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Decoding Strategies for Turbo Product Codes in Frequency-Hop Wireless Communications. Michael B. Pursley and Jason S. Skinner. Department of Electrical ...
Decoding Strategies for Turbo Product Codes in Frequency-Hop Wireless Communications Michael B. Pursley and Jason S. Skinner Department of Electrical and Computer Engineering 303 Fluor Daniel Engineering Innovation Building Clemson University, Clemson, SC 29634

Abstract— Alternative decoding methods are described and evaluated for use with a commercial turbo product code chip that is employed for error control in slow-frequency-hop spreadspectrum wireless communications. One strategy is based on parallel decoding in two independent decoders, and another employs a single decoder with adaptive scaling of the soft decisions at its input. Performance results for such decoding strategies are compared for packet transmissions over channels with partialband interference. When coupled with the development and use of side information in the frequency-hop receiver, these decoding strategies greatly enhance the performance of turbo product codes and make them competitive with parallel-concatenated turbo convolutional codes.

I. I NTRODUCTION For packet transmission in slow-frequency-hop communication systems, each packet is divided into several segments, the segments are transmitted in different intervals of time known as dwell intervals, and different dwell intervals may use different frequency slots. Because the hopping rate is not greater than the symbol transmission rate in slow-frequencyhop communications, each segment contains one or more channel symbols. A terminal that is sending a packet employs its frequency hopping pattern to determine the frequency slot to be used for each segment. For the systems considered in this paper, multiple frequency slots are used for each packet, which provides frequency diversity that helps overcome frequencydependent disturbances such as frequency-selective fading or interference that is nonuniformly distributed across the frequency band. SINCGARS [4] is an example of a frequencyhop radio system for which our results are applicable. Recently, there has been considerable interest in turbo codes for slow-frequency-hop communications over channels with partial-band interference. Kang and Stark [6] evaluated the performance of a parallel-concatenated turbo convolutional code, which we refer to as a turbo convolutional code (TCC). In comparisons with previous results on Reed-Solomon codes [3], they showed a TCC is superior to a Reed-Solomon code or a concatenated Reed-Solomon and convolutional code, albeit at a cost of increased decoder complexity. In an earlier paper [5], they demonstrated significant performance advantages of a TCC over a convolutional code. This research was supported by the U.S. Army Research Office under grant DAAD19-99-1-0289 and by the DoD MURI program under Office of Naval Research grant N00014-00-1-0565.

Following the work of Kang and Stark, others have investigated a different class of turbo codes, known as turbo product codes, in the hope of achieving comparable performance with less complexity. Results on the performance of turbo product codes for frequency-hop communications on channels with partial-band interference are reported in [10] and [13]. Methods to exploit side information in turbo product codes are described in [10], and alternative iterative algorithms for soft-decision decoding are examined in [13]. In this paper, we examine the use of parallel decoders and adaptive scaling of soft decisions in a single decoder for frequency-hop packet transmission over channels with partial-band interference. We compare the performance of systems that are supplied perfect side information with systems that derive side information from test symbols and systems that derive side information from the data symbols without the need for additional redundancy. II. S YSTEM D ESCRIPTION As in [6] and [13], our primary results are for noncoherent demodulation of binary orthogonal signals (e.g., binary frequency-shift keying); however, some results are given for coherent demodulation of binary antipodal signals (e.g., binary phase-shift keying). A three-dimensional turbo product code of rate 0.325 and block length 4096 is investigated for both the noncoherent and coherent communication systems. Each of the three components codes is a (16,11) extended Hamming code. Since the (16,11) extended Hamming code has a minimum Hamming distance of 4, the (16,11)×(16,11)×(16,11) product code has a minimum Hamming distance of 64. The turbo product code of comparable rate in [13] is a two-dimensional code for which each component code is a (64,36) BCH code. The resulting product code has rate 0.316. Turbo product codes of higher rates are also investigated in [10] and [13]. We base the soft decisions on ratios of the outputs of noncoherent detectors. For some of our results, test symbols are employed to provide side information. Binary symbols that are known to the receiver are included in each dwell interval, and hard decisions are made on these symbols in the demodulator. If an error occurs in any of the test symbols within a given dwell interval, the soft decisions for all other demodulated symbols in that dwell interval are multiplied by a number λ that is in the range 0 ≤ λ ≤ 1. The scaled soft decisions are the input to a soft-input soft-output iterative

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decoder. The decoder can iterate at most 32 times, but we found that most packets decode in five or fewer iterations. For each of the performance curves for turbo product codes, each packet represents 1331 information bits with a single code word consisting of 4096 binary code symbols. The code symbols are transmitted in a sequence of 128 dwell intervals, so there are 32 binary code symbols per dwell interval. If the test-symbol method is employed to generate side information, binary test symbols are included in each dwell interval with the code symbols. For example, if there are six test symbols per dwell interval, the total number of binary symbols per dwell interval is 38 and the rate is reduced to 0.274. If there are two test symbols per dwell interval, each dwell interval has 34 binary symbols and the overall rate for the coding system is 0.306. For six test symbols per dwell interval, the packet consists of 4096 code symbols and 768 test symbols, for a total of 4864 binary symbols. If there are two test symbols per dwell interval, the packet has 4352 symbols, 256 of which are test symbols. The encoding and decoding functions are performed by a AHA4501 Astro Turbo Product Codec Evaluation Module [2] from Efficient Channel Coding, Inc. This ISA plug-in board contains the AHA4501 Astro [1] a single-chip turbo product codec from Advanced Hardware Architectures, Inc. One of the goals of our investigation is to determine the extent to which such a commercial codec can provide performance that is comparable to that obtained with parallel-concatenated turbo convolutional codes, for which the decoders are considerably more complex. III. C HANNEL M ODEL Our channel model is that same as the model employed in several previous investigations, including [3], [6], and [13]. The primary disturbance is partial-band interference, which is modeled as band-limited white Gaussian noise that is present in a fraction ρ of the frequency slots. The one-sided power spectral density for the band-limited noise is ρ−1 NI in the frequency slots that have interference, and it is zero in the remaining fraction 1 − ρ of the slots. The total power in the interference is proportional to NI but it is independent of ρ. The partial-band noise model is convenient for use in analysis and simulation, and it represents a good approximation to many frequency-hop systems with partial-band interference [12]. The receiver’s thermal noise, white Gaussian noise with one-sided power spectral density N0 in the full band, is added to the interference. For all of the performance results in this paper, the ratio of the energy per information bit to the one-sided thermal noise density is Eb /N0 = 20 dB. Use of this value facilitates comparisons with previous results, since it is the same as in [6], [10], and [13]. The side information provided to the decoder is any information that pertains to the status of the frequency slots used in the transmission of the packet that is being decoded. Define the random variable Zi by Zi = 1 if there is interference in the ith frequency slot and Zi = 0 otherwise. Define EN Ri to be the ratio of the energy per bit to the noise density in

the ith frequency slot. Thus, EN Ri = Eb /η1 if there is interference in the ith frequency slot, and EN Ri = Eb /η0 if there is no interference in the ith slot. The state of the ith frequency slot is the pair (Zi , EN Ri ), and the state of the channel is the collection of all such pairs as the index i ranges over all frequency slots used by the system. We say that the decoder has side information for a packet if it has any information pertaining to the states of at least some of the frequency slots, whether it is provided by the receiver or obtained from some external source. We say the decoder has perfect side information for a packet if it is told which of the frequency slots used in the transmission of the packet have interference, and we say the decoder has perfect channel state information for the packet if it is told the state of the channel. Thus, perfect channel state information requires perfect side information in addition to knowledge of the values for both Eb /η1 and Eb /η0 . The aforementioned definition of perfect side information is consistent with early references on the subject (e.g., [8] and [9]). The term perfect side information as used in [6] is what we refer to as perfect channel state information, since knowledge of the signal-to-noise ratios is assumed in [6]. IV. P ERFORMANCE PARAMETERS The performance curves of Fig. 1 provide illustrations that are helpful in the definitions of the primary performance parameters. The codes for these curves are the turbo product code (TPC) of rate 0.325 and a TCC of rate 1/3, each of which is decoded without the aid of side information. Each curve is a plot of the value of Eb /NI that is required to achieve a packet error probability of 10−3 on a channel in which interference is present in a fraction ρ of the band. There are 32 code symbols per hop for the TPC, and no test symbols are transmitted for either code. The results published in [6] for 20 and 80 symbols per hop were used to estimate the performance of the TCC in a system with 32 symbols per hop. For each curve, we define ρˆ to be the fractional bandwidth for which the required value of Eb /NI is largest. From the point of view of the decoder, ρˆ represents the worst value for ρ. The parameter EN R∗ is defined in [11] as the minimum value of Eb /NI that achieves the specified packet error probability for all values of ρ. In other words, EN R∗ is the value of Eb /NI that is required if ρ = ρˆ. The parameter EN R∗ corresponds to the amount of energy required to guarantee the specified packet error probability is achieved for a channel in which the value of ρ is unknown (which is true of most channels). As illustrated in Fig. 1, ρˆ = 1 and EN R∗ is greater than 9.5 dB for each TCC. For the TPC, ρˆ is approximately 0.75 and EN R∗ is less than 8.9 dB. Thus, the TPC has a performance advantage of more than 0.6 dB over the TCC as measured by EN R∗ . The parameter ρ˜ is defined to be the maximum value of ρ for which Eb /NI = 0 dB achieves the specified packet error probability. The choice of 0 dB as the reference value is somewhat arbitrary, and any other value could be employed. However, a change of ±2 dB in the reference value does not

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Figure 1. Comparison of TPC and TCC, packet error probability 10−3 .

change the value of ρ˜ significantly for any of the curves in Fig. 1. The parameter ρ˜ is a measure of a code’s ability to reject strong interference that occupies a relatively small fraction of the band. If the fractional bandwidth of the interference is less than ρ˜, it has negligible effect on the performance, even if its power is increased by several decibels. For example, for each code evaluated in Fig. 1, if Eb /NI = EN R∗ and ρ ≤ ρ˜, a packet error probability of 10−3 would still be achieved even if the interference is increased by 8 dB. For fractional bandwidths less than approximately 0.45, the TCC gives the best performance. Both codes overcome partialband interference if the fractional bandwidth is less than 0.25. For fractional bandwidths in the range from 0.46 to 1.00, the TPC gives better performance than the TCC. The TPC has a performance advantage in excess of 0.7 dB if ρ = 1. In general, we found that the TCC is superior for moderate values of ρ and the TPC gives better results for large values of ρ. Similar conclusions are given in [13] for a comparison of the same TCC with a TPC that has BCH codes as component codes. Although the TPC gives better performance than the TCC in terms of the parameter EN R∗ , the TCC is superior to the TPC with respect to ρ˜. The TPC achieves ρ˜ = 0.26, whereas our estimated value for the TCC with 32 symbols per hop is ρ˜ ≈ 0.4. The primary goal of our investigation is to increase the value of ρ˜ for the TPC while maintaining both low complexity and a small value of EN R∗ compared to the TCC. Specifically, we would like to have ρ˜ as close to 0.4 as possible without increasing EN R∗ above 9 dB. We accomplish this by developing side information in the receiver and applying it to aid the soft-decision iterative decoder.

makes hard decisions on these test symbols and soft decisions on the remaining symbols. For each dwell interval in which at least one demodulated test symbol is incorrect, all soft decisions in the dwell interval are scaled by a factor λ. If λ = 0, all symbols in such a dwell interval are erased, but if λ = 1, all symbols are passed to the decoder without alteration. For λ < 1, soft-decisions in each dwell interval that has a testsymbol error are weighted less than soft decisions in dwell intervals with no test-symbol errors. Unless stated otherwise, all subsequent results are for binary orthogonal signaling, noncoherent demodulation, and a packet error probability of 10−2 (note that Fig. 1 is for a packet error probability of 10−3 ). Performance curves for a system with six test symbols per dwell interval are given in Fig. 2 for several choices for λ. For λ = 1, the soft decisions and the decoder are the same as for the TPC curve in Fig. 1, but there is now an energy penalty of approximately 0.75 dB that results from the inclusion of six test symbols per dwell interval. The penalty is a consequence of having to transmit more symbols while maintaining the same energy per information bit, and it is equal to the increase in Eb /NI that is required at ρ = 1. For ρ = 1, each dwell interval has interference, so the side information is of no value and the energy in the test symbols is wasted. Of course, the inclusion of test symbols also results in a rate penalty (i.e., a lower information rate for the packet). For λ < 1, the test symbols are beneficial, and the value of ρ˜ decreases as λ is decreased, as illustrated in Fig. 2. In Table I we give the values of EN R∗ and ρ˜ for two systems with λ = 1/2. One uses six test symbols per dwell interval to generate side information and the other is given perfect side information (PSI). The values for ρ˜ differ by only 0.01, and the only difference in the values for EN R∗ is the result of the energy penalty for the inclusion of six test symbols. This shows that six test symbols are enough to provide nearly perfect side information. Included in Table I are the values of EN R∗ and ρ˜ for the system that has no side information and uses no scaling (λ = 1, no TS). The systems with side information that use λ = 1/2 achieve much larger values of ρ˜ than the system with no side information, and the only difference in EN R∗ is due to the energy penalty for the test symbols. VI. PARALLEL D ECODING Although there are some variations among the values of EN R∗ for λ > 0 in Fig. 2, the primary differences are in the

V. S IDE I NFORMATION For systems that employ test symbols to generate side information, a fixed number of binary symbols that are known to the receiver are included in each dwell interval. The demodulator

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TABLE I C OMPARISON OF ρ˜, ρˆ, AND EN R∗ VALUES .

λ = 1, no TS λ = 1/2, 6TS λ = 1/2, PSI λ = 1/2 || λ = 0, 6TS Linear (w0 = 2, w1 = 4) Linear (w0 = 2.5, w1 = 5)

ρ˜ 0.29 0.34 0.35 0.37 0.34 0.37

ρˆ 0.85 0.95 0.90 0.95 0.85 0.85

EN R∗ (dB) 8.70 9.51 8.72 9.51 8.73 8.96

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Figure 3. Parallel decoding.

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Figure 2. Decoding with test symbols, scaling by λ, error probability 10−2 .

values of ρ˜. The largest value of ρ˜ is achieved for λ = 0, but the performance of the decoder for λ = 0 is poor for large values of ρ. In particular, EN R∗ is much too large. The curve for λ = 1/4 achieves EN R∗ = 9.7 dB, and ρ˜ almost as large as for λ = 0. The curves for λ ≥ 1/2 have even better values of EN R∗ , but their values for ρ˜ are smaller than for λ ≤ 1/4. This motivates the use of parallel decoding. By employing parallel decoding with λ = 0 for one stream of soft-decision data and λ > 0 for another stream, we hope to achieve a large value of ρ˜ and a small value of EN R∗ . Furthermore, the use of λ = 0 for one decoder improves the robustness of the system for non-Gaussian narrowband interference. The use of λ > 0 for the other decoder avoids the large energy requirement in the region ρ > ρ˜ that we see in Fig. 2 for λ = 0. Parallel decoding of Reed-Solomon codes was introduced in [7] for frequency-hop systems, and it has been applied in several subsequent publications (e.g., [3] and [11]). For turbo codes, our approach is to use side information provided by test symbols to scale the soft-decision outputs of the demodulator in n ways to produce n streams of scaled soft-decision data. The streams can be applied simultaneously to n separate decoders that operate in parallel, as illustrated in Fig. 3 for n = 2. Each decoder performs iterative decoding and produces its estimate of the original packet of information bits that was encoded by the TPC. In the simplest implementation, no information is exchanged between different decoders, in which case the decoding could also be accomplished in a serial fashion using a single decoder chip. As in most packet communication systems, a cyclic redundancy check (CRC) code or other error-detecting code is applied to the information bits prior to encoding by the TPC. The output of each decoder is verified by the CRC code, and the output packets that fail the check are discarded. A packet error occurs if each of the parallel decoders produces an invalid

packet. CRC codes have very high rates, so the overall rate of the coding system is not reduced significantly. For most applications, CRC codes are essential for other purposes, and our use of them imposes no additional requirements. Performance results are given in Fig. 4 for a system with six test symbols per dwell interval and two parallel decoders. One soft-decision data stream uses λ = 0 (erasures) and the other uses λ = 1/2. Compared to the performance of a single decoder that uses λ = 1/2, the parallel decoder provides an improvement in ρ˜ without sacrificing performance for ρ > ρ˜. In fact, the value of EN R∗ is the same for the two decoding strategies, as indicated in Table I. Also shown in Fig. 4 are the performance curves for a parallel decoder in a system with two test symbols and a single decoder with no scaling and no test symbols (λ = 1, no TS). Comparisons among the curves in Fig. 4 illustrate the tradeoffs between the energy penalty for the inclusion of test symbols and the increase in ρ˜ obtained from the side information. As the number of test symbols is increased from two to six, ρ˜ improves from 0.35 to 0.37 but EN R∗ increases from 9.1 dB to 9.5 dB. The improvement in ρ˜ is due to the increased reliability of the side information. The increased energy requirement is due to the greater energy penalty for more test symbols. In comparison with the other two curves in Fig. 4, there is not a large performance difference for the parallel decoder if the number of test symbols per dwell interval is decreased from six to two. VII. A DAPTATION OF THE S CALE PARAMETER Because the addition of test symbols incurs rate and energy penalties, it is beneficial to examine other methods of generating side information. In one method, a quality measure for the reliability of the symbols in a dwell interval is determined from the outputs of the noncoherent detectors during the demodulation of the symbols in the dwell interval. The quality measure is employed to adjust the scaling of the soft decisions in the dwell interval. Let Zi,j,k denote the output of the detector matched to symbol i (i=0,1) in the kth symbol position of the jth dwell interval. One quality measure that we investigated is given by  max{Z0,j,k , Z1,j,k } k . (1) Wj =  min{Z0,j,k , Z1,j,k }

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Figure 4. Parallel decoding with test symbols.

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Figure 6. Decoding with an adaptive scale factor.

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Figure 5. Adaptive scaling. −6

Each soft decision in the jth dwell interval is multiplied by λj = g(Wj ). Thus, the soft decisions in a packet are scaled adaptively from dwell interval to dwell interval as the packet is received. The system for adaptive scaling according to the quality measure Wj is illustrated in Fig.5. One choice for the function g is the linear ramp from w0 to w1 . That is,   for w ≤ w0 , 0 g(w) = (w − w0 )/(w1 − w0 ) for w0 ≤ w ≤ w1 , (2)   1 for w ≥ w1 .

The performance of a single decoder with adaptive scaling using a linear ramp is shown in Fig. 6 for two choices of the pair (w0 ,w1 ). Shown also in Fig. 6 for comparison are the curves for decoding with no side information and the parallel decoder with six test symbols. From Table I we see that w0 = 2.5 and w1 = 5 gives ρ˜ = 0.37 and EN R∗ < 9 dB. Compared to soft-decision decoding with no side information and no scaling, we have increased ρ˜ by more than 25% without increasing EN R∗ by more than 0.3 dB.

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Figure 7. Decoding with test symbols and scaling, coherent BPSK.

VIII. C OHERENT D EMODULATION The final set of performance curves in Fig. 7 is for coherent demodulation of antipodal signals (e.g., coherent BPSK). For each of the curves, we see that the improvements in the required values of Eb /NI are substantial for antipodal signals and coherent demodulation compared with orthogonal signals and noncoherent demodulation. The increase in EN R∗ is more than 7 dB for each value of λ. We use the reference value of Eb /NI = −8 dB to define ρ˜ for coherent demodulation. For the system with no side information and no scaling (λ = 1, no TS), ρ˜ is 0.25 and EN R∗ is 1.55 dB. For parallel decoding, four test symbols are used to provide side information. Parallel decoding with λ = 0 and λ = 1/2 gives ρ˜ = 0.37 and EN R∗ = 1.91 dB.

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Parallel decoding with λ = 0 and λ = 1/4 gives approximately the same value for ρ˜. Although the value for EN R∗ is 0.29 dB larger, the performance in the region 0.4 ≤ ρ˜ ≤ 0.55 is better for parallel decoding with λ = 0 and λ = 1/4 than for parallel decoding with λ = 0 and λ = 1/2. Each of the parallel decoding systems has a value of ρ˜ that is approximately 50% larger than for decoding without side information. IX. C ONCLUSIONS With a good decoding strategy, the lower-complexity turbo product code performs nearly as well as a turbo convolutional code of comparable rate. Our results demonstrate that the turbo product code benefits greatly from the use of side information, and we showed that test symbols can provide nearly perfect side information for frequency-hop wireless communications over channels with partial-band interference. The use of a parallel decoder enhances the performance of the turbo product code, especially against strong interference that occupies a small to moderate fraction of the band (i.e., for ρ ≤ ρ˜). By adapting the scaling parameter, we achieve the same value of ρ˜ that is provided by parallel decoding, but only a single decoder is used and no test symbols are required. Thus, we eliminate the energy and rate penalties associated with the use of test symbols without sacrificing the partial-band interference rejection performance. R EFERENCES [1] Advanced Hardware Architectures, Inc., Product Specification for AHA4501 Astro 36 Mbits/sec Turbo Product Code Encoder/Decoder. Pullman, WA. Available: http://www.aha.com [2] Efficient Channel Coding, Inc., Product Summary: AHA4501 Astro Turbo Product Code Evaluation Module. Brooklyn Heights, OH. Available: http://www.eccincorp.com

[3] C. D. Frank and M. B. Pursley, “Concatenated coding for frequency-hop spread-spectrum with partial-band interference,” IEEE Transactions on Communications, vol. 44, pp. 377-387, March 1996. [4] B. J. Hamilton, “SINCGARS system improvement package (SIP) specific radio improvements,” Proceedings of the 1986 Tactical Communications Conference, pp. 397-406, April 1996. [5] J. H. Kang and W. E. Stark, “Turbo codes for coherent FH-SS with partial band interference,” Proceedings of the 1997 Military Communications Conference, vol. 1, pp. 5-9, November 1997. [6] J. H. Kang and W. E. Stark, “Turbo codes for noncoherent FH-SS with partial band interference,” IEEE Transactions on Communications, vol. 46, pp. 1451-1458, November 1998. [7] M. B. Pursley, “Coding and diversity for channels with fading and pulsed interference,” Proceedings of the 1982 Conference on Information Sciences and Systems (Princeton Univ.), pp. 413-418, March 1982. [8] M. B. Pursley, “Frequency-hop transmission for satellite packet switching and terrestrial packet radio networks,” IEEE Transactions on Information Theory, vol. IT-32, pp. 652-667, September 1986. [9] M. B. Pursley, “Reed-Solomon codes in frequency-hop communications,” Chapter 8 in Reed-Solomon Codes and Their Applications, S. B. Wicker and V. K. Bhargava (eds.), pp. 150-174, IEEE Press, New York, 1994. [10] M. B. Pursley and J. S. Skinner, “Turbo product coding in frequency-hop wireless communications with partial-band interference,” Proceedings of the 2002 IEEE Military Communications Conference (Anaheim, CA), pp. U405.5.1-6, October 2002. [11] M. B. Pursley and W. E. Stark, “Performance of Reed-Solomon coded frequency-hop spread spectrum communications in partial-band interference,” IEEE Transactions on Communications, vol. COM-33, pp. 767774, August 1985. [12] A. J. Viterbi and I. M. Jacobs, “Advances in coding and modulation for noncoherent channels affected by fading, partial band, and multipleaccess interference,” in Advances in Communication Systems: Theory and Applications, vol. 4, A. J. Viterbi (ed.), pp. 279-308, Academic Press, New York, 1975. [13] Q. Zhang and T. Le-Ngoc, “Turbo product codes for FH-SS with partialband interference,” IEEE Transactions on Wireless Communications, vol. 1, no. 3, pp. 513-520, July 2002.

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