Design of rate-compatible punctured serial ... - IEEE Xplore
Design of Rate-Compatible Punctured Serial Concatenated Convolutional Codes Fulvio Babich
Guido Montorsi
Francesca Vatta
DEEI, Universit`a di Trieste Via A. Valerio, 10 I-34127 Trieste (Italy) Email: [email protected]
Dipartimento di Elettronica Politecnico di Torino Corso Duca degli Abruzzi, 24 I-10129 Torino (Italy) Email: [email protected]
DEEI, Universit`a di Trieste Via A. Valerio, 10 I-34127 Trieste (Italy) Email: [email protected]
Abstract— In this paper, we propose a new design criterion to obtain well performing rate-compatible serial concatenated convolutional codes (SCCC) families. To obtain rate-compatible SCCCs, the puncturing is limited to inner coded bits. However, the puncturing is not restricted to inner parity bits, but extended also to inner systematic bits, thus obtaining high rate SCCCs (i.e., beyond the outer code rate). The considerations presented in [1] to find “best” component encoders for SCCCs construction are extended to find the “best” rate-compatible puncturing patterns for a given input decoding delay I. A rate-compatibility restriction to the puncturing rule is used, implying that all the code bits of a high-rate punctured code are used by the lower rate codes. The two main applications of this technique are its use in hybrid ARQ/FEC schemes and to achieve unequal error protection (UEP) of an information sequence.
I. I NTRODUCTION In this paper, we propose a new design criterion to obtain well performing rate-compatible SCCC families. A family of so-called rate-compatible punctured codes [2] is obtained by adding a rate-compatibility restriction to the puncturing rule. This restriction requires that the rates are organized in a hierarchy, where all coded bits of a higher rate code are used by all lower rate codes. The concept of rate-compatible codes was extended to parallel concatenated convolutional codes (PCCCs) for instance in [3] and [4]. Design criteria for the puncturing patterns have successively appeared in [5] and [6]. Shortly after the discovery of PCCCs, serial concatenated convolutional codes (SCCCs) were proposed as an alternative to them, since SCCCs generally have lower error floors [1]. Thus, the concept of rate-compatible codes was extended to SCCCs in [7] and [8], since SCCCs show less error flooring problems even with small interleavers [1]. The two main applications of this technique are: 1) Modified type-II Automatic Repeat reQuest / ForwardError Correction (ARQ/FEC) schemes. The principle of this hybrid ARQ/FEC scheme [9] is not to repeat information or parity bits if the transmission is unsuccessful, as in previous type II ARQ/FEC schemes, but to transmit additional code bits of a lower rate RCPC code, until the code is powerful enough to enable decoding. Namely, if the higher rate codes are not sufficiently powerful to decode channel errors, only
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supplemental bits, which were previously punctured, have to be transmitted in order to upgrade the code. This includes several decoding attempts on the receive side. 2) Unequal error protection (UEP). Since codes are compatible, rate variation within a data frame is possible to achieve unequal error protection: this is required when different levels of error protection for different parts of an information sequence or block are needed. To obtain rate-compatible SCCCs, the puncturing must be limited to inner coded bits. However, we do not limit the puncturing to inner parity bits only, but we extend it also to inner systematic bits, thus obtaining high rate SCCCs (i.e., beyond the outer code rate). With this assumption, the punctured code performance depends not only from the puncturing pattern, optimized following the design rules presented in [1], but also from the percentage of inner systematic and, consequently, parity bits to be punctured to obtain a given inner rate. Thus, the problem of finding the optimal percentage of inner systematic and, consequently, parity bits to be punctured to obtain a given inner rate must be addressed: it is shown that the best puncturing strategy should be chosen with regard to the application considered (i.e., with regard to the signal-tonoise ratios (SNRs) at which the SCCCs are to be employed). The paper is organized as follows. In Section II the proposed puncturing design rule to obtain good systematic ratecompatible SCCC families is outlined. In Section III, the puncturing methods that can be applied are addressed. Their performance is discussed in Section IV. Finally, Section V summarizes the conclusions. II. T HE NEW DESIGN CRITERION The proposal of rate-compatible serial concatenated convolutional codes (SCCC) has been introduced in [7] and [8]. Here, no design criteria are given. In particular, in [7] and [8], some rate-compatible patterns for inner code parity bits only are presented and evaluated with different puncturing periods, frame sizes and mother codes. The first paper dealing with design criteria is [10] where, as done in [11] for PCCCs, design criteria for rate R = k/(k+1) (3 ≤ k ≤ 15) punctured SCCCs
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are given, deriving high-rate codes via puncturing some basic SCCCs with various memory sizes. However, the problem of finding rate compatible patterns is not specifically addressed in [10]. The design of a turbo-like code using two constituent encoders and one interleaver involves the choice of the interleaver and the constituent encoders. The joint optimization, on the other hand, seems to pose unsolvable complexity problems. To decouple the codes design from the interleaver design, a uniform interleaver approach has been proposed in [12], where the authors suggested replacing the actual interleaver with the average interleaver1 . Following this approach, the constituent encoders are chosen first in order to match every interleaver close enough to the average, i.e., every interleaver possessing some degree of randomness. In this paper we will base our design on the uniform interleaver approach. Using this approach, in order to design a rate-compatible punctured SCCC, the code choice consists essentially in finding the puncturing patterns satisfying some optimality criteria subject to the compatibility constraint. We discuss here a new design criterion for the puncturing patterns based on the considerations presented in [1] to find “best” component encoders for SCCCs construction. These considerations can be extended to find the “best” rate-compatible puncturing patterns for a given input decoding delay I. The criterion obtained applying the rules indicated in [1] is expected to give promising results, like those obtained in [1], where it was applied to find good constituent convolutional codes for serial concatenated codes construction. To show this, the simulated performance of the codes obtained applying the rules indicated in [1], and the theoretical performance limits obtained through the application of the sphere packing bound [13], will be compared in Section IV. Applying the design rules stated in [1], the optimized puncturing pattern can be found considering the inner parity and systematic bits separately. Namely: 1) To find the optimum puncturing pattern for inner code parity bits, start puncturing the inner mother code parity bits one bit at a time, fulfilling the rate-compatibility restriction. Define as dw the minimum weight of inner code words generated by input words with weight w, and by Nw the number of nearest neighbors (multiplicities) with weight dw . Select at each step the candidate puncturing pattern P{Pi } (see Fig. 1) for the inner code parity bits as the one yielding the optimum values for (dw , Nw ) for w = 2, . . . , wmax , i.e., the one which sequentially optimize the pairs (dw , Nw ) (first dw is maximized and then Nw is minimized). 2) Since the systematic bits at the input of the inner encoder are an interleaved version of the outer encoder output bits, select the candidate puncturing pattern for the inner code systematic bits as the one yielding the best outer code output weight enumerating function (OWEF). Namely, to find the optimum puncturing pattern for inner 1 This
III. R ATE - COMPATIBLE PUNCTURED SCCC S CODES : PUNCTURING METHODS
Depending on the puncturing pattern, the resulting code may be systematic (none of the systematic bits are punctured), partially-systematic (some of the systematic bits are punctured) or non-systematic (all systematic bits are punctured). In [7] and [8], to obtain rate-compatible SCCCs, inner code parity bits only were punctured, thus obtaining SCCC rates lower than, or at most equal to, the outer code rate. This choice was made since it was assumed that a systematic inner code performs better than a partially-systematic or a non-systematic one, when used in a serially concatenated scheme with interleaver. This assumption, however, is not valid for all SNRs. Namely, given the role of systematic bits in determining the convergence threshold of an iterative decoding mechanism [14], their presence is very important in the socalled waterfall region of the resulting turbo-like code, i.e., for lower SNRs (which is usually meant for ARQ applications). On the other hand, a systematic code gives rise, with respect to a partially-systematic one, to a performance which is worse in the so-called error floor region of the resulting turbo-like code, i.e., for higher SNRs (which is usually meant for UEP applications), as shown, for instance, in [15]. IV. R ESULTS AND COMPARISONS AMONG THE DIFFERENT PUNCTURING METHODS
Define, as in [15], the systematic permeability rate ρu and the parity permeability rate ρp as the proportion of the inner code systematic bits and of the inner code parity bits which survive (i.e., are not punctured), respectively. Assume an inner 2 This method minimizes the rate of growth of the OWEF with the output distance d.
average interleaver is actually the weighted set of all interleavers.
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code systematic bits, start puncturing the outer mother code output bits one bit at a time, fulfilling the ratecompatibility restriction. Define as A(d) the number of nearest neighbors (multiplicities) with output distance d of the outer code. Select at each step the candidate puncturing pattern P{So +Po } (see Fig. 1) for the outer code information and parity bits as the one yielding the optimum values for A(d), i.e., the one which sequentially optimize the values A(d) for d = dfree , . . . , dmax 2 , where dfree is the outer code free distance. Since also outer code information bits are punctured, the invertibility of the outer code at each step must be guaranteed. At the end, take the best puncturing pattern P{So +Po } and apply its interleaved version π[P{So +Po } ] to inner code systematic bits (see Fig. 1). Being the outer and inner codes systematic, the knowledge of the interleaver-induced permutation permits to identify, within the inner systematic bits, which of them correspond to outer code systematic bits, and which to outer code parity bits. Thus, the best puncturing scheme for outer code output bits can be limited to outer parity bits only. In this case, the invertibility of the whole SCCC scheme is straightforward.
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I
Rate 1/2 encoder
1 1 1 0
So
Interleaver
Fig. 1.
π [P{So+Po}]
Pi
P{Pi}
−1
10
ρp=10/30
−2
10
(1)
3 The above mentioned SCCC mother code was one of the initial candidates for the UMTS standard. Its scheme is depicted, for instance, in Fig. 1 of [8]. 4 The interleaver length N is given by I/R (see [1]). o 5 The optimal puncturing position is the one giving the best code performance from the point of view of the rule applied.
554
Block Error Rate
1 ρu + (n − 1)ρp
From (1), given a certain desired RSCCC , ρu and ρp are related by: Ro ρu = − (n − 1)ρp (2) RSCCC We have compared through simulation some ratecompatible puncturing schemes, obtained starting from a rate 1/3 SCCC mother code formed by serially concatenating a 4-states rate 2/3 outer systematic recursive convolutional code (SRCC) and a 4-states rate 1/2 inner SRCC3 . A non periodic puncturing has been applied on inner output bits only. The frame length, measured in terms of input information bits, has been set to I = 200 using a random interleaver4 . This choice has been made since the uniform interleaver approach has been applied to design the puncturing patterns: thus, the optimality of a certain pattern is not due to its compatibility with a particular interleaving scheme, but to the optimality of the average distance spectrum of the code obtained applying that pattern. Thus, since there is no particular compatibility of the puncturing patterns with a specific interleaver, the simulation is performed using a random interleaver. In Tables I and II we report the optimized rate-compatible puncturing patterns obtained applying the design rules stated in Section II. The algorithm to find the optimal puncturing pattern (where optimal is intended to be according to one of the two rules previously mentioned in Section II) works sequentially, by puncturing one bit at a time in the optimal position5 , subject to the constraints of rate compatibility and code invertibility. This sequential puncturing is performed starting from the lowest rate code (i.e., the mother code) and ending up at the highest possible rate. In Table I we report some rate-compatible puncturing patterns for inner code parity bits that are used to obtain the results shown in the following. The puncturing positions of P{Pi } go from 1 to the interleaver length N (see Fig. 1). In Table II we report some rate-compatible puncturing patterns for inner code systematic bits that are used to obtain the results shown in the following. The puncturing positions of P{So +Po } go from 1 to 2I (see Fig. 1), since the outer
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Si
Block diagram of the transmitter with the turbo encoder.
mother code of rate 1/n. Given the variable inner code rate Ri and the outer code rate Ro , which is fixed, the rate of the resulting SCCC is given by: RSCCC = Ro Ri = Ro
N
Po
Rate 1/2 encoder
To the channel
Information bits
ρ =6/30 p
−3
10
−4
10
ρ =4/30 p
−5
10
ρ =8/30 p
ρp=9/30
−6
10
0.5
1
1.5
2 SNR [dB]
2.5
3
3.5
Fig. 2. Block Error Rate vs. Signal-to-Noise Ratio Es /N0 for different SCCCs with total rate RSCCC = 2/3. Sphere packing bound limiting performance: solid curve. Simulation results obtained applying the puncturing patterns of Tables I and II for some parity permeability rates ρp : dash-dotted curves.
mother code is a rate 2/3 code which is obtained from a rate 1/2 native code by applying the fixed puncturing pattern shown in Fig. 1 and in the first row of Table II. Thus, the puncturing pattern P{So +Po } has been supposed to include these fixed puncturing positions in itself. This is the reason why the puncturing positions of P{So +Po } go from 1 to 2I instead of going from 1 to N = I/Ro (as the puncturing positions of P{Pi } ). The performance of a rate-compatible SCCC depends mainly on its overall rate RSCCC and on the chosen combination of ρu and ρp needed to obtain it (using (2)). To obtain a certain ρp the patterns are applied only to inner code parity check bit positions. To obtain a certain ρu , the listed patterns are applied only to the inner code systematic bit positions through the following steps: 1) find the outer code output bit positions given by the puncturing pattern; 2) interleave those positions using the interleaver; 3) puncture the inner code output bit stream in the corresponding interleaved positions. Fig. 2 shows the performance of the SCCCs with total rate RSCCC = 2/3 on the additive white Gaussian noise (AWGN) channel, in terms of residual Block Error Rate (BLER) vs. Es /N0 in dB, being Es = RSCCC Eb and Eb the energy per
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TABLE I P UNCTURING PATTERNS ( IN OCTAL FORM ) FOR THE INNER CODE PARITY BIT POSITIONS GOING FROM 1 TO N .
ρp 30/30
7 777
19/30
1 663
10/30
1 062
9/30
1 022
8/30
1 022
6/30
0 022
4/30
0 020
1/20
0 000
777 777 347 167 142 122 142 122 102 122 102 022 102 020 000 000
777 777 315 156 114 046 110 044 110 044 010 044 010 040 010 040
777 777 473 363 443 222 443 222 442 222 402 202 002 200 000 000
777 777 334 356 110 112 110 112 110 102 100 102 100 102 000 002
777 777 733 356 211 112 211 102 211 102 211 100 210 000 200 000
777 777 171 356 111 244 111 244 011 244 010 204 010 204 010 004
Puncturing pattern 777 777 777 777 777 777 777 777 777 777 671 671 546 636 354 635 563 546 716 715 211 211 144 422 114 231 062 046 204 510 211 211 104 422 104 221 062 046 204 510 201 211 104 422 104 221 042 042 204 510 201 210 104 420 104 201 042 040 204 410 200 210 100 020 100 001 002 040 204 010 000 200 000 020 100 000 002 000 004 010
777 777 671 467 211 462 211 422 210 422 210 420 210 020 000 000
777 777 671 166 220 144 220 104 220 104 020 104 020 104 000 004
777 777 653 633 451 231 451 221 451 221 441 021 401 001 400 000
777 777 473 154 021 104 021 104 021 104 021 004 020 004 020 000
777 777 157 734 111 510 111 510 101 500 101 100 001 000 000 000
TABLE II P UNCTURING PATTERNS ( IN OCTAL FORM ) FOR THE ENTIRE SET OF INNER CODE SYSTEMATIC BIT POSITIONS GOING FROM 1 TO 2I .
ρu 30/30
16 735 673
26/30
12 725 672
24/30
12 525 672
22/30
12 525 272
21/30
12 525 252
20/30
12 525 252
567 735 567 527 735 567 525 735 527 525 725 525 525 525 525 525 525 525
356 673 356 356 273 256 352 253 252 352 253 252 352 253 252 252 252 252
735 567 735 535 527 725 535 527 725 525 525 725 525 525 725 525 525 525
673 356 673 652 256 273 652 256 273 252 256 253 252 252 252 252 252 252
567 735 567 567 725 565 565 525 525 565 525 525 565 525 525 525 525 525
information bit. Being n = 2, ρu is obtained from ρp using (2). To obtain a better performance in the error floor region (i.e., for higher SNRs), it is advantageous to puncture more severely the inner systematic bits [1]. In fact, the lowest error floors are obtained with ρp = 9/30 and ρp = 10/30 (ρu = 21/30 and ρu = 20/30, respectively). On the other hand, to improve the performance in the waterfall region (i.e., for lower SNRs), it is advantageous to puncture more inner parity bits, since, in this way, the largest number of systematic bits will help the convergence of the iterative decoding algorithm. In fact, the best performance for lower SNRs is obtained with ρp = 6/30 (ρu = 24/30). Puncturing the inner parity check bits cannot go beyond a certain threshold, though, since an insufficient number of inner parity bits would make the inner code so weak that the
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Puncturing pattern 356 735 673 567 356 673 567 356 735 673 356 735 673 567 356 352 735 253 565 256 672 527 352 735 272 356 725 673 527 352 252 735 252 565 256 672 527 352 535 252 356 725 272 527 352 252 725 252 565 252 252 527 252 535 252 252 725 272 525 352 252 525 252 525 252 252 527 252 535 252 252 725 272 525 352 252 525 252 525 252 252 525 252 525 252 252 725 252 525 252
735 567 735 725 567 535 725 567 535 725 565 525 725 525 525 525 525 525
673 356 673 653 352 672 253 252 652 252 252 652 252 252 252 252 252 252
567 735 567 565 725 567 565 725 527 565 525 525 525 525 525 525 525 525
356 673 356 352 672 252 252 672 252 252 672 252 252 652 252 252 252 252
performance of the overall SCCC code becomes bad for all SNRs [16]. To see this, consider, for example, the performance with ρp = 4/30 (ρu = 26/30). Finally, we observe that the code, the resulting performance of which is obtained with ρp = 9/30 and ρu = 21/30, respectively, represents a good compromise between waterfall and error floor performance. Fig 3 shows the simulated performance of the SCCCs with rate RSCCC = 2/3 in terms of residual BLER vs. ρp , for different Eb /N0 in dB. The curves show that the higher the SNR, and hence the lower the target BLER, the heavier should be the puncturing on inner systematic bits. Finally, in Fig 4, theoretical performance limits obtained through the application of the sphere packing bound [13] and simulation results are given in terms of residual BLER vs.
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−1
V. C ONCLUSIONS
10
E \N =2.8 dB b
E \N =3.2 dB
−2
10
b
0
Eb\N0=3.6 dB
−3
10
Eb\N0=4.0 dB −4
10
E \N =4.4 dB b
0
E \N =4.8 dB
−5
10
b
0
E \N =5.2 dB b 0
R EFERENCES
−6
10
0.09
0.13
0.17
0.21 ρ
0.25
0.29
0.33
p
Fig. 3. Block Error Rate vs. ρp for different Eb /N0 in dB and for SCCCs with total rate RSCCC = 2/3.
10
10
Block Error Rate
Block Error Rate
In this paper we have proposed a design method for the construction of well performing families of rate-compatible punctured SCCCs. To obtain rate-compatible SCCCs, the puncturing has not been limited to inner parity bits only, but has been extended also to inner systematic bits, thus obtaining higher rate SCCCs (i.e., beyond the outer code rate). Moreover, the problem of finding the optimal percentage of inner systematic and, consequently, parity bits to be punctured to obtain a given SCCC rate has been addressed: it has been shown that the best puncturing strategy must be chosen with regard to the application considered.
0
10
10
−1
−2
−3
−4
4/5
2/3
1/2 10
8/9
−5
−2
−1
0
1
2 SNR [dB]
3
4
5
6
Fig. 4. Frame Error Rate vs. Signal-to-Noise Ratio Es /N0 for different code rates RSCCC = {1/2, 2/3, 4/5, 8/9}. Simulation results: dash-dotted curves. Sphere-packing bound limiting performance: solid curves.
Es /N0 in dB for rate-compatible SCCCs with rates from 1/2 up to 8/9. All plots are obtained with ρu = 21/30, which was shown in Fig. 2 to give a code representing a reasonable compromise between the waterfall and the error floor performance. For each rate, the condition (2) is satisfied. Puncturing optimization has led to ρp = 19/30 (rate 1/2 code), ρp = 9/30 (rate 2/3 code), ρp = 4/30 (rate 4/5 code), ρp = 1/20 (rate 8/9 code). In most cases, the performance is within less than 1 dB from the sphere packing bound (such as the best codes presented in [13]), despite the random interleaver and its short length.
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[1] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design and iterative decoding”, IEEE Transactions on Information Theory, Vol. 44, n. 3, May 1998, pp. 909-926. [2] J. Hagenauer, “Rate-compatible punctured convolutional codes (RCPC codes) and their applications”, IEEE Transactions on Communications, Vol. 36, No. 4, April 1988, pp. 389-400. [3] A. S. Barbulescu and S. S. Pietrobon, “Rate compatible turbo codes”, IEE Electronics Letters, 30th March 1995, Vol. 31, No. 7, pp. 535-536. [4] D. N. Rowitch and L. B. Milstein, “Rate compatible punctured turbo (RCPT) codes in a hybrid FEC/ARQ system”, Proc. of the IEEE Communication Theory Mini Conference, held in conjunction with GLOBECOM’97, Phoenix, Arizona, November 5-6, 1997, pp. 55-59. [5] P. Jung and J. Plechinger, “Performance of rate compatible punctured turbo codes for mobile radio applications”, IEE Electronics Letters, 4th December 1997, Vol. 33, No. 25, pp. 2102-2103. [6] D. N. Rowitch and L. B. Milstein, “On the performance of hybrid FEC/ARQ systems using rate compatible punctured turbo (RCPT) codes”, IEEE Transactions on Communications, Vol. 48, No. 6, June 2000, pp. 948-959. [7] N. Chandran and M. C. Valenti, “Hybrid ARQ using serial concatenated convolutional codes over fading channels”, Proc. of the 2001 IEEE 53rd Vehicular Technology Conference - VTC’01 Spring, Vol. 2, Rhodes, Greece, 6-9 May 2001, pp. 1410-1414. [8] H. Kim and G. L. St¨uber, “Rate compatible punctured SCCC”, Proc. of the 2001 IEEE 54th Vehicular Technology Conference - VTC’01 Fall, Vol. 4, Atlantic City, NJ, U.S.A., 7-11 October 2001, pp. 2399-2403. [9] S. Kallel and D. Haccoun, “Generalized type II hybrid ARQ scheme using punctured convolutional coding”, IEEE Transactions on Communications, Vol. 38, No. 11, November 1990, pp. 1938-1946. ¨ F. Ac¸ikel and W. E. Ryan, “Punctured high rate SCCCs for [10] O. BPSK/QPSK channels”, Proc. of the 2000 IEEE International Conference on Communications - ICC 2000, New Orleans, U.S.A, 18-22 June 2000, pp. 434-438. ¨ F. Ac¸ikel and W. E. Ryan, “Punctured turbo-codes for BPSK/QPSK [11] O. channels”, IEEE Transactions on Communications, Vol. 47, No. 9, September 1999, pp. 1315-1323. [12] S. Benedetto and G. Montorsi, “Design of parallel concatenated convolutional codes”, IEEE Transactions on Communications, Vol. 44, No. 5, May 1996, pp. 591-600. [13] C. Schlegel and L. Perez, “On error bounds and turbo-codes”, IEEE Communications Letters, Vol. 3, n. 7, July 1999, pp. 205-207. [14] D. Divsalar, S. Dolinar and F. Pollara, “Iterative turbo decoder analysis based on density evolution”, IEEE Journal on Selected Areas in Communications, Vol. 19, No. 5, May 2001, pp. 891-907. [15] I. Land and P. Hoeher, “Partially systematic rate 1/2 Turbo codes”, Proc. of the II International Symposium on Turbo codes and related topics, Brest, 4-7 September 2000, pp. 287-290. [16] M. A. Herro, D. J. Costello and L. Hu, “Capacity and cutoff rate calculations for a concatenated coding system”, IEEE Transactions on Information Theory, Vol. 34, n. 2, March 1988, pp. 212-222. [17] S. Benedetto, R. Garello and G. Montorsi, “A search for good convolutional codes to be used in the construction of turbo codes”, IEEE Transactions on Communications, Vol. 46, n. 9, September 1998, pp. 1101-1105.
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Guido Montorsi
Francesca Vatta
DEEI, Universit`a di Trieste Via A. Valerio, 10 I-34127 Trieste (Italy) Email: [email protected]
Dipartimento di Elettronica Politecnico di Torino Corso Duca degli Abruzzi, 24 I-10129 Torino (Italy) Email: [email protected]
DEEI, Universit`a di Trieste Via A. Valerio, 10 I-34127 Trieste (Italy) Email: [email protected]
Abstract— In this paper, we propose a new design criterion to obtain well performing rate-compatible serial concatenated convolutional codes (SCCC) families. To obtain rate-compatible SCCCs, the puncturing is limited to inner coded bits. However, the puncturing is not restricted to inner parity bits, but extended also to inner systematic bits, thus obtaining high rate SCCCs (i.e., beyond the outer code rate). The considerations presented in [1] to find “best” component encoders for SCCCs construction are extended to find the “best” rate-compatible puncturing patterns for a given input decoding delay I. A rate-compatibility restriction to the puncturing rule is used, implying that all the code bits of a high-rate punctured code are used by the lower rate codes. The two main applications of this technique are its use in hybrid ARQ/FEC schemes and to achieve unequal error protection (UEP) of an information sequence.
I. I NTRODUCTION In this paper, we propose a new design criterion to obtain well performing rate-compatible SCCC families. A family of so-called rate-compatible punctured codes [2] is obtained by adding a rate-compatibility restriction to the puncturing rule. This restriction requires that the rates are organized in a hierarchy, where all coded bits of a higher rate code are used by all lower rate codes. The concept of rate-compatible codes was extended to parallel concatenated convolutional codes (PCCCs) for instance in [3] and [4]. Design criteria for the puncturing patterns have successively appeared in [5] and [6]. Shortly after the discovery of PCCCs, serial concatenated convolutional codes (SCCCs) were proposed as an alternative to them, since SCCCs generally have lower error floors [1]. Thus, the concept of rate-compatible codes was extended to SCCCs in [7] and [8], since SCCCs show less error flooring problems even with small interleavers [1]. The two main applications of this technique are: 1) Modified type-II Automatic Repeat reQuest / ForwardError Correction (ARQ/FEC) schemes. The principle of this hybrid ARQ/FEC scheme [9] is not to repeat information or parity bits if the transmission is unsuccessful, as in previous type II ARQ/FEC schemes, but to transmit additional code bits of a lower rate RCPC code, until the code is powerful enough to enable decoding. Namely, if the higher rate codes are not sufficiently powerful to decode channel errors, only
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supplemental bits, which were previously punctured, have to be transmitted in order to upgrade the code. This includes several decoding attempts on the receive side. 2) Unequal error protection (UEP). Since codes are compatible, rate variation within a data frame is possible to achieve unequal error protection: this is required when different levels of error protection for different parts of an information sequence or block are needed. To obtain rate-compatible SCCCs, the puncturing must be limited to inner coded bits. However, we do not limit the puncturing to inner parity bits only, but we extend it also to inner systematic bits, thus obtaining high rate SCCCs (i.e., beyond the outer code rate). With this assumption, the punctured code performance depends not only from the puncturing pattern, optimized following the design rules presented in [1], but also from the percentage of inner systematic and, consequently, parity bits to be punctured to obtain a given inner rate. Thus, the problem of finding the optimal percentage of inner systematic and, consequently, parity bits to be punctured to obtain a given inner rate must be addressed: it is shown that the best puncturing strategy should be chosen with regard to the application considered (i.e., with regard to the signal-tonoise ratios (SNRs) at which the SCCCs are to be employed). The paper is organized as follows. In Section II the proposed puncturing design rule to obtain good systematic ratecompatible SCCC families is outlined. In Section III, the puncturing methods that can be applied are addressed. Their performance is discussed in Section IV. Finally, Section V summarizes the conclusions. II. T HE NEW DESIGN CRITERION The proposal of rate-compatible serial concatenated convolutional codes (SCCC) has been introduced in [7] and [8]. Here, no design criteria are given. In particular, in [7] and [8], some rate-compatible patterns for inner code parity bits only are presented and evaluated with different puncturing periods, frame sizes and mother codes. The first paper dealing with design criteria is [10] where, as done in [11] for PCCCs, design criteria for rate R = k/(k+1) (3 ≤ k ≤ 15) punctured SCCCs
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are given, deriving high-rate codes via puncturing some basic SCCCs with various memory sizes. However, the problem of finding rate compatible patterns is not specifically addressed in [10]. The design of a turbo-like code using two constituent encoders and one interleaver involves the choice of the interleaver and the constituent encoders. The joint optimization, on the other hand, seems to pose unsolvable complexity problems. To decouple the codes design from the interleaver design, a uniform interleaver approach has been proposed in [12], where the authors suggested replacing the actual interleaver with the average interleaver1 . Following this approach, the constituent encoders are chosen first in order to match every interleaver close enough to the average, i.e., every interleaver possessing some degree of randomness. In this paper we will base our design on the uniform interleaver approach. Using this approach, in order to design a rate-compatible punctured SCCC, the code choice consists essentially in finding the puncturing patterns satisfying some optimality criteria subject to the compatibility constraint. We discuss here a new design criterion for the puncturing patterns based on the considerations presented in [1] to find “best” component encoders for SCCCs construction. These considerations can be extended to find the “best” rate-compatible puncturing patterns for a given input decoding delay I. The criterion obtained applying the rules indicated in [1] is expected to give promising results, like those obtained in [1], where it was applied to find good constituent convolutional codes for serial concatenated codes construction. To show this, the simulated performance of the codes obtained applying the rules indicated in [1], and the theoretical performance limits obtained through the application of the sphere packing bound [13], will be compared in Section IV. Applying the design rules stated in [1], the optimized puncturing pattern can be found considering the inner parity and systematic bits separately. Namely: 1) To find the optimum puncturing pattern for inner code parity bits, start puncturing the inner mother code parity bits one bit at a time, fulfilling the rate-compatibility restriction. Define as dw the minimum weight of inner code words generated by input words with weight w, and by Nw the number of nearest neighbors (multiplicities) with weight dw . Select at each step the candidate puncturing pattern P{Pi } (see Fig. 1) for the inner code parity bits as the one yielding the optimum values for (dw , Nw ) for w = 2, . . . , wmax , i.e., the one which sequentially optimize the pairs (dw , Nw ) (first dw is maximized and then Nw is minimized). 2) Since the systematic bits at the input of the inner encoder are an interleaved version of the outer encoder output bits, select the candidate puncturing pattern for the inner code systematic bits as the one yielding the best outer code output weight enumerating function (OWEF). Namely, to find the optimum puncturing pattern for inner 1 This
III. R ATE - COMPATIBLE PUNCTURED SCCC S CODES : PUNCTURING METHODS
Depending on the puncturing pattern, the resulting code may be systematic (none of the systematic bits are punctured), partially-systematic (some of the systematic bits are punctured) or non-systematic (all systematic bits are punctured). In [7] and [8], to obtain rate-compatible SCCCs, inner code parity bits only were punctured, thus obtaining SCCC rates lower than, or at most equal to, the outer code rate. This choice was made since it was assumed that a systematic inner code performs better than a partially-systematic or a non-systematic one, when used in a serially concatenated scheme with interleaver. This assumption, however, is not valid for all SNRs. Namely, given the role of systematic bits in determining the convergence threshold of an iterative decoding mechanism [14], their presence is very important in the socalled waterfall region of the resulting turbo-like code, i.e., for lower SNRs (which is usually meant for ARQ applications). On the other hand, a systematic code gives rise, with respect to a partially-systematic one, to a performance which is worse in the so-called error floor region of the resulting turbo-like code, i.e., for higher SNRs (which is usually meant for UEP applications), as shown, for instance, in [15]. IV. R ESULTS AND COMPARISONS AMONG THE DIFFERENT PUNCTURING METHODS
Define, as in [15], the systematic permeability rate ρu and the parity permeability rate ρp as the proportion of the inner code systematic bits and of the inner code parity bits which survive (i.e., are not punctured), respectively. Assume an inner 2 This method minimizes the rate of growth of the OWEF with the output distance d.
average interleaver is actually the weighted set of all interleavers.
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code systematic bits, start puncturing the outer mother code output bits one bit at a time, fulfilling the ratecompatibility restriction. Define as A(d) the number of nearest neighbors (multiplicities) with output distance d of the outer code. Select at each step the candidate puncturing pattern P{So +Po } (see Fig. 1) for the outer code information and parity bits as the one yielding the optimum values for A(d), i.e., the one which sequentially optimize the values A(d) for d = dfree , . . . , dmax 2 , where dfree is the outer code free distance. Since also outer code information bits are punctured, the invertibility of the outer code at each step must be guaranteed. At the end, take the best puncturing pattern P{So +Po } and apply its interleaved version π[P{So +Po } ] to inner code systematic bits (see Fig. 1). Being the outer and inner codes systematic, the knowledge of the interleaver-induced permutation permits to identify, within the inner systematic bits, which of them correspond to outer code systematic bits, and which to outer code parity bits. Thus, the best puncturing scheme for outer code output bits can be limited to outer parity bits only. In this case, the invertibility of the whole SCCC scheme is straightforward.
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I
Rate 1/2 encoder
1 1 1 0
So
Interleaver
Fig. 1.
π [P{So+Po}]
Pi
P{Pi}
−1
10
ρp=10/30
−2
10
(1)
3 The above mentioned SCCC mother code was one of the initial candidates for the UMTS standard. Its scheme is depicted, for instance, in Fig. 1 of [8]. 4 The interleaver length N is given by I/R (see [1]). o 5 The optimal puncturing position is the one giving the best code performance from the point of view of the rule applied.
554
Block Error Rate
1 ρu + (n − 1)ρp
From (1), given a certain desired RSCCC , ρu and ρp are related by: Ro ρu = − (n − 1)ρp (2) RSCCC We have compared through simulation some ratecompatible puncturing schemes, obtained starting from a rate 1/3 SCCC mother code formed by serially concatenating a 4-states rate 2/3 outer systematic recursive convolutional code (SRCC) and a 4-states rate 1/2 inner SRCC3 . A non periodic puncturing has been applied on inner output bits only. The frame length, measured in terms of input information bits, has been set to I = 200 using a random interleaver4 . This choice has been made since the uniform interleaver approach has been applied to design the puncturing patterns: thus, the optimality of a certain pattern is not due to its compatibility with a particular interleaving scheme, but to the optimality of the average distance spectrum of the code obtained applying that pattern. Thus, since there is no particular compatibility of the puncturing patterns with a specific interleaver, the simulation is performed using a random interleaver. In Tables I and II we report the optimized rate-compatible puncturing patterns obtained applying the design rules stated in Section II. The algorithm to find the optimal puncturing pattern (where optimal is intended to be according to one of the two rules previously mentioned in Section II) works sequentially, by puncturing one bit at a time in the optimal position5 , subject to the constraints of rate compatibility and code invertibility. This sequential puncturing is performed starting from the lowest rate code (i.e., the mother code) and ending up at the highest possible rate. In Table I we report some rate-compatible puncturing patterns for inner code parity bits that are used to obtain the results shown in the following. The puncturing positions of P{Pi } go from 1 to the interleaver length N (see Fig. 1). In Table II we report some rate-compatible puncturing patterns for inner code systematic bits that are used to obtain the results shown in the following. The puncturing positions of P{So +Po } go from 1 to 2I (see Fig. 1), since the outer
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Si
Block diagram of the transmitter with the turbo encoder.
mother code of rate 1/n. Given the variable inner code rate Ri and the outer code rate Ro , which is fixed, the rate of the resulting SCCC is given by: RSCCC = Ro Ri = Ro
N
Po
Rate 1/2 encoder
To the channel
Information bits
ρ =6/30 p
−3
10
−4
10
ρ =4/30 p
−5
10
ρ =8/30 p
ρp=9/30
−6
10
0.5
1
1.5
2 SNR [dB]
2.5
3
3.5
Fig. 2. Block Error Rate vs. Signal-to-Noise Ratio Es /N0 for different SCCCs with total rate RSCCC = 2/3. Sphere packing bound limiting performance: solid curve. Simulation results obtained applying the puncturing patterns of Tables I and II for some parity permeability rates ρp : dash-dotted curves.
mother code is a rate 2/3 code which is obtained from a rate 1/2 native code by applying the fixed puncturing pattern shown in Fig. 1 and in the first row of Table II. Thus, the puncturing pattern P{So +Po } has been supposed to include these fixed puncturing positions in itself. This is the reason why the puncturing positions of P{So +Po } go from 1 to 2I instead of going from 1 to N = I/Ro (as the puncturing positions of P{Pi } ). The performance of a rate-compatible SCCC depends mainly on its overall rate RSCCC and on the chosen combination of ρu and ρp needed to obtain it (using (2)). To obtain a certain ρp the patterns are applied only to inner code parity check bit positions. To obtain a certain ρu , the listed patterns are applied only to the inner code systematic bit positions through the following steps: 1) find the outer code output bit positions given by the puncturing pattern; 2) interleave those positions using the interleaver; 3) puncture the inner code output bit stream in the corresponding interleaved positions. Fig. 2 shows the performance of the SCCCs with total rate RSCCC = 2/3 on the additive white Gaussian noise (AWGN) channel, in terms of residual Block Error Rate (BLER) vs. Es /N0 in dB, being Es = RSCCC Eb and Eb the energy per
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TABLE I P UNCTURING PATTERNS ( IN OCTAL FORM ) FOR THE INNER CODE PARITY BIT POSITIONS GOING FROM 1 TO N .
ρp 30/30
7 777
19/30
1 663
10/30
1 062
9/30
1 022
8/30
1 022
6/30
0 022
4/30
0 020
1/20
0 000
777 777 347 167 142 122 142 122 102 122 102 022 102 020 000 000
777 777 315 156 114 046 110 044 110 044 010 044 010 040 010 040
777 777 473 363 443 222 443 222 442 222 402 202 002 200 000 000
777 777 334 356 110 112 110 112 110 102 100 102 100 102 000 002
777 777 733 356 211 112 211 102 211 102 211 100 210 000 200 000
777 777 171 356 111 244 111 244 011 244 010 204 010 204 010 004
Puncturing pattern 777 777 777 777 777 777 777 777 777 777 671 671 546 636 354 635 563 546 716 715 211 211 144 422 114 231 062 046 204 510 211 211 104 422 104 221 062 046 204 510 201 211 104 422 104 221 042 042 204 510 201 210 104 420 104 201 042 040 204 410 200 210 100 020 100 001 002 040 204 010 000 200 000 020 100 000 002 000 004 010
777 777 671 467 211 462 211 422 210 422 210 420 210 020 000 000
777 777 671 166 220 144 220 104 220 104 020 104 020 104 000 004
777 777 653 633 451 231 451 221 451 221 441 021 401 001 400 000
777 777 473 154 021 104 021 104 021 104 021 004 020 004 020 000
777 777 157 734 111 510 111 510 101 500 101 100 001 000 000 000
TABLE II P UNCTURING PATTERNS ( IN OCTAL FORM ) FOR THE ENTIRE SET OF INNER CODE SYSTEMATIC BIT POSITIONS GOING FROM 1 TO 2I .
ρu 30/30
16 735 673
26/30
12 725 672
24/30
12 525 672
22/30
12 525 272
21/30
12 525 252
20/30
12 525 252
567 735 567 527 735 567 525 735 527 525 725 525 525 525 525 525 525 525
356 673 356 356 273 256 352 253 252 352 253 252 352 253 252 252 252 252
735 567 735 535 527 725 535 527 725 525 525 725 525 525 725 525 525 525
673 356 673 652 256 273 652 256 273 252 256 253 252 252 252 252 252 252
567 735 567 567 725 565 565 525 525 565 525 525 565 525 525 525 525 525
information bit. Being n = 2, ρu is obtained from ρp using (2). To obtain a better performance in the error floor region (i.e., for higher SNRs), it is advantageous to puncture more severely the inner systematic bits [1]. In fact, the lowest error floors are obtained with ρp = 9/30 and ρp = 10/30 (ρu = 21/30 and ρu = 20/30, respectively). On the other hand, to improve the performance in the waterfall region (i.e., for lower SNRs), it is advantageous to puncture more inner parity bits, since, in this way, the largest number of systematic bits will help the convergence of the iterative decoding algorithm. In fact, the best performance for lower SNRs is obtained with ρp = 6/30 (ρu = 24/30). Puncturing the inner parity check bits cannot go beyond a certain threshold, though, since an insufficient number of inner parity bits would make the inner code so weak that the
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Puncturing pattern 356 735 673 567 356 673 567 356 735 673 356 735 673 567 356 352 735 253 565 256 672 527 352 735 272 356 725 673 527 352 252 735 252 565 256 672 527 352 535 252 356 725 272 527 352 252 725 252 565 252 252 527 252 535 252 252 725 272 525 352 252 525 252 525 252 252 527 252 535 252 252 725 272 525 352 252 525 252 525 252 252 525 252 525 252 252 725 252 525 252
735 567 735 725 567 535 725 567 535 725 565 525 725 525 525 525 525 525
673 356 673 653 352 672 253 252 652 252 252 652 252 252 252 252 252 252
567 735 567 565 725 567 565 725 527 565 525 525 525 525 525 525 525 525
356 673 356 352 672 252 252 672 252 252 672 252 252 652 252 252 252 252
performance of the overall SCCC code becomes bad for all SNRs [16]. To see this, consider, for example, the performance with ρp = 4/30 (ρu = 26/30). Finally, we observe that the code, the resulting performance of which is obtained with ρp = 9/30 and ρu = 21/30, respectively, represents a good compromise between waterfall and error floor performance. Fig 3 shows the simulated performance of the SCCCs with rate RSCCC = 2/3 in terms of residual BLER vs. ρp , for different Eb /N0 in dB. The curves show that the higher the SNR, and hence the lower the target BLER, the heavier should be the puncturing on inner systematic bits. Finally, in Fig 4, theoretical performance limits obtained through the application of the sphere packing bound [13] and simulation results are given in terms of residual BLER vs.
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−1
V. C ONCLUSIONS
10
E \N =2.8 dB b
E \N =3.2 dB
−2
10
b
0
Eb\N0=3.6 dB
−3
10
Eb\N0=4.0 dB −4
10
E \N =4.4 dB b
0
E \N =4.8 dB
−5
10
b
0
E \N =5.2 dB b 0
R EFERENCES
−6
10
0.09
0.13
0.17
0.21 ρ
0.25
0.29
0.33
p
Fig. 3. Block Error Rate vs. ρp for different Eb /N0 in dB and for SCCCs with total rate RSCCC = 2/3.
10
10
Block Error Rate
Block Error Rate
In this paper we have proposed a design method for the construction of well performing families of rate-compatible punctured SCCCs. To obtain rate-compatible SCCCs, the puncturing has not been limited to inner parity bits only, but has been extended also to inner systematic bits, thus obtaining higher rate SCCCs (i.e., beyond the outer code rate). Moreover, the problem of finding the optimal percentage of inner systematic and, consequently, parity bits to be punctured to obtain a given SCCC rate has been addressed: it has been shown that the best puncturing strategy must be chosen with regard to the application considered.
0
10
10
−1
−2
−3
−4
4/5
2/3
1/2 10
8/9
−5
−2
−1
0
1
2 SNR [dB]
3
4
5
6
Fig. 4. Frame Error Rate vs. Signal-to-Noise Ratio Es /N0 for different code rates RSCCC = {1/2, 2/3, 4/5, 8/9}. Simulation results: dash-dotted curves. Sphere-packing bound limiting performance: solid curves.
Es /N0 in dB for rate-compatible SCCCs with rates from 1/2 up to 8/9. All plots are obtained with ρu = 21/30, which was shown in Fig. 2 to give a code representing a reasonable compromise between the waterfall and the error floor performance. For each rate, the condition (2) is satisfied. Puncturing optimization has led to ρp = 19/30 (rate 1/2 code), ρp = 9/30 (rate 2/3 code), ρp = 4/30 (rate 4/5 code), ρp = 1/20 (rate 8/9 code). In most cases, the performance is within less than 1 dB from the sphere packing bound (such as the best codes presented in [13]), despite the random interleaver and its short length.
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[1] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design and iterative decoding”, IEEE Transactions on Information Theory, Vol. 44, n. 3, May 1998, pp. 909-926. [2] J. Hagenauer, “Rate-compatible punctured convolutional codes (RCPC codes) and their applications”, IEEE Transactions on Communications, Vol. 36, No. 4, April 1988, pp. 389-400. [3] A. S. Barbulescu and S. S. Pietrobon, “Rate compatible turbo codes”, IEE Electronics Letters, 30th March 1995, Vol. 31, No. 7, pp. 535-536. [4] D. N. Rowitch and L. B. Milstein, “Rate compatible punctured turbo (RCPT) codes in a hybrid FEC/ARQ system”, Proc. of the IEEE Communication Theory Mini Conference, held in conjunction with GLOBECOM’97, Phoenix, Arizona, November 5-6, 1997, pp. 55-59. [5] P. Jung and J. Plechinger, “Performance of rate compatible punctured turbo codes for mobile radio applications”, IEE Electronics Letters, 4th December 1997, Vol. 33, No. 25, pp. 2102-2103. [6] D. N. Rowitch and L. B. Milstein, “On the performance of hybrid FEC/ARQ systems using rate compatible punctured turbo (RCPT) codes”, IEEE Transactions on Communications, Vol. 48, No. 6, June 2000, pp. 948-959. [7] N. Chandran and M. C. Valenti, “Hybrid ARQ using serial concatenated convolutional codes over fading channels”, Proc. of the 2001 IEEE 53rd Vehicular Technology Conference - VTC’01 Spring, Vol. 2, Rhodes, Greece, 6-9 May 2001, pp. 1410-1414. [8] H. Kim and G. L. St¨uber, “Rate compatible punctured SCCC”, Proc. of the 2001 IEEE 54th Vehicular Technology Conference - VTC’01 Fall, Vol. 4, Atlantic City, NJ, U.S.A., 7-11 October 2001, pp. 2399-2403. [9] S. Kallel and D. Haccoun, “Generalized type II hybrid ARQ scheme using punctured convolutional coding”, IEEE Transactions on Communications, Vol. 38, No. 11, November 1990, pp. 1938-1946. ¨ F. Ac¸ikel and W. E. Ryan, “Punctured high rate SCCCs for [10] O. BPSK/QPSK channels”, Proc. of the 2000 IEEE International Conference on Communications - ICC 2000, New Orleans, U.S.A, 18-22 June 2000, pp. 434-438. ¨ F. Ac¸ikel and W. E. Ryan, “Punctured turbo-codes for BPSK/QPSK [11] O. channels”, IEEE Transactions on Communications, Vol. 47, No. 9, September 1999, pp. 1315-1323. [12] S. Benedetto and G. Montorsi, “Design of parallel concatenated convolutional codes”, IEEE Transactions on Communications, Vol. 44, No. 5, May 1996, pp. 591-600. [13] C. Schlegel and L. Perez, “On error bounds and turbo-codes”, IEEE Communications Letters, Vol. 3, n. 7, July 1999, pp. 205-207. [14] D. Divsalar, S. Dolinar and F. Pollara, “Iterative turbo decoder analysis based on density evolution”, IEEE Journal on Selected Areas in Communications, Vol. 19, No. 5, May 2001, pp. 891-907. [15] I. Land and P. Hoeher, “Partially systematic rate 1/2 Turbo codes”, Proc. of the II International Symposium on Turbo codes and related topics, Brest, 4-7 September 2000, pp. 287-290. [16] M. A. Herro, D. J. Costello and L. Hu, “Capacity and cutoff rate calculations for a concatenated coding system”, IEEE Transactions on Information Theory, Vol. 34, n. 2, March 1988, pp. 212-222. [17] S. Benedetto, R. Garello and G. Montorsi, “A search for good convolutional codes to be used in the construction of turbo codes”, IEEE Transactions on Communications, Vol. 46, n. 9, September 1998, pp. 1101-1105.
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