IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001 765
Noncoherent Detection for Trellis-Coded MPSK
Ruey-Yi Wei and Mao-Chao Lin
Abstract—This letter shows several methods for improving
the decoding of noncoherent trellis-coded M-ary phase-shift keying. We propose two new metrics for basic decision-feedback algorithm. These metrics are derived based on the idea of reducing the effect of incorrect amplitude of the reference signal. We also propose a new decoding algorithm that uses a simple way to estimate the metric of the future sequence.
Index Terms—Coded modulation, decoding, noncoherent
detec-tion.
I. INTRODUCTION
N
ONCOHERENT detection is a detection technique that can be implemented without using carrier recovery. For uncoded -ary phase-shift keying (MPSK) signals, noncoherent detection is restricted to differential phase-shift keying (DPSK) signals, and is called differential detection. For coded MPSK signals, noncoherent detection is not necessarily restricted to differentially encoded symbols. Noncoherent trellis-coded MPSK, for which differential encoding is not used, was proposed by Raphaeli in 1996 [1]. The optimal decoding trellis based on the Viterbi algorithm for nonco-herent trellis-coded MPSK can be implemented by using an augmented trellis, for which the number of states grows exponentially with the observation length. Using the optimal decoding trellis is impractical when the observation length is not small. Hence, decoding algorithms which use the orig-inal trellis are preferred. The trivial algorithm, called basic decision-feedback algorithm (BDFA) [2], is the simplest one. However, the error performance of BDFA using the metric in [1] and [2] is poor. Thus, modified decision-feedback algorithm (MDFA) that takes future metrics into account was proposed in [2]. Although MDFA proposed in [2] uses the metric in [1], MDFA using other metrics is possible. This letter shows that MDFA using the metric in [3] can be somewhat modified to achieve the advantage of lower complexity and slightly better error performance.Different decoding metrics for noncoherent coded MPSK were proposed in [3] and [4]. For decoding using the optimal trellis, error performances of various metrics proposed in [1], [3], and [4], respectively, are similar. However, for BDFA,
Paper approved by O. Andrisano, the Editor for Modulation for Fading Chan-nels of the IEEE Communications Society. Manuscript received November 15, 1998; revised May 15, 2000 and September 15, 2000. This work was supported in part by the National Science Council of the R.O.C. under Grant NSC 87-2213-E-002-095.
R.-Y. Wei is with the Department of Electrical Engineering, National Central University, Taiwan 320, R.O.C. (e-mail: rywei@ee.ncu.edu.tw).
M.-C. Lin is with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: mclin@cc.ee.ntu.edu.tw).
Publisher Item Identifier S 0090-6778(01)04089-2.
using different metrics will result in significantly different error performances. By observing that for BDFA, incorrect amplitude of the reference signal used in the metrics proposed in [1] and [3] will deteriorate the error performance, we propose new metrics by modifying the metrics in [1] and [3].
There is a significant performance gap between BDFA and coherent decoding if the observation length is small. MDFA can narrow the gap for some metrics. But there is still much room for improvement. The drawback of MDFA is that the estima-tion of future metrics is not accurate enough. In [2], a compli-cated algorithm called the estimated-future decision-feedback algorithm (EFDFA) was proposed. To estimate future metrics more accurately, in addition to the original forward decoding process, EFDFA performs a backward decoding in advance to recover the future symbols which will be used in the forward de-coding. This letter proposes a new algorithm, called improved decision-feedback algorithm (IDFA). Instead of using the av-erage of past metrics in MDFA or using backward decoding as in EFDFA, the proposed IDFA obtains the estimate for the future sequence by primitively detecting future signals like detecting uncoded signals with a signal reference obtained by past signals. In this way, the increased complexity of detecting future signals is small. In case the observation length is small, error perfor-mance of the proposed IDFA is better than that of MDFA or BDFA and is close to that of decoding using the optimal trellis.
II. BDFAANDMDFA
Consider a stream of trellis-coded -ary PSK signals, which are transmitted over an additive white Gaussian noise channel. For each time unit, all the symbols of a certain trellis branch are transmitted. For , the th signal of a trellis branch which is transmitted at time unit
is , where is the signal
power and is the modulation phase corresponding to the coded symbol. The corresponding signal at the receiving
side is , where is
an arbitrary, constant, uniformly distributed phase shift and is a zero-mean white complex Gaussian noise. In [1], the decoding metric for a possible code sequence
is
(1)
where is the number of branches which are covered by one observation and is the complex conjugate of . For
convenience, we define and
. Thus, .
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766 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001
In [3], a different metric was proposed as
(2)
Recently, a new metric was proposed in [4] as
(3)
for which each time unit corresponds to only one symbol. We can modify (3) to so that each time unit corresponds to to one branch ( symbols), where
(4)
We now briefly describe BDFA and MDFA, which were proposed in [2]. Either BDFA or MDFA keeps track of the associated survivor for each state. Let be the accumu-lated metric for the survivor path of state at time unit . Let denote the number of branches which connect the states at time unit and state at time unit . Let
represent the th branch that connect
state at time unit and state at time unit . Note
that is the same as for if and
represent two parallel branches connecting (or ) and . Let represent the path which is the union of and the survivor path of state at time unit . Let be the branch metric for . For BDFA, the estimated metric for the path is computed as
(5) The index that maximizes , denoted as , is chosen to update the survivor of state at time unit and
.
Since the symbols of the selected path will be used later as decision feedback, when is detected, all the later metrics which are related to should be considered in advance. Let
be the estimated future metric at time
for , where
(6)
where is the estimated branch metric at time . However,
when computing for cannot be
obtained. For MDFA, with is
estimated by , which is the average of the past values, i.e.,
(7)
For MDFA, the index that maximizes , denoted as , is chosen to update the survivor of state at time unit and . In [2], either BDFA or MDFA uses
as the branch metric . Clearly, we can also use or as the metric .
MDFA using can be further improved and simplified as follows. The estimated future branch metrics in (6) is
(8)
For contains
no information of and hence should be omitted. Then, we have
(9)
Replacing by the estimate , we then
have
(10)
By setting and , (10) becomes
(11) where
Consider the rate- 16-state trellis-coded quadrature phase-shift keying (built over the modulo 4 group) used in [1]–[4], for which the generator is (in base four). Simulation results for BDFA with and are given in Figs. 1 and 2, respectively. For BDFA, using metric yields error performance better than using and , and using metric yields the worst error performance. Simulation results for MDFA with are given in Fig. 3. For MDFA using , the error performance of the modified version using (11) is slightly better than that of original version using (8). Note that the error performance of MDFA using is worse than that of BDFA using . Hence, MDFA does
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001 767
Fig. 1. Error performance of BDFA withL = 4 (using a 16-state decoding trellis) and decoding using the optimal trellis withL = 4 (using a 1024-state decoding trellis): (a) BDFA, ; (b) BDFA, ; (c) BDFA, ; (d) BDFA, ; (e) BDFA, ; (f) BDFA, ; (g) optimal trellis, ; (h) optimal trellis, ; (i) optimal trellis, ; (j) coherent detection.
Fig. 2. Error performance of BDFA withL = 8 (using a 16-state decoding trellis): (a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g) coherent detection.
not always provide error performance better than BDFA even though extra complexity is needed in MDFA.
III. NEWMETRICS ANDIDFA
In Fig. 1, we find that although BDFA with using and , respectively, will yield significantly dif-ferent error performances, the decoding with the optimal trellis using these metrics will yield similar error performances. This
Fig. 3. Error performance of MDFA and IDFA withL = 4 (using a 16-state decoding trellis): (a) MDFA, ; (b) MDFA, , using (11); (c) MDFA, , using (8); (d) MDFA, ; (e) IDFA, ; (f) IDFA, ; (g) IDFA, ; (h) coherent detection.
phenomenon can be interpreted as that these metrics are derived for the decoding with the optimal trellis and not for BDFA. In the following, we show new decoding metrics which are suit-able for BDFA.
From (2) and (5), we note that if BDFA using has an ideal signal reference , i.e., , then its error performance is exactly the same as that of coherent decoding. However, is not accurate for BDFA, especially when there are decision-feedback errors. If the observation length is not small, and there are only a few decision-feedback errors, the phase of is likely to be close to , while the amplitude of is likely to be lessened. Hence, for BDFA, this letter proposes to modify the metric into by reducing the effect of the incorrect amplitude of , where
(12)
The metric is similar to the metric proposed for QAM constellation in [5].
With a similar idea, we may modify by totally ignoring the amplitude of . Let
(13)
Although this modified form has satisfactory error performance for BDFA, it is impractical since the signal power is unknown at the noncoherent receiver end. Note that
. By removing the term , we have a new metric
(14)
768 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 5, MAY 2001
for BDFA. In , removing the term , to a certain de-gree, reduces the effect of incorrect amplitude of . Figs. 1
and 2 show that and outperform and for
BDFA. The error performance of is inferior to that of
ei-ther or . The advantage of over and
is that for there is no need for either the operation of taking square root or the operation of division by the amplitude of . Since and are designed only for BDFA, the error performances of and for other decoding algorithms are not satisfactory and are not shown in this letter. By checking (4), we see that the amplitude of does not affect the value of if has the same phase as . Thus, there is no advan-tage in modifying for BDFA.
In Figs. 1 and 2, the curves of (impractical case), , and are close. When the observation length is long, the per-formance gap between BDFA and coherent decoding is small. But for short observation length, there is much room to improve for BDFA. Fig. 3 already shows that MDFA is not a good alter-native for BDFA. In the following, we propose a new decoding algorithm, which is suitable for short obsevation length.
The drawback of MDFA is that the estimate of
for by is not
accurate enough. Since contains decoded symbols, decision errors would affect its accuracy. Thus, to lower error probability, an algorithm for reliably finding the future code
sequence is necessary. The
EFDFA proposed in [2] satisfies this demand by performing backward decoding. However, EFDFA is quite complicated. As an alternative, we propose IDFA which provides a simple way to detect the future sequence and provides estimate for with better than the estimate by . For IDFA, detecting
is treated like detecting uncoded MPSK sequences according to the estimate of reference given by
(15) In other words, for , IDFA decides
, as the one among possible , which maximizes
(16)
The complexity of IDFA is slightly higher than that of MDFA, and is significantly lower than that of EFDFA. IDFA can use each of the metric and . Simulation results shown in Fig. 3 indicate that for all the three metrics with , the error performance of IDFA is better than that of BDFA or MDFA and is close to that of decoding using optimal trellis. In the simulation, the results for IDFA using is obtained by using the version shown in (9).
Simulation for the rate- 64-state trellis-coded 8PSK (built over the modulo 8 group) with generator (in base eight) [1] has been made. The results are not shown in this letter. The advantage and disadvantage for various metrics and de-coding algorithms for the 64-state example is similar to that for the 16-state example.
IV. CONCLUSION
We have shown mothods to enhance the noncoherent decoding of trellis-coded MPSK. In the aspect of decoding metric, we propose two new metrics which can reduce the effect of incorrect amplitude of the reference signal. In the aspect of decoding algorithm, we propose a new docoding algorithm IDFA. When the observation length is small, the error performance of IDFA is better than that of BDFA and MDFA and is close to that of decoding using the optimal trellis.
REFERENCES
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