processing - September 2017 - 40
Signal Processing - September 2017 - 41
Signal Processing - September 2017 - 42
Signal Processing - September 2017 - 43
Signal Processing - September 2017 - 44
Signal Processing - September 2017 - 45
Signal Processing - September 2017 - 46
Signal Processing - September 2017 - 47
Signal Processing - September 2017 - 48
Signal Processing - September 2017 - 49
Signal Processing - September 2017 - 50
Signal Processing - September 2017 - 51
Signal Processing - September 2017 - 52
Signal Processing - September 2017 - 53
Signal Processing - September 2017 - 54
Signal Processing - September 2017 - 55
Signal Processing - September 2017 - 56
Signal Processing - September 2017 - 57
Signal Processing - September 2017 - 58
Signal Processing - September 2017 - 59
Signal Processing - September 2017 - 60
Signal Processing - September 2017 - 61
Signal Processing - September 2017 - 62
Signal Processing - September 2017 - 63
Signal Processing - September 2017 - 64
Signal Processing - September 2017 - 65
Signal Processing - September 2017 - 66
Signal Processing - September 2017 - 67
Signal Processing - September 2017 - 68
Signal Processing - September 2017 - 69
Signal Processing - September 2017 - 70
Signal Processing - September 2017 - 71
Signal Processing - September 2017 - 72
Signal Processing - September 2017 - 73
Signal Processing - September 2017 - 74
Signal Processing - September 2017 - 75
Signal Processing - September 2017 - 76
Signal Processing - September 2017 - 77
Signal Processing - September 2017 - 78
Signal Processing - September 2017 - 79
Signal Processing - September 2017 - 80
Signal Processing - September 2017 - 81
Signal Processing - September 2017 - 82
Signal Processing - September 2017 - 83
Signal Processing - September 2017 - 84
Signal Processing - September 2017 - 85
Signal Processing - September 2017 - 86
Signal Processing - September 2017 - 87
Signal Processing - September 2017 - 88
Signal Processing - September 2017 - 89
Signal Processing - September 2017 - 90
Signal Processing - September 2017 - 91
Signal Processing - September 2017 - 92
Signal Processing - September 2017 - 93
Signal Processing - September 2017 - 94
Signal Processing - September 2017 - 95
Signal Processing - September 2017 - 96
Signal Processing - September 2017 - 97
Signal Processing - September 2017 - 98
Signal Processing - September 2017 - 99
Signal Processing - September 2017 - 100
Signal Processing - September 2017 - 101
Signal Processing - September 2017 - 102
Signal Processing - September 2017 - 103
Signal Processing - September 2017 - 104
Signal Processing - September 2017 - 105
Signal Processing - September 2017 - 106
Signal Processing - September 2017 - 107
Signal Processing - September 2017 - 108
Signal Processing - September 2017 - 109
Signal Processing - September 2017 - 110
Signal Processing - September 2017 - 111
Signal Processing - September 2017 - 112
Signal Processing - September 2017 - 113
Signal Processing - September 2017 - 114
Signal Processing - September 2017 - 115
Signal Processing - September 2017 - 116
Signal Processing - September 2017 - 117
Signal Processing - September 2017 - 118
Signal Processing - September 2017 - 119
Signal Processing - September 2017 - 120
Signal Processing - September 2017 - 121
Signal Processing - September 2017 - 122
Signal Processing - September 2017 - 123
Signal Processing - September 2017 - 124
Signal Processing - September 2017 - 125
Signal Processing - September 2017 - 126
Signal Processing - September 2017 - 127
Signal Processing - September 2017 - 128
Signal Processing - September 2017 - 129
Signal Processing - September 2017 - 130
Signal Processing - September 2017 - 131
Signal Processing - September 2017 - 132
Signal Processing - September 2017 - 133
Signal Processing - September 2017 - 134
Signal Processing - September 2017 - 135
Signal Processing - September 2017 - 136
Signal Processing - September 2017 - 137
Signal Processing - September 2017 - 138
Signal Processing - September 2017 - 139
Signal Processing - September 2017 - 140
Signal Processing - September 2017 - 141
Signal Processing - September 2017 - 142
Signal Processing - September 2017 - 143
Signal Processing - September 2017 - 144
Signal Processing - September 2017 - 145
Signal Processing - September 2017 - 146
Signal Processing - September 2017 - 147
Signal Processing - September 2017 - 148
Signal Processing - September 2017 - 149
Signal Processing - September 2017 - 150
Signal Processing - September 2017 - 151
Signal Processing - September 2017 - 152
Signal Processing - September 2017 - 153
Signal Processing - September 2017 - 154
Signal Processing - September 2017 - 155
Signal Processing - September 2017 - 156
Signal Processing - September 2017 - 157
Signal Processing - September 2017 - 158
Signal Processing - September 2017 - 159
Signal Processing - September 2017 - 160
Signal Processing - September 2017 - 161
Signal Processing - September 2017 - 162
Signal Processing - September 2017 - 163
Signal Processing - September 2017 - 164
Signal Processing - September 2017 - 165
Signal Processing - September 2017 - 166
Signal Processing - September 2017 - 167
Signal Processing - September 2017 - 168
Signal Processing - September 2017 - 169
Signal Processing - September 2017 - 170
Signal Processing - September 2017 - 171
Signal Processing - September 2017 - 172
Signal Processing - September 2017 - 173
Signal Processing - September 2017 - 174
Signal Processing - September 2017 - 175
Signal Processing - September 2017 - 176
Signal Processing - September 2017 - 177
Signal Processing - September 2017 - 178
Signal Processing - September 2017 - 179
Signal Processing - September 2017 - 180
Signal Processing - September 2017 - 181
Signal Processing - September 2017 - 182
Signal Processing - September 2017 - 183
Signal Processing - September 2017 - 184
Signal Processing - September 2017 - 185
Signal Processing - September 2017 - 186
Signal Processing - September 2017 - 187
Signal Processing - September 2017 - 188
Signal Processing - September 2017 - 189
Signal Processing - September 2017 - 190
Signal Processing - September 2017 - 191
Signal Processing - September 2017 - 192
Signal Processing - September 2017 - 193
Signal Processing - September 2017 - 194
Signal Processing - September 2017 - 195
Signal Processing - September 2017 - 196
Signal Processing - September 2017 - Cover3
Signal Processing - September 2017 - Cover4
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