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Transactions Letters________________________________________________________________ Iterative Decoding and Channel Parameter Estimation Algorithms for Repeat–Accumulate Codes Wangrok Oh, Member, IEEE, and Kyungwhoon Cheun, Member, IEEE
Abstract—The sensitivity of the iterative decoder for repeat–accumulate (RA) codes to carrier phase and channel signal-to-noise ratio estimation errors is investigated, and efficient algorithms to estimate and correct these errors are developed. The behavior of RA codes with imperfect channel estimation is different from that of turbo codes, and correction algorithms specific to RA codes must be formulated. The proposed algorithms use the soft information generated within the iterative decoder, and thus, are not only hardware-efficient, but also offer excellent performance. Index Terms—Iterative decoding, phase synchronization, repeat–accumulate (RA) codes, signal-to-noise ratio (SNR) estimation.
I. INTRODUCTION
P
REVIOUS works [1]–[3] have demonstrated that repeat–accumulate (RA) codes, one of the simplest serial concatenated codes, can achieve remarkable performance over additive white Gaussian noise channels with iterative decoding. These results were based on the assumption of perfect estimation of channel parameters at the decoder, such as the channel signal-to-noise ratio (SNR) and carrier phase offset. This assumption may be unrealistic in practical implementations, due to the extremely low operating SNR range of RA codes. Simulation studies indicate that the performance of RA codes is very sensitive to errors in channel SNR and carrier phase offset estimates. Also, the behavior of RA codes as a function of these measurement errors may be quite different from that corresponding to turbo codes. Hence, in order to fully realize the exceptional performance of RA codes, it is crucial to formulate proper estimation and correction algorithms specific to RA codes, giving satisfying performance even at very low SNRs. Naturally, it would also be desirable to achieve this with minimum hardware overhead. In this letter, we investigate the sensitivity of the iterative decoder for RA codes to channel SNR and carrier phase offset estimation errors, and propose very simple correction algorithms. Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received April 30, 2004; revised January 27, 2005 and May 4, 2005. This work was supported by the Center for Broadband OFDM Mobile Access (BrOMA) at POSTECH, supported by the ITRC Program of the Ministry of Information and Communication, Korea, supervised by the Institute of Information Technology Assessment (IITA). The authors are with the Division of Electrical and Computer Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea (email:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857150
The algorithms exploit the properties of the log-likelihood ratios (LLRs) generated within the iterative decoder which are similar to the algorithms proposed for turbo codes [4], [5]. The carrier phase offset correction algorithm also exploits the differential encoding characteristic specific to RA codes. Since the estimation is performed using the soft information already available within the iterative decoder, the required additional hardware complexity is minimal. There are two notable differences in the behaviors of RA and turbo codes under channel parameter estimation errors. A more subtle difference is that unlike turbo codes [4], [6], RA codes are much less lenient to overestimation of the channel SNR. A more significant difference is that due to inherent differential inner encoding, RA codes exhibit relatively good performance under carrier phase offset in the vicinity of 180 . This inhibits straightforward application of stochastic gradient-type carrier phase offset correction algorithms, due to the presence of a local maximum. Hence, in order to fully realize the exceptional performance of RA codes, it is crucial to develop channel parameter estimation and correction algorithms specific to RA codes. The remainder of the letter is organized as follows. In Section II, we present the system model and investigate the sensitivity of RA codes to channel SNR and carrier phase offset estimation errors. In Section III, channel parameter estimation and correction algorithms appropriate for RA codes are described and their performances are evaluated. Finally, conclusions are drawn in Section IV. II. SYSTEM MODEL The system model under consideration is shown in Fig. 1. Biare grouped into frames of size nary information bits and encoded by an RA encoder. The RA encoder consists of a serial concatenation of a –times repeater and a rate–1 accu. The encoded mulator, separated by an interleaver of size are binary phase-shift keying modulated, symbols with a double–sided power specand white Gaussian noise tral density of is added before being presented to the demodulator. The corresponding demodulator output with a car, is prerier phase offset of , denoted is a zero-mean comsented to the iterative decoder, where plex Gaussian random variable (r.v.) with variance per dibe the channel SNR, where mension. Let is the code rate and is the received energy per information th decoder bit. The resulting LLRs for at the end of the
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Fig. 1. System model. u are the input information bits and c are the RA encoded symbols, y is the demodulator output with a carrier phase offset , u ^ denotes the decoder output for u , L (u ) is the resulting LLR for u at the end of the lth decoder iteration, ~ is the initial carrier phase offset estimate, and ^ and ^ are the channel SNR and carrier phase offset estimates used in the lth decoder iteration.
^ and ^ are the channel reliability value and carrier phase offset estimates used in the lth decoder iteration. Fig. 2. Factor graph of an RA code. L
iteration, denoted , are used to generate the estimates and of and to be used in the th decoder iteration. The th iteration of the iterative decoder is carried out using the demodulator outputs compensated by , and using on the factor graph [7] shown in Fig. 2, where the dependence of the messages on was suppressed to simplify the figure. The and using the iterative decoder computes the messages following recursions:
(3)
(4) Here
(1)
(2)
[7], denotes the real part of , is the estimated channel reliability value [8], and denotes the th output poare the messages sition of the interleaver (Fig. 2). Also,
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Fig. 3. BER performance of the iterative decoder for an RA code versus the ^
dB, assumed to be fixed for all l. Frame channel SNR estimation error size is 512, q = 3, l = 10, and perfect carrier phase offset estimation is assumed.
Fig. 4. BER performance of the iterative decoder for an RA code versus the carrier phase offset estimation error , assumed to be fixed for all l. Frame size = 10, and perfect channel SNR estimation is assumed. is 512, q = 3 , l
0
passed from the information nodes to the check nodes, initialized to zero and computed as (5) where denotes the integer part of , and denotes the th output position of the deinterleaver (Fig. 2). The messages are the messages originating from the check nodes destined for and as the information nodes, computed using (6) with . Finally, the LLRs for the information bits the end of the th iteration are then given as
at
with turbo codes [4], [6], a slight underestimation of the channel SNR optimizes the BER performance of the iterative decoder for RA codes.2 In Fig. 4, the performance of the standard iterative RA dewith , versus , assumed to be coder with and 2 dB. This graph fixed for all , is shown for clearly shows the detrimental effect of carrier phase offset estimation errors. However, note that the iterative decoder for RA codes shows relatively good performance, with carrier phase offset estimation errors in the vicinity of 180 . This is due to the differential encoding of the accumulator inner code. This characteristic is unique to RA codes, and must be taken into account when designing carrier phase offset correction algorithms. III. PROPOSED CHANNEL PARAMETERESTIMATION ALGORITHMS
(7) The decoder iteration stops when it reaches a preset maximum , or when all bits are successfully decoded, which number can be verified using a cyclic redundancy check [9]. plays It is clear that the estimated channel reliability value a key role in the decoding of RA codes. The bit-error rate (BER) for performance of a standard iterative decoder with and versus the channel SNRan RA code with estimation error is shown in Fig. 3 for and 2 dB.1 Here, the assumed channel SNR-estimation error was held fixed throughout the decoder iterations, and the carrier phase offset , is assumed to be zero estimation error, denoted for all decoder iterations. We observe that, unlike turbo codes, both over- and underestimation of the channel SNR results in a sizable degradation. However, it is interesting to note that as 1For all numerical results, we assume that a random interleaver pattern is generated pseudorandomly for each frame, i.e., the uniform interleaver. However, proper operation of the proposed algorithms was verified under various fixed interleavers. Also, 1000 frame errors were observed for each BER simulation point.
A. Channel SNR Estimation Similar to the channel SNR estimation algorithms for turbo codes developed in [4], we investigate the behavior of the variance of the absolute values of the LLRs, computed as follows: (8) where denotes the LLR for the th bit after the th iteragiven by (8) may be tion of the th frame. Also, as in [4], replaced with the following simplified version: (9) The ensemble averaged behavior of simulation runs given by
and
using 100 ,
2This trend may be slightly changed by employing decoder-scheduling schemes different from the one described in Section II, e.g., full flooding [10]. However, the proposed algorithms are directly applicable to iterative decoders employing any decoder-scheduling scheme.
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Behavior of v , i = 1; 2 versus the channel SNR estimation error for l = 10. Frame size is 512 and q = 3. The channel SNR estimation error is
^ with v and = BER performance of the iterative decoder using = 10, and perfect carrier phase offset estimation is assumed.
Fig. 5.
assume to be fixed for all decoder iterations, and perfect carrier phase offset estimation is assumed.
versus the channel SNR estimation error is shown in Fig. 5 for . Here, we assume perfect carrier phase offset estimation and the fixed channel SNR estimation error for all decoder iterations. Comparing this with Fig. 3, we observe that the BER of the iterative decoder for RA codes may be minimized by minior . Based on this observation, we propose the mizing following stochastic gradient-type recursive update equation for : (10) is the estimated channel SNR for the th iteration of where the th frame, and is the update gain. Also, (10) can be further simplified by replacing with its polarity, resulting in
(11) if , and , otherwise. The BER where performance of the iterative decoder using the proposed channel given by (9), is shown SNR estimate given by (11), with in Fig. 6 with perfect carrier phase offset estimation. We observe that, as with turbo codes [4], the iterative decoder using the proposed channel SNR estimate gives BER performance slightly better than that using the exact channel SNR. B. Carrier Phase Estimation The presence of a carrier phase offset estimation error results in an effective reduction in the received signal power, as seen by the matched-filter receiver. This will, in effect, reduce the average power of the LLRs of the information bits within the iterative decoder. This indicates that the measured power of the LLRs of the information bits may be used for carrier phase offset estimation and correction. As in [5], we consider the following
Fig. 6.
2:5. Frame size is 512, and q = 3, l
simplified measure in place of the more complex power estimator: (12) We may also consider the following much simpler measure [5] that may further reduce the hardware complexity, with minimal loss in performance: (13) In Fig. 7, the ensemble averaged behavior of and , using 100 simulation runs given by , , versus the carrier phase offset estimation error are shown for . Here, we assume perfect channel SNR estimation and the fixed carrier phase offset estimation error for all decoder iterations. Note the existence of a prominent local maximum at 180 carrier phase offset estimation error. This indicates that unlike turbo codes, straightforward application of a stochastic gradient update algorithm of the following type may converge to the undesirable local maximum at 180 , depending on the initial carrier phase offset:3
(14) Here, is the estimated carrier phase offset for the th iteration of the th frame. However, if a coarse initial carrier phase recovery is available which guarantees carrier phase offset in the vicinity of 0 , (14) may successfully be used to compensate and track the carrier phase offset. If the coarse initial carrier phase 3The update term in (14) has polarity opposite to that of (11), due to the fact that s , j = 1; 2 are convex functions of the carrier phase offset estimation error, while v , j = 1; 2 are concave function of the channel SNR-estimation error.
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Fig. 7. Behavior of s , i = 1; 2 versus the carrier phase offset estimation error for l = 10. Frame size is 512 and q = 3. The carrier phase offset estimation error is assumed to be fixed for all decoder iterations, and perfect channel SNR estimation is assumed.
Fig. 8. BER performance of the proposed iterative decoder using ~ and ^ with s and = 0:5, versus the channel SNRs for p = 1; 5. Frame size is ^ = 4 . = 10, and L 512, q = 3, l
recovery algorithm is nondata aided and thus possesses a 180 phase ambiguity [11], we may use the following observation to easily detect and compensate for the 180 phase ambiguity within the iterative decoder before the tracking operation with (14). generated under a carrier phase Note that the messages offset of 180 are identical to those generated under a zero car, which has a reversed polarity. rier phase offset, except for This is due to the differential encoding inherent in RA codes. is alHence, neglecting the effect of noise, the polarity of ways opposite to that of under a 180 carrier phase offset. Based on this observation, we may modify the iterative decoder to detect and compensate for carrier phase offsets near 180 by and after the comparing the polarity of the messages first decoder iteration. Let denote the initial carrier phase offset estimation with a 180 resolution, given as if otherwise.
(15)
If is 0 , the proposed iterative decoder moves on with the remaining decoder iterations. Otherwise, the decoder negates and the message , and recalthe demodulator outputs before culates the messages affected by the negation of performing the remaining decoder iterations. Note that only out of , ’s need be recalculated as a result of negating . Hence, the additional complexity required to detect and compensate for carrier phase offset near 180 is much less than of that required for one decoder iteration. We have also verified that this trend is maintained and reinforced even after iterations. Hence, in order to increase the reliability of the estimate of , the detection operation described above may be performed after decoder iterations. The BER performance of the proposed iterative decoder and 180 with versus the channel SNR under
is shown in Fig. 8, assuming perfect channel SNR estimation.4 Note that the proposed iterative decoder under a carrier phase offset of 0 and 180 offers performance within 0.03 and 0.1 at BER dB of the standard iterative decoder, with , respectively. The BER performance curve of the with proposed iterative decoder under an initial carrier phase offset of 45 is also included in Fig. 8, and lies between those correand 180 . Although not shown in Fig. 8, sponding to we have carried out extensive simulations, and verified that the initial carrier phase offset of 180 results in the largest SNR loss for the proposed iterative decoder. We have also verified that the increase in the required average number of decoder iterations, due to the increase in , is slightly less than . In most packet-based networks, a preamble is usually available, and a crude initial carrier phase recovery may be performed using the preamble. In continuous transmission systems, the stochastic gradient update algorithm of (14) may first be used to let the residual carrier phase offset converge to the vicinity of either 0 or 180 . A decision on may then be made using (15), after which the tracking operation may be resumed. IV. CONCLUSIONS In this letter, we investigated the sensitivity of the iterative decoder for RA codes to carrier phase and channel SNR estimation errors, and developed algorithms to accurately estimate and correct these errors. The characteristics specific to RA codes were identified and fully exploited. The proposed algorithms, using the soft information generated within the iterative decoder, are not only very hardware-efficient, but also provide excellent performance.
4Note that due to the fact that s takes on values much larger than those of v , the update gain used for the channel SNR estimation is much larger than
that used for the carrier phase offset compensation.
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REFERENCES [1] D. Divsalar, H. Jin, and R. McEliece, “Coding theorems for “turbo-like” codes,” in Proc. 36th Allerton Conf. Commun., Control, Comput., Sep. 1998, pp. 201–210. [2] H. Jin and R. J. McEliece, “RA codes achieve AWGN channel capacity,” in Proc. 13th Symp. Appl. Algebra, Algebraic Algorithms, Error Correcting Codes, 1999, pp. 10–18. [3] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat-accumulate codes,” in Proc. 2nd Int. Symp. Turbo Codes, Sep. 2000, pp. 1–8. [4] W. Oh and K. Cheun, “Adaptive channel SNR estimation algorithm for turbo decoder,” IEEE Commun. Lett., vol. 4, no. 8, pp. 255–257, Aug. 2000. [5] , “Joint decoding and carrier phase recovery algorithm for turbo codes,” IEEE Commun. Lett., vol. 5, no. 9, pp. 375–377, Sep. 2001. [6] T. A. Summers and S. G. Wilson, “SNR mismatch and online estimation in turbo decoding,” IEEE Trans. Commun., vol. 46, no. 4, pp. 421–423, Apr. 1998.
[7] F. R. Kschischang, B. J. Frey, and H. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [8] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 429–445, Mar. 1996. [9] A. Shibutani, H. Suda, and F. Adachi, “Reducing average number of turbo decoding iterations,” Electron. Lett., vol. 35, pp. 701–702, Apr. 1999. [10] F. R. Kschischang and B. J. Frey, “Iterative decoding of compound codes by probability propagation in graphical models,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 219–230, Feb. 1998. [11] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers. New York: Wiley, 1998.
