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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 4, APRIL 2003

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On the Error Probability of Decision-Feedback Differential Detection Robert Schober, Member, IEEE, Yao Ma, Member, IEEE, and Subbarayan Pasupathy, Fellow, IEEE

Abstract—In this letter, we give a tight approximation for the bit-error rate (BER) of decision-feedback differential detection (DF-DD). The influence of error propagation is modeled by a Markov chain. A simple state reduction method is proposed to limit computational complexity. Our results show that error propagation strongly depends on the chosen feedback filter. In particular, the popular assumption that error propagation increases BER by a factor of two is not always justified. Index Terms—Bit-error rate (BER) analysis, decision-feedback differential detection (DF-DD), Markov chains.

Markov model might require a large number of states, we propose an efficient method for state reduction. We also show that error propagation strongly depends on the adopted DF-DD feedback filter coefficients. In some cases, there is no error propagation at high SNRs. This letter is organized as follows. In Section II, DF-DD is briefly reviewed. An approximation for the BER for DF-DD is given in Section III. A comparison of the approximation with simulation results is carried out in Section IV, while some conclusions are drawn in Section V.

I. INTRODUCTION

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N SITUATIONS where carrier phase recovery is difficult, differential -ary phase-shift keying ( PSK) with noncoherent detection is often applied. In particular, various decision-feedback differential detection (DF-DD) schemes have been proposed in the literature [1]–[7]. DF-DD is very attractive for implementation, since it achieves a higher power efficiency than conventional differential detection (DD) [8], while the required computational complexity is lower than for multiple-symbol detection [9]. Despite the popularity of DF-DD, a rigorous error probability analysis has not been presented yet. In most contributions, the bit-error rate (BER) of DF-DD is analyzed under the assumption of error-free feedback, cf. e.g., [1], [2], [5], [7]. The effect of error propagation was studied in [3] and [10], and it was found that at high signal-to-noise ratios (SNRs), error propagation increases BER by a factor of two. In [11], it is claimed that the exact BER of DF-DD can be obtained by using a finite-state Markov chain to model error propagation, and a formula for BER was given for the most simple case where only one feedback symbol is used to aid the decision process.1 In this letter, we calculate a tight approximation for the true BER of DF-DD with an arbitrary number of feedback symbols using a Markov chain to model error propagation. We show that this technique, which has already been applied successfully to evaluate the error rate of other DF schemes, e.g., [12], does not allow us to calculate the exact BER of DF-DD. Since the Paper approved by R. A. Kennedy, the Editor for Data Communications Modulation and Signal Design of the IEEE Communications Society. Manuscript received September 11, 2001; revised August 19, 2002. R. Schober is with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]). Y. Ma is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA (e-mail: [email protected]). S. Pasupathy is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2003.810857

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1In [11] minimum shift keying (MSK) was considered. However, the adopted differential phase-shift DF-DD scheme is similar to the schemes used for keying (DPSK) modulation.

II. DF-DD In this letter, we assume DPSK transmission over an additive white Gaussian noise (AWGN) channel with unknown but constant phase . Therefore, in complex baseband representation, the discrete-time received signal can be modeled as [8] (1) , , and denote the imaginary where PSK symbols, and AWGN, reunit, the transmitted is obtained from the DPSK symbol spectively. by differential , where is an independent and encoding bits are Gray identically distributed (i.i.d.) sequence. mapped to each of the elements of , respectively. The AWGN has variance , where and refer to the received energy per bit and the single-sided noise power spectral density of the underlying continuous-time passband process, respectively. is given by For DF-DD, the decision variable (2) with phase reference (3) refer to the observation window size, where , , and the feedback filter coefficients, and the estimates of the transand , mitted PSK symbols, respectively. For DF-DD is identical to conventional DD. However, in the folsince the BER lowing, we exclusively consider the case of conventional DD has been studied extensively in the literature [8]. For our numerical results, we consider two special cases for , . • DF-DD proposed by Leib and Pasupathy [1] (LP-DF-DD) For LP-DF-DD, which was proposed first in [1] and , , revisited in [2], [3], and [10],

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is valid. This choice maximizes the SNR in the phase . Note that for the AWGN channel reference for prediction-based DF-DD, which was reported in [7], , results. Since for DPSK the magnitude of the decision variable is not important, prediction-based DF-DD yields the same performance as LP-DF-DD. • DF-DD proposed by Bin and Ho [6] (BH-DF-DD) BH-DF-DD was designed to achieve robustness against frequency offset. The feedback filter coefficients are given by , . It is worth mentioning that recursive DF-DD [4], [5] can also be analyzed with the technique presented here. In this case, , , is valid, where , , is a forgetting factor. In order to obtain a Markov chain with a finite number of states to model the error propagation, the feedback filter has to . The BER results obtained be truncated to a finite length for the truncated filter will approach the results for the infinteis chosen large enough to make length filter, provided that negligible. III. BER CALCULATION Using (1)–(3) and can be shown that

A. Approximation for BER , correLet us define Markov states , possible values of . Now, for sponding to the , the random process will take on state with the limiting state occupancy probability . Therefore, an approximation for the BER of DF-DD for DPSK is given by

(8) ,

, it

can be expressed as

with the AWGN noise process tions

is clear that both and , , are necessary to describe the memory inherent to the decision , , do not process. Unfortunately, the belong to a finite set. Consequently, the memory of the decision process cannot be described with a finite-state Markov model, and the exact BER cannot be calculated using this approach. Nevertheless, a tight approximation for the true BER can be obtained if the memory introduced by the noise samples , , is neglected and only the influence of is accounted for.

(4) and the defini-

where we have again exploited the fact that , , is true because of the symmetry of the signal constellation. [ denotes transposition The vector of a vector] of limiting state occupancy probabilities can be obtained by solving the eigenvalue problem (9)

(5) (6) Using (4) and, e.g., the generic method of Pawula et al. [13], it is straightforward to calculate the probability , i.e., the probability that is transmitted and is detected conditioned on . Because the vector of the symmetry of the adopted signal constellation, is identical for all . Therefore, for error-free feedback, i.e., genie-aided DF-DD, BER is given by

(7) and refer to the all-ones row vector elements, and the number of bits in which and differ, respectively. From (4), it can be observed that the decision variable depends on both and , . Since depends on and , , it

where with

is the state transition matrix with where in row and column . has only nonzero element entries which are given by , such that is an allowed with state transition. If (9) has to be solved by a computer, numerical problems may arise. In this case, it may be more convenient , , to calculate from the recursion . In general, this recursion converges very is smaller than fast and can be stopped if denotes the -norm of a some predefined small value ( vector). and small , a closed-form expression for For small can be given. For binary DPSK (BDPSK) we discuss two special cases. • Using (8) and (9) it is easy to verify that for (10) results, where for notational simplicity, the short-hand is used. The notation same formula was derived in [11], however, there it was claimed to give the exact BER. For LP-DF-DD and


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(i.e., erroneous feedback), the decision is variable is obtained from (4). Since the transmitted symbol results. multiplied by zero-mean noise, Therefore, is true. At high holds, and therefore, SNR is obtained. and On the other hand, for BH-DF-DD with for high SNR results. Therefore, is obtained. Note, however, in this case, for BH-DF-DD is still higher than for that LP-DF-DD. • In this case, we obtain (11), as shown at the bottom of the page, where for convenience, the same short-hand notation as above is used. For high SNR, from (4) it can be observed that for LP-DF-DD and , is follows. valid. Therefore, from (11), For BH-DF-DD, it can be shown that for high SNR , , holds, and is obtained. therefore, again and suggest Our considerations for BPSK with that the commonly adopted assumption that error propagation increases BER by a factor of two is not always justified. Obviously, error propagation strongly depends on the adopted feedback filter. This claim is supported by the results presented in Section IV. B. Complexity Reduction For the method described in Section III-A, a large number of states is necessary for high-level DPSK and/or large observation , windows. An approximation for (8) can be obtained if only , states are considered, where is a design parameter. Of course, for a good approximation, the states have to be selected carefully. Here, we use the most recent entries , , of vector for definition of the states. For calculation of the associated nonzero state transition probabilities, also the remaining entries of have to be specified. For this, we divide the states into two sets. and , • Set I (containing states): . , , i.e., Here, we choose for these states correspond to the important case when the previous decision was wrong but all other were correct. As discussed, e.g., in decisions affecting [3] and [10], for LP-DF-DD the associated error probabilities are relatively high, leading to so-called double error events. Therefore, this case should be explicitly taken into account in our model.

Fig. 1. BER versus 10 log (E =N ) for LP-DF-DD and BH-DF-DD with N = 4, respectively. The proposed approximation for BER is compared to simulation results and the BER for error-free feedback.

• Set II (containing the remaining states). These states can only be occupied if an error had occurred at least two time steps before, i.e., if at least one of , , differs from 1. Our investigations the have shown that, in general, the best approximation of the , , actual BER is achieved if the , are chosen to yield , i.e., the influence of decision errors is limited to , .2 Now, the nonzero entries of the state transition matrix can be obtained analogous to can be calculated, and Section III-A. IV. COMPARISON WITH SIMULATIONS In this section, we compare the approximation proposed in Section III with simulation results and the BER for error-free feedback. Quaternary DPSK (QDPSK) is adopted, and both LP-DF-DD and BH-DF-DD are considered. In Fig. 1, the results for an observation window size of are depicted. For the proposed approximation for BER is adopted, i.e., a full-state Markov model is used. As can be observed, there is a very good agreement between and the simulation points ( ). For LP-DF-DD, approaches for high SNRs, whereas is a better approximation for the true BER for low SNRs. For BH-DF-DD, is practically identical with .

0 0

0 0 0 0 0

2Note that the condition y [k ] = 1, Z + 2 N 1, does not uniquely specify x[k ], Z + 1 N 2. However, this is not necessary since the decision variable directly depends on y [k ], 1 N 1, cf. (4), and any vector x [k 1] which fulfills y [k ] = 1, Z + 2 N 1, can be used.

0

0

(11)

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In order to limit computational complexity, a simple state reduction technique has been given. A comparison with simulation results has confirmed the tightness of the approximation. In addition, it has been shown that error propagation for DF-DD strongly depends on the chosen feedback filter. REFERENCES

BER versus 10 log (E =N ) for LP-DF-DD and BH-DF-DD with = 10, respectively. The proposed approximation for BER is compared to

Fig. 2.

N

simulation results and the BER for error-free feedback.

Similar observations can be made in Fig. 2, which is valid for . In order to limit complexity, again is used, i.e., the state reduction technique discussed in Section III-B is is a good approximation for applied. Also, in this case, the true BER. Both figures show that LP-DF-DD is more power efficient than BH-DF-DD if the same observation window size is used. Note, however, that BH-DF-DD is more robust against frequency offset [6]. V. CONCLUSION In this letter, we have used Markov chains to calculate the BER of DF-DD. Although this approach does not allow us to calculate the exact BER, a tight approximation can be obtained.

[1] H. Leib and S. Pasupathy, “The phase of a vector perturbed by Gaussian noise and differentially coherent receivers,” IEEE Trans. Inform. Theory, vol. IT-34, pp. 1491–1501, Nov. 1988. [2] F. Edbauer, “Bit-error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun., vol. 40, pp. 457–460, Mar. 1992. [3] F. Adachi and M. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett., vol. 29, no. 15, pp. 1385–1387, 1993. [4] H. Leib, “Data-aided noncoherent demodulation of DPSK,” IEEE Trans. Commun., vol. 43, pp. 722–725, Feb.–Apr. 1995. [5] N. Hamamoto, “Differential detection with IIR filter for improving DPSK detection performance,” IEEE Trans. Commun., vol. 44, pp. 959–966, Aug. 1996. [6] L. Bin and P. Ho, “Data-aided linear prediction receiver for coherent DPSK and CPM transmitted over Rayleigh flat-fading channels,” IEEE Trans. Veh. Technol., vol. 48, pp. 1229–1236, July 1999. [7] R. Schober and W. H. Gerstacker, “Decision-feedback differential detection based on linear prediction for MDPSK signals transmitted over Ricean fading channels,” IEEE J. Select. Areas Commun.: Wireless Commun. Ser., vol. 18, pp. 391–402, Mar. 2000. [8] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [9] D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [10] J. Liu, S. C. Kwatra, and J. Kim, “An analysis of decision-feedback detection of differentially encoded MPSK signals,” IEEE Trans. Veh. Technol., vol. 44, pp. 261–267, May 1995. [11] L. Bin, “Decision-feedback detection of minimum shift keying,” IEEE Trans. Commun., vol. 44, pp. 1073–1076, Sept. 1996. [12] P. Monsen, “Adaptive equalization of the slow fading channel,” IEEE Trans. Commun., vol. COM-22, pp. 1064–1075, Aug. 1974. [13] R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun., vol. COM-30, pp. 1828–1841, Aug. 1982.