A Novel Turbo Coded Modulation Scheme for Deep

IEICE TRANS. COMMUN., VOL.Exx–??, NO.xx XXXX 200x

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LETTER

A Novel Turbo Coded Modulation Scheme for Deep Space Optical Communications∗∗ Sangmok OH† , Member, Inho HWANG† , Adrish BANERJEE†† , and Jeong Woo LEE†∗ , Nonmembers

SUMMARY A novel turbo coded modulation scheme, called the turbo-APPM, for deep space optical communications is proposed. The proposed turbo-APPM is a serial concatenation of turbo codes, an accumulator and a pulse position modulation (PPM), where turbo codes act as an outer code while the accumulator and the PPM act together as an inner code. The generator polynomial and the puncturing rule for generating turbo codes are chosen to lower the bit error rate. At the receiver, the joint iterative decoding is performed between the inner decoder and the outer turbo decoder. In the outer decoder, local iterative decoding for turbo codes is conducted. Simulation results are presented showing that the proposed turbo-APPM outperforms all previously proposed schemes such as LDPC-APPM, RS-PPM and SCPPM reported in the literature. key words: Deep space optical communications, turbo coded modulation, iterative decoding, pulse position modulation, Poisson channel.

1.

Fig. 1

The transmitter of the proposed turbo-APPM.

Introduction

Many efforts have been made to improve the performance of the deep space optical communication system [1]-[5]. One of these efforts is to find a modulation scheme proper for optical communications [1]. Due to its good receiver sensitivity, the pulse position modulation (PPM) has been widely considered for use in deep space optical communications. Another effort is the use of error correction codes (ECC) in the optical communications. The serial concatenation of Reed Solomon (RS) codes and PPM [2], and that of turbo codes and PPM [3] were proposed to provide coding gains, where the demodulation and the decoding are performed separately. Recently, coded modulation techniques using the serial concatenation of ECC, accumulator and PPM have been proposed to provide further coding gains [4][5]. In these coded modulation schemes, ECC is an outer code and the cascade of accumulator and PPM is an inner code, and the joint decoding between the inner decoder and the outer decoder is performed in an iterative manner. The serially concatenated PPM (SCPPM) [4] uses convolutional codes serially concatenated with an accumulator and a PPM, whereas the LDPC-APPM † The authors are with the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul, Korea. †† The author is with the department of Electrical Engineering, Indian Institute of Technology, Kanpur, India. ∗ Corresponding author ∗∗ This research was supported by the Chung-Ang University Research Scholarship Grants in 2008.

Fig. 2

The turbo encoder of the proposed turbo-APPM.

[5] uses low density parity check (LDPC) codes serially concatenated with an accumulator and a PPM. In this letter, a coded modulation scheme using the serial concatenation of turbo codes, accumulator and PPM is proposed, which will be called the turboAPPM. Turbo codes introduced by Berrou et. al. [6] in 1993 show capacity-approaching error correction capability over the additive white Gaussian noise (AWGN) channel. Turbo codes refer to a class of parallel concatenation of two component encoders separated by an interleaver and are decoded sub-optimaly using iterative decoding. Each of two component decoders employs maximum a posteriori (MAP) decoding by using the BCJR algorithm [7] and exchange soft information between them. Turbo codes can be easily punctured to obtain higher rate codes. The proposed turbo-APPM uses the iterative joint decoding between the inner decoder and the outer turbo decoder. The outer turbo decoder performs some local iterations between component decoders before passing the soft information to the inner decoder. We search for good component encoders as well as a puncturing pattern for turbo codes. Simulation studies show that the proposed turbo-APPM provides coding gains over SCPPM, LDPC-APPM and RS-PPM techniques [2].


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Fig. 3

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The receiver of the proposed turbo-APPM.

System Model

The transmitter of the proposed turbo APPM scheme is shown in Fig. 1. It consists of a serial concatenation of turbo codes with an accumulator and a PPM. Turbo codes can be viewed as an outer code while the cascade of the accumulator and the PPM forms a single inner code. A random interleaver Π is placed between the inner and the outer code. Let m denote the user data bits, c denote the coded bits generated by the turbo encoder and d denote the scrambled version of c by the interleaver Π, where c and d have a blocklength of 𝑁 bits. The structure of the turbo encoder is shown in Fig. 2. It consists of parallel concatenation of two identical component encoders. We use nonsystematic and nonrecursive encoders with a generator polynomial given by (5, 7)8 , as shown in Fig. 2. Let u and v be the outputs of two component encoders, where c is obtained by concatenating u and v through the puncturing block. In a 𝑀 -ary signaling, the log2 𝑀 neighboring bits form a symbol, so 𝐾 = 𝑁/ log2 𝑀 symbols are generated by the 𝑀 -ary PPM from d following the accumulator. Let 𝑠𝑘 denote the symbol transmitted at the time instance 𝑘, 𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾, and we define a vector s = [𝑠1 , ⋅ ⋅ ⋅ , 𝑠𝐾 ], where 𝑠𝑘 has a decimal value 𝑗 ∈ {0, 1, ⋅ ⋅ ⋅ , 𝑀 − 1}. In the 𝑀 -ary PPM, one symbol period is divided into 𝑀 time slots, and a pulse is generated only in the 𝑗-th time slot if the symbol to be transmitted is 𝑠𝑘 = 𝑗. The optical channel is modeled as Poisson distribution [1], [?, ]. Let 𝑛𝑏 be the mean number of noise photons in one PPM time slot and 𝑛𝑠 be the mean number of signal photons in a pulsed time slot. The probability mass function of the number of photons detected at the receiver is given by (𝑛𝑠 + 𝑛𝑏 )ℓ exp[−(𝑛𝑠 + 𝑛𝑏 )] ℓ! 𝑛ℓ𝑏 exp[−𝑛𝑏 ] 𝑝(ℓ ∣ 0) = , ℓ! 𝑝(ℓ ∣ 1) =

(1) (2)

where 𝑝(ℓ ∣ 1) and 𝑝(ℓ ∣ 0) are the probabilities that ℓ photons are detected in a certain time slot at the

receiver in case a pulse is generated and a pulse is not generated, respectively, in this time slot. As shown in Fig. 3, the receiver is composed of a soft output demapper, an inner decoder (APPM decoder) and an outer decoder (turbo decoder), where an interleaver and a deinterleaver are placed between the inner decoder and the outer decoder. The inner decoder and the outer decoder perform joint decoding by exchanging iteratively soft information between each other. The outer decoder is composed of two component MAP decoders and conducts a local iterative decoding by exchanging soft information between these two component decoders. After a pre-defined number of local iterations between two component MAP decoders, the global iteration for joint decoding of the inner decoder and the outer decoder is performed. 3.

Iterative Decoding

Let y = [y1 , ⋅ ⋅ ⋅ , y𝐾 ] be the input of the receiver, where y𝑘 , 𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾, is the 𝑀 -tuple column vector whose 𝑗-th element, denoted by 𝑦𝑘𝑗 , is the number of photons detected in the 𝑗-th PPM time slot corresponding to 𝑠𝑘 . Let 𝐿𝑐 (y) denote the channel information. Let 𝐿𝑎 (s) and 𝐿(s) denote the log-likelihood (LL) of the a priori probabilities and the LL of the a posteriori probabilities, respectively, of the transmitted symbol s. Note that 𝐿𝑎 (s) and 𝐿(s) are 𝑀 × 𝐾 matrices and the (𝑗, 𝑘)th element of 𝐿𝑎 (s) is defined by log 𝑝(𝑠𝑘 = 𝑗). We let 𝐿𝑎 (d), 𝐿𝑎 (c), 𝐿𝑎 (m), 𝐿𝑎 (u) and 𝐿𝑎 (v) be the loglikelihood ratio (LLR) of the a priori probabilities of d, c, m, u and v, respectively, where the 𝑖-th element of 𝑖 =0) 𝐿𝑎 (d) is defined by log 𝑝(𝑑 𝑝(𝑑𝑖 =1) . Likewise, 𝐿𝑒 (d), 𝐿𝑒 (c), 𝐿𝑒 (m), 𝐿𝑒 (u) and 𝐿𝑒 (v) denote the extrinsic information of d, c, m, u and v, respectively, and 𝐿(d), 𝐿(c), 𝐿(m), 𝐿(u) and 𝐿(v) denote the LLR of the a posteriori probabilities of d, c, m, u and v, respectively. The operation of iterative decoding is described in Fig. 3. The soft symbol demapper computes 𝐿𝑐 (y) by [4] ) ( 𝑛𝑠 + 𝑛𝑏 𝑗 + constant. (3) 𝐿𝑐 (y) = 𝑦𝑘 ⋅ log 𝑛𝑠 The inner decoder computes 𝐿(s) by the symbol-based

LETTER

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BCJR algorithm using 𝐿𝑐 (y) as channel information and 𝐿𝑎 (s) as a priori information [4]. The outer decoder computes 𝐿(c) and 𝐿𝑒 (c) as well as 𝐿(m) by the bit-based BCJR algorithm [7] using 𝐿𝑎 (c) as channel information. 𝐿𝑒 (c) is interleaved by Π to 𝐿𝑎 (d) and sent to the inner decoder, and 𝐿𝑒 (d) is deinterleaved by Π−1 to 𝐿𝑎 (c) and sent to the outer decoder. Note that 𝐿𝑎 (d) and 𝐿(d) are the LLR information for bits while 𝐿𝑎 (s) and 𝐿(s) are the LL information for symbols. Thus, 𝐿𝑎 (d) needs to be converted to 𝐿𝑎 (s) at the input and 𝐿(s) needs to be converted to 𝐿(d) and consequently to 𝐿𝑒 (d) at the output, respectively, of the inner decoder [4]. Let d𝑘 be the log2 𝑀 -bit stream forming the symbol 𝑠𝑘 , and 𝑑𝑘,𝑖 be the 𝑖-th bit of d𝑘 . Let 𝐿𝑎 (𝑑𝑘,𝑖 ) and 𝐿(𝑑𝑘,𝑖 ) be the LLR of the a priori probabilities and the LLR of the a posteriori probabilities of 𝑑𝑘,𝑖 , respectively. We also let 𝐿𝑎 (𝑠𝑘 ) and 𝐿(𝑠𝑘 ) be the LL of the a priori probabilities and the LL of the a posteriori probabilities of 𝑠𝑘 , respectively. Then, 𝐿𝑎 (𝑠𝑘 ) is obtained by log2 𝑀

𝐿𝑎 (𝑠𝑘 = 𝑗) =

∑

log[𝑝(𝑑𝑘,𝑖 = 𝑎𝑗𝑖 )]

𝑖=1 log2 𝑀

∑ 1 𝑗 = (−1)𝑎𝑖 𝐿𝑎 (𝑑𝑘,𝑖 ) + constant, 2 𝑖=1 where the last equality comes from 𝑗 1 log[𝑝(𝑑𝑘,𝑖 = 𝑎𝑗𝑖 )] = (−1)𝑎𝑖 𝐿𝑎 (𝑑𝑘,𝑖 ) 2 1 + log[𝑝(𝑑𝑘,𝑖 = 0)𝑝(𝑑𝑘,𝑖 = 1)] 2

(4)

(5)

and 𝑎𝑗𝑖 is the value of 𝑑𝑘,𝑖 corresponding to 𝑠𝑘 = 𝑗. After 𝐿(s) is computed by the inner decoder using symbolbased BCJR algorithm, it is converted to 𝐿(d) by ⎡ ⎤ ⎡ ⎤ ∑ ∑ 𝐿(𝑑𝑘,𝑖 ) = log ⎣ 𝐿(𝑠𝑘 = 𝑗)⎦−log ⎣ 𝐿(𝑠𝑘 = 𝑗)⎦ . 𝑗:𝑎𝑗𝑖 =0

𝑗:𝑎𝑗𝑖 =1

(6) Then, 𝐿𝑒 (d) is obtained by subtracting 𝐿𝑎 (d) from 𝐿(d). At the input of the outer decoder, 𝐿𝑎 (c) is demultiplexed into 𝐿𝑎 (u) and 𝐿𝑎 (v) according to the puncturing pattern, where 𝐿𝑎 (u) and 𝐿𝑎 (v) are used as the channel information at the component MAP decoder 1 and the component MAP decoder 2, respectively. Each component MAP decoder computes 𝐿(m) using bitbased BCJR algorithm and sends 𝐿𝑒 (m) to the other component MAP decoder to be used as 𝐿𝑎 (m) after interleaved by 𝜋 or deinterleaved by 𝜋 −1 . At the last local iteration for each global iteration, 𝐿𝑒 (u) and 𝐿𝑒 (v) are obtained at two component MAP decoders by the BCJR algorithm, and are multiplexed to 𝐿𝑒 (c) according to the puncturing pattern. After a certain number of global and local iterations, the hard decision for m is performed by using the signs of 𝐿(m).

Table 1 Puncturing patterns of turbo codes used in the turboAPPM, where the overall code rate is 1/2.

4.

Numerical Results

In our simulation studies, the optical channel is modeled as (1) and (2) with 𝑛𝑏 = 0.2, and 64-ary PPM is used with the time slot duration 𝑇𝑠 = 32 ns [4]. The overall code rate is chosen to be 1/2. First, we tested various puncturing patterns for the proposed turbo-APPM, where some sample puncturing patterns are listed in Table 1. Puncturing pattern 1 and 2 results in assigning different code rates to the two component MAP decoders, whereas puncturing pattern 3 and 4 results in assigning the same code rates to the two component MAP decoders. As shown in Fig. 4, the BER performance is sensitive to the choice of puncturing pattern and the uneven assignment of channel information to MAP decoders results in better performance. We propose to use puncturing pattern 1 for the turbo-APPM scheme. Fig. 5 shows the BER performance of the proposed turbo-APPM, and compares it with the BER performances of SCPPM, LDPC-APPM, RS-PPM and double binary turbo-APPM with systematic turbo codes used in the IEEE 802.16e standard. 𝑁 is set to 4603 for all schemes except for RS-PPM, where 𝑁 = 4085. As shown in Fig. 5, the proposed turboAPPM outperforms all other schemes. We examined the two best schemes, namely, the turbo-APPM and the SCPPM, for longer blocklength 𝑁 = 15120. In this comparison, we used the interleaver Π proposed in [4] while a random interleaver 𝜋 was used for turbo codes. The number of decoding iterations of the SCPPM is set to 20, while for the turbo-APPM, the number of local iterations for outer turbo decoding is set to 3 and the number of global iterations for joint decoding between the inner decoder and the outer decoder is set to 15. The numbers of decoding iterations for SCPPM and turbo-APPM are chosen such that the required computational complexities for these two schemes are similar. As shown in Fig. 6, the proposed turbo-APPM shows a coding gain of almost 0.1 dB over SCPPM at BER=10−5 . The latency of coded modulation scheme is highly dependent on the hardware algorithm and architecture of the decoder. The proposed turbo-APPM scheme implemented by high-speed decoding techniques [8]–[10] may attain low latency and high throughput enough to be used for high-speed deep-space optical commu-


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Fig. 4 The BER performances of the proposed turbo-APPM with various puncturing patterns, where 𝑁 = 4608

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Fig. 6 The BER and FER performances of the proposed turboAPPM and the SCPPM with 𝑁 = 15120.

PPM, LDPC-APPM and SCPPM. The performance of the turbo APPM scheme is very sensitive to the puncturing pattern. The performance of proposed turboAPPM may improve further by using specialized interleavers and using puncturing patterns of larger period.

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Fig. 5 The BER performances of the proposed turbo-APPM, RS-PPM, SCPPM, LDPC-APPM and double binary turboAPPM with 𝑁 = 4608. (For RS-PPM, 𝑁 = 4085.)

nications. The turbo-APPM decoder may achieve the throughput of 4.7 Mbps with five global iterations and three local iterations of Max-log-MAP decoding with scaling, where the BER performance is 1 dB away from the capacity. It is reported by [8] that the SCPPM decoder with equivalent computational complexity achieves the throughput of 3.36 Mbps and the BER performance of 1.1 dB away from the capacity. 5.

Conclusion

We proposed a new coded modulation scheme for deep space optical communications that uses a serial concatenation of turbo codes, accumulator and PPM. The simulation results show that the proposed scheme outperforms all published coding schemes such as RS-

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