AWGN Performance of Superothogonal Convolutional ... - IEEE Xplore
AWGN Performance of Superorthogonal Convolutional Codes Brian K. Butler, Senior Member, IEEE, QUALCOMM, Incorporated, San Diego, CA
Abstract—The bit error rate performance of superorthogonal convolutional codes is examined for the AWGN channel with an emphasis on graphical error rate curves. The little-noticed generalization of the class of superorthogonal codes that allows for more parameter combinations is examined and found to have error rate performance than is primarily a function of constraint length and not code rate. Symbol puncturing of the codes is also briefly examined. Index Terms—CDMA, binary orthogonal, low-rate convolutional codes, orthogonal codes, superorthogonal codes.
I. INTRODUCTION This short paper serves to correct and extend the performance of the superorthogonal convolutional codes published in [1]. We believe such performance curves may put superorthogonal convolutional codes in a more favorable light with system designers and provide a useful benchmark. Superorthogonal codes (SOC) were preceded by orthogonal convolutional codes which were first published in 1967 in [2]. The superorthogonal convolutional codes, which required less bandwidth expansion, were described in the 1989–1990 period by [3], [4]. A careful explanation of superorthogonal codes was presented again in [1] with graphs of error rate upper bounds for the additive white Gaussian noise (AWGN) channel. This remains the most common reference for superorthogonal convolutional codes. Regrettably, a publishing mistake produced a graph with the bit error probability bounds (Fig. 5.21 of [1]) a factor ten too high. With reasonably good performance at very low rates, the superorthogonal codes can be suitable to systems that enjoy significant bandwidth to bit-rate ratios, e.g., spread-spectrum or ultra-wideband (UWB) applications. Given the relatively low complexity to implement the corresponding soft decision Viterbi decoder for these codes, a recent resumption of interest in these codes has been witnessed in the UWB community. The orthogonal and superorthogonal codes serve to provide a construction technique alternative to the more common search technique to find good generator polynomials for convolutional coding. As the search techniques can be quite lengthy at low-rates these classes provide a rapid benchmark for convolutional coding performance. Using search techniques (in the range 3 ≤ K ≤ 7 and 2 ≤ N ≤ 8), the best known, rate, R = 1/N, low-rate convolutional codes have been published in [5]. Utilizing a nested search technique, PaperID 1516 for MILCOM 2006
maximum free distance convolutional codes down to very low rates (in the range 7 ≤ K ≤ 11 and 4 ≤ N ≤ 512) have been found by [6]. One possible difficulty in finding application of the superorthogonal codes is the strict relationship between rate and constraint length as presented in [1], [3], in which the code rate, R, of superorthogonal is fixed at R = 2–(K–2). Reference [4] extended the definition of the superorthogonal code to produce higher constraint length codes at a given rate. These extensions are not widely recognized in the literature This short paper documents the AWGN (viz. [6]). performance of those extensions and briefly examines code symbol puncturing of superorthogonal codes. II. BASIC SUPERORTHOGONAL CODES The development in this section of the orthogonal and superorthogonal classes of low-rate binary convolutional codes follows the detailed description of [1]. These codes are both characterized by a linear feed-forward linear function of the current input bit, and the previous K–1 input bits. The orthogonal convolutional encoder uses all K bits to select an orthogonal Walsh function of length 2K (i.e., a row from the Hadamard matrix of order K, H2^K,) for each bit time interval. The superorthogonal convolutional encoder uses just the middle K–2 bits to select an orthogonal sequence of length 2K-2 and inverts it or not depending upon the exclusive OR (XOR) of the outer two bits. Hence, the code rates of the orthogonal and superorthogonal encoders are 2–K and 2–(K–2), respectively. A trellis of 2K–1 states fully captures the encoder’s transitions in either case. Encoder block diagrams are illustrated in Figs. 1 and 2. One can think of the orthogonal convolutional encoder as choosing one orthogonal sequence every bit time, while the superorthogonal encoder chooses one sequence from the set of biorthogonal sequences every bit time. Hence, superorthogonal convolutional coding requires that the receiver be coherent. This paper will focus only on the coherent case. As there is a great deal of uniformity of the distance within the orthogonal and biorthogonal signal sets used by the codes, a reduced state diagram is used for analysis in [1], [3] to create the generating functions. Within the orthogonal signal set, every branch differs from every incorrect branch by exactly w=2(K–1) symbols. Similarly, within the superorthogonal, signal set, every branch differs from every incorrect branch by exactly w=2(K–3), with just two exceptions. Those exceptions are during the first step of divergence and final reemergence 1 of 5
TSO (W ,1) =
K–1 delay latches, thus 2K–1 states D
Input bit
D
(3)
and 2
∂ W K +2 ⎛ 1 − W ⎞ TSO (W , β ) = ⎜ ⎟ . ∂β (1 − 2W )2 ⎝ 1 − W K − 2 ⎠ β =1
Select row of H2^K ⎡H4 H4 ⎤ e.g., H8= ⎢ ⎥ ⎣⎢ H 4 H 4 ⎦⎥
2K symbols per input bit.
D
Select row of H2^(K–2) ⎡0 0 0 0⎤ ⎢0 1 0 1 ⎥ ⎥ e.g., H4= ⎢ ⎢0 0 1 1 ⎥ ⎢ ⎥ ⎣0 1 1 0⎦
0
and
+
D
+
When 1, inverts all symbols
+ K–2
2 symbols per input bit.
Fig. 2. Superorthogonal convolutional encoder (basic, r =1). The XOR operation is shown here as addition with implied modulo 2.
with respect to the correct path. These exceptions are triggered by the compete inversion of the correct symbol generated by the outer XOR function. Hence, the initial and final branches of a superorthogonal code’s error event differ in 2w symbols. For both codes, there is the simple relation between w and code’s rate, R, 1 w= . (1) 2R This then defines the labeling of the reduced state diagram in [1]. A capital W is used to indicate any pattern of w binary symbol errors and W2 the pattern of all 2w binary symbols in error for superorthogonal. The relationship to the placeholder for a coded symbol error Z, is simply W 2 = Z1 R . (2) The state transition diagram can be greatly simplified as elegantly explained in [3]. These codes are deemed to belong to a larger class of codes called zero-run length (ZRL) convolutional codes. The zero-run length of the encoder state is defined as the number of consecutive leading zeros in the binary state representation. A rate 1/N convolutional code is said to be a ZRL code if the output weight depends only on the input and the zero-run length of the state. The tools developed in [3] greatly simplify finding the state transition diagram and the generating function. Below are two useful forms of the superorthogonal code’s generating function as presented in [1]:
PaperID 1516 for MILCOM 2006
Once the generating function of the code, T(δ,β), is derived, error rate bounds may be computed. As shown in [1], the coded B-bit block error probability, PE, and bit error probability, Pb, can be upper-bounded, respectively, by (5) PE < B T (W ,1) W = exp[ − E /(2 N )] b
K–1 delay latches, thus 2K–1 states D
(4)
III. ERROR BOUNDS
Fig. 1. Orthogonal convolutional encoder block diagram.
Input bit
W K +2 ⎛ 1 − W ⎞ ⎜ ⎟ 1 − 2W ⎝ 1 − W K − 2 ⎠
Pb
1 these superorthogonal codes are no longer ZRL codes. This fact makes finding an elegant form of the transfer function difficult. We have chosen to find PaperID 1516 for MILCOM 2006
We have used this puncturing strategy to create Fig. 7. We found it easier to use numeric techniques as in Section IV than solving the problem analytically. It is interesting to note that the performance is actually improved by puncturing for code rates < ¼. As expected, at very low rates the performance difference to the unpunctured code is negligible.
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VI. CONCLUSION Updated AWGN performance curves were presented for the original superorthogonal codes. Performance curves were introduced for Zehavi and Viterbi’s generalized superorthogonal codes for the unquantized AWGN channel. A richer variety of rate and constraint length combinations are available with these variations, at seemingly reasonable performance. Further, the code rate may be fine-tuned through symbol puncturing. No loss, and in fact a gain in performance was witnessed for the light levels of puncturing studied. Table I summarizes the Eb/No required to reach a 10-6 bit error rate bound for a variety of superorthogonal codes and compares that to the Eb/No requirements found through searching for the best convolutional codes at 10-6 in [5]. Except for the outlier case of r = 3 and R = 1/2, all SOC codes appear to be no worse than 0.12 dB of those codes in [5]. In fact, the punctured superorthogonal codes of Table I are the same codes identified by [5]. Table II similarly summarizes the Eb/No required to reach a 10-6 bit error rate bound of superorthogonal codes for 7 ≤ K ≤ 9 and compares that to the Eb/No requirements produce by the nested searching technique for maximum free distance (MFD) convolutional codes [6]. As is well known, MFD guarantees the best asymptotic performance as Eb/No approaches infinity. In spite of the MFD codes better free distance, the superorthogonal codes are seen to generally be perform better at the finite Eb/No corresponding to 10-6 bit error rates due to better properties of the distance spectrum.. Superothogonal codes provide a quick way to derive the encoder’s generating polynomials and produce very acceptable performance compared with search techniques over the space of low-rate feed-forward convolutional codes. ACKNOWLEDGMENT The author gratefully acknowledges the assistance of Drs. Andrew J. Viterbi and Ephriam Zehavi in verifying the integrity of the first error rate figure and the encouragement given by Dr. Jack K. Wolf. REFERENCES [1] A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication, Reading, Mass.:Addison-Wesley, 1995, ch 5. [2] A. J. Viterbi, “Orthogonal tree codes for communication in the presence of white Gaussian noise,” IEEE Trans Commun. Technol., vol. COM-15, pp. 238-242, Apr. 1967. [3] R. McEliece, S. Dolinar, F. Pollara, and H. Van Tilborg, “Some easily analyzable convolutional codes,” TDA Progress Report 42-99, Jet Propulsion Laboratory, Pasadena, Calif., Nov. 15, 1989, pp. 105-114. Available: http://tmo.jpl.nasa.gov/. [4] E. Zehavi and A. J. Viterbi, “On new classes of orthogonal convolutional codes,” in Proceedings of Bilkent International Conference on New Trends in Communication, Control, and Signal Processing, (Ankara, Turkey), pp. 257-263, July 1990. [5] P. J. Lee, “New short constraint length, rate 1/N convolutional codes which minimize the required SNR for given desired bit
PaperID 1516 for MILCOM 2006
error rates,” IEEE Trans. on Comm., vol. 33, pp. 171-177, Feb. 1985. [6] P. Frenger, P. Orten, and T. Ottosson, “Code-spread CDMA using maximum free distance low-rate convolutional codes,” IEEE Trans. on Comm., vol. 48, pp. 135-144, Jan. 2000. TABLE I COMPARISON OF REQUIRED EB/NO FOR BER OF 10-6 TO [5] K 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7
Rate 1/2 1/2 1/2 1/3 1/4 1/2 1/4 1/6 1/7 1/8 1/2 1/4 1/8 1/14 1/15 1/16 1/4 1/8
SOC order 1 2 1b 1b 1 3 2 1b 1b 1 4 3 2 1b 1b 1 4 3
SOC Required Eb/No (dB) 6.705 6.180 6.180 5.958 5.906 6.066 5.354 5.211 5.211 5.280 5.238 4.791 4.772 4.680 4.704 4.756 4.377 4.369
Reference [5] Eb/No (dB) 6.706 a 6.180 a 6.180 a 5.958 a 5.906 a 5.745 5.298 5.211 a 5.211 a 5.211 5.238 4.779 4.693
Ratio (dB) -0.001 0.000 0.000 0.000 0.000 0.321 0.056 0.000 0.000 0.069 0.000 0.012 0.079
c
c
c
c
4.367 4.251
0.010 0.118
c
c
Required bit error probability of SOC is computed by upper union bound. For unpunctured SOC codes of order r =1, the bound is computed exactly by (4) and (8). For other SOC codes, (12) is used with truncated generating function. a Generator polynomials are identical. b SOC codes of order r =1 are punctured 1 or 2 symbols per bit to achieve the rate indicated. c No convenient code of comparison. TABLE II COMPARISON OF REQUIRED EB/NO FOR BER OF 10-6 TO [6] K 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9
Rate 1/4 1/8 1/16 1/30 1/31 1/32 1/8 1/16 1/32 1/62 1/63 1/64 1/16 1/32 1/64 1/126 1/127 1/128
SOC order 4 3 2 1b 1b 1 4 3 2 1b 1b 1 4 3 2 1b 1b 1
SOC Required Eb/No (dB) 4.377 4.369 4.312 4.259 4.277 4.307 3.925 3.936 3.918 3.890 3.901 3.916 3.623 3.580 3.575 3.560 3.566 3.574
Reference [6] Eb/No (dB) 4.375 4.310 4.300 4.305 4.307 4.299 3.976 3.991 4.022 4.058 4.063 4.066 3.636 a 3.639 a 3.663 a 3.622 a 3.622 a 3.623 a
Ratio (dB) 0.002 0.058 0.012 -0.047 -0.029 0.008 -0.051 -0.056 -0.103 -0.168 -0.162 -0.150 -0.012 -0.059 -0.089 -0.062 -0.056 -0.049
In this table, SOC codes are treated similarly to Table I. The bit error probability union bound for the Maximum Free Distance codes with Optimal Distance Spectrum [6] is found by (12) with truncated generating function. a Generator list for K=9 [6] corrected by original authors to be {g0, g1, ..., g17} = {453, 455, 457, 467, 473, 475, 513, 517, 551, 557, 573, 657, 671, 675, 735, 745, 753, 765} b SOC codes of order r =1 are punctured 1 or 2 symbols per bit to achieve the rate indicated.
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Eb/No (dB) for tight upper bound of BER = 1E-6
7.0
orthogonal code basic SOC SOC r=2 SOC r=3 SOC r=4
6.0
5.0
K=7 4.0
K=9 3.0
2.0
1
10
100
1000
Bandwidth Expansion (1 / Rate) Fig. 6. Required Eb/No (dB) for the tight upper bound on bit error probability to achieve 1E-6.
Eb/No (dB) for tight upper bound of BER = 1E-6
7.0
SOC without puncturing SOC with p=1 SOC with p=2
6.0
K=5 K=6
5.0
K=7 K=8 4.0
3.0
1
10
Bandwidth Expansion (1 / Rate) Fig. 7. Required Eb/No (dB) for several puncturing schemes for the basic SOC (r =1) are shown.
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Abstract—The bit error rate performance of superorthogonal convolutional codes is examined for the AWGN channel with an emphasis on graphical error rate curves. The little-noticed generalization of the class of superorthogonal codes that allows for more parameter combinations is examined and found to have error rate performance than is primarily a function of constraint length and not code rate. Symbol puncturing of the codes is also briefly examined. Index Terms—CDMA, binary orthogonal, low-rate convolutional codes, orthogonal codes, superorthogonal codes.
I. INTRODUCTION This short paper serves to correct and extend the performance of the superorthogonal convolutional codes published in [1]. We believe such performance curves may put superorthogonal convolutional codes in a more favorable light with system designers and provide a useful benchmark. Superorthogonal codes (SOC) were preceded by orthogonal convolutional codes which were first published in 1967 in [2]. The superorthogonal convolutional codes, which required less bandwidth expansion, were described in the 1989–1990 period by [3], [4]. A careful explanation of superorthogonal codes was presented again in [1] with graphs of error rate upper bounds for the additive white Gaussian noise (AWGN) channel. This remains the most common reference for superorthogonal convolutional codes. Regrettably, a publishing mistake produced a graph with the bit error probability bounds (Fig. 5.21 of [1]) a factor ten too high. With reasonably good performance at very low rates, the superorthogonal codes can be suitable to systems that enjoy significant bandwidth to bit-rate ratios, e.g., spread-spectrum or ultra-wideband (UWB) applications. Given the relatively low complexity to implement the corresponding soft decision Viterbi decoder for these codes, a recent resumption of interest in these codes has been witnessed in the UWB community. The orthogonal and superorthogonal codes serve to provide a construction technique alternative to the more common search technique to find good generator polynomials for convolutional coding. As the search techniques can be quite lengthy at low-rates these classes provide a rapid benchmark for convolutional coding performance. Using search techniques (in the range 3 ≤ K ≤ 7 and 2 ≤ N ≤ 8), the best known, rate, R = 1/N, low-rate convolutional codes have been published in [5]. Utilizing a nested search technique, PaperID 1516 for MILCOM 2006
maximum free distance convolutional codes down to very low rates (in the range 7 ≤ K ≤ 11 and 4 ≤ N ≤ 512) have been found by [6]. One possible difficulty in finding application of the superorthogonal codes is the strict relationship between rate and constraint length as presented in [1], [3], in which the code rate, R, of superorthogonal is fixed at R = 2–(K–2). Reference [4] extended the definition of the superorthogonal code to produce higher constraint length codes at a given rate. These extensions are not widely recognized in the literature This short paper documents the AWGN (viz. [6]). performance of those extensions and briefly examines code symbol puncturing of superorthogonal codes. II. BASIC SUPERORTHOGONAL CODES The development in this section of the orthogonal and superorthogonal classes of low-rate binary convolutional codes follows the detailed description of [1]. These codes are both characterized by a linear feed-forward linear function of the current input bit, and the previous K–1 input bits. The orthogonal convolutional encoder uses all K bits to select an orthogonal Walsh function of length 2K (i.e., a row from the Hadamard matrix of order K, H2^K,) for each bit time interval. The superorthogonal convolutional encoder uses just the middle K–2 bits to select an orthogonal sequence of length 2K-2 and inverts it or not depending upon the exclusive OR (XOR) of the outer two bits. Hence, the code rates of the orthogonal and superorthogonal encoders are 2–K and 2–(K–2), respectively. A trellis of 2K–1 states fully captures the encoder’s transitions in either case. Encoder block diagrams are illustrated in Figs. 1 and 2. One can think of the orthogonal convolutional encoder as choosing one orthogonal sequence every bit time, while the superorthogonal encoder chooses one sequence from the set of biorthogonal sequences every bit time. Hence, superorthogonal convolutional coding requires that the receiver be coherent. This paper will focus only on the coherent case. As there is a great deal of uniformity of the distance within the orthogonal and biorthogonal signal sets used by the codes, a reduced state diagram is used for analysis in [1], [3] to create the generating functions. Within the orthogonal signal set, every branch differs from every incorrect branch by exactly w=2(K–1) symbols. Similarly, within the superorthogonal, signal set, every branch differs from every incorrect branch by exactly w=2(K–3), with just two exceptions. Those exceptions are during the first step of divergence and final reemergence 1 of 5
TSO (W ,1) =
K–1 delay latches, thus 2K–1 states D
Input bit
D
(3)
and 2
∂ W K +2 ⎛ 1 − W ⎞ TSO (W , β ) = ⎜ ⎟ . ∂β (1 − 2W )2 ⎝ 1 − W K − 2 ⎠ β =1
Select row of H2^K ⎡H4 H4 ⎤ e.g., H8= ⎢ ⎥ ⎣⎢ H 4 H 4 ⎦⎥
2K symbols per input bit.
D
Select row of H2^(K–2) ⎡0 0 0 0⎤ ⎢0 1 0 1 ⎥ ⎥ e.g., H4= ⎢ ⎢0 0 1 1 ⎥ ⎢ ⎥ ⎣0 1 1 0⎦
0
and
+
D
+
When 1, inverts all symbols
+ K–2
2 symbols per input bit.
Fig. 2. Superorthogonal convolutional encoder (basic, r =1). The XOR operation is shown here as addition with implied modulo 2.
with respect to the correct path. These exceptions are triggered by the compete inversion of the correct symbol generated by the outer XOR function. Hence, the initial and final branches of a superorthogonal code’s error event differ in 2w symbols. For both codes, there is the simple relation between w and code’s rate, R, 1 w= . (1) 2R This then defines the labeling of the reduced state diagram in [1]. A capital W is used to indicate any pattern of w binary symbol errors and W2 the pattern of all 2w binary symbols in error for superorthogonal. The relationship to the placeholder for a coded symbol error Z, is simply W 2 = Z1 R . (2) The state transition diagram can be greatly simplified as elegantly explained in [3]. These codes are deemed to belong to a larger class of codes called zero-run length (ZRL) convolutional codes. The zero-run length of the encoder state is defined as the number of consecutive leading zeros in the binary state representation. A rate 1/N convolutional code is said to be a ZRL code if the output weight depends only on the input and the zero-run length of the state. The tools developed in [3] greatly simplify finding the state transition diagram and the generating function. Below are two useful forms of the superorthogonal code’s generating function as presented in [1]:
PaperID 1516 for MILCOM 2006
Once the generating function of the code, T(δ,β), is derived, error rate bounds may be computed. As shown in [1], the coded B-bit block error probability, PE, and bit error probability, Pb, can be upper-bounded, respectively, by (5) PE < B T (W ,1) W = exp[ − E /(2 N )] b
K–1 delay latches, thus 2K–1 states D
(4)
III. ERROR BOUNDS
Fig. 1. Orthogonal convolutional encoder block diagram.
Input bit
W K +2 ⎛ 1 − W ⎞ ⎜ ⎟ 1 − 2W ⎝ 1 − W K − 2 ⎠
Pb
1 these superorthogonal codes are no longer ZRL codes. This fact makes finding an elegant form of the transfer function difficult. We have chosen to find PaperID 1516 for MILCOM 2006
We have used this puncturing strategy to create Fig. 7. We found it easier to use numeric techniques as in Section IV than solving the problem analytically. It is interesting to note that the performance is actually improved by puncturing for code rates < ¼. As expected, at very low rates the performance difference to the unpunctured code is negligible.
3 of 5
VI. CONCLUSION Updated AWGN performance curves were presented for the original superorthogonal codes. Performance curves were introduced for Zehavi and Viterbi’s generalized superorthogonal codes for the unquantized AWGN channel. A richer variety of rate and constraint length combinations are available with these variations, at seemingly reasonable performance. Further, the code rate may be fine-tuned through symbol puncturing. No loss, and in fact a gain in performance was witnessed for the light levels of puncturing studied. Table I summarizes the Eb/No required to reach a 10-6 bit error rate bound for a variety of superorthogonal codes and compares that to the Eb/No requirements found through searching for the best convolutional codes at 10-6 in [5]. Except for the outlier case of r = 3 and R = 1/2, all SOC codes appear to be no worse than 0.12 dB of those codes in [5]. In fact, the punctured superorthogonal codes of Table I are the same codes identified by [5]. Table II similarly summarizes the Eb/No required to reach a 10-6 bit error rate bound of superorthogonal codes for 7 ≤ K ≤ 9 and compares that to the Eb/No requirements produce by the nested searching technique for maximum free distance (MFD) convolutional codes [6]. As is well known, MFD guarantees the best asymptotic performance as Eb/No approaches infinity. In spite of the MFD codes better free distance, the superorthogonal codes are seen to generally be perform better at the finite Eb/No corresponding to 10-6 bit error rates due to better properties of the distance spectrum.. Superothogonal codes provide a quick way to derive the encoder’s generating polynomials and produce very acceptable performance compared with search techniques over the space of low-rate feed-forward convolutional codes. ACKNOWLEDGMENT The author gratefully acknowledges the assistance of Drs. Andrew J. Viterbi and Ephriam Zehavi in verifying the integrity of the first error rate figure and the encouragement given by Dr. Jack K. Wolf. REFERENCES [1] A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication, Reading, Mass.:Addison-Wesley, 1995, ch 5. [2] A. J. Viterbi, “Orthogonal tree codes for communication in the presence of white Gaussian noise,” IEEE Trans Commun. Technol., vol. COM-15, pp. 238-242, Apr. 1967. [3] R. McEliece, S. Dolinar, F. Pollara, and H. Van Tilborg, “Some easily analyzable convolutional codes,” TDA Progress Report 42-99, Jet Propulsion Laboratory, Pasadena, Calif., Nov. 15, 1989, pp. 105-114. Available: http://tmo.jpl.nasa.gov/. [4] E. Zehavi and A. J. Viterbi, “On new classes of orthogonal convolutional codes,” in Proceedings of Bilkent International Conference on New Trends in Communication, Control, and Signal Processing, (Ankara, Turkey), pp. 257-263, July 1990. [5] P. J. Lee, “New short constraint length, rate 1/N convolutional codes which minimize the required SNR for given desired bit
PaperID 1516 for MILCOM 2006
error rates,” IEEE Trans. on Comm., vol. 33, pp. 171-177, Feb. 1985. [6] P. Frenger, P. Orten, and T. Ottosson, “Code-spread CDMA using maximum free distance low-rate convolutional codes,” IEEE Trans. on Comm., vol. 48, pp. 135-144, Jan. 2000. TABLE I COMPARISON OF REQUIRED EB/NO FOR BER OF 10-6 TO [5] K 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7
Rate 1/2 1/2 1/2 1/3 1/4 1/2 1/4 1/6 1/7 1/8 1/2 1/4 1/8 1/14 1/15 1/16 1/4 1/8
SOC order 1 2 1b 1b 1 3 2 1b 1b 1 4 3 2 1b 1b 1 4 3
SOC Required Eb/No (dB) 6.705 6.180 6.180 5.958 5.906 6.066 5.354 5.211 5.211 5.280 5.238 4.791 4.772 4.680 4.704 4.756 4.377 4.369
Reference [5] Eb/No (dB) 6.706 a 6.180 a 6.180 a 5.958 a 5.906 a 5.745 5.298 5.211 a 5.211 a 5.211 5.238 4.779 4.693
Ratio (dB) -0.001 0.000 0.000 0.000 0.000 0.321 0.056 0.000 0.000 0.069 0.000 0.012 0.079
c
c
c
c
4.367 4.251
0.010 0.118
c
c
Required bit error probability of SOC is computed by upper union bound. For unpunctured SOC codes of order r =1, the bound is computed exactly by (4) and (8). For other SOC codes, (12) is used with truncated generating function. a Generator polynomials are identical. b SOC codes of order r =1 are punctured 1 or 2 symbols per bit to achieve the rate indicated. c No convenient code of comparison. TABLE II COMPARISON OF REQUIRED EB/NO FOR BER OF 10-6 TO [6] K 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9
Rate 1/4 1/8 1/16 1/30 1/31 1/32 1/8 1/16 1/32 1/62 1/63 1/64 1/16 1/32 1/64 1/126 1/127 1/128
SOC order 4 3 2 1b 1b 1 4 3 2 1b 1b 1 4 3 2 1b 1b 1
SOC Required Eb/No (dB) 4.377 4.369 4.312 4.259 4.277 4.307 3.925 3.936 3.918 3.890 3.901 3.916 3.623 3.580 3.575 3.560 3.566 3.574
Reference [6] Eb/No (dB) 4.375 4.310 4.300 4.305 4.307 4.299 3.976 3.991 4.022 4.058 4.063 4.066 3.636 a 3.639 a 3.663 a 3.622 a 3.622 a 3.623 a
Ratio (dB) 0.002 0.058 0.012 -0.047 -0.029 0.008 -0.051 -0.056 -0.103 -0.168 -0.162 -0.150 -0.012 -0.059 -0.089 -0.062 -0.056 -0.049
In this table, SOC codes are treated similarly to Table I. The bit error probability union bound for the Maximum Free Distance codes with Optimal Distance Spectrum [6] is found by (12) with truncated generating function. a Generator list for K=9 [6] corrected by original authors to be {g0, g1, ..., g17} = {453, 455, 457, 467, 473, 475, 513, 517, 551, 557, 573, 657, 671, 675, 735, 745, 753, 765} b SOC codes of order r =1 are punctured 1 or 2 symbols per bit to achieve the rate indicated.
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Eb/No (dB) for tight upper bound of BER = 1E-6
7.0
orthogonal code basic SOC SOC r=2 SOC r=3 SOC r=4
6.0
5.0
K=7 4.0
K=9 3.0
2.0
1
10
100
1000
Bandwidth Expansion (1 / Rate) Fig. 6. Required Eb/No (dB) for the tight upper bound on bit error probability to achieve 1E-6.
Eb/No (dB) for tight upper bound of BER = 1E-6
7.0
SOC without puncturing SOC with p=1 SOC with p=2
6.0
K=5 K=6
5.0
K=7 K=8 4.0
3.0
1
10
Bandwidth Expansion (1 / Rate) Fig. 7. Required Eb/No (dB) for several puncturing schemes for the basic SOC (r =1) are shown.
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