A Novel Constellation Reshaping Method for PAPR ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011

A Novel Constellation Reshaping Method for PAPR Reduction of OFDM Signals Cai Li, Tao Jiang, Senior Member, IEEE, Yang Zhou, and Haibo Li

Abstract—In this paper, we propose a constellation reshaping method for peak-to-average power ratio (PAPR) reduction in orthogonal frequency division multiplexing (OFDM) systems. The key idea of the proposed method is to generate a reshaped quadrature amplitude modulation (R-QAM) constellation, in which the minimum distance of the R-QAM is the same as that of the traditional QAM constellation. When the novel R-QAM constellation is employed with the traditional partial transmit sequence (PTS) and selected mapping (SLM) methods in OFDM systems, called as the R-PTS and R-SLM methods in this paper, respectively, the mean square errors between the constellations of the received data and each R-QAM rotated by different phase rotation factors are calculated at the receiver, and the phase rotation factor with the minimum mean square error is selected to recover the original data without side information. Therefore, the flexibility in phase rotation choice and the ability to avoid data rate loss make the R-PTS and R-SLM methods more suitable for good bit-error-rate performance in OFDM systems. Theoretical analysis and simulation results show that the R-PTS and R-SLM methods could provide the same PAPR reduction and bit-error-rate as those of the conventional PTS and SLM methods with perfect side information, respectively. Index Terms—Orthogonal frequency division multiplexing (OFDM), peak-to-average power ratio (PAPR), constellation reshaping, partial transmit sequence, selected mapping, symbol error rate (SER).

I. INTRODUCTION RTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) technique is widely used in broadband wireless communication systems due to its high data rate and immunity to the frequency selective fading channels. However, one major drawback is the high peak-to-average power ratio (PAPR) of OFDM signals. As a result, the complexity of digital-to-analog converters (DAC) and the linear

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Manuscript received July 27, 2010; revised November 10, 2010; accepted January 17, 2011. Date of publication February 04, 2011; date of current version May 18, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Roberto Lopez-Valcarce. This work was supported by the National Science Foundation of China with Grant 60872008, the Program for New Century Excellent Talents in University of China under Grant NCET-08-0217, the Research Fund for the Doctoral Program of Higher Education of the Ministry of Education of China under Grant 200804871142, the Science Found for Distinguished Young Scholars of Hubei in China with Grant 2010CDA083, and the National High Technology Development 863 Program of China under Grant 2009AA011803. The authors are with the Wuhan National Laboratory For Optoelectronics, Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TSP.2011.2109955

range of high power amplifiers (HPA) have to be increased to avoid signal distortion, severely reducing system performance. Recently, various methods have been proposed to reduce the PAPR of OFDM signals in the literature [1], including clipping [2], [3], nonlinear companding transforms [4]–[6], coding technique [7], selected mapping (SLM) [8]–[10], partial transmit sequence (PTS) [11]–[16], time-domain symbol combining [17], adaptive projected subgradient method [18], constellation modification [19], and tone reservation [20]. Moreover, some methods are proposed to control the PAPR in OFDM systems, such as using fountain codes [21]. Among these methods, the PTS and SLM methods are the most attractive and significant techniques due to their good performance in terms of PAPR reduction. Their basic idea is to generate several alternative OFDM signals by multiplying every original data subblock (i.e., cluster of some subcarriers) with different phase rotation vectors. Then, the generated signal with the minimum PAPR is selected to be transmitted, and its corresponding phase rotation vectors for subblocks is called as optimal phase rotation combination. Therefore, parts of subcarriers are occupied to transmit the optimal phase rotation combination as side information to recover the original data at the receiver in the traditional PTS and SLM methods. Obviously, when the side information is corrupted due to fading channel and noise, etc., the original data cannot be correctly recovered. As a result, the bit-error-rate (BER) performance is much degraded. Hence, some extra protection bits have to be employed for the protective transmission of the side information to guarantee a reasonable BER performance, resulting in a decrease in spectrum efficiency. Therefore, some extensions of the SLM and PTS methods have been proposed to reduce the amount of the occupied subcarriers for the side information transmission [15]. In this paper, we propose a constellation reshaping method to reduce the PAPR of OFDM signals without side information. For the proposed method, a reshaped quadrature amplitude modulation (R-QAM) constellation is generated, then, the inherent diversity of the R-QAM constellation with different phase rotation factors could be exploited at the receiver. In other words, the mean square errors between the constellations of the received data and the rotated R-QAM constellations are calculated, and the phase rotation factor with the minimum mean square error is employed to recover the original data at the receiver. Moreover, the average power of the R-QAM constellation is almost the same as that of the traditional square quadrature amplitude modulation (S-QAM) constellation because the minimum distance of the R-QAM constellation is the same as that of the S-QAM constellation, resulting in good BER performance of OFDM systems. Simulation results also show that the

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LI et al.: NOVEL CONSTELLATION RESHAPING METHOD

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proposed method has the same ability of the PAPR reduction as that of the conventional PTS and SLM methods with perfect side information, and could provide good BER performance. The rest of this paper is organized as follows. We review the typical OFDM system and introduce the traditional PTS and SLM methods for the PAPR reduction in Section II. In Section III, we propose the constellation reshaping method in detail, and also integrate it with the PTS and SLM methods without side information. Performance analysis is given in Section IV and simulation results are presented in Section V, followed by conclusions in Section VI. II. PAPR FORMULATION IN OFDM SYSTEMS To better approximate the transmitted OFDM signal, the discrete-time signal is obtained by employing the operation -point inverse discrete Fourier transform (IDFT) to of zero-padding, the input original data block with where the oversampling factor is a positive integer. Then, the OFDM signal is expressed as (1) where is the input data block. is defined as the Generally, the PAPR of OFDM signals ratio between the maximum instantaneous power and its average power, i.e.

. Meanwhile, the the total number of the combinations is receiver must have the knowledge about the generation process of the transmitted OFDM symbol. Therefore, the optimal phase rotation combination with the minimum PAPR must be transmitted as the side information. B. Selected Mapping For the SLM method,

phase rotation vectors are generated

as (5) where data block quences as

and is multiplied by each

. Therefore, the input to obtain alternative se-

(6) Then, the alternative sequences are transformed into the time via IDFT operation, and the signal with domain signals the minimum PAPR is searched to be transmitted. Similar to the PTS method, it is common to restrict the phase elements in the SLM method. rotation factors to a set with Generally, it is reasonable to assume that the phase rotation vectors are known for both the transmitter and receiver, and thus, of the phase rotation vector as side information, the index with the minimum PAPR needs to be transmitted for the receiver to recover the data sequence .

(2) C. Side Information where

is the average power.

A. Partial Transmit Sequence For the traditional PTS method, the input data block is partitioned into disjointed subblocks , where and is an integer. Obviously, the size of subblocks is the same as that of the original data block. Moreover, it satisfies (3) with or where . Therefore, the number of nonzero elements in is for . With IDFT operation, the partitioned subare transformed into time domain signals . Then, blocks with the alternative signal could be obtained by multiplying phase rotation factors as (4) , and . Therewhere fore, the objective of the PTS method is to find the optimal comof the phase rotation factors to bination minimize the PAPR of the alternative signal . Commonly, the phase rotation factors are restricted to a elements, i.e., . Therefore, set with

For the traditional PTS and SLM methods, the side information is required to inform the receiver of the optimal phase rotation vector. In order to illustrate why the side information needs to be transmitted in the PTS and SLM methods, we give some intuitive results of the traditional -ary S-QAM de. As shown in Fig. 1, noted by , in which and . the minimum distance between the constellation points is multiplied by a phase rotation factor , we have When

. Similarly, is . Obviously, obtained when the phase rotation factor is equals to since is an integer multiple , which implies that we can not of determine which value the phase rotation factor is without the side information. Therefore, for the conventional PTS and SLM methods, the side information of the phase rotation factors has to be transmitted and protected by extra bits. III. PROPOSED CONSTELLATION RESHAPING METHOD In this section, we propose a constellation reshaping method to generate novel R-QAM constellations, and its big contribution is that the PTS and SLM methods do not need to transmit the side information when it combines with the PTS or SLM

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Fig. 1. The 4-ary S-QAM and 4-ary R-QAM constellations.

methods to reduce the PAPR of OFDM signals. Based on the conventional -ary S-QAM constellation, the reshaped constellation of the -ary R-QAM could be formulated as (7) where and imaginary part. As shown in Fig. 1, the 4-ary R-QAM

denotes the

Fig. 2. Constellations of the received data subblocks multiplied by different phase rotation factors when the R-PTS method is employed. (a) Phase rotation factor b = 1. (b) Phase rotation factor b = j.

0

Denote the data after the multiplication by the factor

When

, where is multiplied by a phase rotation factor

.

. Similarly, . Obviously, is the same as if and only if is an integer multiple of . However, the phase rotation vectors ( , 2), which means . hold in Therefore, is the same as if and only if , which implies that a rotated R-QAM constellation corresponds to only one phase rotation factor. Therefore, when the R-QAM constellations are employed with the traditional PTS and SLM methods, called as the R-PTS and R-SLM methods in this paper, respectively, the side information does not need to be transmitted in OFDM systems.

as . (correAs shown in Fig. 2(a), the constellation points of sponding phase rotation factor is ) are around . (corresponding phase Similarly, the constellation points of ) are around . Therefore, the oprotation factor is timal phase rotation vectors could be recovered at the receiver according to the following steps. is partitioned into vectors 1) , where . For , execute the following steps; each vector . 2) A hard decision is made for each data in is determined as the nearest constellation point in by comparing with . Denote the distance between and the nearest constellation point as , i.e.

A. The R-PTS Method (9)

With the R-PTS method, the received signals could be expressed as

(8) where and denote the channel frequency response and the channel noise at the th subcarrier, respectively. Suppose that the channel estimation is perfect, then, , where and .

where and 3) Calculate the mean square error of the decisions

;

(10) 4) Select the phase rotation factor whose corresponding mean square error is minimum to recover the original data.


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Fig. 4. Comparison of the PAPR reduction with the conventional and proposed and 64, respectively. methods when

K = 4; 16;

Fig. 3. Constellations of the received data block demodulated with different phase rotation vectors when the R-SLM method is employed. (a) The correct phase rotation vector. (b) The incorrect phase rotation vector.

B. The R-SLM Method For the R-SLM method, the received signals are denoted as . Suppose the optimal phase rotation , then we have vector is

Therefore, the optimal phase rotation vector for the SLM method could be estimated according to the following steps. with 1) According to (12), calculate the and each phase rotation vector . For each , execute the following steps; . is de2) A hard decision is made for each data in termined as the nearest constellation point in , which is with . obtained by comparing and the nearest constelDenote the distance between lation point as , i.e. (13)

(11) where and are the channel frequency response and the channel noise at the th subcarrier. For simplicity, the channel estimation is assumed as perfect, then, the recovered symbol could be calculated as

where ; 3) Calculate the mean square error of the decisions

(14) 4)

(12)

where . for When , i.e., the constellation points of are around . Otherwise, when , where does not . As a result, the identically equal 1 for constellation points of are around of both and . For example, Fig. 3 shows some intuitive results when and , respectively, where the phase rotation factors for .

with the minimum mean square error is determined as is its corresponding phase rotathe original data, and tion vector. For convenience, the S-PTS method is denoted as that the PTS method combines with the conventional S-QAM constellation in this paper. Similarly, the S-SLM method is defined as that the SLM method combines with the conventional S-QAM constellation. As shown in Fig. 4, the R-PTS and R-SLM methods have the same performance of the PAPR reduction as that of , the S-PTS and S-SLM methods, respectively, in which for the S-PTS and 16, and 64, respectively, for the S-SLM and R-SLM methods. R-PTS methods, Moreover, it is obvious that the order of the QAM modulation does not have any effect on the PAPR reduction performance of the R-PTS and R-SLM methods.

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IV. PERFORMANCES ANALYSIS A. Symbol Error Rate (SER) of R-QAM In this subsection, we detailedly derive out the formulation of the SER when the R-QAM constellation is employed in OFDM systems. For simplicity, we consider 4-ary R-QAM constellation as an example. As shown in Fig. 5, the signal space is partitioned into four parts, denoted as part-I, part-II, part-III, and part-IV, respectively, and their corresponding data are denoted , and , respectively, since the hard decision is as used for the demodulation at the receiver. of . SupFirst, we calculate the error probability pose that the transmitted data is , but the decision is incorrect. is not in the part-I, In other words, the received signal where denotes the standard deviation of the additive white and follow Gaussian noise (AWGN). Suppose both the Gaussian distribution with a zero mean and a variance of , distribution respectively. Then, the distribution of follows . The probability denwith 2 degree of freedom and distribution with degree of freedom 2 sity function (pdf) of is written as

Fig. 5. Hard decision of the 4-ary R-QAM modulation.

(15) otherwise where . Therefore, the pdf of the noise amplitude

Fig. 6. Schematic diagram with the analysis of .

is (16)

As shown in Fig. 6(a) and (b), there are two cases of that does not lie in the part-I, i.e., and , for . Note respectively, where and is as shown in that, the maximum of both Fig. 6(b). Therefore, the error probability of is Fig. 7. Schematic diagram with the analysis of .

(17)

Moreover, it is obvious that the error probability of is the is the same same as that of , and the error probability of as that of , respectively. Hence, the error probability of each R-QAM symbol could be expressed as follows:

Similar to

, Fig. 7 shows the case of , where for . Note that, the for also is . Therefore, the maximum of is error probability of

(19) (18)

Obviously, the bit signal-to-noise ratio (SNR) is for


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, where the average power of signal is . According to (15), (16), and (19), we have the SER of the 4-ary R-QAM constellation as

(20) Obviously, the analysis of the 4-ary R-QAM constellation could be extended to other QAM constellations. For example, the SER of the 4-ary S-QAM constellation could be expressed as

Fig. 8. SER performance of the R-QAM and S-QAM modulations.

(21) Similarly, the SER of the 16-ary R-QAM constellation could be calculated as

TABLE I MEAN POWER OF QAM CONSTELLATIONS

(23) (22) and the SER of the 64-ary R-QAM constellation is

As shown in Fig. 8, the theoretical results of the SER performance are great agreement with the simulation results for the -ary S-QAM and R-QAM constellations when , 16, 64, respectively. Moreover, it is obvious that the 4-ary R-QAM constellation requires about 0.7 dB bit SNR more than the 4-ary S-QAM to achieve the same SER performance. However, the SER performance of the 64-ary R-QAM constellation is almost the same as that of the 64-ary S-QAM constellation. That is because that the mean power of the 64-ary R-QAM constellation is only 1.3% smaller than that of the 64-ary S-QAM constellation, which is shown in Table I. B. Error Probability of Side Information As analyzed above, one of the key factors is the error probability of the side information to affect the BER performance when the PTS and SLM methods are employed in OFDM systems. Therefore, it is indeed needed to pay attention to the error probability of the side information for the R-PTS and R-SLM methods. For simplicity, the channel estimation is perfect and . the symbol SNR is denoted as For the R-SLM method, the received symbol is deand scribed as (11), where

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are the original data and the phase rotation vector, respectively. The probability denotes the probability of the incorrect side information . Then, we have

and (31) Since generating function of

are independent, the moment is (32)

(24) Set where Set

, we rewrite the probability

as

.

(33) and we define

where denotes the pole in left half plane. Obviously, Hence, we have

.

(25) . Then, . Obviously, where and are zero mean. For simplicity, suppose , . Therefore, we have the variance and then and as covariance of

(34) According to (26), (27), (28), (30), and (31), we have (26)

(27)

(35) (28) According to [22], the moment generating function of

is (29)

According to (24), (34) and (35), we finally obtain the error probability . Similarly, for the R-PTS method, the probability denotes the probability of incorrect phase rotation factor for the th subblock , and we have

where

(36) (30)


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Fig. 9. Error probability of the side information with R-SLM and R-PTS methods.

where . Then, the error probability is obtained as . As shown in Fig. 9, the error probability of the side information with the R-SLM method is better than that of the R-PTS for the R-SLM method, method, in which for the R-PTS method, and the phase rotation factors are chosen from . However, the error probability of the R-PTS when dB, which is small enough to method is be accepted in practice.

Fig. 10. 64-ary S-QAM and 64-ary R-QAM constellations.

V. SIMULATION RESULTS In this section, simulations have been conducted to evaluate the BER performance of the proposed R-PTS and R-SLM methods in the uncoded OFDM systems over AWGN channel and the coded OFDM systems over fading channel, respectively, in which the number of subcarries , and the . Since most radio systems often oversampling factor employ HPA at the transmitter to obtain sufficient transmit power, one well-known HPA, solid-state power amplifier (SSPA), with its saturation point 13.0 is employed in OFDM system in this paper. Moreover, the convolutional coding with coding rate is also used in coded OFDM system. Note that, the curve marked with “Ideal” means that the original signals are affected only with the channel noise including fading and AWGN channels and without any HPA, resulting in the best BER performance. Fig. 10 shows different 64-ary constellations, in which “ ” and “ ” denote the point of the S-QAM and R-QAM constellations, respectively. Fig. 11 shows the BER performances of the S-PTS and R-PTS methods with the employed SSPA over AWGN and fading channels, respectively. Obviously, with the perfect side information, the BER performance of the S-PTS method is almost the same as that of the R-PTS method. However, the BER performance of the proposed R-PTS method without side information is a little worse than that of the perfect side information when . For example, when , the SNR for the R-PTS

Fig. 11. BER performance of the S-PTS and R-PTS methods with AWGN and fading channels, respectively.

M = 4 over

method without side information is 1.1 dB and 1.9 dB larger than that of its perfect side information over AWGN channel and fading channel, respectively. This is because the recovered phase rotation factor may not be correct when the SNR is large. in practice. However, the BER is generally smaller than Therefore, the proposed methods are applicable in a real OFDM system. With the decrease of the BER, the error rate of phase rotation factors becomes very small, resulting in that the BER performance of the R-PTS method without side information is the same as that of the S-PTS method with perfect side information. Moreover, it is clear that the proposed R-PTS method could still be directly employed in the coded OFDM system with good BER performance.

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REFERENCES

Fig. 12. BER performance of the S-SLM and R-SLM methods with U over AWGN and fading channels, respectively.

= 16

The performances of the S-SLM method and the R-SLM and . method are shown in Fig. 12, where As shown in Fig. 12, the BER performance of the R-SLM method without side information is almost the same as that of the S-SLM method with perfect side information over AWGN channel. In addition, the BER performance of the R-SLM method without side information has a little degradation compared with the R-SLM method with perfect side information over fading channel. Similarly, the proposed R-SLM method also could be directly employed in the coded OFDM system with good BER performance. In summary, the conducted simulation results show that the constellation reshaping method could help the conventional PTS and SLM methods to eliminate side information when they are employed for the PAPR reduction while keeping good BER performance in both uncoded and coded OFDM systems, i.e., the BER performances of the R-PTS and R-SLM methods over AWGN and fading channels are almost the same as those of the conventional PTS and SLM methods with perfect side information, respectively. Furthermore, the possible values of phase rotation factors for the R-PTS and R-SLM methods are not limited , and the proposed constellation reshaping method is suitto able for all square constellations. VI. CONCLUSION In this paper, we proposed a constellation reshaping method, which could be directly integrated with the conventional PTS or SLM methods to reduce the PAPR without side information in OFDM systems. Theoretical analysis and simulation results showed that the proposed R-PTS and R-SLM methods could achieve the same PAPR reduction as that of the S-PTS and S-SLM methods, respectively, and the BER performance of the proposed R-PTS and R-SLM methods is also almost identical to that of the S-PTS and S-SLM methods with perfect side information in uncoded and coded OFDM systems over AWGN and fading channels, respectively.

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Cai Li received the B.S. degree from Huazhong University of Science and Technology, Wuhan, P. R. China, in 2009. She is currently working toward the M.S. degree at Huazhong University of Science and Technology, Wuhan. Her current research interests include the areas of wireless communications, especially for OFDM systems with emphasis on research of PAPR reduction.

Tao Jiang (M’06–SM’10) received the B.S. and M.S. degrees in applied geophysics from the China University of Geosciences, Wuhan, in 1997 and 2000, respectively, and the Ph.D. degree in information and communication engineering from Huazhong University of Science and Technology, Wuhan, in April 2004. He is currently a full Professor with Wuhan National Laboratory for Optoelectronics, Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan. From August 2004 to December 2007, he was with some universities, such as Brunel University and the University of Michigan. He has authored or coauthored more than 70 technical papers in major journals and conferences and five books/chapters in the areas of communications. His current research interests include the areas of wireless communications and corresponding signal processing, especially for cognitive wireless access, vehicular technology, OFDM, UWB and MIMO, cooperative networks, nano networks, and wireless sensor networks. Dr. Jian served or is serving as Symposium Technical Program Committee member of many major IEEE conferences, including INFOCOM, ICC, and GLOBECOM, etc. He served as TPC Symposium Chair for the International Wireless Communications and Mobile Computing Conference 2010, and as a General Co-Chair for the workshop of Machine-to-Machine Communications and Networking in conjunction with IEEE INFOCOM 2011. He served or is serving as Editor of some technical journals in communications, including in the IEEE Communications Surveys and Tutorials, and Wiley’s Wireless Communications and Mobile Computing (WCMC), etc. He is a recipient of the Best Paper Awards in IEEE CHINACOM09 and WCSP09. He is a senior member of the IEEE Communication Society, IEEE Broadcasting Society, and IEEE Signal Processing Society.

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Yang Zhou received the B.Eng. degree in information engineering from the Wuhan University of Technology, Wuhan, P. R. China, in 2008. He is currently pursuing the Ph.D. degree at the Huazhong University of Science and Technology, Wuhan. Since September 2010, he is also with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, as a joint Ph.D. candidate. His research interests include communication theory and signal processing for wireless communications.

Haibo Li received the B.S. degree from Huazhong University of Science and Technology, Wuhan, P. R. China, in 2009. He is currently working toward the M.S. degree at Huazhong University of Science and Technology, Wuhan. His current research interests include the areas of wireless communications, especially for OFDM systems with emphasis on research of PAPR reduction.