SER analysis of PTS based techniques for PAPR reduction in OFDM systems

Digital Signal Processing 23 (2013) 302–313

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Digital Signal Processing www.elsevier.com/locate/dsp

SER analysis of PTS based techniques for PAPR reduction in OFDM systems Ashish Goel a,∗ , Prerana Gupta a , Monika Agrawal b a b

Electronics & Communication Engineering Department, Jaypee Institute of Information Technology, A-10, Sector-62, Noida (U.P.), India Centre for Advance Research in Electronics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India

a r t i c l e

i n f o

Article history: Available online 5 September 2012 Keywords: OFDM PAPR PTS

a b s t r a c t A major drawback of orthogonal frequency division multiplexing (OFDM) is high peak-to-average power ratio (PAPR). An OFDM signal with high PAPR requires power amplifier’s (PAs) with large linear operating ranges but such PAs are difficult to design and costly to manufacture. Therefore, to reduce PAPR various methods have been proposed. One of the existing technique to reduce PAPR is partial transmit sequences (PTS). The major drawback of this technique is that it requires transmission of side information (SI) with each OFDM symbol, which results in low bandwidth efficiency. It is hard to recover the side information from the OFDM signal received at the receiver. The two methods, which do not require SI to decode the OFDM symbol at the receiver, are multipoint square mapping combined with PTS (M-PTS) and concentric circle mapping based PTS (CCM-PTS). In this paper, the SER performance of PTS based methods namely CCM-PTS and M-PTS over AWGN channel is mathematically analyzed. The SER performance of CCM-PTS over AWGN is analyzed using two decoding techniques, namely minimum distance decoding and circular boundary decoding, whereas M-PTS is analyzed using minimum distance decoding. The simulation results for SER performance of CCM-PTS and M-PTS, over fading channel, have been presented using computer simulations and the SER performance of CCM-PTS by both the decoding techniques is compared with M-PTS. Also, a comparison of PAPR reduction capability and computational complexity of CCM-PTS and M-PTS has been presented. CCM-PTS method almost has the same PAPR reduction capability as M-PTS, but its SER performance is better than M-PTS and uses a simpler method to decode the data symbols. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Orthogonal frequency division multiplexing is now being used for data transmission in a number of wireless communication systems, which includes digital audio broadcasting (DAB) and digital video broadcasting (DVB) systems. OFDM is a multicarrier system, with various advantages over single carrier systems, such as ISI mitigation, robustness to multipath fading by the use of cyclic prefix, high bandwidth efficiency and low implementation complexity [1]. As a result, OFDM is a good choice for high data rate broadband communication. An OFDM signal results in large peak to average power ratio (PAPR), especially when the total number of subcarriers is large and all the subcarriers with same initial phase are added. In an OFDM transmitter, power amplification is performed by power amplifier (PA). If the OFDM signal has high envelope fluctuations then PA should have very large linear range of operation, which makes it very expensive. If PA has limited linear range then its operation

*

Corresponding author. E-mail addresses: [email protected] (A. Goel), [email protected] (P. Gupta), [email protected] (M. Agrawal). 1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.dsp.2012.08.018

in non-linear mode introduces out-of-band radiation and in-band distortion. A large number of solution have been proposed in the literature [2] to reduce the PAPR of the OFDM signal. One of the simple solutions is to clip [3] the OFDM signal before amplification but such non-linear processing causes signal distortion and also introduces in-band distortion and out-of-band radiation. The out-of-band radiation can be eliminated by using frequency domain filtering operation, but it is likely to re-grow the peaks that were originally reduced and as a result, PAPR of OFDM signal will be increased again. Therefore, if clipping and filtering are repeated several times, then both out-of-band radiation and PAPR can be reduced, as discussed in [4]. However, it [4] does not eliminate the in-band distortion. Another distortion technique to reduce the PAPR is companding. Many companding techniques like μ-law companding, nonlinear companding transform and exponential companding, etc. [5,6] have been proposed by researchers. It is shown in [5], that μlaw companding technique can reduce PAPR more effectively than clipping approach but the average signal power after companding the OFDM signal becomes more than the original OFDM signal power, on the other hand, in other companding techniques like non-linear companding transform, exponential companding, etc.

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

The average power of OFDM signal before and after companding remains same. But all the companding techniques distort the shape of original OFDM signal and result in BER performance degradation. Two more distortion-less PAPR reduction techniques namely selective mapping (SLM) [7] and partial transmit sequence (PTS) [8] are also proposed in the literature. In SLM, parallel data signal of length N is multiplied by a predetermined set of U phase vectors of length N, and generate U alternative signals. Out of U alternate signals, one of them with the least PAPR is selected for transmission. In PTS scheme all the subcarriers are partitioned into multiple disjoint sub-blocks and then each of the sub-blocks are multiplied by a set of rotating phase factors and combined to achieve PAPR as low as possible. The information about the phase factors by which these sub-blocks are multiplied needs to be conveyed to the receiver, which is known as SI. The SI has the highest importance because it is required to recover the original transmitted data signal. If SI gets corrupted then the entire OFDM data block can get damaged and by which error performance of PTS based OFDM system degrades severely. In PTS technique, if the number of sub-blocks or the number of phase factors in the phase set increases then it not only increases the computational complexity for selecting the optimum set (provide least possible PAPR) of phase sequence but also increases the amount of SI to be conveyed to the receiver. The SI results in data rate loss in OFDM system. Many PTS based schemes for eliminating the requirement of SI transmission have been proposed in [9–15]. In [9], an SI embedding scheme has been proposed by Cimini and Sollenberger based on a marking algorithm and decision statistic at transmitting and receiving end, respectively. The scheme in [9] may not be reliable for large constellation size and applicable only for M-ary PSK modulation. Jayalath and Tellambura proposed a maximumlikelihood decoding [10] for eliminating the requirement of side information. In this scheme [10] the modulation symbols of given constellation and multiple signals generated by multiplication of phase factors have sufficient Hamming distance to decode the original signal, but the SI detection capability of this scheme degrades at lower SNR values. The methods [9,10] embed SI but suffer from the problem of peak re-growth [9], increased decoding complexity [9,10] and incapable of general search algorithm [9,10]. In [11] trellis shaping system design, based on control of autocorrelation side lobes of an OFDM data sequence is proposed by Ochiai et al., but it [11] also results in loss of data rate. In [12] Nguyen and Lampe proposed a combinatorial optimization based search algorithm, to find the optimal phase factors and precoding of data stream with small redundancy is used to embed SI, but it [12] requires one bit per sub-block and hence transmission of SI is not completely eliminated. In [13], Zhou et al. proposed M-PTS scheme which extends the QPSK constellation points to disjoint points of 16-QAM constellation and eliminates the requirement of side information. In [14], we have proposed CCM-PTS scheme, which takes data in quaternary form and map these data points to disjoint points lying on concentric circles through phase rotations. In [15], Yang et al. proposed an SI embedding scheme, in which the candidate signals are obtained by cyclically shifting each of the sub-blocks in time domain and combining them in recursive order, but the receiver design of such a system is complex. The schemes proposed in [9–12,15] embed SI in the transmitted signal and extracts SI from the received signal at the receiver, whereas the schemes proposed in [13,14] are completely free from SI, i.e. extraction of SI from the received signal is not required. In [9,10,12,15], the information about the phase factors used at the transmitter for minimizing the PAPR is recovered from the extracted SI. The reciprocal of recovered phase factors are further used to multiply the demodulated signal at the receiver to recover the original data signal, but this operation increases the computational complexity at the receiving end, whereas the schemes

303

proposed in [13,14] do not require SI and therefore, no such multiplication operation needs to be performed at the receiver, hence the receiver structure of the scheme proposed in [13,14] is computationally less complex. In many of the SI embedding schemes [9,10,12,15], the SI detection at low SNR is very poor, and due to which error performance of the OFDM system degrades severely. In wireless standards like LTE, OFDM is used in downlink, where mobile station acts as receiver. The mobile stations have limited computational resources; therefore, a PAPR reduction scheme with less computational complexity at receiving end will be beneficial. As discussed above, the schemes proposed in [9,10, 12,15] have computationally complex receiver in comparison to the schemes proposed in [13,14]. Hence, CCM-PTS and M-PTS schemes are the viable choices for PTS-OFDM system. Based on our review of the existing literature, the M-PTS method of [13] is a suitable scheme to eliminate the requirement of SI in PTS-OFDM system, in terms of complexity and performance. Motivated by M-PTS, in this paper, we discuss a method namely concentric circle mapping based PTS (CCM-PTS), to completely eliminate the requirement of SI in a PTS-OFDM system. Theoretical results for SER performance of CCM-PTS-OFDM over AWGN channel is derived using minimum distance decoding and circular boundary decoding algorithms. The SER analysis of MPTS scheme over AWGN channel is also carried out by minimum distance decoding and verified by simulation results. The simulation results for SER performance of CCM-PTS-OFDM over fading channel using minimum distance decoding and circular boundary decoding have been found out and compared with M-PTS OFDM system. The PAPR and symbol error rate (SER) performances of the CCM-PTS method [14] are compared with the existing M-PTS method of [13]. The advantages of CCM-PTS over M-PTS in terms of SER performance and computational complexity are shown. Since the methods [13,14] are SI-free, the drawbacks associated with SI transmission, as discussed above, are automatically removed. The remaining paper is organized as follows. In Section 2, we briefly discuss the OFDM system model and PAPR of OFDM signal. The conventional PTS scheme and PTS based methods (M-PTS and CCM-PTS) are discussed in Section 3. In Section 4, we present theoretical analysis of SER and computational complexity of PTS based methods. To confirm the validity of the analytical expressions obtained in Section 3, computer simulations are also performed; these results are presented and discussed in Section 5. Finally, we conclude in Section 6. In this paper, time domain signals are represented by lower case letters and frequency domain signals are represented by upper case letters. Here E {.} and max{.} denotes the expectation and the maximum value operator, respectively. 2. OFDM system model and PAPR of OFDM signal In OFDM system, serial binary data sequence is converted into N parallel data sequences and then it is mapped to desired constellation points (M-PSK or M-QAM). Let { X q,k }kN=−01 be the N complex symbols to be transmitted at qth OFDM block, then corresponding time domain OFDM signal can be represented as N −1 1  xq (t ) = √ X q,k exp( j2π k f t ), N k =0

0  t  Ts

(1)

where T s ,  f and N are the OFDM symbol duration, the subcarrier frequency spacing and the total number of subcarriers, respectively. The sampled version of time domain OFDM signal xq (t ) given by (1) can be expressed as

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A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

Fig. 1. Block diagram of conventional PTS-OFDM scheme.

 xq [n] = xq

nT s N



N −1 1 

=√

N k =0

N −1 1 

=√

N k =0

phase factors b(m) followed by N point IFFT and finally combining them.

X q,k exp( j2π  f knT s / N )

 X q,k exp

j2π kn



N

(2)

Here, T s  f = 1. Eq. (2) represents the discrete time OFDM signal, which is obtained by taking the IDFT of the transmitted symbols { X q,k }kN=−01 and can be implemented efficiently by using IFFT algorithm. PAPR is the ratio of peak power to the average power of OFDM signal, for discrete-time OFDM signal xq [n], it may be expressed as

 PAPR(dB) = 10 log10

max{|xq [n]|2 }



E {|xq [n]|2 }

(3)

If a discrete-time OFDM signal (xq [n]) is over sampled by a factor L  4, then its PAPR is a good approximation of the continuoustime OFDM signal xq (t ) [5]. The complementary cumulative distribution function (CCDF) is a parameter to characterize the peak power statistics of a digitally modulated OFDM signal. The CCDF of PAPR provides information about the percentage of OFDM signals that have PAPR above a particular level. 3. PTS based methods for PAPR reduction 3.1. Conventional PTS scheme and existing SI embedding schemes The block diagram of OFDM transmitter with PTS technique is shown in Fig. 1. In PTS approach, N input complex symbols { X q,k }kN=−01 are partitioned into M disjoint sub-blocks of equal M −1 size { X q }m =0 . The various partitioning methods to divide complex symbols into disjoints sub-blocks are proposed in [16], these include pseudorandom, adjacent and interleaved partitioning schemes. Out of these, pseudorandom technique has the best performance, whereas adjacent partitioning, with less computational complexity, can be used because its PAPR reduction capability is very close to pseudorandom partitioning. The qth OFDM symbol (xq ) can be obtained by multiplying each of these sub-blocks with

(m)

xq =

M −1 

(m)

b(m)xq

(4)

m =0

where b(m) is the phase factor for mth partition. The phase factors (m) are assumed to be pure rotational in time domain. xq the IFFT (m)

of X q is called the partial transmit sequence. The set of phase factors are defined as



B= e

j2π l W

   l = 0, 1 , 2 , 3 , . . . , W − 1

(5)

where, W is the number of phase factors, b(m) ∈ B. The commutation of IFFT operation and multiplication by rotating phase factors is noteworthy, because both the operations are linear and can be mutually interchanged. Therefore, without any loss it can be assumed that multiplication with phase factor can be applied before IFFT operation, i.e. directly on the constellation symbols. This observation helps us in simplifying the structure of OFDM system and for suggesting any modification. As mentioned earlier, conventional PTS scheme requires the transmission of SI with the transmitted OFDM signal, which not only results in loss of data rate but also results degradation in error performance, in case of erroneous detection of SI. Therefore, in literature many SI embedding schemes [9,10,12,15] have been proposed. In [9], Cimini and Sollenberger embedded a marker on the transmitted data sequence and at the receiver; SI is extracted from embedded marking sequence. The scheme proposed in [9] uses 16 partitions (M) and Walsh Hadamard codes of length 16 are used as phase factors to multiply the sub-blocks. In this scheme, if any of the sub-block needs to multiply by a phase factor 1, then we keep that sub-block as it is, otherwise we rotate the data on every other subcarrier in that sub-block by π /4. At the receiver, after performing the subcarrier demodulation, raise the fourth power of demodulated data symbols (for QPSK modulation) and detect them differentially for extracting the side information without using quantization. The Euclidean distance of the detected SI is calculated with each of the Walsh Hadamard code words and one of

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

305

Fig. 2. (a) Quaternary data. (b) Mapping of quaternary data to 16-QAM constellation using 4 phase factors in M-PTS. Table 1 Mapping of quaternary data points to 16-QAM constellation. Quaternary data

Initially mapped quaternary data points to 16 QAM constellation

Constellation points after multiplication with phase factors in B 1

j

−1

−j

0

3+3j

3+3j

−3 + 3 j

−3 − 3 j

3−3j

G# 1

1

−3 + j

−3 + j

−1 − 3 j

3− j

1+3j

G# 2

Group number

2

−1 − j

−1 − j

1− j

1+ j

−1 + j

G# 3

3

1−3j

1−3j

3+ j

−1 + 3 j

−3 − j

G# 4

the codes with minimum Hamming distance is chosen as the recovered phase factors. In this decoding algorithm, the SI detection capability depends on the Hamming distance of the code words. In [9], the length of Walsh Hadamard code words is 16, which has a minimum Hamming distance of 8. In [9], the error in SI detection has been evaluated using Walsh Hadamard code, over AWGN and fading channel. It has been shown in [9] that over AWGN channel, the probability of error in SI detection at 3.2 dB SNR, is 10−3 , whereas, over fading channel it requires about 8 dB SNR for the same probability of error in SI detection. It is noteworthy that when a PTS-OFDM system uses 4 subblocks then to multiply these sub-blocks with phase factors, we require Walsh Hadamard codes of length 4, which have a minimum Hamming distance of 2, due the smaller value of Hamming distance there are more chances of performance loss in SI detection. The lesser number of code words in Walsh Hadamard matrix also has significant effect on PAPR performance because for M = 4, we can have only 4 candidates for searching the OFDM signal with lowest PAPR, due to which the PAPR reduction capability of the scheme proposed in [9] is very limited. Hence, for smaller number of sub-blocks, it is not considered as a good PAPR reduction scheme. As mentioned earlier, in [10], Jayalath and Tellambura proposed a receiver structure based on ML decoding algorithm for eliminating the requirement of SI. In this type of PTS based system with 4 partitions and 4 phase factors, 64 searches are required to found out the optimal set of phase factors, which requires a receiver structure with 64 branches to decode the side information. The computational complexity of such a receiver is very high and is found to be unsuitable for practical implementation. As seen from Fig. 1 shown in [10], Jayalath and Tellambura proposed to use a receiver structure with less number of branches, to limit the computational complexity of the receiver, but by limiting the number of branches in the receiver structure, the numbers of candidate OFDM signals for PAPR reduction also get reduced. In [10] only

six signals for M = 4 have been used for searching an OFDM signal with lowest PAPR. Therefore, by having very limited number of candidate signals, the PAPR reduction capability of the scheme proposed in [10] is very limited in comparison to conventional PTS scheme. Hence, the scheme proposed in [10] is not a good choice from PAPR reduction and computational complexity point of view. Nguyen and Lampe proposed a trellis shaping based SI embedding scheme [12], which requires pre-processing of data stream before PAPR reduction. As mentioned in [12], it requires few redundant bits per sub-block to embed the SI. As seen from Fig. 6 shown in [12], the PAPR reduction capability of this scheme is very close to the conventional PTS scheme, but it has been found that its SI detection capability is not significant. Based on the results given in [12], over AWGN and fading channel it requires 7.8 dB and 11.7 dB SNR, respectively, for achieving 10−3 probability of error in SI detection, which seems to degrade the error performance of the overall OFDM system at low SNR. Hence, the requirement of redundant bits for embedding the SI and its poor SI detection capability at low SNR makes it unattractive for PTS-OFDM system. In [15], Yang et al. proposed a SI embedding scheme, which generates the candidate OFDM signals after cyclically shifting the sub-blocks and combining them in recursive order. As seen from the Fig. 2 shown in [15], at the receiving end, we require M detectors to retrieve the original data signal. As mentioned in [15], this type of the detector requires K (number of cyclic shifts) times more additions and multiplications in comparison to conventional PTS receiver, which makes this type of receiver computationally complex. Hence, this type of receiver is not found suitable for wireless standard like LTE in downlink. Also, based on the results given in [15], over AWGN and fading channel, it requires 5 dB and 9 dB SNR respectively, for achieving 10−3 probability of error in SI detection. Hence, receiver computational complexity and poor SI detection performance at low SNR makes this scheme unsuitable for wireless standard like LTE.

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A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

Based on the above discussion, we can now conclude that SI embedding schemes [9,10,12,15] are not suitable because of their poor SI detection capability at low SNR and increased computational complexity of the receiver. Therefore, we focus on two SI free schemes [13,14] described in following subsections. 3.2. Modified PTS (M-PTS) In [13] a multipoint square mapping based PTS technique has been proposed by Zhou et al. to reduce the PAPR of the OFDM signal. In this method, receiver does not require any SI to retrieve the original OFDM signal. In this scheme, the input bit stream is first converted into quaternary data points (0, 1, 2 and 3) lying in four different quadrants and then obtained quaternary data points are initially mapped to four different points of 16-QAM constellation as per Table 1. As seen from Fig. 2(a) the initially mapped constellation points lie in four different quadrants and cover all 16 points of 16-QAM constellation after multiplication with 4 phase factors (1, j , −1, − j ). In Fig. 2(b) the constellation points generated from 3 + 3 j, −3 + j, −1 − j, 1 − 3 j, after multiplication with phase factors (1, j , −1, − j ), are denoted by the same color and style of marker. All data points of 16-QAM constellation are divided into four different groups. Each of these groups has four constellation points lying in four different quadrants. Each of these groups correspond to one quaternary data point. The complete mapping scheme used in M-PTS [13] is given in Table 1. After performing quaternary to 16-QAM mapping, conventional PTS scheme is used to reduce the PAPR of the OFDM signal. At the receiver, after subcarrier demodulation, a de-mapping scheme given in Table 2 is used to recover the quaternary data points. According to the Table 2, if any of the data point is decoded as {3 + 3 j , −3 + 3 j , −3 − 3 j or 3 − 3 j }, {−3 + j , −1 − 3 j , 3 − j or 1 + 3 j }, {−1 − j , 1 − j , 1 + j or −1 + j } or {1 − 3 j , 3 + j , −1 + 3 j or −3 − j }, then it is de-mapped to quaternary data 0, 1, 2 or 3, respectively. Decoding does not require any information about the phase factors thus the major constraints of PTS technique, i.e. the need of SI is completely eliminated by this scheme [13].

of radius 2d and 4d. As shown in Fig. 3 and Table 3, the quaternary data points 0, 1, 2 and 3 are initially mapped to √four different points and these are located at 0, j2d, −4d and 2 2(1 − j )d respectively. The quaternary data points 0 and 1 are located at the origin and on a circle of radius 2d, respectively, whereas, the quaternary data points 2 and 3 are located on a circle of radius 4d. As shown in Fig. 3, the quaternary data points 0, 1, 2 and 3, after initial mapping are denoted by diamond, triangle, circle and square, respectively. The constellation points lying on concentric circles of radius 2d and 4d ensures a minimum Euclidean distance of 2d between any two constellation points. In QPSK constellation (±1 ± j), the minimum Euclidean distance between two constellation points is 2. To maintain the same Euclidean distance in the CCM-PTS constellation, √ we can take d = 1. The constellation points 0, j2d, −4d and 2 2(1 − j )d after multiplication with phase factors chosen from set B, defined in Eq. (5) are converted to the points lying on the circles to which they originally belong. In √ Fig. 3, the constellation points 0, j2d, −4d and 2 2(1 − j ) d, after multiplication with 4 phase factors {1, j , −1, − j } are mapped √ to {0}, √{ j2d, −2d, − √j2d, 2d}, {−4d √, − j4d, 4d, j4d} and {2 2(1 − j )d, 2 2(1 + j )d, 2 2(−1 + j )d, 2 2(−1 − j )d} respectively. At the receiver the decoding of constellation symbols are performed by using Table 4 to obtain the quaternary data points. All the constellation points are divided into four different groups. These groups are located: (i) at origin, (G 1 ) (ii) on a circle of radius 2d, (G 2 ) (iii) on a circle of radius 4d with phase angles (0, π2 , π or 32π ),

(G 3 ) and (iv) on a circle of radius 4d with phase angles ( π4 , 34π , 54π or 74π ), (G 4 ). Any constellation point after multiplication with four

3.3. Concentric circle mapping based PTS (CCM-PTS) CCM-PTS [14] is another PTS based scheme to reduce the PAPR of the OFDM signal without SI. In this technique, like M-PTS, the original bit stream is converted into quaternary data and then obtained quaternary data points are mapped to the concentric circle constellation points lying at the origin and two concentric circles Table 2 De-mapping of 16-QAM constellation symbols to quaternary data points. Demodulated constellation symbols belonging to group

De-mapped constellation point

Recovered quaternary data

G# 1

3+3j

0

G# 2

−3 + j

1

G# 3

−1 − j

2

G# 4

1−3j

3

Fig. 3. CCM-PTS constellation mapping and effect of phase rotation on data symbols. Table 4 De-mapping of concentric circle constellation symbols to quaternary data points. Demodulated constellation symbols belonging to group

De-mapped constellation point

Recovered quaternary data

G1 G2 G3 G4

0 + j0 0 + j2 −√ 4 + j0 2 2(1 − j )

0 1 2 3

Table 3 Mapping of quaternary data points to concentric circle constellation for d = 1. Quaternary symbol

Initially mapped quaternary data points to concentric circle constellation

Constellation points after multiplication with the phase factors in B 1

j

−1

−j

Group number

0 1 2

0 + j0 0 + j2 −4 + j0

0 + j0 0 + j2 −4 + j0

0 + j0 −2 + j0 0 − j4

0 + j0 0 − j2 4 + j0

0 + j0 2 + j0 0 + j4

G1 G2 G3

3

2 2(1 − j )

2 2(1 − j )

2 2(1 + j )

2 2(−1 + j )

2 2(−1 − j )

G4

√

√

√

√

√

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

phase factors either remains at same location or lies on the circle of the same radius.

In this paper two decoding techniques for CCM-PTS have been considered, namely minimum distance decoding rule and circular boundary decoding [14]. The SER performance of CCM-PTS over AWGN is mathematically analyzed by both the decoding schemes. Here, we also present the SER analysis of M-PTS scheme over AWGN channel by using minimum distance decoding rule. The mathematical analysis for SER performance of the schemes under consideration over fading channel is not carried out in this paper because it is not possible to derive their close form expressions, however to confirm their effectiveness computer simulations have been performed. 4.1. Calculation of probability of error using minimum distance decoding rule 1 Let P el denote the conditional probability of making an error when the transmitted point belongs to group G l . The superscript 1 is used for the case of minimum distance decoding and subscript l denotes the group number. In CCM-PTS, the constellation points are divided into 4 different groups G 1 to G 4 . Therefore, in order to calculate average probability of error, we need to calculate 4 1 1 different error probabilities P e1 to P e4 . The constellation points in group G 1 have four nearest neighbours, i.e. the four points belonging to group G 2 , (Fig. 3) at a distance of 2d. The probability of making an error can be calculated using union bound approximation [17] as

1 P e1

4×

1 2

 erfc

d2

 (6)

η

Here, η is the single sided PSD of complex Gaussian white noise and erfc(.) is the complementary error function [17]. Similarly, for the constellation points belonging to group G 2 , there are four nearest neighbours, one of them belongs to group G 1 and is at a distance of 2d, one belongs to group G 3 and is also at a distance of 2d, while the remaining two belong to the group G 4 and are at distance of 2d 5 − calculated as 1 P e2

2×

1 2



erfc

d2

 +2×

η

√

1 2

1 2. The error probability P e2 is

 erfc



(5 −

√

2)d2

η

 (7)

Similarly, for group G 3 , there are three nearest neighbours, one of them belongs to group G 2 and is at a distance of 2d and the remaining two belong to group G 4 and are at a distance of 8d sin( π8 ).

1 The error probability P e3 can be calculated as follows

1 P e3



1 2



erfc

d2

 +2×

η

1 2



erfc

{8d sin( π8 )}2 4η

 (8)

Similarly, for group G 4 , there are four nearest neighbours, two of

√

them belong to group G 2 and are at a distance of 2d 5 − 2, while the remaining two belong to group G 3 and are at a distance 1 of 8d sin( π8 ). The error probability P e4 is given by 1 P e4 2×

1 2





erfc

+2×

1 2

(5 −

 erfc

√

2)d2



4η

P e1 



1

4 1

1 1 1 1 P e1 + P e2 + P e3 + P e4



4

7 2



erfc



(10)

 

 + 2 × erfc

η

{8d sin( π8 )}2

(5 −

√

2)d2



η

 (11)

4η

4.2. Calculation of probability of error using circular boundary decoding rule In this method, the decoding of constellation points are performed circle wise as discussed in [14]. Following decoding rule is utilized to decode the received constellation points. For a complex-valued received data point of the form re j φ , we can decode as

r
⇒

quaternary data 0

d  r < 3d ⇒ quaternary data 1 nπ π nπ π r  3d, − φ< + , 2 8 2 8 0  n  3 ⇒ quaternary data 2 Otherwise it is decoded as quaternary data 3. Here r and φ are the magnitude and the phase angle of complex valued received data point. In this decoding scheme the same four different groups G 1 , G 2 , G 3 and G 4 considered in previous section are utilized for calculat2 2 2 2 , P e2 , P e3 and P e4 represents the ing the error probabilities. Let P e1 error probabilities of signal points belonging to groups G 1 , G 2 , G 3 and G 4 respectively. 2 of received signal with ampliTo calculate error probability P el tude r (t ) (transmitted symbol plus complex Gaussian white noise n(t )) and phase φ(t ), let R and φ be the random variables corresponding to r and φ respectively and their probability density functions are f R (r ) and f φ (φ) respectively. Here, the superscript 2 denotes the circular boundary decoding rule and subscript l denotes the group number. The constellation point belonging to group G 1 , is located at origin. In this particular case, the magnitude and phase of r (t ) will denote nothing but the magnitude and phase of noise. Therefore, the probability density functions of magnitude and phase of r (t ) will be same as that of noise. So, in this particular case the probability density of phase, considered as a random variable φ , has a uniform distribution in the interval [0, 2π ] and the probability density function of the magnitude of r (t ) taken as a random variable R has a Rayleigh distribution, given by

f 1 R (r ) =

r

σn2

 exp −

r2



2σn2

r0

,

(12)

where σn2 = η/2 is the variance of complex Gaussian white noise n(t ). If any of the constellation point belongs to G 1 is transmitted then it will be detected incorrectly if |r (t )|  d, and the probability 2 can be calculated as follows of error P e1

d =1− 

(9)

d2



+ 2 × erfc

2 P e1

η {8d sin( π8 )}2

The overall error probability ( P e1 ) is calculated as follows

P e1 

4. SER and computational complexity analysis

307

0

= exp −

r

σn2 d2 2σn2

 exp −

r2 2σn2



 dr

(13)

 (14)

308

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

For any transmitted point belonging to group G 2 , the constellation points are located on the circle of radius 2d, the joint probability density function f 2 R ,φ (r , φ) of magnitude (r ) and phase (φ) for the received signal [18] (transmitted symbol plus noise) is given by



r

f 2 R ,φ (r , φ) =

2πσn2

exp −

r + 4d − 4d cos(φ) 2

2

+

2 P e2

=1− 0

=1−



r 2πσn2

d

3d

2σn2

+

dr dφ



r

(16)

exp −

 

r2

I0

2σn2

2rd

+

dr

σn2

(17)

where I 0 ( z) is modified Bessel function of the first kind zero order, I 0 ( z) can be expanded in term of following series [19]

I 0 ( z) =

∞ 

z2m

(18)

22m × (m!)2

m =0

=1−

1

σn2

3d

2πσ d

2 P e2 =1−

1

σn2

  ∞ r2 exp − 2 2σn

m =0

2rd 2m

3d ×

r

2m+1

 exp −

d

Let u =

r2 2σn2

σn2

22m

× (m!)2

dr

2σn2

dr

(20)

2 P e3

=1−

(u ) exp(−u ) du





2σn2

dr dφ

exp −

2πσn2

×

16d2 sin2 (φ) 2σn2

− π8

 r exp −



(r − 4d cos(φ))2 2σn2







dr dφ

(25)

2 P e3 =1−

π



8

1 2πσn2

exp − − π8

∞



16d2 sin2 (φ)



2σn2

    u2 u + 4d cos(φ) exp − du dφ 2 2σn

(26)

Using the standard result for integration in Eq. (26), we get π

8  (21)

Using the standard result for integration [19], Eq. (21) is given by

× exp(−u ) um

(23)

r 2 + 16d2 − 8d cos(φ)

3d−4d cos(φ)

d2 2σn2

  ∞  m 2d2  (d)2m 2 2 P e2 = 1 − exp − 2 σn m=0 (m!)2 σn2  

exp −



8

1

∞

 m



2πσn2



2σn2



r

×

m =0

×

exp −

2πσn2

r 2 + 16d2 − 8d cos(φ)

3d

  ∞  m 2d2  (d)2m 2 = 1 − exp − 2 (m!)2 σn2 σn



r

π

and by substituting the value of u in Eq. (20), we have

9d2 2σn2

(22)

2σn

By substituting u = r − 4d cos(φ) in Eq. (25), we have





2σn2

 2 m−k  9d m(m − 1) · · · (m − k + 1) 2

(19)

(d)2m (σn2 )2m × (m!)2 m =0

r2

9d

2 m

(24)



2 P e2

9d

2σn

 2  

2σn2

=1−



  ∞ 2d2  exp − 2

σn



f 3 R ,φ (r , φ) =

By interchanging the order of the summation and integration, Eq. (19) can be written as



2σn2

− π8 3d

σn

2 n

2σn2

The constellation points of group G 3 are located on the circle of radius 4d and at an angle (0, π2 , π or 32π ). The joint probability density function f 3 R ,φ (r , φ) for received signal (transmitted symbol plus noise) is given by

2 P e3

  2d2 exp − 2

r

×

k =1

d2

π8 ∞

From Eqs. (17) and (18) we have 2 P e2

m 

d2



 2 m−k  d m(m − 1) · · · (m − k + 1) 2

− exp −





m  k =1

σn

2πσn2

d





  1 2d2 − exp 2 2

σn

×

exp −

× exp −

d2 2σn2 9d2 2σn2

  ∞  m 2d2  (d)2m 2 = 1 − exp − 2 σn m=0 (m!)2 σn2    m

(15)

r 2 + 4d2 − 4d cos(φ)

m(m − 1)(m − 2) · · · (m − k + 1)u



According to the circular boundary decoding rule for the quaternary data 1, we have

2π 3d

 m−k

k =1



2σn2

m 

2 P e3

=1− − π8

1 2π

 exp −

25d2 + 24d2 cos(φ) 2σn2



4d cos(φ)

+

2πσn2

     16d2 sin2 (φ) 3d − 4d cos(φ) × exp − Q dφ 2 2σn

σn

(27)

In general the close form solution of the integration of Eq. (27) is not possible and it must be evaluated numerically. At high SNR and for |φ|  π , the first term in the above integration can be neglected and value of Q (.) function is approximately 1, so Eq. (27) reduces to

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313 π

8  2 P e3

0

Let

4d sin(φ)

σn



4d cos(φ)



≈1−2

exp −

2πσn2

16d2 sin2 (φ)



 dφ

2σn2

4.3. SER analysis of M-PTS

(28)

= λ, Eq. (28) can now be written as

 2 P e3 ≈1−

4d sin(π /8) σn

2

e−

π 

2 P e3 ≈ erfc

0

2d sin( π8 )

λ2 2

dλ

(29)

 (30)

σn

The constellation points belonging to group G 4 are located on circle of radius 4d and at an angle ( π4 , 34π , 54π or 74π ), the error 2 2 is same as P e3 because decision boundaries of conprobability P e4 stellation points in group G 4 and G 3 are similar.

∴

2 P e4

2 P e3

=

(31)

In M-PTS, the quaternary data points are mapped to 16-QAM constellation by using four phase factors. As seen from Fig. 2(b), the constellation points of 16-QAM are divided into four groups G # 1 # # # to G # 4 and each of them has 4 points. The groups G 1 , G 2 , G 3 and G# 4 have 4 constellation points and the coordinates of these points are {3 + 3 j , −3 + 3 j , −3 − 3 j , 3 − 3 j }, {−3 + j , −1 − 3 j , 3 − j , 1 + 3 j }, {−1 − j , 1 − j , 1 + j , −1 + j } and {1 − 3 j , 3 + j , −1 + 3 j , −3 − j } respectively. The minimum Euclidean distance between two constellation points is 2, same as in CCM-PTS constellation. The con# # # stellation points in groups G # 1 , G 2 , G 3 and G 4 have 2, 3, 2 and 3 nearest neighbours with an Euclidean distance of 2. The average symbol energy of 16-QAM constellation shown in Fig. 2(b) is E s = 10. The average SER ( P e# ) of M-PTS scheme can be calculated using union bound approximation [17] as follows

P e#

=

The overall probability of symbol error in CCM-PTS when circular boundary decoding is used may be calculated from Eqs. (14), (21), (30) and (31). The error probability P e2 can be calculated as follows

P e2 = P e2 =

1

4 1 4

2 2 2 2 P e1 + P e2 + P e3 + P e4

+

1 4



 exp −

(32)



d2

2σn

− exp − +

m  k =1

+

1 4

2σn

k =1



9d

 2  

2σn2

9d

2 m

2σn2

 2 m−k  9d m(m − 1) · · · (m − k + 1) 2 

2σn

 2d sin( π ) 8

× 2 × erfc

(33)

σn

By substituting the variance of AWGN to the form

P e2

=

1



4



exp −

d2





η



+ 1 − exp −

σn2 = η/2, Eq. (33), reduces

4d2

 ∞

η

m =0

 2  2 m d d × exp − 

η

+

m  k =1

 m  (d)2m 4 (m!)2 η

η

− exp −

9d

9d

η

+ 2 erfc 2d



η

4

erfc

2



erfc

0.1E s



η 0.1E s

η

+4×3×

 +4×3×

1 2

1 2



erfc



erfc

0.1E s



η 0.1E s



η



η

(35)

 sin

In order to calculate the total computational complexity of CCM-PTS and M-PTS schemes, we take both transmitter and receiver into consideration. As discussed in Section 3.1, we used adjacent partitioning scheme to divide the entire data block into M sub-blocks. If we use oversampling by a factor L then we have to calculate LN point IFFT of each sub-block. This requires LN log2 ( LN ) complex additions and LN log2 ( LN ) complex 2 multiplications. For M sub-blocks, we require M LN log2 ( LN ) and M LN log2 ( LN ) complex additions and multiplications respectively. 2 In order to combine M partial transmit sequences obtained from M IFFT operations, ( M − 1) LN complex additions are required. Here, we have to perform W M −1 number of searches to get the OFDM signal with minimum PAPR; therefore we require W M −1 ( M − 1) LN numbers of complex additions. At the receiver, we have to perform one LN point FFT for the subcarrier demodulation and for decoding each of the symbols we require 2D N additions and D N multiplications, where D is the number of decision regions. The overall computational complexity can be calculated as

nadd = Total No. of complex additions at Tx.

+ Total No. of complex additions at Rx.

M LN 2

log2 ( LN ) + D N

5. Performance evaluation

η

2

5

=

2 m

η



=



1

erfc

0.1E s

+ Total No. of complex multiplications at Rx.

 2 m−k  m  9d + m(m − 1) · · · (m − k + 1) k =1

+4×2×

2



nmul = Total No. of complex multiplications at Tx.

η

 2 

16

4×2×

1

= M LN log2 ( LN ) + W M −1 ( M − 1) LN + 2D N

 2 m−k  d m(m − 1) · · · (m − k + 1) 



1

4.4. Computational complexity

2σn2

  ∞  m    2d2  (d)2m 2 d2 exp − 2 exp − − 4 σn m=0 (m!)2 σn2 2σn2  m   2 m−k  m d2 d m(m − 1) · · · (m − k + 1) × 2 2 1





309

π 8

 (34)

Here we consider an OFDM system with N = 256 subcarriers. To evaluate the CCDF of PAPR and SER of the schemes under consideration, 20,000 OFDM symbols are used. These subcarriers are (m) M −1 divided into 4 equal partitions { X q }m =0 using adjacent partitioning technique. To compare the performance of the proposed scheme with M-PTS, complex AWGN with zero mean is assumed

310

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

Fig. 4. Comparison of CCDF of PAPR. Table 5 Probability of error in SI detection. Scheme

Cimini et al. [9] Nguyen et al. [12] Yang et al. [15] M-PTS CCM-PTS

Table 6 Computational complexity of CCM-PTS and M-PTS. Probability of error in SI detection

CCM-PTS

AWGN channel (SNR = 3 dB)

Fading channel (SNR = 8 dB)

Minimum distance decoding (D = 13)

Circular boundary decoding (D = 10)

1.2 × 10−3 2.3 × 10−2 10−2 0 0

10−3 4.8 × 10−1 2 × 10−3 0 0

244 224 23 808

242 688 23 040

in the simulation. In CCM-PTS we take the value of d as 1 and therefore the coordinates√of initially mapped constellation points becomes {0, 2 j , −4 and 2 2(1 − j )}. The CCDFs of original OFDM signal (without PAPR reduction), scheme proposed in [9,10,12,15], CCM-PTS with two phase factors, CCM-PTS with four phase factors and M-PTS with four phase factors are shown in Fig. 4. The complementary cumulative distribution function (CCDF) of PAPR specifies the percentage of OFDM symbols that have PAPR greater than a value (PAPR0). It can be observed from Fig. 4 that for a CCDF of PAPR = 0.001, i.e. for 0.1% of the OFDM symbols, the original OFDM signal (without PAPR reduction) has a PAPR of 11 dB or more, whereas the scheme proposed in [9,10] and CCM-PTS with two phase factors have PAPR of 9.1 dB, 8.65 dB and 8.5 dB respectively and therefore achieves a PAPR reduction capability of 1.9 dB, 2.35 dB and 2.5 dB respectively. The PAPR performance of the schemes proposed in [12,15], CCM-PTS and M-PTS with four phase factors are very close and have at least 1 dB more PAPR reduction capability in comparison to the scheme proposed in [9,10]. The scheme proposed in [9, 10] have limited number of candidate signals for PAPR reduction, which is the main reason behind the loss in PAPR performance and hence, these schemes [9,10] are not good choices in comparison to the remaining PAPR reduction schemes under consideration. In Table 5, we have presented a comparison of probability of error in SI detection for the schemes under consideration, over

nadd nmul

M-PTS (D = 13)

244 224 23 808

AWGN and fading channel. It has been found out from the results presented in Table 5 that the schemes proposed in [9,12,15] have a non-zero probability of error in SI detection whereas, the scheme proposed in [13,14] has no error in SI detection. It can be observed from Table 5 that the SI detection capability of the scheme proposed by Nguyen et al. is least in comparison to the remaining scheme under consideration. M-PTS and CCM-PTS have best performance. Based on the PAPR performance and SI detection capability, the SI embedding schemes proposed in [9,10,12,15] are not good choices in comparison to the schemes proposed in [13,14]. As we have seen that the PAPR reduction capability of CCM-PTS and M-PTS is almost same and both the schemes are SI free. Now, we compare the computational complexity of CCM-PTS and M-PTS. In CCM-PTS with circular boundary decoding, we have 10 decision regions, one each for decoding the constellation point located at zero and on a circle of radius 2. Further to decode the eight constellation points located on the circle of radius 4 we require 8 decision regions. So, in this scheme we require a total of 10 different decoding regions (D ). In CCM-PTS with minimum distance decoding rule, we have only 13 points and we require one separate decision region for each of them, so we have a total of 13 decoding regions (D). In M-PTS with four phase factors quaternary data is mapped to a 16-QAM constellation which has 16 distinct points. To decode all of them, we require only 13 decision regions (D ). A comparison of computational complexity of CCM-PTS and M-PTS schemes for M = W = L = 4 and N = 256 is given in Table 6. As seen from Table 6, the CCM-PTS with circular boundary decoding

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

311

Fig. 5. SER performance comparison over AWGN channel.

Fig. 6. BER performance comparison over AWGN channel.

requires least computational complexity in comparison to all the schemes under consideration. The plot of the analytical results for SER performance of CCM-PTS scheme, using minimum distance decoding and circular boundary decoding, and M-PTS scheme over AWGN channel, discussed in Section 4 are shown in Fig. 5. These results, in Fig. 5 are denoted by ‘CCM-PTS(1) Theoretical’, ‘CCM-PTS(2) Theoretical’ and ‘M-PTS Theoretical’ respectively. Also, to confirm the

validity of these results, computer simulations using MATLAB are performed for the schemes under consideration. In Fig. 5, the simulation results for SER performance of CCM-PTS, using minimum distance decoding and circular boundary decoding, and M-PTS scheme over AWGN channel are denoted by ‘CCM-PTS(1) Simulated’, ‘CCM-PTS(2) Simulated’ and ‘M-PTS Simulated’ respectively. All simulation results are coinciding with their analytical results and therefore confirm the validity of the theoretical results. The

312

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

Fig. 7. SER performance comparison over fading channel.

results denoted by ‘CCM-PTS(1) theoretical’ is the upper bound of SER and the results denoted by ‘CCM-PTS(1) Simulated’ will never exceed it. It can be seen from Fig. 5 that SER performances of CCM-PTS using both the decoding schemes are better than M-PTS. If we compare the two decoding techniques of CCM-PTS then minimum distance decoding technique provides the lower SER. To achieve a SER = 10−5 , CCM-PTS with circular boundary decoding and minimum distance decoding, we require a SNR of 19.5 and 19.2 dB. To achieve the same SER performance M-PTS requires a SNR of 20 dB. Hence for same SER performance, M-PTS requires a SNR which is about 0.5 dB and 0.8 dB more, as compared to that required by CCM-PTS with circular boundary decoding and with minimum distance decoding, respectively. The CCM-PTS with circular boundary decoding requires a SNR which is 0.3 dB more as compared to CCM-PTS with minimum distance decoding. CCM-PTS with minimum distance decoding requires the least SNR amongst all the schemes under consideration, but CCM-PTS with circular boundary decoding is simplest to decode and its computational complexity is least among all the three. In Fig. 6, comparison of BER performance of M-PTS and CCMPTS using both the decoding techniques with original OFDM signal (without PAPR reduction) has been shown. It can be observed from Fig. 6 that to achieve a BER = 10−5 , CCM-PTS using minimum distance decoding requires about 3.3 dB more E b / N o in comparison to original OFDM signal, which is the only performance loss to achieve a gain of 3.4 dB in PAPR reduction capability of PTS-OFDM system without SI. But CCM-PTS using both the decoding schemes has better BER performance in comparison to M-PTS scheme. In Fig. 7, comparison of SER performance of CCM-PTS scheme using minimum distance decoding, circular boundary decoding and M-PTS with minimum distance decoding schemes over three tap Rayleigh fading channel is presented. Here, we have evaluated the SER performance of the PTS based scheme over fading channel using computer simulation in MATLAB. We have considered a 3 tap Stanford University Interim (SUI) 5-multipath fad-

ing channel with average path gains [0 dB, −5 dB, −10 dB], and path delays [0 μs, 4 μs, 10 μs]. In multipath fading channel the SER performance of the proposed CCM-PTS scheme with circular boundary decoding and minimum distance decoding are very close. The proposed CCM-PTS scheme with circular boundary decoding outperform in comparison to existing M-PTS scheme in terms of SER and computational complexity. The SER performance of the methods under consideration over fading channel diverges in comparison to their respective SER performance over AWGN because the intra symbol interference cannot be avoided using cyclic prefix of sufficient length (CP only eliminates inter symbol interference (ISI)). 6. Conclusion In PTS based methods, SER performance depends on how SI is encoded with the OFDM symbol, and if it gets corrupted, the entire OFDM symbol may be erroneous. Existing SI embedding schemes eliminate the requirement of SI transmission but these suffer from one drawback or the other, whether in terms of computational complexity, poor PAPR reduction capability or incorrect SI detection. In this paper, we have considered the PTS based methods (M-PTS, CCM-PTS) that do not utilize SI at the receiver. The SI free schemes are found to be the good alternative of SI embedding schemes. The SER performance of such PAPR reduction techniques over AWGN is derived analytically and their performance over fading channel is evaluated using simulations. The PAPR reduction capabilities of CCM-PTS and M-PTS are found to be almost the same. The SER performance of CCM-PTS over AWGN channel is analyzed using minimum distance decoding and circular boundary decoding and is compared with M-PTS. The CCM-PTS with minimum distance decoding performs the best among them. In CCM-PTS, with circular boundary decoding, we require only 10 decoding regions, whereas in others we require 13 decoding regions. Therefore, the decoding complexity of CCM-PTS with circular boundary decoding is the least. At the same time, it is also seen that, this method results in a SER loss of merely 0.3 dB as compared to CCM-PTS with

A. Goel et al. / Digital Signal Processing 23 (2013) 302–313

minimum distance decoding. In fading channel also, the SER performance of CCM-PTS by both the decoding schemes is better than that of M-PTS. Acknowledgments The authors would like to thank the anonymous reviewers for their constructive comments, which were of great help to improve the quality of this paper. References [1] Y. Wu, W.Y. Zou, Orthogonal frequency division multiplexing: A multi-carrier modulation scheme, IEEE Trans. Consumer Electron. 41 (1995) 392–399. [2] T. Jiang, Y. Wu, An overview: Peak-to-average power ratio reduction techniques for OFDM signals, IEEE Trans. Broadcast. 54 (2008) 257–268. [3] X. Li, L.J. Cimini, Effects of clipping and filtering on the performance of OFDM, IEEE Commun. Lett. 2 (1998) 131–133. [4] J. Armstrong, Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering, Electron. Lett. 38 (2002) 246–247. [5] X.B. Wang, T.T. Tjhung, C.S. Ng, Reduction of peak-to-average power ratio of OFDM system using a companding technique, IEEE Trans. Broadcast. 45 (1999) 303–307. [6] T. Jiang, Y. Yang, Y. Song, Exponential companding transform for PAPR reduction in OFDM systems, IEEE Trans. Broadcast. 51 (2005) 244–248. [7] R.W. Baüml, R.F.H. Fischer, J.B. Hüber, Reducing the peak-to-average power ratio of multicarrier modulation by selective mapping, Electron. Lett. 32 (1996) 2056–2057. [8] S.H. Muller, J.B. Huber, OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences, Electron. Lett. 33 (1997) 36–69. [9] Leonard J. Cimini Jr., Nelson R. Sollenberger, Peak-to-average power ratio reduction of an OFDM signal using partial transmit sequences with embedded side information, in: Proc. of IEEE GlobeComm’00, November 2000, pp. 740–750. [10] A.D.S. Jayalath, C. Tellambura, SLM and PTS peak-power reduction of OFDM signals without side information, IEEE Trans. Wireless Commun. 4 (5) (2005) 2006–2013. [11] H. Ochiai, A novel trellis-shaping design with both peak and average power reduction for OFDM systems, IEEE Trans. Commun. 52 (2004) 1916–1926. [12] T.T. Nguyen, L. Lampe, On partial transmit sequences for PAR reduction in OFDM systems, IEEE Trans. Wireless Commun. 2 (2008) 746–755. [13] Y. Zhou, T. Jiang, A novel multi-point square mapping combined with PTS to reduce PAPR of OFDM signals without side information, IEEE Trans. Broadcast. 55 (2009) 831–835. [14] A. Goel, P. Gupta, M. Agrawal, Concentric circle mapping based PTS for PAPR reduction in OFDM without side information, in: Proceedings of 6th IEEE Conference on Wireless Communication and Sensor Networks, Allahabad, India, 2010, pp. 201–204. [15] L. Yang, K.K. Soo, S.Q. Li, Y.M. Siu, PAPR reduction using low complexity PTS to construct of OFDM signals without side information, IEEE Trans. Broadcast. 57 (2011) 284–290.

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Ashish Goel was born in Rampur, India, on January 1, 1979. He received his B.Tech. degree in Electronics and Communications Engineering and M.Tech. in Information and Communication Technology from Institute of Engineering Technology Bareilly, India, and Jayapee Institute of Information Technology, Noida, in year 2002 and 2004 respectively. He joined ECE department of Jayapee Institute of Information Technology, Noida, as a faculty member in year 2004. He is currently pursuing his Ph.D. at the Department of Electronics and Computer Engineering, Jayapee Institute of Information Technology, Noida. His research interests include high speed wireless communication systems: channel coding, MIMO-OFDM, mobility management and cryptography. Prerana Gupta was born in Delhi, India, on October 14, 1981. She received her B.Tech. degree in Electronics and Communications Engineering from Indira Gandhi Institute of Technology (GGSIPU), New Delhi, India. She received her Ph.D. degree from Indian Institute of Technology Roorkee. She joined Electronics and Communication Engineering Department of Jaypee Institute of Information Technology, Noida, as Assistant Professor in year 2009. Her research interests include channel estimation and equalization techniques in doubly selective systems, channel coding and MIMO-OFDM. Dr. Monika Agrawal was born in Dehradun, India. She received the B.Tech. degree in Electrical Engineering and the M.Tech. degree in Electronics and Communication Engineering from the Regional Engineering College, Kurukshetra, India, and the Ph.D. degree from the Indian Institute of Technology, New Delhi, India, in 1993, 1995, and 2000, respectively. She was employed with Hughes Software Systems (HSS), Gurgaon, India, from 1999 to 2002. During 2001 she was a visiting researcher in the Dept. of Systems and Control, Uppsala University, Uppsala, Sweden. She joined C.A.R.E, I.I.T Delhi, as Assistant Professor in January 2003.