Rate Allocation for Serial Concatenated Block Codes

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1 1 Rate Allocation for Serial Concatenated Block Codes Maja Bystrom and Robert A. Coury Abstract While serial concatenated codes were designed to provide good overall performance with reasonable system complexity, they may arise naturally in certain cases, such as the interface between two networks. In this work we consider the problem of constrained rate allocation between nonsystematic block codes in a serial concatenated coding system with either ideal or no interleaving between the codes. Given constraints on system parameters, such as a limit on the overall rate, analytic guidelines for the selection of good inner code rates are found by using an upper bound on the average system block error rate. Keywords: channel coding, concatenated codes I. INTRODUCTION Serial concatenated block codes were developed to achieve a low average system error rate, without the decoding expense of applying a single long-block-length code [1]. However, a serial concatenated coding system may naturally occur in a heterogeneous network at the interface of two subnetworks if no transcoding is applied. In practice this may be realized when data either are coded separately from a transmission protocol or are transmitted between networks employing different protocols, and are not decoded and then re-encoded at the network interface. In these cases there may be constraints either on the total available rate or the set of available codes, and it may be useful to adapt code rate in the face of varying source rates, bandwidth, and/or channel conditions. This work was supported in part by NSF award #870418, and was presented in part at the 2002 International Symposium on Information Theory. M. Bystrom is with the ECE Department, Boston University.

2 2 This work addresses this possibility in one manner, by developing an analytical approximation to appropriate channel code rates, given selected system constraints. Consider the serial concatenated block coding scheme illustrated in Fig. 1. This serial coder consists of two nonsystematic block codes with a binary symmetric channel (BSC) as the inner (physical layer) channel. The inner coder together with the BSC form the outer channel which is presented to the outer coder. Between the two coders is an optional interleaver/deinterleaver. When an interleaver with sufficient depth is employed, the outer channel appears to be a BSC with crossover probability equal to the decoded bit error rate out of the inner decoder. If the code concatenation represents a system such as the heterogeneous network described previously, possible constraints on a concatenated coding scheme could be the overall code rate and the block lengths of the constituent codes. Typically, appropriate code rates for the constituent codes in the concatenated system have been found through heuristic methods and simulations. In 188 Herro, et al. [2] first employed information-theoretic techniques to conclude that there is an optimal rate for the inner code of a concatenated block coding system, and that this optimum depends on the overall system capacity or cutoff rate. Furthermore, they concluded that given a fixed overall rate there exists an optimum rate distribution which minimizes the overall decoded bit error rate, and that this distribution is a function of the inner code block length. However, they did not present a method for finding the optimal rates, nor did they consider the effects of rate constraints. Similar results for concatenated coding systems using inner convolutional codes can be found in an earlier paper [3]. The rate tradeoff is further illustrated in Lin and Costello [4] with an example based on the work of Odenwalder [5]. In more recent work on turbo codes by Schlegel and Perez [6] it can be seen that performance varies by adjusting the constituent codes and interleaver. In this work we expand upon the results presented in [2] by assuming selected constraints on system parameters, and develop a tractable method for determining appropriate code rates as functions

3 3 of these constraints. We consider the effect of both the inner and outer coders, as well as the channel SNR, on the overall block error rate and develop an analytic guideline for selection of appropriate inner code rates, extending the work in [2] to supply a guideline for appropriate rate selection that can readily be used in an adaptive system or can serve as the starting point for a heuristic or exhaustive search for optimal allocations. Given the computational complexity that would be involved in determining optimal operating points for wide ranges of physical channel qualities and system parameters, it would be useful to have an initial point in order to constrain searches. In Section II we further discuss the concatenated block coding system and the ensemble performance bound employed. This upper bound on overall block error rate is used to determine a guideline for selection of the rates of the constituent binary codes as a function of the selected constraints. The analytic guidelines are given in Section III and are compared to a concatenated block coding scheme with BCH constituent codes in Section IV. II. SERIAL CONCATENATED BLOCK CODES We denote the inner and outer code rates of the serial concatenated block coding scheme of Fig. 1 as and, respectively, and assume that the block lengths, and, of the codes are predetermined, as might occur in a practical coding system, while the information block lengths, and, are permitted to vary. The overall system code rate,, is the product of the rates of the two constituent codes, i.e.,. A constraint on the overall system rate,, will be imposed, so that must be satisfied. In general, the subscript will denote a parameter which is associated with the inner channel, the physical channel, while the subscript indicates an association with the outer channel. Thus, the inner channel, respectively, while "! "!# and $&% and $&%!!# denote the cutoff rate and bit error rate of denote the cutoff rate and bit error rate of the outer channel. An exception to this convention is that the block error rate of the outer channel is denoted by $'!#, while the overall block error rate of the system, that is, the decoded block error

4 + rate after the outer decoder is then denoted as $('!. 4!# $&% In the system shown in Fig. 1, the inner (physical) channel is a BSC with crossover probability, thus the cutoff rate )"!# is given by "! *,+.-0/21, $%! *:,+;$%! 2EFHG (1) The channel cutoff rate can be viewed to be the largest practical rate at which arbitrarily small probability of error can be assumed for many codes. Assuming ideal interleaving is employed, the outer channel is a BSC as well, and thus bit error rate seen by the outer decoder. "! is also given by (1) with $I%! being the decoded In [7] the average block error probability for a block code on a discrete memoryless channel is shown to satisfy where X* Z*=< is the code block length and Y! $&' LKLMONQPSRUT8NWV EFHG (2) is the code-rate-dependent random coding exponent. In [8] Viterbi and Omura gave a lower bound on the random coding exponent as Y * for ] system block error rate J^*5_"! Finally, note that the bit error rate of the inner channel, $% Z *=<[\"! * EFHGLb This bound is then employed in (2) to bound the overall! J $&' dc"e K8THfhg e M e b (3)!# is dictated by the channel SNR, while for nonsystematic binary codes the bit error rate out of the inner decoder is taken to be $&%! i $&'!# b (4) Note that since an ensemble of codes is considered, one can do no worse, on average, than assuming that half of the bits in the block are in error [8], []. As shown in [2] one can use (1), (3), and (4) (assuming ideal interleaving) to obtain a bound on

5 + t < u 5 the cutoff rate of the outer channel "! `j,+;-w/1 )k 54 7 cl K8T fhg l M l <m )+n cl K8T fhg l M l <po b (5) If interleaving is not used then the channel is not memoryless. However, as shown in [2], based on the block interference channel of McEliece and Stark [10], it can be modeled as a memoryless cl -ary channel. The cutoff rate of the inner channel and the bit error rate seen by the outer decoder remain the same as given in (1) and (4). However, the cutoff of the outer channel now becomes "! `j,+ 8I-W/1 q 7,+n c l K8T8fhg l M l 4 c l +r K8T8fhg l M lzs,+n K c l b (6) In the following section we approximate the cutoff rates of (5) and (6), and then use these in (3) to minimize approximations to the bound with respect to v and. The results will serve as guidelines for determining the optimal inner code rates of the concatenated system with ideal or no interleaving as functions of selected system parameters. To permit an analytical solution of the minimum points on the approximations of the bounds, we treat the H* s as real-valued and round the resulting values to obtain the guidelines to the optimal code rates. III. RATE SELECTION GUIDELINES In the previous section we introduced an upper bound on the overall system error rate. In this section we solve for the minimum point on an approximation to the bound as a function of selected system parameters. The minimizing inner code rate will serve as a guideline for reasonable inner code rate selection in a practical system. A. Ideal Interleaving To find the minimum point on the bound on $('! given in (3), where )"! is given by (5), we minimize the exponent XF"! (7)

6 z z ] ] Œ t 7 t 6 with respect to and, subject to the rate constraints while assuming that & and are fixed. J n_w"!# Jr_w"! ` _ r_ (8) () (10) Because we are seeking only an approximation to the optimal inner and outer rates, we first! approximate $&% by c l K8THfhg l M l K. Next, we bound and perform a multi-stage approximation of "!. We first approximate )"! by its upper value, and then take the first term in a Taylor series expansion of this approximation. Finally, we change the base the of logarithm and employ the approximation ln yxx<bz{x +r for x z "! }z,+;-0/1,+;-0/1,+ -0ˆ, ~ This sequence of steps yields! $%! 7 $&%,LR c l K8THfhg l M l V s The tightness of the result will be discussed in Section IV.,+ cfl K8T fhg l M lƒ b (11) Since "! does not depend on L, minimizing (7) is a separable problem. Clearly H must be made as small as possible while still satisfying (10), i.e., H F, 8. Then we can minimize (7) with respect to to find where Z < 8B -0ˆ`S -0ˆ` ` "Ž THfhg l M lzs is the Lambert -function, the solution to an equation of the form x. [11], [12]. It is interesting to note that, in this case, the approximation to the optimum value of is independent of S. Fig. 2 shows the inner code rate guideline as a function of inner code block length for selected rates and channel SNRs with S. Also shown in this figure are the inner channel cutoff rates (12)

7 z z t b Œ Œ 4 7 Ž t 4 E 4 for Ỳ % i ] and db. For short inner code block lengths or poor channels, a greater percentage 7 of the total rate must be allocated to the inner code in order to clean up the channel. If the channel is good, however, a powerful inner code is not required. Note that for both channels shown the inner rate converges for large inner code block lengths, as will be shown below. It can be surmised that for shorter inner code block lengths or poorer channels, the inner code must be made more powerful, that is, assigned a lower code rate, in order to clean up the channel, but if the channel is good a powerful inner code is not required. We now examine the behavior of the inner rate for long block lengths. As shown in [11] an expansion of the Lambert ž <" -Wˆ` 7 ` T8fhg l M lzs Ÿš < =š < -0ˆ +D-0ˆ -function can be written by letting b Then from [11] we have =š << -0ˆ 54-0ˆ -0ˆ Ÿš <"< -0ˆ Ÿš < 4 =š <I -0ˆ -0ˆ -0ˆ Z < 4, where šœ L4 Ÿš <"<< Ÿš < 4 6b (13), Employing the first term of the above expansion, for long block lengths we have -0 W M lz, -0 W M l= ` -0 W M l= ` -0 W M l ` "!# Z < -Wˆ, -Wˆ` -0ˆ -Wˆ` Œ ˆ -0ˆ` -0ˆ` `X Œ "Ž T8fhg l M lzs -Wˆ``4-0ˆ ` 4n-Wˆ "!# -0ˆ` Ž(Ž (14) So, as the block length increases, the optimum inner rate tends to the cutoff rate. This conclusion can also be drawn from the results of [2]. B. No Interleaving In the case of no interleaving the objective remains to minimize $B'!, c"e KLM e T8fhg e subject to the constraints of (8) - (10), however the inequality of (8) is now considered to be a strict inequality to permit the approximations below. It is assumed that ) E E and the channel SNR are fixed.

8 < z z Œ Œ E t < Ž 8 Again, we first approximate the outer channel cutoff rate as follows "! `+ `+ `+ &-W/1 &-W/1 7,+n c l K8T8fhg l M l 4 54 cl K8T fhg l M l s < c l K8T8fhg l M lzs -0ˆ, c l +a K8THfhg l M lÿs,+n c l b (15) Thus we can now minimize XF` + with respect to, for fixed parameters & E Ž `+ < c l K8T8fhg l M lÿs -0ˆ` X `, and "!#, which yields (16) 8B Z < 4w <" -Wˆ, (17) with ª `I -Wˆ` T8fhg l M lzs. As was concluded in [2], and can be seen in Fig. 3, the guideline inner code rate rapidly converges to "!#. We can again use the approximation to the Lambert -function of (13) to show this convergence as -0 W M l ` M l, z"!# Z < 4w -0ˆ \b (18) Therefore, a short block length suffices in this case to provide the optimum results. Compared with the case of ideal interleaving, the indicated inner code rate is significantly lower. Thus, this system is optimized when the outer channel is made as narrow as possible, in the sense that is reduced as far as possible and the inner channel is efficiently cleaned. IV. COMPARISON TO SELECTED CODES Fig. 4 illustrates the upper bound on overall block error rate as a function of inner code rate for a BSC with Ỳ % db, `«z ] b 8E and ideal interleaving. The block length of the inner code is

9 fixed at i ^ while is varied. Results of simulating transmission using a serial concatenated coding system with constituent BCH codes with generators as given in [4, Appendix C] are shown in Fig. 4. Since non-shortened codes are employed, available block lengths are limited, yielding few simulation points, which are indicated by stars. The bound is a smooth curve for each set of block lengths and the minimum point found by the analytic solution of the approximation to the bound is indicated by a diamond. Since the upper bound of (2) is a bound over an ensemble of codes, it is not expected that the bounds would be tight; hence, the large distance, in terms of block error rate between the bounds and the simulations. In this work, however, we are only concerned with the distance, in terms of between the minima on the simulations and the guidelines to which are indicated by the diamonds. These results illustrate that the optimal inner code rate is independent of S and that the bound underestimates the optimal inner rate for the selected codes. However, the results suggest that analytic solution can serve as an acceptable guideline for selection of the optimal inner code rate. Related results are given for the no interleaving case in Fig. 5. In this figure, the block lengths are fixed to while two selections for physical channel quality and rate are shown. Again, BCH codes are used, the bound is a smooth curve for each choice of channel quality, the guideline to the inner code rate is shown by a diamond. and the simulation results are given by stars. In this case the approximation to the minima on the bounds are not as tight as for the interleaved case. As before we are interested only in the location of the minima on corresponding curves, and not in the distance between the curves. Again, the guideline can assist in selection of good rate allocations. Future work will involve extending these results to AWGN and fading inner channels with softdecision decoders In addition, a serial concatenated coder with an inner convolutional code can be easily implemented with the rate-compatible punctured convolutional codes introduced in [13]. While the case of an outer block code and inner convolutional code was briefly examined in [3] it would be of interest to revisit this work and develop an analytic guideline for the optimal inner

10 10 and outer code rates. Finally, for scenarios other than described in the introduction, that is, when iterative decoding is feasible, a capacity bound would likely provide more useful results than the cutoff rate bound. V. ACKNOWLEDGMENTS The authors wish to thank Prof. Daniel J. Costello, Jr. for his excellent suggestions for improving this paper. REFERENCES [1] G.D. Forney, Concatenated Codes. MIT Press, 166. [2] M.A. Herro, D.J. Costello, Jr., and L. Hu, Capacity and Cutoff Rate Calculations for a Concatenated Coding System, IEEE Trans. Inform. Theory, vol. 34, pp , Mar [3] M.A. Herro, D.J. Costello, Jr., and L. Hu, Capacity and Cutoff Rate of a Concatenated Coding System with an Inner Convolutional Code, in Proc. Conf. Information Sciences and Systems, pp , Mar [4] S. Lin and D.J. Costello, Jr., Error Control Coding: Fundamentals and Applications. Englewood Cliffs, N.J.: Prentice-Hall, 183. [5] J.P. Odenwalder, Optimal Decoding of Convolutional Codes. PhD thesis, Department of Systems Sciences, UCLA, 170. [6] C. Schlegel and L. Perez, On Error Bounds and Turbo-Codes, IEEE Communications Letters, vol. 3, pp , July 1. [7] R. G. Gallager, Information Theory and Reliable Communication. John Wiley and Sons, 168. [8] A.J. Viterbi and J.K. Omura, Principles of Digital Communication and Coding. New York: McGraw-Hill, 17. [] J. G. Proakis, Digital Communications. McGraw-Hill Book Company, New York, 18. [10] R.J. McEliece and W.L. Stark, Channels with Block Interference, IEEE Trans. Inform. Theory, vol. 30, Jan [11] J.P. Boyd, Global Approximations to the Principal Real-Valued Branch of the Lambert -function, Appl. Math. Lett., vol. 11, no. 6, pp , 18. [12] D.A. Barry, Y.-Y. Parlange, L. Li, H. Prommer, C.J. Cunningham, and F. Stagnitti, Analytical Approximations for Real Values of the Lambert -function, Mathematics and Computers in Simulation, vol. 53, pp , [13] J. Hagenauer, Rate-Compatible Punctured Convolutional Codes (RCPC Codes) and their Applications, IEEE Trans. Comm., vol. COM-36, pp , April 188.

11 ¼ 11 Outer Encoder Interleaver Inner Encoder Inner Channel (BSC) Outer Decoder Deinterleaver Inner Decoder Outer Channel Fig. 1. A serial concatenated coding scheme. 1 R 0,1 Eb /N 0 =10 db 0. E b /N 0 = 10 db R 0,1 Eb /N 0 =4 db R E b /N 0 = 4 db = 0.75 = 0.5 = 0.25 = n 1 Fig. 2. Guideline inner code rate as a function of inner code block length for ±Ÿ²h³ fi.µ? and db, total overall rate Ş¹»º F½¾µ: "ÀÁZ ½ "ÀÁh F½ ÀFÁŸ ½ ÂÀ ÃÁ and ideal interleaving.

12 R 0,1 /2 Eb /N 0 =10 db R R 0,1 /2 Eb /N 0 =4 db = 0.45 = 0.25 = = n 1 Fig. 3. Optimum inner code rate as a function of inner code block length for SNRs of 10 and 4 db, total overall rate ¹ º ¼ F½ "Ä ÀFÁh ½¾µ: ÀFÁZ ½ ÀFÁZ ½ À"Ã, Å e Ǽµ: Â, and no interleaving P B n 1 = 63, n 2 = 31 n 1 = 63, n 2 = 63 n 1 = 63, n 2 = R 1 Fig. 4. The performance of a concatenated coder with BCH codes as compared with that predicted by the upper bound for ±Z²Z³ f œ db, ¹,Ç ½ È, and ideal interleaving. Simulated points for the selected BCH codes are indicated by stars.

13 =0.2, SNR=2dB =0.27, SNR=5dB P B, R 1 Fig. 5. The performance of a concatenated coder with BCH codes as compared with that predicted by the upper bound for varying ±Z²Z³ f, ¹, and no interleaving. Simulated points for the selected BCH codes are indicated by stars.

Performance Evaluation and Comparative Analysis of Various Concatenated Error Correcting Codes Using BPSK Modulation for AWGN Channel

Performance Evaluation and Comparative Analysis of Various Concatenated Error Correcting Codes Using BPSK Modulation for AWGN Channel International Journal of Electronics and Communication Engineering. ISSN 0974-2166 Volume 5, Number 3 (2012), pp. 235-244 International Research Publication House http://www.irphouse.com Performance Evaluation