likewise contribute in improving the bit error rate [3][4][5][6]. The theoretical aspects of the relation between the code rate,. Paper approved by H. Leib, the Editor ...
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 8, AUGUST 2011
Transactions Letters Near Shannon Limit and Low Peak to Average Power Ratio Turbo Block Coded OFDM Maryam Sabbaghian, Member, IEEE, Yongjun Kwak, Student Member, IEEE, Besma Smida, Member, IEEE, and Vahid Tarokh, Fellow, IEEE
Abstract—In this paper, we present an advanced solution for the long standing problem of large peak to average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems. Although the design of low PAPR codewords has been extensively studied and the existence of asymptotically good codes with low PAPR is also proven, still no code has been constructed to satisfy all requirements. The main goal of the paper is to develop a coding scheme that not only generates low PAPR codewords, but it also performs relatively close to the Shannon limit. We achieve this goal by implementing a timefrequency turbo block coded OFDM. In this scheme, we design the frequency domain component to have a tightly bounded PAPR. The time domain component code is designed to obtain good performance while the decoding algorithm has reasonable complexity. Through comparative performance evaluation we show that utilizing the proposed method, we achieve considerable improvement in terms of PAPR while we slightly loose the performance compared to capacity achieving codes with similar overall block length.
minimum Euclidean distance of the code and its block length is provided in [7] as two fundamental theorems. The first one proves a lower bound for PAPR based on the three aforementioned parameters. The second theorem provides a lower bound for the code rate as a function of maximum acceptable PAPR, code block length and code minimum distance. Despite all the research on this subject, the error correction capability of coding-based PAPR reduction methods has not been paid the attention it deserves. One of the works which considers this issue, in addition to the PAPR of the code, is [8] where a new class of Reed Muller (RM) code with a very low PAPR is proposed. However, the performance of this code is quite far from the Shannon limit and can not compete with that of the capacity achieving codes. Therefore the problem of designing codes that perform closely to the Shannon limit while having low PAPR has remained unsolved.
Index Terms—Time-frequency turbo block code, Chase algorithm, OFDM, PAPR, Reed-Muller code.
In this paper, we propose a time-frequency turbo block code (TBC) to solve the problem of achieving good BER performance with a low PAPR. The frequency domain component code is designed such that it provides codewords with low PAPR. To obtain this goal, we have used the approach proposed in [8] to exploit Golay complementary sequences as the generalized coset for first order RM codes within the second order RM codes [9]. The design based on this approach guarantees low PAPR. Through comparative simulations, we demonstrate that for the choice of time domain code, low density parity check (LDPC) codes and Bose-Ray-Chaudhuri (BCH) codes are two appropriate candidates. The algorithms utilized to decode LDPC code and BCH code have reasonable complexity. Thereby, the good performance is obtained at the cost of a practical moderate computational complexity. We exhibit the superiority of this scheme by examining the performance and amplitude distribution of other codes with similar code rates.
I. I NTRODUCTION
H
IGH peak to average power ratio (PAPR) is considered as the principal drawback of multi-carrier systems [1]. To alleviate this problem, amplifiers having large power backoff values are required, otherwise, the nonlinear characteristics of the amplifier distorts the in-band signal and splatters the out-of-band spectrum. However, increasing the back-off of the amplifier, reduces the power efficiency and rises the amplifier cost significantly. This is an issue for both user terminals and base stations in which the amplifier cost is a significant portion of the overall expense. Several PAPR reduction schemes have been extensively studied for multi-carrier systems. Most of these methods have adverse effects such as transmission of side information or spectral regrowth [1][2]. A prominent solution to reduce PAPR is utilizing coding techniques which likewise contribute in improving the bit error rate [3][4][5][6]. The theoretical aspects of the relation between the code rate, Paper approved by H. Leib, the Editor for Communication and Information Theory of the IEEE Communications Society. Manuscript received June 30, 2009; revised April 9, 2010. The authors are with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA (e-mail: {maryam, ykwak, bsmida, vahid}@seas.harvard.edu). The work of the first and third authors was partly funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Digital Object Identifier 10.1109/TCOMM.2011.080111.090356
We show that using the suggested scheme we can obtain bit error rate relatively close to the Shannon limit while maintaining PAPR bounded by 3 dB, 6 dB and 9 dB for OFDM block lengths of 16, 32 and 64 respectively. The block lengths are in terms of symbols and we use BPSK. The PAPR is not bounded for LDPC code or turbo code, but to have a comparable value for their PAPR, we consider the level that the probability of PAPR exceeding which is 10−6 . This value for block lengths of 16, 32 and 64 is 9 dB, 7 dB and 4 dB larger than the maximum PAPR of the proposed TBC.
c 2011 IEEE 0090-6778/11$25.00 ⃝
SABBAGHIAN et al.: NEAR SHANNON LIMIT AND LOW PEAK TO AVERAGE POWER RATIO TURBO BLOCK CODED OFDM
The paper is organized as follows: Section II describes the fundamentals of PAPR problem in OFDM systems. The details of time-frequency turbo block coded OFDM is explained in Section III. The simulation results are presented in Section IV, and Section V concludes the paper. II. PAPR P ROBLEM IN OFDM S YSTEMS The OFDM signal which is a linear combination of equally weighted data symbols, might exhibit a large magnitude. The fluctuations of the signal are usually quantified by PAPR 𝑃 (𝑡) , in which 𝑃 (𝑡) and 𝑃𝑎𝑣 denote defined as 𝑃 𝐴𝑃 𝑅 = max 𝑃𝑎𝑣 the instantaneous and average power over one OFDM block respectively. To satisfy the spectral masks imposed by regulatory agencies and to overcome the implementational constraints, investigating methods to generate a signal with favorable PAPR properties is of vital importance. Various methods have been proposed to solve this problem. An interesting approach is to deal with the problem in the framework of channel coding. This approach, was first proposed in [3]. The method of [3] entails an exhaustive search to determine the low PAPR codewords. This is extremely complex for large number of sub-carriers. It also requires a huge memory to encode/decode large data blocks. In [10], the selection of the appropriate codewords is developed based on specific sequences such as Shapiro-Rudin and Golay sequences. However, neither in [3] nor in [10] the error correcting issue has been addressed. In [4], first a powerful error correcting code is selected and then by using a weight vector the PAPR of the codeword is decreased. In [7] the existence of asymptotically good codes with PAPR of at most 8 log 𝑁 is proven where 𝑁 is the block length. In [11] it is shown that the PAPR of a code is associated with minimum distance decoding of the code. Despite extensive research on PAPR reduction using coding algorithms, none of the proposed methods have been utilized in practice. The main reason is the loss in spectral efficiency as OFDM block length increases and the poor error correcting capability of the proposed schemes. Therefore, the design of a capacity achieving code for which all codewords generate low PAPR OFDM signals, is still an important open problem. III. T IME -F REQUENCY T URBO B LOCK C ODE Turbo block codes were introduced in [12] and developed in [13][14]. As opposed to the traditional turbo codes, in TBC the original data block consists of a 𝑘1 𝑘2 rectangular matrix. In the encoding process, the horizontal and vertical block codes are applied to generate first a 𝑘1 𝑛2 and then a 𝑛1 𝑛2 coded data block. These codes are decoded by exchanging soft information between the component decoders iteratively. Some component decoders are inherently capable of generating soft information. If the component decoder generates only hard output, we can exploit the trellis based decoding of the block codes to have soft output [12]. To prevent high complexity of the trellis based decoding, we employ the Chase algorithm which is significantly simpler [15]. In this paper, we propose a time-frequency OFDM turbo block code to achieve low PAPR while maintaining the performance relatively close to the Shannon limit. This
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scheme is generated when horizontal and vertical codes in a TBC are exploited in frequency and time domain respectively. In other words, the row codewords are transmitted using OFDM method. To obtain low PAPR, we confine the frequency domain component code to a set of codes with theoretically proven good PAPR properties. To improve the BER performance, we utilize powerful codes such as LDPC codes and BCH codes in the time domain. A. Frequency Domain Component Code For the choice of frequency domain component code we use a class of Reed-Muller codes, introduced in [8], which are the cosets of generalized first order Reed-Muller codes in the second order Reed-Muller codes. This approach is based on the Golay complementary sequences which are defined as two sequences whose auto correlation functions satisfy the following equation 𝑅𝑎 (𝑛) + 𝑅𝑏 (𝑛) = (∣a∣2 + ∣b∣2 )𝛿(𝑛), where 𝛿(𝑛) is the Dirac delta function and the∑ auto correlation ∗ function, 𝑅(𝑛), is defined as 𝑅𝑎 (𝑛) = 𝑖 𝑎𝑖 𝑎𝑖+𝑛 . The relationship of 𝑅𝑎 and 𝑅𝑏 leads to an appealing property between the instantaneous power of the Fourier transform of a and b. Assume 𝑃𝑎 and 𝐴𝑘 denote the instantaneous power and Fourier transform of sequence a. Since 𝑃𝑎 (𝑘) = ∑ −𝑗 2𝜋𝑛𝑘 𝑁 , we have: 𝑅 𝑛 𝑎 (𝑛)𝑒 ∑ 2𝜋𝑛𝑘 [∣a∣2 + ∣b∣2 ]𝛿(𝑛)𝑒−𝑗 𝑁 𝑃𝑎 (𝑘) + 𝑃𝑏 (𝑘) = An alternative interpretation of the above equation can be presented in the following theorem [16]: Theorem: For unit power constellations such as BPSK and QPSK, the PAPR of any complementary Golay sequence is bounded by 2. This theorem, as the fundamental idea based on which the entire approach is developed, motivates to utilize Golay complementary sequences for OFDM systems. To understand the link between the RM codes and Golay complementary pairs we review what is proven in [8]: Let 𝜋 denote any permutation of the set {1, . . . , 𝑚}. For any 𝑐𝑘 , 𝑐, 𝑐′ ∈ ℤ2ℎ , the sequences generated by 𝑢(x) = 2ℎ−1
𝑚−1 ∑
𝑥𝜋(𝑘) 𝑥𝜋(𝑘+1) +
𝑘=1 ℎ−1
𝑣(x) = 𝑢(x) + 2
𝑥𝜋(𝜇) + 𝑐′ ,
𝑚 ∑
𝑐𝑘 𝑥𝑘 + 𝑐
𝑘=1
are Golay complementary over ℤ2ℎ . Noting that RM codes are generated by a linear combination of Boolean monomials of 𝑥𝑖 ’s, we can conclude that each coset of RM(1,∑ 𝑚) in RM(2, 𝑚) having a coset representative of 𝑚−1 the form 𝑘=1 𝑥𝜋(𝑘) 𝑥𝜋(𝑘+1) comprises 2𝑚+1 binary Golay sequences of length 2𝑚 . Henceforth, for generalized RM codes over ℤ2ℎ , the PAPR of the codewords in the cosets does not exceed 3 dB [8]. This justifies the use of RM codes as the frequency domain component code in TBC. A critical issue to develop a feasible iterative structure is to utilize component codes with good performance and tolerable complexity. Among the various methods proposed for decoding of RM codes, the MAP decoder outperforms the others [17][18][19]. This decoder by utilizing fast Hadamard
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transform develops an efficient decoding method which inherently generates log-likelihood ratios (LLR) [19].
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B. Time Domain Component Code Although RM codes possess the property of having the maximum possible minimum distance for their affordable code rates, their performance is far from the Shannon limit. Therefore, despite their appropriate PAPR properties, they are not as efficient as we need performance-wise. This is the motivation to use them in a turbo structure to improve the overall performance. To select the time domain component code, we take these issues into account; efficiency in terms of BER performance, moderate decoding complexity and feasibility of generating soft output. Considering these requirements, LDPC codes seem as natural candidates for time domain component code. In addition to LDPC codes, we examine BCH codes since other turbo block codes based on BCH codes are reported to perform close to the Shannon limit[13]. The simplicity of BCH decoder is also appealing although it only generates hard output. To implement a soft-input soft-output decoder for the BCH, we utilize the Chase algorithm which uses a near ML approach to calculate LLRs [13].
1 + 𝐷 + 𝐷3 1 + 𝐷 + 𝐷2 + 𝐷3 , ]. (1) 1 + 𝐷2 + 𝐷3 1 + 𝐷2 + 𝐷3 The outputs of the constituent encoders are punctured and repeated to achieve the desired rate of 14 . Fig. 1 depicts the complementary cumulative distribution function (ccdf) of PAPR for turbo block coded OFDM, LDPC code and convolutional based turbo code for different OFDM block lengths. For all TBC cases, we have used RM(1,4) with length 16. OFDM symbols with larger block lengths consist of several blocks of RM(1,4). For LDPC and turbo code we divide the entire coded data block (length 𝑛1 𝑛2 ) into smaller block with the desired length and transmit each sub-block using OFDM. To generate the ccdf, the signal is oversampled [1,
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Fig. 1. Complementary cumulative distribution function of PAPR for TBC, LDPC and turbo code (𝑁 =OFDM block length). 0
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BTC (LDPC & RM), Itr#1 BTC (LDPC & RM), Itr#2 BTC (LDPC & RM), Itr#4 BTC (LDPC & RM), Itr#6 BTC (LDPC & RM), Itr#10 Turbo code, R=0.25 LDPC code, R=0.25
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In comparative performance evaluation presented in this section, we restrict the frequency domain component code to a RM(1,4) code (𝑛1 = 16) which is decoded by the MAP algorithm [19]. For the time domain component code, we consider LDPC and BCH codes. The LDPC code is a rate 45 left-regular code with block length of 𝑛2 = 125 and variable node degree of 3 which is decoded by belief propagation algorithm with 10 iterations. The other considered time domain component code is a BCH(51,63) code (𝑛2 = 63). To have an insight of how the PAPR changes for capacity approaching codes, we examine LDPC code and conventional turbo code. For a fair comparison we set the block length of these codes to 𝑛1 𝑛2 = 2000. The LDPC code has the following degree distribution: Λ2 = 0.59, Λ3 = 0.23, Λ5 = 0.07, Λ6 = 0.05, Λ15 = 0.06, where Λ𝑖 depicts the fraction of the variable nodes of degree 𝑖. The average check node degree is 𝑑¯𝑐 = 3.42. The other code considered for comparison is a turbo code which includes two recursive convolutional codes. The transfer function of the turbo code is as follows
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Fig. 2. Performance comparison of turbo block code (LDPC and RM), convolutional turbo code and LDPC code.
by a factor of four. The unusual form of ccdf for length of 16 is related to its short block length. The presented results are for BPSK constellation. To improve the bandwidth efficiency, we can utilize higher order constellations and use 𝑅𝑀2ℎ (𝑟, 𝑚) which generalizes 𝑅𝑀 (𝑟, 𝑚) from alphabet 𝑍2 to alphabet 𝑍2ℎ and its cosets are Golay pairs in 𝑍𝑅𝑀2ℎ (2, 𝑚). As can be seen in Fig. 1, the threshold from which the PAPR exceeds with probability of 10−6 , has been decreased from 12 dB in LDPC code and convolutional turbo code to 3 dB in TBC for 𝑁 =16. For block lengths of 32 and 64, the ccdf is improved by 7dB and 4dB at probability of 10−6 . We should note that this significant improvement in the PAPR distribution is obtained at no extra cost such as decreasing the transmission rate or distorting the transmitted signal. For larger block codes, there is a trade-off between the code rate and peak to average ratio. One can use sub-blocks of the RM code to preserve the code rate at the expense of having PAPR larger than 3 dB. Alternatively, we can use RM codes with larger code length and confine the PAPR below 3 dB at the cost of reducing the code rate.
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examined LDPC code and BCH code due to their appropriate error correcting properties and simplicity of their decoding scheme. The turbo block code is also utilized for error correcting purpose. We demonstrate that using the proposed TBC, we can significantly reduce the PAPR with a small performance loss compared to best coding schemes with similar code block length. R EFERENCES
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BTC (BCH & RM), Itr#1 BTC (BCH & RM), Itr#2 BTC (BCH & RM), Itr#3 BTC (BCH & RM), Itr#4 Turbo Code, R=0.25 LDPC code, R=0.25
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Fig. 3. Performance comparison of turbo block code (BCH and RM), convolutional turbo code and LDPC code.
The proposed TBC not only provides a low PAPR but its error correcting capability is also comparable with that of LDPC and convolutional based turbo codes. Fig. 2 and Fig. 3 present the BER performance of the proposed TBC where LDPC code and BCH code are time domain components respectively. As expected the performance improves as iterations continue. For both TBCs the code rate is 14 whose corresponding Shannon limit is −.81 dB. Both TBC schemes achieve BER of 10−5 at 2.6 dB which is 3.4 dB away from Shannon limit. The performance of the code might look far from the Shannon limit but we have to note that other capacity achieving codes approach the Shannon limit asymptotically when the code block length approaches to infinity. For a fair comparison we compare TBC performance with two capacity approaching codes when the overall block length of TBC is similar to the length of those codes. The performance of an LDPC code and a turbo code are also presented in Fig. 2 and Fig. 3. As illustrated, these codes outperform TBC by about 1 dB as they obtain BER of 10−5 at 1.5 dB and 1.65 dB. The performance loss of the proposed TBC has been compromised by the significant improvement in PAPR. It is noteworthy that the proposed decoding scheme is about two orders of magnitude less complex than the considered LDPC and turbo code. V. C ONCLUSION In this paper, we proposed a time-frequency turbo block code to solve the PAPR problem of OFDM systems. To obtain low PAPR, we restrict the frequency domain component code to realizations of Golay sequences as cosets of the generalized first order RM codes. For the time domain components we
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