Signal Processing - September 2017 - 20
respectively. Strictly speaking, this is not
or quasi-structured approaches, in which
The auxiliary component
exactly equivalent to solving (4). However,
the form of most of the auxiliary compointroduced to maintain the
as the penalty factors are increased, the
constant envelope occupies nents is presupposed to be a weighted sum
solution of this approximation problem is
of input signals' third-order cross products
some of the transmitted
s k 1 s k 2 s k 3, where k 1, k 2, and k 3 are not
forced to approach the solution of (4).
power,
without
contributing
The use of the scalar form of constraints
equal to each other.
to the received power of the
and the penalty function has an additional
Zhang et al. [15] extend the third-order
advantage over directly solving (4) in that
cross product to a more general case. They
desired component signal.
not all angle and amplitude relationships
defined the kth-order intermodulation (IM)
between any two signal components must
term as the product of any k signals in the
be declared. For those components without power and phase
whole set of component signals and proved that, for bipolar
restrictions, the optimization routine will lead to the optimum
signals, those zeroth-order and 2 + N th-order IM terms form
power and phase relationship that has the best combining effiexactly a set of orthogonal bases of W =, i.e., C can be directciency. BDS B3 signal design is a classic example of using this
ly constructed by the equation shown in the box at the botapproach. By relaxing the phase relation between the opentom of the page, where p^k 1, k 2, f, k jh is the Hadamard product
service signals and the authorized service signals, a relative
of m k 1, m k 2, f, m k j, and 1 = (1, 1, f, 1) T. Such a result not
phase angle of r/4 is found by optimization, under which a
only guarantees the feasibility of some of the quasi-structured
high multiplexing efficiency 85.36% is obtained for four signal
WDP design methods who use third-order IM terms p^k 1, k 2, k 3h
components of equal power.
as bases, but also reveals that those quasi-structured approaches with incomplete bases have limited flexibility in the power
WDP
and phase relationship between components.
In addition to constructing phase mapping, an intuitive idea
In recent studies [17], it has been further proved that the
to obtaining CEM is adding an auxiliary component s AUX (t)
cross products form is not a necessary assumption of the bases
to the direct superposition signal s DS (t) to make the composof W = . s AUX ^ t h can have a more general form, which further
ite signal's envelope constant, i.e., s MUX (t) = s DS (t) + s AUX (t).
widens the application range of WDP-based CEM to multilevel
Visually, one can think of this process as bringing the phase
and multicarrier cases, as discussed in the section "CEM for
points of s DS (t) to a circle on a complex plane.
Multilevel and Multicarrier Signals."
Since, at any moment, the value of the auxiliary component
should be completely determined by the value combination of
Optimization-based WDP approaches
the component signals, the main concept of WDP is specifying
The auxiliary component introduced to maintain the cona mapping rule s AUX ^s 1, f, s N h from the value combination of
stant envelope occupies some of the transmitted power,
s 1, f, s N to the value of s AUX . At a time when the value comwithout contributing to the received power of the desired
bination of s 1, f, s N corresponds to the kth row of the LUT as
component signal. In the WDP view, if all of the bases are
in Figure 3, s AUX takes a complex value J k and the composite
orthonormal, the power efficiency (1) can be customized to
signal takes s M, k = R iN= 1 Pi e jzi m ki + J k . Similar to the dish = r 2 ^ r 2 + w 2 h . Therefore, in the maximum multicussion of PDP, this relation can be represented in a compact
plexing efficiency sense, the optimal w is the minimum norm
T
N
s
=
Mr
+
m
,
matrix form MUX
where m = 6J 1, f, J 2 @ .
in its feasible region. Furthermore, the WDP-based CEM
Substituting s MUX = Mr + m into (2) and using the properdesign can be equivalent to solving the following nonlinear
ty M T M $ 2 -N = I, we can see that in the WDP view, the transprogramming problem:
parency constraint is equated with M T m ) = 0, or m ! W =, as
min
w 2, s.t. s M, 1 = s M, 2 = g = s M, 2 N .
(5)
proved in [17], where W = span " m 1, f, m N , . Thus, m can
w
=
be linearly represented by a set of bases of W (denoted as
the column vectors of C = 6c 1, f, c 2 -N@ ), i.e., m = Cw, where
Yao et al. [17] studied the solution of this general optimization
problem and derived a numerical optimization-based method
w = 6w 1, f, w 2 N - N@T the ith entry of which corresponds to the
weight of c i in m. Consequently, in the WDP view, designing
termed CEM via IM construction (CEMIC). Unlike quasi-struca CEM solution is equivalent to finding a proper weight vector
tured methods, CEMIC can be applied to any number of signals
w of a set of bases under a given design constraint r to make
with arbitrary power and phase relationships and can achieve
each complex entry's modulus of s MUX (w) = Mr + Cw equal.
optimal power efficiency; see the next section for an example.
N
Quasi-structured WDP approaches
Case study: CEM of four bipolar signals
In most of the existing WDP-based CEM techniques, such as
[10], [11], and [23], the construction of m is based on structured
Consider the CEM of four bipolar signals as an example,
where s 1, s 2, and s 3 are located in the in-phase (I) portion
C = ;p^1, 2h, p^1, 3h, f, p^ N - 1, N h, f, p^k 1, k 2, f, k jh, f, p^1, 2, f, N h, 1 E
144444
42444443
1442443
> 5
second IM
20
jth IM
N th IM
|
|
IEEE SIGNAL PROCESSING MAGAZINE
September 2017
zeroth-IM
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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