P. Hoeher and J. Lodge, “Turbo DPSK: Iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. 47, pp. 837–843, June 1999.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 6, JUNE 1999
837
“Turbo DPSK”: Iterative Differential PSK
Demodulation and Channel Decoding
Peter Hoeher, Senior Member, IEEE, and John Lodge, Senior Member, IEEE
Abstract— Multiple-symbol differential detection is known to
fill the gap between conventional differential detection of
PSK ( -DPSK) and coherent detection of
-PSK with differential encoding ( -DEPSK). Emphasis has been so far on
soft-input/hard-output detectors applied in uncoded systems. In
this paper, we investigate a receiver structure suitable for coded
DPSK signals on static and time-varying channels. The kernel
is an a posteriori probability (APP) DPSK demodulator. This
demodulator accepts a priori information and produces reliability
outputs. Due to the availability of reliability outputs, an outer
soft-decision channel decoder can be applied. Due to the acceptance of a priori information, if the outer channel decoder also
outputs reliability information, iterative (“turbo”) processing can
be done. The proposed “APP DPSK demodulator” uses linear
prediction and per-survivor processing to estimate the channel
response. The overall transmission scheme represents a type of
serial “turbo code,” with a differential encoder concatenated with
a convolutional code, separated by interleaving. The investigated
system has the potential to improve the performance of coherent
PSK without differential encoding and perfect channel estimation
for fading cases! Only a small number of iterations are required.
The receiver under investigation can be applied to several existing
standards without changing the transmission format. Results are
presented for uncoded and convolutionally coded 4-DPSK modulation transmitted over the Gaussian channel and the Rayleigh
flat-fading channel, respectively.
Index Terms— Communication systems, convolutional codes,
differential phase shift keying, MAP estimation, modulation/
demodulation.
I. INTRODUCTION
M
-ARY phase shift keying ( PSK) is a popular digital modulation technique. The information signal is
encoded in the phase of the carrier signal. To exploit the
ultimate performance, absolutely encoded coherent detection
is generally thought to be necessary. The optimum receiver
as well as suboptimal receivers are discussed in [1]. The
disadvantages of these techniques include their complexity,
acquisition time, sensitivity to tracking errors (especially when
the carrier phase is rapidly changing), and problems associated
Paper approved by T. Aulin, the Editor for Coding and Communication
Theory of the IEEE Communications Society. Manuscript received November
24, 1997; revised September 17, 1998; November 25, 1998. This paper was
presented in part at the IEEE International Symposium on Information Theory,
Cambridge, MA, USA, August 1998.
P. Hoeher was with the Communications Research Centre, Ottawa, ON
K2H 8S2, Canada, on sabbatical leave from the German Aerospace Center,
Oberpfaffenhofen, Germany. He is now with the Information and Coding
Theory Laboratory, Faculty of Engineering, University of Kiel, D-24143 Kiel,
Germany (e-mail: ph@techfak.uni-kiel.de).
J. Lodge is with the Communications Research Centre (CRC), Ottawa, ON
K2H 8S2, Canada (e-mail: John.Lodge@doc.crc.ca).
Publisher Item Identifier S 0090-6778(99)05008-4.
with an -ary phase ambiguity (such as the possibility of false
locks and error propagation). Some of these problems can be
overcome by inserting training symbols multiplexed into the
data stream, however, the power- and bandwidth efficiencies
are somewhat reduced due to the overhead.
Another common method to overcome the latter problem is
to differentially encode the information before transmission,
i.e., the information is carried in the difference between
adjacent received phases. Coherent detection can still be
applied, resulting in differentially encoded coherently detected
PSK (DEPSK). The bit error rate (BER) of DEPSK is about
twice the BER of PSK.
A robust yet simpler alternative to PSK and DEPSK is
differentially encoded differentially detected PSK (DPSK). A
DPSK receiver recovers the information by subtracting the
phase of the previous symbol sample from the phase of
the current sample. Hence, the observation interval is two
symbols. Compared to DEPSK, an additional degradation
and
occurs, which actually depends on the signal alphabet
on the transmission channel.
Since the early 1970’s, considerable research has been done
to restore the penalty of DPSK compared to DEPSK [1]–[26].
Basically, the observation interval is extended beyond two
symbol intervals. Today, these algorithms are generally known
as multiple-symbol detectors or block demodulators. Specifically, the maximum-likelihood (multiple-symbol) differential
detector derived by Divsalar and Simon [8], [16] and Makrakis
and Feher [9] theoretically fills the gap between DPSK and
DEPSK on static channels. Since the complexity of the ML detector grows exponentially with the sequence length, reduced
complexity block detectors with a similar performance have
also been derived in a structured manner. Further, block demodulators designed for fading channels have been proposed,
since block demodulators specified for static channels usually
perform poorly on time-varying channels. About half of the
potential gain can be resolved in practice. Note that at one
sample per symbol, PSK can be considered to be a discrete
form of continuous phase modulation [10], [27].
In the present paper, we consider multiple-symbol differential demodulation of uncoded and coded systems employing
differential encoding. The receiver is suitable for stationary
and time-varying channels. First, we explore the fact that the
differential encoder can be represented by a trellis diagram.
Hence, standard decoding algorithms, such as the Viterbi
algorithm or the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm
[28], can be applied to this demodulation problem when
designing the transition metrics properly. As opposed to block
0090–6778/99$10.00  1999 IEEE
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 6, JUNE 1999
Fig. 1. Block diagram of the uncoded
-DPSK transmission system with APP DPSK demodulation.
demodulation techniques which make use of the memory of a
small block only, trellis-based approaches make use of all past
and future samples. We propose an “a posteriori probability
(APP) DPSK demodulator,” derived from the BCJR algorithm
but operating in the log-domain (“Log-MAP” [29]) without
loosing optimality. The APP DPSK demodulator accepts soft
channel values plus a priori inputs and outputs a posteriori
probabilities. The soft outputs are useful when the demodulator
is the inner stage of a concatenated system. The proposed APP
DPSK demodulator applies linear prediction [30] in combination with per-survivor processing [31] to estimate the channel
response [10]. Other related trellis-based detection approaches
have been studied in [5], [9], [15], [13], [19], [20], and [23].
However, in those works, the Viterbi algorithm (delivering
hard outputs) was applied. APP DPSK demodulation has only
recently been investigated [21], [26].
Second, we apply iterative processing [32], [33], also called
the “turbo principle” [34]: Reliability information delivered
from the outer decoder is fed back as a priori information
to the inner demodulator. The overall scheme is related to
“turbo equalization” [35] and “turbo demodulation” of coherent systems using training symbols [27]. The most important
observations are that the investigated coded DPSK system
appears to perform asymptotically as well as coded coherent
PSK without differential encoding in the AWGN channel,
while it can marginally outperform the coherent PSK case in
Rayleigh fading. These levels of performance can be achieved
in a few iterations. The possible improvements for fading cases
are due to time diversity induced by the differential encoder.
Hence, in contrast to noniterative schemes, the potential gain
is larger and can essentially be achieved with practical effort.
We give credit to Peleg and Shamai [26], who recently proposed the iterative demodulation/decoding scheme for coded
DPSK signals described above. However, their APP DPSK
demodulator was designed for the Gaussian channel, whereas
our approach is suitable for time-varying channels as well, and
hence applicable to mobile radio systems. We also give credit
to Divsalar and Pollara [36] and Narayanan and Stüber [37],
who derived union bounds for coherent turbo-DPSK systems.
Related work has been published in [38] and [39].
As practical examples, we consider differentially encoded
quaternary PSK (4-DPSK) transmitted over a Gaussian channel and a Rayleigh flat-fading channel, respectively. Simulation results are presented for the uncoded case as well as for
a rate-1/2 convolutionally coded system. Focus is on single
carrier modulation. A training sequence is not required.
Fig. 2. Trellis diagram of 2-DPSK. In this example, the number of information symbols per block is channel or a Rayleigh fading channel. Assume Nyquist signaling (no oversampling) and consider the discrete-time channel
model in Fig. 1.
are obtained
The -ary complex-valued input symbols
by Gray-encoding independent, identically distributed bits and
-PSK constellation.
mapping them onto an
denotes the time index. The differential encoder computes the
-ary symbols
to be transmitted:
(1)
is called the reference symbol. The corresponding received
signal at the matched filter output can be written as
(2)
II. UNCODED SYSTEM AND APP DEMODULATION
depend on the channel model. For
The channel coefficients
the Gaussian channel
, where is a constant or slowly
time varying uniformly distributed phase produced by the
channel. For Rayleigh flat-fading, the quadrature components
are independent, filtered Gaussian random variables with
of
zero mean.
is a sample of a zero-mean complex-valued
white Gaussian noise process with two-sided noise power
, i.e., the noise variance is
spectral density
Assume, for conceptual reasons, that the information seThe reference
quence is divided into blocks of length
initializes the trellis. The first block may be tailed
symbol
, which may simultaneously
by another reference symbol
initialize the second information block, see Fig. 2, etc. Hence,
the differential encoder can be interpreted as a tailed, rate
convolutional code.1 The trellis has
states and
branches per state, and is recursive but not systematic.
(Recursive codes are also used in “turbo codes” [32]). In the
following, we will focus without loss of generality on the first
block. We assume that the length of the received sequence is
, determined by one reference symbol per information
symbols, having in mind that tailing may be applied by the
initial reference symbol of the second trellis segment. The
Let us begin with the uncoded case. Consider the transmission of differentially encoded PSK signals over a Gaussian
1 The differential encoder is similar to a one-symbol parity
code, but is recursive and does not provide a 3-dB asymptotic coding gain.
HOEHER AND LODGE: ITERATIVE DIFFERENTIAL PSK DEMODULATION AND CHANNEL DECODING
reference symbols remove the phase ambiguity and are related
to training symbols in pilot-symbol-assisted coherent schemes.
The optimal decoder in terms of minimizing the bit error
rate, i.e., the symbol-by-symbol MAP decoder,2 can be derived
straightforwardly given the tools known in the area of channel
decoding [28]. We will restrict ourselves here to give an
intuitive explanation of computing the metrics. Assume for
are known to
the moment that the channel coefficients
the receiver. Since the probability density function of the th
conditioned on
and
is
matched filter output
(3)
a proper metric increment in the log-domain given hypothesis
is
(4)
hypotheses exist. The corresponding trellis consists of
states with
branches per state. This coherent receiver serves
as a performance benchmark in the simulations.
are unknown, howIn practice, the channel coefficients
ever. We propose to use linear prediction [30] to obtain channel
estimates. The according metric increment is
839
and per-survivor processing, see [40]). Differential decoding
and
and demodulation is done in the same trellis. For
, the numerator in (7) simplifies to the conventional
decision rule.
Note that our metric, (7), is applicable to all frequency-flat
channels, including AWGN, Rician, and Rayleigh channels.
Denote the autocorrelation coefficients of the channel process
Then, the variance of the prediction error
by
can be written as
[30], [10], where
and
In the
and for a static channel,
For the
limit
example of a Rayleigh fading process with a Doppler power
[41], we have
spectrum
, where
is the Bessel
is the normalized fade rate. For the
function and
example of a static channel, we have
The optimal predictor coefficients
are
obtained by solving the Wiener–Hopf equation
(8)
If we assume (for purposes of illustration) the binary case
and take a priori information into account, denoted
as , we obtain
(5)
is the channel estimate,
is a temporary channel
where
estimate,
is the predictor order,
are the predictor coeffiand
is the variance of the prediction
cients,
error. Note the relationship between the prediction order
and the block length of block demodulators. The coherence
time of the channel should be of order , where typically
The temporary channel estimate
(2) by neglecting the noise term
if
if
(9)
can be obtained from
where
is a log-likelihood ratio
Hence,
and
When deleting common
(6)
denotes the complex conjugate. This operation
where
corresponds to remodulation. When substituting (6) in (5) we
get
terms, we finally obtain
(7)
if
is
where, the transmitted symbol sequence
replaced according to the principle of per-survivor processThe corresponding
ing [31] by hypotheses
states with
branches per state. We
trellis consists of
expanded the number of states, since the APP algorithm has
no concept of sequences. (For a generalization of remodulation
2 We
prefer the notion of an APP algorithm in the following, since no
maxima are taken before the final decision is made.
(10)
if
follows immediately by adding
The quaternary case
the a priori information of the second bit the same way. A
priori information is available after the first sweep.
840
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 6, JUNE 1999
Fig. 3. Block diagram of transmission system with rate-1/2 outer convolutional encoding and inner differential encoding (4-DPSK). The receiver
employs APP DPSK demodulation, APP decoding/filtering, and iterative
processing.
The forward and backward recursions can be performed in
the usual way [28], [29]. Initialization of the forward recursion
is straightforward due to the initial phase reference symbol.
The initial states of backward recursion should be initialized
according to the phase reference symbol of the next block or
should be initialized by the same value when no tailing is
applied. For long block lengths, the difference is negligible.
It should be noted that the backward recursion does not
contribute to the result, when the a priori information is zero
and no tail symbol is sent. This can be seen immediately
from the trellis structure, compare Fig. 2. Hence, the backward
recursion may be dropped in the first sweep. As a consequence,
demodulation has zero delay in this special case. Otherwise,
the delay is determined by the block length.
III. CODED SYSTEM AND ITERATIVE PROCESSING
Now consider the coded system illustrated in Fig. 3. The
transmitter might be interpreted as a serially concatenated
coding system. The outer code is a convolutional code, the
inner code is the differential encoder. Both codes are separated
by a bit interleaver, which can be short in the case of
Gaussian noise. Correspondingly, the receiver consists of an
inner decoder, a deinterleaver, and an outer decoder. The inner
decoder is the APP DPSK demodulator proposed in Section II.
When the outer decoder is an APP decoder/filter [33], [27],
iterative processing can be done. In our terminology, the
APP decoder computes probabilities, likelihood ratios, or log, whereas the
likelihood ratios of the information bits
APP filter computes probabilities, likelihood ratios, or log[33]. In Fig. 3 and
likelihood ratios of the coded bits
in the following, we assume that log-likelihood ratios are
For 2-DPSK, the
are real-valued,
computed, denoted as
are complex valued. The sign of each
for 4-DPSK the
quadrature component is the maximum a posteriori decision,
the magnitude indicates the reliability. The a priori inputs
of the APP DPSK demodulator are obtained by subtracting
from the APP filter outputs
the APP decoder/filter inputs
(
is called extrinsic information in [32], see also
[33]). The a priori inputs
of the APP DPSK demodulator
must be subtracted from the APP DPSK demodulator outputs
as well.
Fig. 4. Differentially encoded 4PSK: Performance for Gaussian noise assuming perfect channel estimation. The two top curves correspond to the
uncoded system, the remaining curves are for a rate-1/2 convolutional coded
system with 64 states.
IV. SIMULATION RESULTS
As practical examples, we investigated differentially
encoded 4-PSK signals transmitted over a Gaussian and a
Rayleigh flat-fading channel, respectively. The uncoded case
treated in Section II as well as the coded system proposed
in Section III were studied. In the coded case, the widely
convolutional code with 64 states and
applied
generators (133, 171) was used. Sufficient interleaving (200
200 block interleaver, interleaving over 20 blocks) was
applied for both channels to avoid any spurious effects with
respect to correlations. The APP DPSK demodulator and the
APP decoder/filter were optimally implemented in the logdomain [29] as shown in Section II. The block length was
info bits. The outer code was zero-tailed with six
bits, the inner code was tailed by one DPSK symbol. The case
of not tailing the inner code has also been investigated, the
performance difference was negligible.
In order to optimize the predictor coefficients, a perfect
knowledge of the signal-to-noise ratio and the fade rate was assumed. Related publications have revealed that the actual performance is insensitive with respect to model errors. Rayleigh
fading process with a Doppler power spectrum
[41] was assumed. Both fading
and
rates under investigation,
, were small enough to prevent decorrelation [42], [43].
To explore the best possible performance of the given scenario,
we first assumed perfect channel estimation, i.e.,
Linear prediction was studied afterwards in order to evaluate
the practical performance.
Let us begin the discussion with the Gaussian channel and
assume perfect channel information at the demodulator. The
results are plotted in Fig. 4. The two top curves correspond
to the uncoded system, the remaining curves are for the
coded system. The top curve within each set is obtained
for the conventional DPSK receiver employing differential
demodulation over a two-symbol interval. In the coded case,
no hard decisions are performed in the demodulator. The next
HOEHER AND LODGE: ITERATIVE DIFFERENTIAL PSK DEMODULATION AND CHANNEL DECODING
Fig. 5. Differentially encoded 4PSK: Performance for Rayleigh fading assuming perfect channel estimation. The four top curves correspond to the
uncoded system, the remaining curves are for a rate-1/2 convolutional coded
system with 64 states.
curve within each set is for coherent demodulation employing
the APP DPSK demodulator, where (4) is applied (augmented
by a priori information when doing iterative processing). The
BER performance corresponds to DEPSK, but soft decisions
are delivered. At 10 , the gain is about 1.25 dB in the
coded case. State-of-the-art block demodulators could only
recover about half of the gain, assuming they could deliver
soft outputs. (Block demodulators perform badly at low signalto-noise ratios, where coded systems would operate.) The
next five curves are for “turbo processing” with one to five
iterations.3 The bottom curve finally represents coherent PSK
which appears to provide a lower bound. Note that coherent
PSK is approached by “turbo processing” for bit error rates
of 10 and less with just two iterations. At 10 , the gain
with respect to conventional DPSK demodulation is about 2.9
dB in the coded case. The potential gain promised by iterative
processing significantly exceeds the potential gain promised
by block demodulation.
Now, consider the Rayleigh fading channel and assume
again perfect channel estimation. The results are plotted in
Fig. 5. The four top curves correspond to the uncoded system,
the remaining curves are for the coded system. At a normalized
, the potential gain of
fading rate of
DEPSK versus DPSK is about 1.3 dB/2.2 dB in the coded case,
With turbo processing, however, the potential
both at 10
gain is about 4.65 dB with respect to DPSK for a fading rate
at 10 , which is about 0.2 dB better
of
than coherent PSK with perfect channel state information.
Coherent PSK with perfect channel state information does
not provide a lower bound. The memory introduced by the
differential encoder provides additional time diversity, which
is properly taken into account here along with the interleaving,
coding, and iterative decoding. As a consequence, the ultimate
performance is not obtained with absolutely encoded signals,
as is often suggested. The diversity gain is achievable even
without proper tailing the inner trellis.
3 In this paper, the first iteration is defined as the first complete demodulation
and decoding cycle with feedback.
841
Fig. 6. Differentially encoded 4PSK: Performance for Rayleigh fading with
fading rate A linear predictor with 64 states is employed.
The two top curves correspond to the uncoded system, the remaining curves
are for the rate-1/2 convolutional coded system with 64 states.
Fig. 7. Differentially encoded 4PSK: Performance for Rayleigh fading with
A linear predictor with 64 states is employed.
fading rate The two top curves correspond to the uncoded system, the remaining curves
are for the rate-1/2 convolutional coded system with 64 states.
Finally, consider Rayleigh fading and nonperfect channel
estimation. The proposed APP DPSK demodulator using metwas chosen,
ric (10) is applied. A predictor order of
states. The results are plotted in
which corresponds to
and in Fig. 7 for a
Fig. 6 for a fading rate of
, respectively. For slow fading,
fading rate of
the results are very promising: At 10 the gain with respect
to two-symbol differential demodulation is about 2.6 dB for
just two iterations, see Fig. 6. Hence, the advantage over
block demodulation is almost as much as the potential gain
promised by perfect channel estimation. More importantly,
most of this gain can be achieved with reasonable effort. In
the uncoded case, the APP DPSK demodulator outperforms
the conventional demodulator at signal-to-noise ratios where
uncoded systems usually operate.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 6, JUNE 1999
With fast fading, the results are less encouraging when
applying linear prediction. At 10 , the gain with respect to
conventional demodulation is about 0.5 dB with two iterations,
see Fig. 7. Possible improvements include APP demodulation
with oversampling [27], [40], time-continuous signal processing or simplifications thereof [43], and a replacement of linear
prediction by filtering or smoothing techniques.
V. CONCLUSIONS
Peleg and Shamai have recently proposed an iterative demodulation/decoding scheme for coded DPSK signals [26].
The kernel is an a posteriori probability demodulator [21],
[26]. This APP DPSK demodulator accepts a priori information and produces (log-) likelihood values. Hence, reliability
values can be passed to an outer channel decoder and iterative
processing can be done. The APP DPSK demodulators in [21]
and [26] are designed for the Gaussian channel.
In this paper, we proposed an APP DPSK demodulator
suitable for time-varying fading channels and static channels,
and we improved the performance by doing iterative processing. Linear prediction together with per-survivor processing
is applied to estimate the channel response. The APP DPSK
demodulator is revealed to outperform state-of-the-art block
demodulators, even in an uncoded system. With turbo processing, on fading channels the performance asymptotically
exceeds that of coherent PSK without differential precoding
due to a time-diversity gain of the differential encoder. Hence,
the ultimate performance for the given convolutional code
is not obtained with absolutely encoded PSK as generally
thought. A few iterations are sufficient.
The proposed receiver is compatible with existing standards
employing differential encoding, such as the North American
IS-54/136 and the Japanese PDC digital cellular systems em-DQPSK, and the European EUREKA-147 Digital
ploying
Audio Broadcasting system using DQPSK in conjunction with
OFDM. A generalization to include equalization appears to
be feasible. An extension to multicarrier modulation (in order
to exploit the frequency-correlation in addition to the timecorrelation) seems to be tricky, but possible. The receiver
studied here could also been applied to continuous phase
modulation [10]. Further improvements are expected when
replacing the convolutional code by a turbo code or by
generalized concatenated codes.
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843
Peter Hoeher (M’90–SM’97) was born in Cologne,
Germany, in 1962. He received the Dipl.-Eng. and
Dr.-Eng. degrees in electrical engineering from the
Technical University of Aachen, Germany, and the
University of Kaiserslautern, Germany, in 1986 and
1990, respectively.
From October 1986 to September 1998, he was
with the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. From December 1991 to
November 1992, he was on leave at AT&T Bell
Laboratories, Murray Hill, NJ. In October 1998, he
joined the University of Kiel, Germany, where he is currently a Professor
in Electrical Engineering. His research interests are in the general area of
communications theory, including digital modulation techniques, soft-output
decoding, equalization and channel estimation, with applications in mobile
radio.
Dr. Hoeher received the Hugo-Denkmeier-Award ’90.
John Lodge (S’80–M’81–SM’90) received the
B.Sc. and Ph.D. degrees in electrical engineering
from Queen’s University, Kingston, ON, Canada,
in 1977 and 1981, respectively.
From September 1981 to April 1984, he was
with the Advanced Systems Division of Miller
Communications Systems Ltd., Kanata, ON, where
he was involved in the analysis and design of
HF, spread-spectrum, and satellite communications
systems. Since May 1984, he has been at the
Communications Research Centre (CRC) in Ottawa,
Canada. He is currently the Research Manager of the Communications Signal
Processing Group in the Satellite Communications Branch. His research
interests pertain to the application of digital signal processing techniques to
wireless communications problems, including data detection techniques in the
presence of fading and the use of iterative (“turbo”) processing for decoding
and detection. He is an Adjunct Professor of the University of Ottawa and
the University of Manitoba.