Signal Processing - September 2017 - 22

form has the greatest realization flexibility. If it is necessary to adjust the
CEM scheme when using this multi~
p
plexer architecture, such as changing
sMUX (t ) = Aexp( jθk(t))
= M-1θ
sMUX (t ) = Aexp j Σi pi si (t )
q
+Σj qj ξj (t )
the power ratio or phase relationship
between some component signals, one
PMB
PSB
p, q
θ
only needs to regenerate the phasemapping rule, replacing the original
r
sin qn . ξn t = ξn t sin qn
~ -1
= M sMUX (θ)
LUT. The WSB form has the clearest
w
cos qn . ξn t = cos qn
r, w
physical meaning for signal characteristic analysis. From this representation,
sMUX (t ) = Σi ri si (t ) + Σj wj ξj (t )
WSB
one can see the number, composition,
and the proportion of IM terms introWaveform Domain Representation
duced by CEM, and their power and
phase relationships with the compoFIGURE 5. The conversion relationship of three representation forms of a CEM scheme.
nent signals. This allows for an easier
analysis of spectrum characteristics
with their own features in realization or analysis, and they can
such as the spectrum occupancy of the composite signal and
be converted into each other.
the spectral compatibility between useful signals and the auxiliary term. Since the frequencies of baseband components are
PMB form
too low to avoid harmonic and intermodulation interference
Regardless of the design method, every CEM scheme can evenwith the desired output during the up-conversion, the PSB
tually correspond to a phase LUT. The LUT stores the mapping
form is conceptually useful but presents a series of limitations
from every possible value combination of component signals
in a real implementation [25].
to a phase angle i k, in which the phase index k is a function
of time. Such phase values are passed to sine and cosine mapCEM for multilevel and multicarrier signals
pers to generate the real and imaginary parts of A exp ^ ji k^ t hh,
With other signal elements becoming more diverse and comand then this complex baseband signal is modulated to radio
plex in future GNSSs, the demand for the flexibility of a CEM
frequency and transmitted.
becomes greater. The high-flexibility multicarrier CEM is
gaining more and more attention in recent years as a new hot
PSB form
topic in the signal multiplexing field. Several CEM design and
Since the phase angle can be completely determined by the valimplementation methods with a higher flexibility for different
ue combination of the component signals, it can be decomposed
input signal types have emerged.
into the linear combination of component signals s n ^ t h and their
N
IM terms p n ^ t h, i.e., i ^ t h = R nN= 1 p n s n ^ t h + R n2 =-1N q n p n ^ t h,
CEM for multilevel signals
where p n and q n are the weighting coefficients of s n
Most of the existing CEM techniques can only apply to biand p n respectively. In matrix representation, that is
polar signals. However, some research has shown that mulu u , where M
u = 6M, C@ is a full-rank mai = Mp + Cq = Mp
tilevel spreading chip waveforms, which have more design
trix, and pu = 6 p 1, f, p N , q 1, f, q 2 N - N@T the entries of which
latitude compared with bipolar waveforms, can bring better
corresponds to the weighting coefficients and can be obtained
radio-frequency compatibility [26] and higher potential rangu -1 i.
by pu = M
ing accuracy [27], and provide the possibility to have a higher
transmission rate per single signal component. The composite
WSB form
BOC (CBOC) signal [6] is a typical multilevel signal, the chip
As discussed in the section "WDP," for any given s MUX, the
waveform of which is the superposition of two square waves of
complex envelope of the composite signal can be decomdifferent frequencies. The multilevel amplitude characteristic
N
posed into s MUX = R iN= 1 ri s i + R 2j =-1 N w j p j, by calculating
of this signal results in the unsuitability of earlier CEM techu -1 s MUX, where the coefficients ri and w j are the
[r T, w T] T = M
niques. In the Galileo E1 signal design, to combine two CBOC
entries of r and w, respectively. On the other hand, for biposignals and another bipolar signal into a constant envelope siglar signals, using the identities cos ^q n $ p n ^ t hh = cos ^q nh and
nal, designers limit these two CBOC signals to be of equal
sin ^q n $ p n ^ t hh = p n ^ t h sin ^q n h, one can also expand the PSB
power, both in phase. The polarities of the high-frequency
form into the WSB form.
square wave components are reversed to make their superposition take on a pseudo-random time multiplexing form so that
Summary
a modified Interplex technique [5] can be applied.
Figure 5 shows the mutual conversion relationship of these
Galileo uses an elaborate design to address a specific multhree forms. Using this diagram, we can convert a CEM scheme
tilevel signal's CEM requirement. However, this case cannot
into a form that is easy to handle: of these three, the PMB
be treated as a more general multilevel signal's multiplexing
Phase Domain Representation

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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