Common detectors for shaped offset QPSK (SOQPSK) and Feher-patented QPSK (FQPSK)

Abstract

Symbol-by-symbol detection of SOQPSK and FQPSK using detectors designed for offset QPSK represents a simple common detector architecture for these two interoperable waveforms. Unfortunately, this detection method results in a 2 dB loss in bit error rate performance. This paper describes detection methods for recovering this loss without the need for knowing which modulation is used by the transmitter. An equivalent cross-correlated trellis-coded quadrature modulation (XTCQM) representation for SOQPSK is developed which forms the basis of a common TCM detector. An equivalent CPM representation for FQPSK is developed which forms the basis for a common CPM detector. The common XTCQM detector performs slightly better than the common CPM detector, but achieves this gain at the expense of higher complexity

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Common Detectors for Shaped Offset QPSK
(SOQPSK) and Feher-patented QPSK (FQPSK)
Tom Nelson∗, Erik Perrins†, and Michael Rice∗
∗Department of Electrical & Computer Engineering, Brigham Young University, Provo, UT 84602
†Department of Electrical Engineering & Computer Science, University of Kansas, Lawrence, KS 66045
Abstract—Symbol-by-symbol
FQPSK using detectors designed for offset QPSK represents a
simple common detector architecture for these two interoperable
waveforms. Unfortunately, this detection method results in a 2 dB
loss in bit error rate performance. This paper describes detection
methods for recovering this loss without the need for knowing
which modulation is used by the transmitter. An equivalent
cross-correlated trellis-coded quadrature modulation (XTCQM)
representation for SOQPSK is developed which forms the basis
of a common TCM detector. An equivalent CPM representation
for FQPSK is developed which forms the basis for a common
CPM detector. The common XTCQM detector performs slightly
better than the common CPM detector, but achieves this gain at
the expense of higher complexity.
detectionofSOQPSKand
I. INTRODUCTION
Power and bandwidth constraints present challenges to
waveform and modulation design. These challenges are exac-
erbated when the power amplifiers must operate in their non-
linear (saturated) mode. In this case, the additional constraint
of constant envelope is often imposed. Usually some form of
continuous phase modulation (CPM) is used since it possesses
both constant envelope and is bandwidth efficient. Examples
include, GMSK [1] in digital mobile telephony and PCM/FM
in aeronautical telemetry [2].
PCM/FM has been the modulation of choice in aeronautical
telemetry since the early 1970s. However, as data rates have
increased and available bandwidth has decreased, the need for
more spectrally efficient modulations has intensified. In 2000,
Feher-patented QPSK (FQPSK) [3] was adopted as a standard
in the aeronautical telemetry standard IRIG 106 [2]. FQPSK
is a form of offset QPSK (OQPSK) where the inphase and
quadrature pulse shapes are selected from a set of 16 pulses.
The selection is determined by the inphase and quadrature
data transitions and constrained to produce a quasi-constant
envelope. This constraint produces an I/Q modulation that
is termed cross correlated since the quadrature waveform is
constrained by the inphase waveform and vice versa. Three
of the pulses were redefined by Jefferis and Formeister [4] to
produce a version of FQPSK, denoted FQPSK-JR, that has a
true constant envelope. FQPSK-JR was adopted as an option
in the IRIG 106 standard in 2004. Simon [5] showed that
FQPSK can be interpreted as a cross-correlated trellis-coded
quadrature modulation (XTCQM).
A competing modulation, known as shaped offset QPSK
(SOQPSK) was also adopted as an option in the 2004 version
of IRIG 106. SOQPSK is a constrained ternary CPM with
modulation index h = 1/2. The IRIG 106 version, known
as SOQPSK-TG, is a partial response version of the full
response SOQPSK defined in MIL-STD 188-181 [6], the
military standard for UHF satellite communications.
SOQPSK-TG and FQPSK-JR were selected as interoperable
standards in aeronautical telemetry because both have approx-
imately the same bandwidth and approximately equivalent bit
error rate performance when detected using a simple offset
QPSK detector. The simple symbol-by-symbol detector has
two attractive features: 1) low complexity, and 2) it does not
have to “know” which modulation is used by the transmitter.
These attractive features are achieved at the expense of detec-
tion efficiency: the bit error rate performance of this simple
detector is about 2 dB worse than what could be achieved with
optimum detection.
Since SOQPSK-TG is a CPM and FQPSK-JR is a XTCQM,
it is natural to assume that the optimal detector must be
equipped with two different detection algorithms and endowed
with the knowledge of which modulation is used by the
transmitter. In this paper we show that a single detection
algorithm can be used for both modulations and that this
algorithm does not have to “know” which modulation is used
(that is, its bit error rate performance is the same for both
SOQPSK-TG and FQPSK-JR and that this performance is
within 0.1 dB of the SOQPSK-TG optimum and equal to the
FQPSK-JR optimum). We refer to such a detector as a common
detector.
This remarkable result is derived as follows. SOQPSK-TG
is shown to have an equivalent XTCQM representation in
Section III. The corresponding maximum likelihood detector,
which follows naturally from this representation, is modified
to form an XTCQM detector that is fully compatible with
both SOQPSK-TG and FQPSK-JR. The common XTCQM
detector performs within 0.1 dB of optimal for SOQPSK-
TG and achieves optimal performance for FQPSK-JR. In
Section IV, an equivalent CPM representation for FQPSK-
JR is derived. This representation is simplified and used to
produce a common CPM detector. The common CPM detector
performs within 0.25 dB of optimal for SOQPSK-TG and
essentially achieves optimal for performance for FQPSK-
JR. While the common XTCQM detector outperforms the
common CPM detector, it does so with greater complexity as
measured by the number of matched filters and trellis states.
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−4−3−2−101234
t/Tb
SOQPSK−TG
MIL−STD SOQPSK
Fig. 1.The frequency pulses for SOQPSK-TG and MIL-STD SOQPSK.
II. INTEROPERABLE MODULATIONS
A. SOQPSK-TG
SOQPSK (both the MIL-STD and -TG versions) is defined
as a CPM. The baseband modulated signal may be expressed
as
s(t) =
Tb
where
?t
n=−∞
for nTb ≤ t ≤ (n + 1)Tb; φ0 is an arbitrary phase which,
without loss of generality, can be set to 0; h = 1/2 is
the modulation index; and αn ∈ {−1,0,1} are the ternary
symbols and are related to the input binary symbols an ∈
{−1,1} by [7]
αn= (−1)n+1an−1(an− an−2)
The frequency pulse for SOQPSK-TG, g(t), is a spectral raised
cosine windowed with a temporal raised cosine with length
L = 8Tb [8] and is shown in Fig. 1. Also shown is the
frequency pulse for MIL-STD SOQPSK which is a 1REC.
The performance of the maximum likelihood CPM detector
was analyzed by Geoghegan [9], [10], [11]. The asymptotic
probability of bit error is given by
??
?2Eb
exp[j (φ(t,α) + φ0)]
(1)
φ(t,α) = 2πh
−∞
∞
?
αng(τ − nTb)dτ
(2)
2
.
(3)
Pb=1
2Q
1.60Eb
N0
?
.
(4)
B. FQPSK-JR
FQPSK-JR is defined as an offset QPSK modulation of the
form
?
with data dependent pulses sI,m(t) and sQ,m(t) each drawn
in a constrained way from a set of 16 waveforms [4]. The 16
pulses are listed in [4] and [5] and, due to space constraints,
s(t) =
k
sI,m(t − kTs) + jsQ,m(t − kTs− Ts/2).
(5)
are not reproduced here. Simon showed that FQPSK has an
XTCQM interpretation from which the optimum maximum
likelihood detector followed [5]. This representation consists
of 16 waveforms for the inphase component and 16 waveforms
for the quadrature component for a total of 32 possible
complex-valued waveforms when the constraints on possible
combinations are taken into account. Due to the symmetries
of the waveforms, 16 real-valued matched filters are required
together with a 16-state trellis. The corresponding performance
analysis showed that the asymptotic bit error probability is
??
Pb=1
2Q
1.56Eb
N0
?
.
(6)
The application to FQPSK-JR is straightforward and the
corresponding optimum XTCQM detector has the same per-
formance as its FQPSK counterpart and is given by (6).
C. Symbol-by-Symbol Detection
SOQPSK-TG and FQPSK-JR are considered to be interop-
erable because of their similar performance with an integrate-
and-dump detector normally used with offset QPSK without
any pulse shaping. Using this detector with FQPSK (and its
variants) is natural since FQPSK is defined as an offset QPSK
with data dependent pulse shapes. The use of this detector
with SOQPSK-TG is motivated by the well established con-
nection between CPM with modulation index h = 1/2 and
offset QPSK [12]–[15]. Symbol-by-symbol detection has been
thoroughly investigated for SOQPSK-TG by Geoghegan [9]
and for FQPSK by Simon [5]. Our own simulation results are
shown in Fig. 2 where we see that SOQPSK-TG performs
about 2.0 dB worse than its optimum CPM detector and that
FQPSK-JR performs about 2.2 dB worse than its optimum
XTCQM detector. Symbol-by-symbol detection with better
detection filters has also been investigated for SOQPSK-TG
in [9] and for FQPSK in [5]. The XTCQM representations for
both modulations can be used to define detection filters for use
with a symbol-by-symbol detector as explained in Section III.
The bit error rate performance of SOQPSK-TG and FQPSK-
JR using the improved detection filter is also plotted in Fig. 2.
Observe that the use of this detection filter improves the bit
error rate performance by 0.5 dB for SOQPSK-TG and 0.6 dB
for FQPSK-JR. This improvement is well short of the 2 dB
loss relative to the bit error rate performance corresponding to
(4) and (6), respectively. This observation motivates the search
for common detectors with improved detection efficiency.
III. XTCQM DETECTOR
Since FQPSK-JR can be interpreted as an XTCQM, we
seek an equivalent XTCQM representation for SOQPSK-TG.
This representation allows the design of a common XTCQM
detector. The XTCQM representation of SOQPSK-TG is de-
rived as follows. Inserting (3) into (2), exchanging the order
of integration and summation and extending the sum to n+1
to get a waveform of duration 2Tb= Tsresults in
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46810 12
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Eb/N0 (dB)
bit error rate
Theory (Tx=SOQPSK−TG)
Theory (Tx=FQPSK−JR)
I&D (Tx=SOQPSK−TG)
Common det. filter (Tx=SOQPSK−TG)
I&D (Tx=FQPSK−JR)
Common det. filter (Tx=FQPSK−JR)
Fig. 2.
and-dump (I&D) detector and the common symbol-by-symbol detector along
with the theoretical curves for each modulation. The I&D detector performs
about 2.0 dB worse than optimum for SOQPSK-TG and about 2.2 dB worse
than optimum for FQPSK-JR at Pb= 10−5while the average matched filter
detector performs about 0.5 dB better than I&D for SOQPSK-TG and about
0.6 dB better for FQPSK-JR.
Bit error rates for SOQPSK-TG and FQPSK-JR for the integrate-
φ(t,an) =
θn+ 2πh
n+1
?
i=n−L+1
(−1)i+1ai−1(ai− ai−2)
2
q(t − iTb)
(7)
where
an=?an−L−1
θn= πh
an−L
···
an+1
?,
(8)
n−L
?
i=−∞
αi,
(9)
and
q(t) =
?t
−∞
g(x)dx.
(10)
With L = 8 there are 9 terms in the sum and 11 bits that
contribute to φ(t,an) during the interval nTb≤ t ≤ (n+2)Tb.
There are 4 phase states, but they do not increase the number of
waveforms because the phase state θnis a function of the last
two bits in an, an−Land an−L−1. As a result, 2048 complex
waveforms are needed to exactly represent SOQPSK-TG. The
I and Q waveforms are given by
sI,m(t) = cos(φ(t,an)),sQ,m(t) = sin(φ(t,an))
(11)
where the index m is given by
m = 210bn+1+ 29bn+ ... + 20bn−L−1
and bn∈ {0,1} is related to the bipolar bits an∈ {−1,1} by
bn= (an+ 1)/2. Then the modulated signal takes the form
?
(12)
s(t) =
k
sI,m(t − kTs) + jsQ,m(t − kTs)
(13)
with sI,m(t) and sQ,m(t) defined in (11). Thus, even though
SOQPSK-TG is defined as a constrained ternary CPM, it can
also be viewed as a XTCQM consisting of 2048 waveforms.
This view suggests an alternate form for the optimal detector.
The fact that the XTCQM representation of SOQPSK-
TG uses 2048 waveforms while FQPSK-JR uses 32 wave-
forms presents a difficulty in defining a common XTCQM
detector. The number of waveforms required by the XTCQM
representation of SOQPSK-TG can be reduced by averaging
the time-domain waveforms that differ in the first and last
bits. (This technique was used by Simon [16] to reduce the
number of waveforms required to represent FQPSK.) Applying
this technique once reduces the number of waveforms from
2048 to 512. Repeating two more times reduces the number
of waveforms to 32. (Simulations, not presented here, show
that an XTCQM detector for SOQPSK-TG based on this
approximation performs within 0.1 dB of (4).) Since the 32
waveforms used in this approximate XTCQM representation
of SOQPSK-TG are different from the 32 waveforms used by
the XTCQM representation of FQPSK-JR, the two sets can
be averaged to obtain a set of 32 waveforms which form the
basis of a common XTCQM detector. This detector requires 16
real-valued length-2Tbmatched filters together with a 16-state
trellis.
The bit error rate performance of the common XTCQM
detector is shown in Fig. 3. A plot of (4) and (6) are also
inlcuded for reference. The bit error rate performances of both
modulations with this detector are equivalent. Note that the
BER curves coincide with the FQPSK-JR bound (6) and are
0.1 dB worse than the SOQPSK-TG bound (4).
Note that the process for reducing the number of wave-
forms in the XTCQM approximation for SOQPSK-TG can be
continued until only one waveform remains. This waveform
can be averaged with the average of the FQPSK waveforms
to produce a common representation consisting of only one
waveform. A detection filter matched to this waveform forms
the basis of the common symbol-by-symbol detector whose
bit error rate performance is summarized in Fig. 2.
IV. CPM DETECTOR
Another candidate for the common detector is the CPM
detector. SOQPSK-TG is defined as a CPM, as explained in
Subsection II-A. The CPM approximation of FQPSK-JR is
obtained by determining the phase pulse φ(t) as a function of
the XTCQM waveforms. The length of φ(t) is L = 2 because
the waveforms are defined over a two bit interval. The phase
of the signal as it transitions from one constellation point to
an adjacent point determines φ(t).
For example, consider the case where the initial phase state
is π/4 and αn = −1. In that case the I and Q waveforms
(sI(t) and sQ(t), respectively) are given by
?
sI(t)=1 − A2cos2
?π(t − Tb)
?πt
2Tb
?
sQ(t)=
−Asin
2Tb
?
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34567891011
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Eb/N0 (dB)
bit error rate
Theory (Tx=SOQPSK−TG)
Theory (Tx=FQPSK−JR)
Common XTCQM det. (Tx=SOQPSK−TG)
Common XTCQM det. (Tx=FQPSK−JR)
Fig. 3.
XTCQM detector along with the theoretical curves for each modulation.
Bit error rates for SOQPSK-TG and FQPSK-JR for the common
for 0 ≤ t ≤ 2Tb. (A is an adjustable parameter that appears
in the definitions of the 16 waveforms for both FQPSK and
FQPSK-JR. This parameter is set to 1/√2 to produce quasi-
constant envelope FQPSK or constant envelope FQPSK-JR.)
The phase pulse in this case is given by




The frequency pulse g(t) is then the derivative of φ(t) and is
given by
Aπ
2Tb
?
φ(t)=tan−1

−Asin
?
?π(t − Tb)
2Tb
?πt
?πt
?πt
?
?
1 − A2cos2
2Tb
?





= tan−1

−Acos
2Tb
?
1 − A2cos2
2Tb
?

.
(14)
g(t) =
sin
?πt
2Tb
?πt
?
1 − A2cos2
2Tb
?
(15)
and is plotted in Fig. 4. It is easy to show that starting with
the three other phase states this approach produces the same
g(t). The same is true for all four phase states when αn= 1.
When αn= 0 no phase transition occurs and the g(t) in (15)
can be assumed. Thus g(t) in (15) is the frequency pulse for
the CPM approximation of FQPSK-JR.
Since g(t) is partial response, sequences of αnreveal how
the CPM representation is an approximation to the XTCQM
representation. An example of such a sequence is illustrated
in Fig. 5 which shows the instantaneous frequencies of the
two representations with the αn overlaid on the plot. When
the frequency pulses do not overlap (i.e. when consecutive
00.5 11.52
t/Tb
Fig. 4.The frequency pulse for the CPM approximation of FQPSK-JR.
02468 10 1214 16
−2
−1
0
1
2
3
+1 0 −1 0 +1 0 0 +1+1 0 0 +1+1+1 0
t/Tb
Normalized Instantaneous Frequency
CPM FQPSK−JR
FQPSK−JR
Fig. 5.
mation. The ternary data symbols are as shown. The two waveforms match
exactly except for when two or more adjacent non-zero symbols occur.
The instantaneous frequency of FQPSK-JR and its CPM approxi-
non-zero αn do not occur), the CPM signal exactly equals
the XTCQM signal. On the other hand, when two or more
consecutive non-zero αndo occur, the FQPSK-JR waveform
becomes sin(πt/2Tb) (i.e. the instantaneuos frequency is
constant) while in the CPM approximation, the instantaneous
frequency is determined by the overlap of shifted versions of
the partial response frequency pulse g(t). As a consequence,
the CPM representation is only an apprximation1although the
approximation is quite good. Even though the instantaneous
frequencies are different, the accumulated phases are equal
at the symbol boundaries. Since the phase shift due to non-
zero values of αnis ±π/2, the modulation index of the CPM
approximation is h = 1/2.
Several options exist for forming the frequency pulse to
be used in the common CPM detector. These options include
using a truncated version of the SOQPSK-TG pulse (truncated
to L = 2 because that is the length of the FQPSK-JR pulse),
using the FQPSK-JR pulse, and using an L = 2 pulse that is
the average of the first two pulses. A mismatched receiver
1Note that an exact CPM representation could be envisioned as a multi-
pulse CPM consisting of two frequency pulses that are selected by the data.
Space limitations do not allow us to pursue this observation here.
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345678910 11
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Eb/N0 (dB)
bit error rate
Theory (Tx=SOQPSK−TG)
Theory (Tx=FQPSK−JR)
Common CPM det. (Tx=SOQPSK−TG)
Common CPM det. (Tx=FQPSK−JR)
Fig. 6. Bit error rates for SOQPSK-TG and FQPSK-JR for the common CPM
detector (based on the FQPSK-JR frequency pulse) along with the theoretical
curves for each modulation. SOQPSK-TG performs about 0.25 dB worse than
optimum and FQPSK-JR performs about 0.05 dB worse than the optimum for
that waveform
analysis similar to [17], [18] for SOQPSK-TG shows that
among the three options listed, the FQPSK-JR pulse pro-
vides the best theoretical performance for the CPM detector.
Since this pulse is also the closest match to the FQPSK-
JR modulation, this is the pulse used in the common CPM
detector. The common detector requires 8 real-valued length-
Tbmatched filters together with an 8-state trellis. The bit error
rate performance of this detector with the two modulations
is illustrated in Fig. 6. Plots of (4) and (6) are included for
reference. As expected, the performance of SOQPSK-TG is
slightly worse than that of FQPSK-JR since the matched filters
and phase trellis are derived from the CPM approximation
to FQPSK-JR. However, the difference is small. Note also
that the bit error rate performance of FQPSK-JR and (6)
are very close. This reinforces the observation that the CPM
approximation of FQPSK-JR is quite good.
V. CONCLUSIONS
SOQPSK-TG and FQPSK-JR share many similarities. We
have shown that both may be represented as cross-correlated
trellis-coded quadrature modulations; and both may be rep-
resented as continuous phase modulations (although the CPM
interpretation for FQPSK-JR is only an approximation). These
common views of these modulations confirm their interoper-
ability and suggest architectures for common detectors. We
have shown that common detectors based on both views
produce good performance for both modulations and that
the performance is a great improvement over the existing
common detector based on symbol-by-symbol detection. The
attractive feature of these common detectors is that they do
not require knowledge of which modulation is employed by
the transmitter. The common XTCQM detector has a slightly
better bit error rate performance than the common CPM
detector. However, the common XTCQM detector requires
16 real-valued, length-2Tbmatched filters together with a 16-
state trellis while the common CPM detector requires 8 real-
valued length-Tbmatched filters and an 8-state trellis. In this
way, these two architectures provide a performance/complexity
trade-off.
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matter experts for publication in the IEEE GLOBECOM 2005 proceedings.This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2005 proceedings. This full text paper was peer reviewed at the direction of IEEE Communications Society subject
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