IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 9, SEPTEMBER 2001
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Joint Decoding and Carrier Phase Recovery Algorithm for Turbo Codes Wangrok Oh and Kyungwhoon Cheun, Member, IEEE
Abstract—In this letter, we investigate the sensitivity of the iterative maximum a posteriori probability (MAP) decoder to carrier phase offsets and propose simple carrier phase recovery algorithms operating within the iterative MAP decoding iterations. The algorithms exploit the information contained in the extrinsic values generated within the iterative MAP decoder to perform carrier recovery, thus requiring low hardware complexity. Index Terms—Iterative decoding, phase synchronization, turbo codes.
I. INTRODUCTION
S
INCE the turbo code was introduced, several studies have demonstrated that turbo codes with iterative maximum a posteriori probability (MAP) decoding can achieve remarkable performance over AWGN channels [1], [2]. These works were based on the assumption that perfect carrier phase recovery is achieved which may be less than realistic for system without a pilot channel since the operating signal-to-noise ratios (SNRs) are usually extremely low. Unfortunately, iterative MAP decoders are quite sensitive to carrier phase offsets and proper phase recovery algorithms are needed to fully realize the exceptional performance of the turbo codes [3]. In this letter, we investigate the sensitivity of the iterative MAP decoder to carrier phase offsets and propose simple carrier phase recovery algorithms operating within the decoder iterations. The proposed algorithms exploit the properties of the extrinsic values generated within the iterative MAP decoder as with the adaptive channel SNR estimation algorithm proposed in [4], thus requiring low hardware complexity. The remainder of this letter is organized as follows. In Section II, we present the system model and briefly investigate the sensitivity of the iterative MAP decoder to carrier phase offsets. In Section III, the proposed carrier recovery algorithms are described and simulation results are presented. II. SYSTEM MODEL The system model considered is shown in Fig. 1. Information are grouped into frames of size and encoded with bits a turbo encoder consisting of a parallel concatenation of two recursive systematic convolutional (RSC) encoders separated and by an interleaver where puncturing of parity bits can be performed to achieve the desired code rate. The encoded Manuscript received May 21, 2001. The associate editor coordinating the review of this letter and approving it for publication was Prof. A. Haimovich. This work was supported by the Korea Research Foundation under Grant KRF2001-13-E00048. The authors are with the Division of Electrical and Computer Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea (e-mail:
[email protected]). Publisher Item Identifier S 1089-7798(01)09050-0.
bits are modulated using binary phase-shift keying (BPSK) and with a double-sided power spectral white Gaussian noise is added to the modulated signal before density of being presented to the demodulator. The demodulator output is first compensated with a very crude estimate of the initial phase offset in order to bring the residual carrier phase offset within the pull-in range of the proposed tracking algorithms. Then, the th symbol of the th code frame with residual carrier phase offset , denoted e , is multiplied by the correction term where is the estimated residual carrier phase offset for e is multiplied the th code frame whose real part, where is by the channel reliability value the received energy per coded symbol and then presented to the iterative MAP decoder for processing. Bit error rate (BER) performance of the iterative MAP decoder with carrier phase offsets is shown in Figs. 2 and 31 which show that the presence of a carrier phase offset can cause severe performancedegradations.Hence,inordertoachieveclosetotheoretical performance using the iterative MAP decoder, it is crucial that the carrier phase offset be properly estimated and compensated for. III. PROPOSED CARRIER RECOVERY ALGORITHMS Since the presence of a carrier phase offset effectively results in a reduction of power at the input to the iterative MAP decoder, the carrier phase offset in turn reduces the power of the decoder extrinsic values. 2 This allows us to use the estimated power of the decoder extrinsic values for carrier phase offset compensation. Here, in order to reduce the implementation complexity, we adopt the following simplified measure for the power of the decoder extrinsic values: (1) is the extrinsic value of the th bit of the th frame where computed by Decoder 2 on the final decoder iteration. We may also consider yet another, much simpler measure as follows:
(2) and given by The average behavior of , versus the carrier phase offset is . Note that the carrier phase offset shown in Fig. 4 for or can be compensated for by adjusting to maximize 1For all numerical results, we assume generator polynomials g
=
(111) and = (101) for RSC1 and RSC2 without puncturing (rate = 1=3) with frame size N = 128 and assume a random interleaver pattern generated randomly for
g
each frame. Also, 10 decoder iterations are performed for each code frame. 2We may also use the log-likelihood ratios in place of extrinsic values with almost identical results.
1089–7798/01$10.00 © 2001 IEEE
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IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 9, SEPTEMBER 2001
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Fig. 1. System model: u and u are the parity bits generated by RSC 1 and RSC 2, respectively. y cos( ^ ) is the received systematic information bit cos( ^ ) is the interleaved version of y cos( ^ ), y cos( ^ ) and y cos( ^ ) are the received parity bits from RSC 1 and and y RSC 2, respectively. ^ , i = 1; 2 is the initial phase offset estimate and is the residual carrier phase offset remaining after initial phase offset compensation and ^ is the estimated residual carrier phase offset for the lth code frame.
0
9
0
0
0
Fig. 2. BER performance of the iterative MAP decoder versus carrier phase offset ( ^ ).
Fig. 3. BER performance of the iterative MAP decoder versus E =N for carrier phase offsets of 10, 20 and 30 degrees.
and the pull-in range of the algorithms are approximately degrees, requiring some form of crude initial phase acquisition. A simple initial carrier phase acquisition algorithm can be devised by selecting an initial carrier phase offset estimate, as follows:
consider replacing
0
in (3) by
resulting in a simpler initial carrier phase acquisition algorithm as follows:
(3) , , and is the where extrinsic value generated by Decoder 2 for the th bit on th iteration of the th frame with the input signal the . The acquisition algorithm performs compensated with decoder iterations to test each of the four possible phases , . Hence, decoder iterations are required to complete the initial phase acquisition. We may again
(4) The acquisition failure probabilities of the algorithms given by and are shown in Table I. Note (3) and (4) with that a very low acquisition failure probability can be guaranteed with only a small number of decoder iteration overhead. After initial carrier phase acquisition is successfully completed, the absolute residual carrier phase offset is guaranteed
OH AND CHEUN: JOINT DECODING AND CARRIER PHASE RECOVERY ALGORITHM FOR TURBO CODES
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TABLE I FAILURE PROBABILITIES VERSUS E =N FOR INITIAL PHASE ACQUISITION. ESTIMATED VALUES WERE OBTAINED USING 100 000 TRIAL RUNS
Fig. 6. Steady-state BER performances of the proposed carrier phase recovery algorithms with = 0:069.
Fig. 4.
, i = 1; 2 versus the carrier phase offset for P = 100. Behavior of M
Fig. 7. BER performance of the proposed carrier frequency recovery algorithms versus the carrier frequency offset with = 0:001 and = 0:1.
Fig. 5. Convergence properties of the proposed algorithms with an initial carrier phase offset of 50 deg and = 0:069. Each curve represents an ensemble average of 1,000 independent simulation runs.
to be less than and we may use the following stochastic gradient type update algorithm to continuously track the carrier phase: (5) . Note that the where is the update step size and is required to properly set the direction term of the phase updates. The convergence and the steady-state BER performance of the proposed algorithms using a sample update chosen to give a reasonable tradeoff begain of tween the convergence rate and the steady state jitter are shown in Figs. 5 and 6. As can be seen, both algorithms converge within equal to 3 dB and the SNR approximately 12 frames at loss compared to perfect phase estimation is less than 0.1 dB at 10 BER. The convergence rate and the steady-state jitter can be controlled by adjusting the update step size and additional
gear shifting of can be used to optimize convergence time and steady-state jitter. A carrier frequency offset, if existent, can be more effectively compensated using a second order update where loop and , are the proportional and integral path gains. The BER performance of the proposed algorithms versus the carrier frequency offset normalized by the symbol rate using a sample loop filter with and for SNR values of 1 and 2 dB are shown in Fig. 7. We note that the proposed algorithms can track frequency offsets approximately less than 100 ppm of the symbol rate with minimal performance degradation. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo codes,” in IEEE Int. Conf. Commun., May 1993, pp. 1064–1070. [2] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inform. Theory, vol. 42, pp. 429–445, Mar. 1996. [3] B. Mielczarek and A. Svensson, “Improved MAP decoders for turbo codes with nonperfect timing and phase synchronization,” in Proc. IEEE Vehicular Technology Conf., Sep. 1999, pp. 1590–1594. [4] W. Oh and K. Cheun, “Adaptive channel SNR estimation algorithm for turbo decoder,” IEEE Commun. Lett., vol. 4, pp. 255–257, Aug. 2000.
