29-10-2012, 10:56 AM
A Cross-Correlated Trellis-Coded Quadrature Modulation Representation of MIL-STD Shaped Offset Quadrature Phase-Shift Keying
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ABSTRACT
We show that MIL-STD shaped offset quadrature phase-shift keying (SOQPSK),
a highly bandwidth-efficient constant-envelope modulation, can be represented
in the form of a cross-correlated trellis-coded quadrature modulation, a generic
structure containing both memory and cross-correlation between the in-phase and
quadrature-phase channels. Such a representation allows identification of the optimum
form of receiver for MIL-STD SOQPSK and at the same time, through
modification of the equivalent I and Q encoders to recursive types, allows for it
to be embedded as the inner code of a serial or parallel (turbo-like) concatenated
coding structure together with iterative decoding.
Introduction
Shaped offset quadrature phase-shift keying (SOQPSK) is a constant-envelope modulation scheme
that was developed as a variant of shaped binary phase-shift keying (SBPSK), introduced by Dapper and
Hill [1] in the early 1980s as a means of bandlimiting a BPSK signal while, at the same time, keeping
its envelope constant. The initial version of SOQPSK, which became adopted as part of a military
standard, was referred to as MIL-STD SOQPSK and assumed a rectangular frequency-shaping pulse of
duration equal to a bit time interval in its continuous phase modulation (CPM) representation. Other
more spectrally efficient versions of SOQPSK were introduced later on by Hill [2] with frequency-shaping
pulses that extend over several bit intervals. These variants offer spectral containment and power efficiency
comparable to or better than Feher-patented quadrature phase-shift keying (FQPSK) [3], depending on
the specifics of the comparison.
A Time-Invariant Trellis Representation Based on Pairs of Bits
Suppose now that we consider transitions between phase states corresponding to a pair of input bits.
Without loss of generality, assume that the first bit of the input pair is always an I bit. Then, the trellis
between the four phase states π/4 (00), 3π/4 (10), π/4 (11), and 7π/4 (01) can be easily derived from the
eight-state trellis diagram of OQPSK shown in Fig. 2. Figure 4 illustrates such a trellis, where each branch
is now labeled with a pair of output α values, say αi, αi+1. The corresponding two input bits are the
same as the two bits representing the terminating phase state. For convenience, we have drawn the trellis
in expanded form with each transition interval (now 2 bits in duration) showing the transitions leaving
from one of the four states. One can associate a pair of waveforms that corresponds to each transition,
namely, sI (t) = cos
representing what would be
time-synchronously transmitted as symbols (of 2-bit duration) on the I and Q channels. Here φ0 is the
initial phase indicated by the starting phase state of each transition, and from Eqs. (2) and (7),
Transmitter Implementation
Based upon the foregoing trellis representation of MIL-STD SOQPSK and the labeling of the waveforms
illustrated in Figs. 5 and 6, it is possible to express the indices of the specific waveforms transmitted for
sI (t) and sQ (t) in a given symbol (two-bit) interval, say iTb ≤ t ≤ (i + 2) Tb (i even), in terms of two
α values in this interval and the phase state at the beginning of the interval (which itself depends on the
previous values of α). Specifically, corresponding to αi and αi+1 in the above interval and φi at the start
of this interval, we have sI (t) = sn (t), where n has the binary-coded decimal (BCD) representation.
Conclusion
Based on the full-response CPM representation of offset QPSK and MIL-STD shaped offset QPSK
(SOQPSK) modulations, we have shown that the latter can be represented in the form of a cross-correlated
trellis-coded quadrature modulation (XTCQM). Such a representation is analogous to a similar form
previously obtained for Feher-patented QPSK (FQPSK) and, as was the case there, will be particularly
useful when concatenating this modulation with an outer turbo code coupled with iterative decoding.