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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.

detection of SOQPSKand

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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468 10 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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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

0 0.51 1.52

t/Tb

Fig. 4.The frequency pulse for the CPM approximation of FQPSK-JR.

02468 1012 1416

−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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10

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10

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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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(Also available online:

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