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July 2005
MMSE-optimal approximation of continuous-phase modulated signal as MMSE-optimal approximation of continuous-phase modulated signal as
superposition of linearly modulated pulses superposition of linearly modulated pulses
Xiaojing Huang
University of Wollongong
, huang@uow.edu.au
Y. Li
University of Wollongong
Follow this and additional works at: https://ro.uow.edu.au/infopapers
Part of the Physical Sciences and Mathematics Commons
Recommended Citation Recommended Citation
Huang, Xiaojing and Li, Y.: MMSE-optimal approximation of continuous-phase modulated signal as
superposition of linearly modulated pulses 2005.
https://ro.uow.edu.au/infopapers/165
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contact the UOW Library: research-pubs@uow.edu.au
MMSE-optimal approximation of continuous-phase modulated signal as MMSE-optimal approximation of continuous-phase modulated signal as
superposition of linearly modulated pulses superposition of linearly modulated pulses
Abstract Abstract
The optimal linear modulation approximation of any M-ary continuous-phase modulated (CPM) signal
under the minimum mean-square error (MMSE) criterion is presented in this paper. With the introduction
of the MMSE signal component, an M-ary CPM signal is exactly represented as the superposition of a
?nite number of MMSE incremental pulses, resulting in the novel switched linear modulation CPM signal
models. Then, the MMSE incremental pulse is further decomposed into a ?nite number of MMSE pulse-
amplitude modulated (PAM) pulses, so that an M-ary CPM signal is alternatively expressed as the
superposition of a ?nite number of MMSE PAM components, similar to the Laurent representation.
Advantageously, these MMSE PAM components are mutually independent for any modulation index. The
optimal CPM signal approximation using lower order MMSE incremental pulses, or alternatively, using a
small number of MMSE PAM pulses, is also made possible, since the approximation error is minimized in
the MMSE sense. Finally, examples of the MMSE-optimal CPM signal approximation and its comparison
with the Laurent approximation approach are given using raised-cosine frequency-pulse CPM schemes.
Keywords Keywords
continuous phase modulation, least squares approximations, pulse amplitude modulation, signal
processing
Disciplines Disciplines
Physical Sciences and Mathematics
Publication Details Publication Details
This paper originally appeared as: Huang, X and Li, Y, MSE-optimal approximation of continuous-phase
modulated signal as superposition of linearly modulated pulses, IEEE Transactions on Communications,
July 2005, 53(7), 1166-1177. Copyright IEEE 2005.
This journal article is available at Research Online: https://ro.uow.edu.au/infopapers/165
1166 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 7, JULY 2005
MMSE-Optimal Approximation of Continuous-Phase
Modulated Signal as Superposition of
Linearly Modulated Pulses
Xiaojing Huang, Member, IEEE, and Yunxin Li, Senior Member, IEEE
Abstract—The optimal linear modulation approximation of
any
-ary continuous-phase modulated (CPM) signal under the
minimum mean-square error (MMSE) criterion is presented in
this paper. With the introduction of the MMSE signal component,
an
-ary CPM signal is exactly represented as the superposition
of a finite number of MMSE incremental pulses, resulting in the
novel switched linear modulation CPM signal models. Then, the
MMSE incremental pulse is further decomposed into a finite
number of MMSE pulse-amplitude modulated (PAM) pulses,
so that an
-ary CPM signal is alternatively expressed as the
superposition of a finite number of MMSE PAM components,
similar to the Laurent representation. Advantageously, these
MMSE PAM components are mutually independent for any
modulation index. The optimal CPM signal approximation using
lower order MMSE incremental pulses, or alternatively, using a
small number of MMSE PAM pulses, is also made possible, since
the approximation error is minimized in the MMSE sense. Finally,
examples of the MMSE-optimal CPM signal approximation and
its comparison with the Laurent approximation approach are
given using raised-cosine frequency-pulse CPM schemes.
Index Terms—Continuous-phase modulation (CPM), Laurent
representation, minimum mean-square error (MMSE).
I. INTRODUCTION
C
ONTINUOUS-PHASE modulation (CPM) [1]–[4] is a
nonlinear modulation scheme, although the phase re-
sponds linearly to the input data. CPM has the advantage of
excellent power and bandwidth efficiency with constant signal
envelope. Its disadvantage is its nonlinear nature, which results
in complexity in receiver implementation and difficulty in
signal analysis. For these reasons, the CPM scheme has been
mostly used for binary signaling with special modulation in-
dexes in many power- and bandwidth-efficient communication
systems. To combat the inherent nonlinearity, Laurent discov-
ered that a binary CPM signal with any noninteger modulation
index can be exactly decomposed as a sum of a finite number
of pulse-amplitude modulated (PAM) signal components [5].
Mengali and Morelli extended this binary PAM decomposition
to multilevel cases by expressing a noninteger modulation
Paper approved by G. M. Vitetta, the Editor for Equalization and Fading
Channels of the IEEE Communications Society. Manuscript received March
11, 2003; revised May 30, 2003; June 29, 2004; July 28, 2004; and February
10, 2005.
X. Huang is with the School of Electrical, Computer, and Telecommunica-
tions Engineering, Faculty of Informatics, University of Wollongong, Wollon-
gong, NSW 2522, Australia (e-mail: huang@uow.edu.au).
Y. Li is with the Twincall Education Center, Twincall Pty Ltd., Ryde, NSW
2112, Australia (e-mail: jeff@twincall.com).
Digital Object Identifier 10.1109/TCOMM.2005.851625
index
-ary CPM signal as the product of several constitu-
tional binary CPM signals [6]. A complementary solution for
decomposing the integer modulation index CPM signal was
also proposed recently [7]. Thus, the theory of CPM signal
PAM decomposition under the banner of Laurent representation
seemed to be complete.
However, the Laurent representation has two major draw-
backs. First, the PAM components of a CPM signal are generally
not mutually independent, except when the modulation index is
a multiple of 0.5. As a consequence, the autocorrelation func-
tion of a CPM signal is not, in general, a sum of all PAM pulses’
autocorrelation functions. Cross-correlations between different
PAM components exist, which still causes difficulty in CPM
signal analysis. Second, when a small number of PAM pulses
are used to approximate a CPM signal, the approximation error
is not minimized in the minimum mean-square error (MMSE)
sense, except for some special modulation indexes. Other draw-
backs include the inefficiency in practical application, due to
the large number of pseudosymbols in which the data symbols
are hidden, and the complicated decomposition procedure, es-
pecially for multilevel CPM signals.
In this paper, a new approach, directly applicable to any
-ary CPM signal, is taken to represent the CPM signal as the
superposition of linearly modulated pulses. By this approach,
novel switched linear modulation models are derived for the
CPM signal with either an integer or noninteger modulation
index. Then, the linearly modulated pulses in the models are
further decomposed into PAM pulses, similar to Laurent rep-
resentation. However, the PAM components are now always
mutually independent. When fewer linearly modulated pulses
are used to approximate the CPM signal, with a simultaneous
reduction in signal memory, the approximation error is always
minimized in the MMSE sense. Other advantages include the
explicit closed-form expressions for all functions and coeffi-
cients in the representation. Thus, the drawbacks associated
with the Laurent representation as mentioned above are all
solved.
As has been known to us, many existing simplified CPM
models, such as Laurent’s PAM decomposition and Rimoldi’s
continuous-phase encoder (CPE) plus memoryless modulator
(MM) decomposition [13], have found wide application in
CPM signal analysis, receiver complexity reduction, signal
synchronization, and parameter estimation [8]–[13]. The pro-
posed switched linear modulation models can be also exploited
to construct simple CPM receivers, reducing both the number
of matched filters and trellis states at the same time [14].
0090-6778/$20.00 © 2005 IEEE
HUANG AND LI: MMSE OPTIMAL APPROXIMATION OF CPM SIGNAL 1167
It is worth noting that in terms of matched-filter reduction
for the maximum-likelihood-type CPM receivers, Huber and
Liu took a signal space dimension reduction approach [15].
Moqvist and Aulin showed that the optimal lower dimensional
approximation was given by the principal components method
under the criterion of minimum residual error [16]. However,
this approach does not reduce the memory of the approximated
CPM signal, so the reduction in trellis states has to be treated
separately.
The rest of this paper is organized as follows. Analogous to
the Laurent signal component defined in Appendix A, the no-
tion of an MMSE signal component for an
-ary CPM signal is
first introduced in Section II. Then, the switched linear modula-
tion models are derived in Section III by decomposing the CPM
signal as a superposition of the MMSE incremental pulses. In
Section IV, the MMSE incremental pulses are further decom-
posed into a finite number of MMSE PAM pulses associated
with independent pseudosymbols. In Section V, the autocorre-
lation function of the CPM signal approximated by lower order
MMSE incremental pulses is formulated, and the approxima-
tion error is proven to be minimized. In Section VI, examples of
the MMSE-optimal CPM signal approximation and the compar-
ison with the Laurent approximation approach are given using
the raised-cosine frequency pulse (LRC) CPM scheme. Finally,
conclusions are drawn in Section VII. A reformulated Laurent
representation is also provided in Appendix B.
II. MMSE S
IGNAL COMPONENT OF
CPM SIGNAL
The equivalent lowpass envelope of an
-ary CPM signal
[1]–[4] with unity signal power can be expressed as
(1)
where the data symbol
with symbol interval belongs to the
-ary alphabet , and the phase shift
function
is assumed to be zero for a negative value of time
and
( denotes the modulation index) for time greater than
symbol intervals, i.e.,
for
for
(2)
For
can be any monotonic function. Ac-
cording to the Laurent representation [5]–[7],
for nonin-
teger
can be decomposed into a sum of a finite number of
PAM components. The coefficient, denoted as
, associated
with the
th Laurent PAM pulse at time is referred to as
the pseudosymbol [6]. Especially, the pseudosymbol associated
with the first Laurent PAM pulse
can be expressed as
(3)
which represents the CPM signal’s accumulative phase rotation
contributed by all previously transmitted data symbols up to
time
. Since all the possible phases of constitute
the phase states of the CPM signal, we simply refer to it as the
phase state symbol.
Now let us consider the contribution of
consecutively trans-
mitted data symbols
, starting from
time
where stands for a specific time
index, to the transmitted CPM signal waveform. We refer to this
contribution as a kind of signal component associated with the
data symbols. The expression of this signal component must
meet the following requirements. First, this signal component
is a function of the
data symbols. Second, due to the CPM’s
phase continuity, this signal component should contain a factor
to account for the influence (phase rotation) of all pre-
viously transmitted data symbols up to time
.
In Appendix A, we have shown that the Laurent signal compo-
nent
is exactly the product of
the Laurent complex pulse
, which is a
function of
data symbols, with (for noninteger )or
(for integer ), see (A4) and (A5).
Thus, analogous to the Laurent signal component, we define
the MMSE signal component
in
the form of
for noninteger
(4)
for integer
(5)
but determine the function
by
the MMSE criterion. Under this criterion, we first ex-
press the mean-square error (MSE) between the
-ary
CPM signal
and its MMSE signal component
as
, where
denotes the ensemble averaging over all
symbols other than
. Then, letting
the functional derivative
to minimize the MSE, we find
for noninteger (6)
where
(7)
We see that
is also a windowed com-
plex exponential similar to the Laurent complex pulse derived
in Appendix A, and is therefore referred to as the MMSE com-
plex pulse. Accordingly,
is called the MMSE window func-
tion. Applying the same MMSE criterion for an
-ary CPM
signal with integer
, the MMSE complex pulse and the MMSE
window function have the same expressions as (6) and (7), re-
spectively, after ignoring the sign ambiguity for odd
due to
an undetermined number of factors
( for odd )
involved in the derivation.
1168 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 7, JULY 2005
Derived from (7) and (2), demonstrates some inter-
esting properties, such as
(8)
(9)
for (10)
generally has an infinite duration (except for ,
where
is any integer), but will attenuate to zero for noninteger
as . For integer oscillates as .If
with integer becomes a periodic function, so that
it is alternatively expressed as
for integer (11)
where
elsewhere
is one (
even) or half ( odd) period of .
III. CPM S
IGNAL REPRESENTATION BY
MMSE I
NCREMENTAL PULSES
To decompose an
-ary CPM signal into additive linearly
modulated pulses, let us first analyze the difference between
the MMSE signal component associated with
data sym-
bols
and the one associated with
data symbols , which in-
dicates an increment to the waveform of the MMSE signal
component when the data symbol
is transmitted after
. Assuming a noninteger and
using the property of
given in (9), this difference is found
to be
(12)
where
(13)
is a causal function (i.e.,
for )
and is referred to as the MMSE incremental pulse of order
.
Note that for any
, the th-order MMSE incremental pulse
will be the same as the th-order one
, since and
for and , according to (10) and (2).
Then we rewrite the MMSE signal component in terms of the
MMSE incremental pulses of orders from 1 to
as
(14)
Letting
, the MMSE signal component will be expressed
as the convolution of only the
th-order MMSE incremental
pulse with the phase state symbol, since the first
terms on
the right-hand side (RHS) of (14) will decay to zero because of
the attenuated window functions, that is
(15)
Finally, letting
, the MMSE signal component becomes
, which is the -ary
CPM signal itself (assuming an initial value of the phase state
symbol
), i.e.,
(16)
In this way, the
-ary CPM signal is decomposed exactly
into a sum of the
th-order MMSE incremental pulses
modulated by the phase state
symbols
.
Following a similar procedure as described above and care-
fully dealing with the sign ambiguity, we can also decompose
the
-ary CPM signal with integer into a sum of the th-order
MMSE incremental pulses. However, since
becomes a pe-
riodic function with integer
[see (11)], it will not decay to zero,
but will appear in the final decomposition expression. Thus
for integer (17)
