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Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems
Optical Fiber Technology xxx (2014) xxx–xxx
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Optical Fiber Technology www.elsevier.com/locate/yofte
Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems Luo Renze a, Li Rui a,⇑, Dang Yupu b, Yang Jiao a, Liu Weihong c a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu 610500, China Department of Well Tech, China Oilfield Services Limited, Beijing 101149, China c Turpan Hami Oilfield Company, Xinjiang 838202, China b
a r t i c l e
i n f o
Article history: Received 16 April 2014 Revised 12 July 2014 Available online xxxx Keywords: Orthogonal frequency division multiplexing–radio over fiber (OFDM–ROF) Peak-to-average power ratio (PAPR) Selected mapping (SLM) Bit error ratio (BER)
a b s t r a c t OFDM–ROF (orthogonal frequency division multiplexing–radio over fiber) system has high spectral efficiency and high transmission rate. However, due to the non-linear power amplifier, the high peakto-average power ratio (PAPR) of the transmission signals is one of the major setbacks in the OFDM– ROF transmission systems. In order to reduce PAPR, this paper proposes two improved selected mapping (SLM) methods without explicit side information (SI): Proposed ED-SLM and Proposed SLM. In the Proposed ED-SLM method, signals will be transmitted without side information which is scrambled by special initial phase sequences and at the receiver an Euclidean phase distance detection (EPD) method is used to detect the SI and finishes the demodulation. In the proposed SLM method, the signals are transformed by Hadamard matrix to reduce PAPR. Then the row vectors of Hadamard matrix are used as phase sequences and superimposed on the data signals. The theory and simulation results show that the two improved SLM methods perform better in PAPR and BER reduction than the conventional SLM method in the OFDM–ROF transmission systems. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction With the construction of the experimental networks such as Long Term Evolution (LTE), and Worldwide Interoperability for Microwave Access (WiMAX), wireless communication has entered a high-speed data era. In order to solve the problem of insufficient spectrum resources, ROF wireless network and OFDM–ROF systems have been proposed [1]. Because of its low transmission loss and ultra-wide bandwidth, the ROF system proved to be a key technology for next generation network (NGN) access applications [2]. The benefits that OFDM brings to wireless communication systems include high transmission capacity, resistance to interference and robustness to multipath fading. Thus, the combination of orthogonal frequency division multiplexing and radio over fiber systems (OFDM–ROF) has attracted much attention for future gigabit broadband wireless communication [3]. However, due to the drawbacks of fiber-optic such as dispersion, nonlinearity, the PAPR of OFDM signals cannot be ignored when data are transmitted over the fiber. The drawbacks of high peak-average power ratio (PAPR) ⇑ Corresponding author. E-mail addresses: [email protected] (R. Luo), [email protected] (R. Li), [email protected] (Y. Dang), [email protected] (J. Yang), 153691280@qq. com (W. Liu).
may outweigh all the potential advantages of the OFDM–ROF transmission systems [4,5], so the system must overcome the defect. Various methods for PAPR reduction are proposed, including clipping [6], coding [7], selected mapping (SLM) [8–12], partial transmit sequence (PTS) [13], and active constellation (ACE) [14]. In these methods, selective mapping (SLM) is effective and promising to reduce PAPR. Ref. [8] introduces a modified SLM scheme which generates alternative signal sequences by adding mapping signal sequences to an OFDM signal sequence for the purpose of reducing computational complexity at the cost of PAPR performance. Ref. [10] explains a scheme to reduce PAPR without transmitting side information, and in the receiver a minimum Euclidian distance (MED) decoder is used to recover the side information. Nevertheless, the computational complexity seems larger and BER performance is not satisfactory. In Ref. [15], the inverse discrete Fourier transform (IDFT) is replaced by a conversion matrix in the time domain and it significantly reduces the computational complexity but bit error ratio (BER) degradation cannot be avoided. To remove a great deal of side information and further reduce PAPR in OFDM–ROF systems, this paper introduces two effective methods based on SLM algorithm: Proposed ED-SLM and Proposed SLM. In Proposed ED-SLM algorithm, signals will be transmitted without side information which is scrambled by special initial phase sequences and at the receiver an Euclidean phase distance
http://dx.doi.org/10.1016/j.yofte.2014.07.007 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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detection method is used to detect SI and finish demodulation. In order to have better PAPR and BER performance, another improved method is proposed. In the Proposed SLM method, signal is transformed by Hadamard matrix to reduce PAPR. Then the row vectors of Hadamard matrix are used as phase sequences and superimposed on data signals. After channel estimation, the superimposed Hadamard sequence on the signals will be removed at the receiver and the SI will be recovered at the same time. The method has a good BER performance after the OFDM–ROF demodulation. The paper is organized as follows: firstly, the OFDM–ROF transmission systems are introduced. Secondly, the Proposed ED-SLM and Proposed SLM schemes are presented. Thirdly, the performance analysis of Proposed ED-SLM and Proposed SLM methods are shown. Fourthly, the PAPR and BER simulation results are compared with different schemes. Conclusion is presented in the last part. 2. The OFDM–ROF transmission systems Fig. 1 shows the block diagram of the OFDM–ROF transmission system. ROF is a technology which takes optical fiber as transmission medium, light wave as carrier, and high-frequency microwave as modulation wave. As is shown in Fig. 1, ROF combines wireless communications and optical fiber transmission, where the wireless access system is composed of the central station (CS), optical fiber link, base stations (BS) and mobile terminals. Shortened as OFDM–ROF system, OFDM–ROF optical wireless communication system combines OFDM modulation and ROF technology. Central Station is mainly responsible for generation of the downlink radio-on-fiber OFDM wireless signals, and the inspection and receiving of the uplink data. Base station is mainly responsible for the detection of the downlink radio-on-fiber OFDM wireless signals and converting them into electricity OFDM wireless signals which will be transmitted through antennas. At the same time, it also receives uplink signals from the antennas and modulates the optical carrier signals to generate upward radio-on-fiber signals. Mobile terminal, called subscriber unit, receives wireless OFDM signals by antennas and demodulate to baseband signals, and at the same time, it launches uplink signals. In practical communication network, a central station is connected to multiple base stations, and a base station is connected to more than one subscriber units. In order to save costs and facilitate the management of the whole communication network, the processing of complex and expensive equipment is usually placed in the central station. 3. Proposed ED-SLM and Proposed SLM schemes
sub-carriers are used, and each sub-carrier is modulated using symbols, the OFDM symbol sðtÞ is expressed as N1 1 X n sðtÞ ¼ pffiffiffiffi Sn ej2pNt ; 0 6 t 6 N 1; N n¼0
ð1Þ
where Sn are the data symbols, and N is the number of sub-carriers. The PAPR of an OFDM signal is defined as
PAPRðSÞD
max06t6N1 jsðtÞj2 EfjsðtÞj2 g
ð2Þ
where E½ denotes the expected value. The OFDM–ROF systems have high PAPR, which requires linear, large-dynamic-range amplifiers that are inefficient and expensive. This paper focuses on improving traditional SLM scheme to reduce PAPR without distortion, and achieve good BER performance. In the conventional SLM scheme, the symbol sequences multiply the phase sequences to generate alternative symbol sequences. After the IFFT blocks, the OFDM–ROF signal with the lowest PAPR is selected to transmit but the side information is also transmitted. Therefore, the bandwidth occupied by the redundant side information is one of the main bottlenecks of the traditional SLM method. 3.1. Proposed ED-SLM The signals will be launched without side information in Proposed ED-SLM algorithm, and the receiver will take advantage of the Euclidean phase distance judgment method to detect side information. The ED-SLM algorithm can obtain certain improvement on PAPR performance and the BER isn’t worse than that of the traditional SLM method. Fig. 2 is the block diagram of the Proposed ED-SLM scheme. As shown in Fig. 2, the binary data bit is put into the transmitter through OFDM–ROF signal transmission systems, and frequency OFDM modulation signal can be obtained through the mapping and series-to-parallel conversion: Y n ¼ ½Y 1 ; Y 2 ; Y N , n 2 ½1; N. Then initialize the phase sequences according to Eq. (3):
u u u Lu ¼ diag ejhp0 ; ejhp1 ; ; ejhpN1
ð3Þ
u
Here, L represents the u-th phase sequence u 2 ð1; UÞ. The rotation phase h 2 ð0; pÞ is obtained randomly and the scrambler coefficient is pui 2 f1; 0; 1g, i 2 ð0; N 1Þ, j ¼ sqrtð1Þ. U initialization phase sequences will be used to scramble the transmitting signals, signals after IFFT can be expressed as:
yun ¼ IFFTðY n Lu Þ
ð4Þ ym n
In the OFDM–ROF transmission systems, signals are demodulated onto the carriers which are orthogonal to each other. If N
Then the signals with a minimum PAPR will be launched, and m 2 ð1; UÞ. Since the Euclidean phase distance judgment method will be used to detect phase sequence at the receiver, the transmit-
Fig. 1. Block diagram of the OFDM–ROF transmission systems.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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IFFT
IFFT Input data
S/P ...
...
Phase initiating Phase optimal
Select the sequence with lowest PAPR to transmit
IFFT
Receive data
Phase Detection select The min Euclidean phase distance
decison
Input data
Hadamard transform
S/P
wireless channel
Receive the Signals and have OFDM demodulat -ion
Fig. 2. The block diagram of Proposed ED-SLM scheme. Fig. 3. The block diagram of the Proposed SLM scheme.
ter will not transmit side information. Transmitting signals will be demodulated at receiver after they pass through wireless channels. The signals are descrambled with all phase sequences after FFT and ^ u , which is written as Eq. (5): restored to U demodulation signals Y n u Y^ un ¼ FFTðym n Þ ðL Þ
ð5Þ u
^ u can be The phase Dem phase of the demodulated signals Y n worked out. In the Euclidean phase distance judgment method, Eq. (6) will calculate the difference of two squares of the phase between receiver signal and initial phase:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N1 uX u DifðuÞ ¼ t jDem phase ðkÞ hj2
ð6Þ
k¼0
where DifðuÞ represents U differences of all the Euclidean phase distances. As is shown in Eq. (7), the indexing of the phase sequence with minimum difference is named temp:
temp ¼ minðDifðuÞÞ
ð7Þ
temp Signals ym n are descrambled with the conjugation sequence L of the indexed phase sequence to obtain output data bits.
3.2. Proposed SLM The Proposed SLM methods using Hadamard were introduced in [11, 12], but the new method presented in Fig. 3 has better performance on PAPR reduction than the schemes in [11, 12]. Previous researches show that if the signals are transformed by the Hadamard matrix before IFFT blocks, PAPR and BER of OFDM–ROF signals can be reduced. In the diagram shown in Fig. 3, the input OFDM–ROF data sequences are converted from series to parallel and the signals X n ¼ ½X 1 ; X 2 ; X N are generated. Then each signal X n is multiplied by an Hadamard matrix H, which is called Hadamard transform. The row vectors of Hadamard matrix Hu are set as phase sequences. To reduce bandwidth efficiency reduction, the phase sequences Hu ; u ¼ 1; 2; U are superimposed onto signal sequences X n . It is called the superimposed process. At the same time, the PAPR reduction should be guaranteed. So the phase sequences should be added as follows:
X un ¼
pffiffiffi u pffiffiffiffiffiffiffiffiffiffiffiffi bH þ 1 bX n
ð8Þ
Here b is a power allocation factor, 0 6 b 6 0:1. When the power ratio factor b is between 0 and 0.1, the systems have a good performance without hardware change at receiver [15]. The phase sequences, according to Eq. (8), can effectively lower the band radiation of the systems and omit the transmission of explicit side information. The channel estimation should be used to restore the transmitted signals at the receiver port. After the superimposed process, the processed signals multiply u the phase sequences h i H to produce alternative signal sequences u 1 2 U X n ¼ X n ; X n ; ; X n ; u ¼ 1; 2; U. xun ¼ x1n ; x2n ; ; xUn with the lowest PAPR Finally, the sequence ~ is transmitted after IFFT block. From what’s been discussed above, it is clear that the SLM scheme proposed improves the traditional one in reduction PAPR and side information.
4. The performance analysis of Proposed ED-SLM and Proposed SLM method 4.1. Proposed ED-SLM In order to have correct demodulation at the receiver, the Proposed ED-SLM algorithm will adopt Euclidean phase distance judgment method to test side information and complete the demodulation of data symbols. Euclidean distance is a common definition of the distance, and it is the real distance between two points in the n dimensional space. It denotes as Eq. (9):
d ¼ sqrt j Si1
X
j
j 2
ðSi1 Si2 Þ
ð9Þ j Si2
is the j dimensional coordinates for the first point, and is the j dimensional coordinates for the second point. Euclidean distance can be viewed as the degree of similarity among signals. The proposed ED-SLM algorithm analogy the distance of two points in the space for phase distance, so the Euclidean phase distance judgment method is put forward. According to the Euclidean phase distance judgment method, u the phases Dem phase of the descrambled signals ym n at the receiver have relation to initial phases at the transmitter as is shown in Fig. 4 below:
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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Nomalized autocorrelation function
1
Fig. 4. The deviation of the phases at the receiver.
Fig. 4 shows that the U phase vectors of the phase sequences distribute around the initialization phase, with deviation of distances, which is written as cu in Eq. (10):
cu ¼ jDem phaseu hj
Original QPSK sequence Hadamard transformed sequence
0.8
0.6
0.4
0.2
0
-0.2
ð10Þ
0
20
40
60
80 100 Subcarrier
120
140
160
The deviation of the phase factors in each phase sequence will be accumulated and the minimum deviation will be chosen to apply into Eq. (6) to compute indexed phase sequence.
Fig. 5. The normalized autocorrelation function of the Original QPSK sequence and Hadamard transformed sequence.
4.2. Proposed SLM
4.3. Theoretical analysis of PAPR
In order to further reduce the PAPR of the signal and achieve good BER performance, a Proposed SLM method with the superimposed training sequence is put forward, in which the superimposed training sequence is chosen from Hadamard matrix. A Hadamard matrix is the matrix of 1’s and 1’s, whose columns are orthogonal. So a Hadamard matrix H of order n satisfies the following equation:
In order to have a comprehensive description of algorithm performance, we will make a full derivation of the PAPR performance in the two algorithms above. The signal in the proposed ED-SLM scheme is YðkÞ, k 2 ½0; N 1. The discrete-time-domain OFDM– ROF signal sð nÞ is obtained by taking the normalized IDFT of SðkÞ:
0
HH ¼ nIn
ð11Þ 0
where In is the n n identity matrix and H is the transposition of H. The Walsh matrix [9] can be applied in the systems proposed. Then H H the partitioned matrix is a Hadamard matrix of order 2n. H H So there are:
H1 ¼ ½1; 1 1 H2 ¼ ; 1 1 .. . H2w1 H2w1 ¼ H2 H2w1 H2w ¼ H2w1 H2w1
N1 1 X n sðnÞ ¼ pffiffiffiffi Sð kÞej2pNk N k¼0
ð13Þ
The OFDM–ROF signals are subject to the distribution for v2 with mean value being zero and the degree of freedom two. The cumulative distribution function of v2 ð2Þ can be expressed as: N
Pr ðPAPR 6 cÞ ¼ ð1 ec Þ
ð14Þ
where, P r ðÞ is probability function. Conventionally, the probability function with PAPR over a certain threshold value-the complementary cumulative distribution function (Complementary CDF, CCDF) is adopted to represent the distribution of PAPR. It is written as:
Pr ðPAPR > cÞ ¼ 1 ð1 ec Þ ð12Þ
where denotes the Kronecker product. It is noted that Hadamard transform can be implemented by a butterfly structure as FFT, so the transform will not cause a sharp increase in the system complexity. For demodulation receiver, it is also very simple, with FFT data multiplied by the inverse Hadamard matrix. Because Hadamard matrix is orthogonal, it just need multiply the matrix transpose H0 . And PAPR values of OFDM signals relate to the autocorrelation of the input sequences, and the greater the input sequences’ autocorrelation is, the lower the sidelobe value of its spectrum function is. The proposed SLM uses Hadamard Matrix to lower the autocorrelation of the input signals in OFDM–ROF systems. Fig. 5 compares the normalized autocorrelation functions of the Original QPSK sequence and Hadamard transformed sequence. From Fig. 5, the maximum value is 1 in both sequences. But the Hadamard transformed sequence has lower sidelobe value than the Original QPSK sequence. It means that applying the Hadamard transformation to the IFFT input sequence can reduce PAPR values of OFDM–ROF signals.
N
ð15Þ
c is threshold. The peak-to-average power (PAR) of a symbol block sðnÞ should be defined as Eq. (16): PAR½sðnÞ ¼
maxo6n6N1 jsðnÞj2 PN1 2 1 n¼0 jsðnÞj N
ð16Þ
P 2 large N, the N1 N1 can be replaced by n¼0 jsðnÞj P N1 h i 2 E½jsðnÞj 2 n¼0 jsðnÞj E , so PAR of sðnÞ should be represented by N For
Eq. (17):
PAR½sðnÞ ¼
maxo6n6N1 jsðnÞj2 h i jsðnÞj2 E
ð17Þ
The signal in Proposed SLM scheme is XðkÞ ¼ SðkÞ, assume that the frequency-domain OFDM–ROF signal SðkÞ is i.i.d. and has the known constellation with variance r2s . According to the Central Limit Theorem, when the number of subcarriers N is large, the time-domain OFDM–ROF signal sðnÞ shown in (13) is approximately i.i.d. complex Gaussian distributing with zero-mean and average power r2s . For the Proposed SLM scheme with the superimposed training framework, we add row vectors of Hadamard
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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matrix denoted by dðnÞ onto sðnÞ to obtain SxðnÞ ¼ sðnÞ þ dðnÞ. Hence, SxðnÞ is independent complex Gaussian distributing with time-varying mean dðnÞ, the average power of pðnÞ is r2d .
Pr ðPAR > cÞ ¼ 1
N 1 Y
fR ½cðr2s þ r2d Þ
Pe ¼
where f R ðrÞ is the CDF of the noncentral follows:
"
jdðnÞj2 þr r2s
v distribution given as 2
pffiffiffi #
k 1 X jdðnÞj 2jdðnÞj r pffiffiffi Bk r2s r k¼0
ð19Þ
where Bk ðÞ is the kth-order modified Bessel function of the first kind, which can be represented by the infinite series, and g > 0:
Bk ðgÞ ¼
ð20Þ
n!ðn þ kÞ!
Because of the Hadamard matrices used as superimposed training sequence, it can be viewed as a special periodic dðnÞ with the period D ¼ N. Because the noncentrality parameter is jdðnÞj2 and dðnÞ is periodic, the CDF fR ðrÞ in (19) is also periodic in N. The CCDF of SxðnÞ in (18) can thus be simplified to Eq. (21): D1 Y
Pr ðPAR > cÞ ¼ 1
fR cðr2s þ r2d Þ
!M ð21Þ
n¼0
where M ¼ ND, and M is an integer. Add Eqs. (19) and (20) into Eq. (21), and the simplified expression for the CCDF is shown as Eq. (22):
8 k 0 l 19M jdðnÞj2 c > > N1 1 jdðnÞj2 X N1 < = Y X Nc 1b r2s B 2 C Pr ðPAR > cÞ ¼ 1 e1b @e rs A > > l! : n¼0 k¼1 k! ; l¼0 ð22Þ where b is power allocation factor defined below:
b¼
r2d r þ r2d
ð23Þ
2 S
Cðn; aÞ ¼ ðn 1Þ!ea
n1 l X a
l!
l¼0
where nbit represents error code number, N bit represents the total code number in the transmission. Suppose that in the wireless channels along which signals travel in OFDM–ROF systems, the noise is band-limited gaussian white noise with the mean value being zero, unilateral power spectral density n0 , binary code ‘‘1’’ and ‘‘0’’, and the prior probability Pð0Þ and Pð1Þ respectively, and we derive:
ð28Þ
where Pð0Þ represents the probability of transmission code ‘‘0’’ and Pð1Þ represents the probability of transmission code ‘‘1’’. Then the total error rate in Eq. (27) can be expressed as:
Pe ¼ Pð0ÞPe0 þ Pð1ÞP e1
ð29Þ
where Pe0 ¼ Pð1=0Þ, which is the conditional probability when ‘‘0’’ is sent and ‘‘1’’ is received;
1 Pe1 ¼ Pðn < aÞ ¼ pffiffiffiffiffiffiffi 2prn
a¼
Z
n0 Pð0Þ 1 ln Pð1Þ 2 2
r2n ¼ DðnÞ ¼
n0 2
Z
Ts
After simplification:
0 k 1M c b 1 C k; 1b c NM BX 1b C 1 e1b @ A k!ðk 1Þ! k¼1 ð26Þ
When b ¼ 0, the value of the signal power allocation factor is 0, which means no training sequence is superimposed onto signals. When pðnÞ ¼ 0, SxðnÞ ¼ sðnÞ, and the CCDF expressions of the PAR for OFDM–ROF signal sðnÞ will be shown as Eq. (15), which is the same as the PAPR expression in Proposed ED-SLM method.
2
a
e
x2 2r n
dx
ð30Þ
1
rf are shown as follows:
½s1 ðtÞ s0 ðtÞ2 dt
ð31Þ
0 Ts
½s1 ðtÞ s0 ðtÞ2 dt
ð32Þ
0
where, s0 ðtÞ is the signal waveform of the transmission code ‘‘0’’, s1 ðtÞ is the signal waveform of the transmission code ‘‘1’’. T s is the length of the code, r2n is the power of the noise. Similarly, we can get the conditions error probability when sending s0 ðtÞ and receiving s1 ðtÞ:
b¼
ð25Þ
Z
The parameters of a and
ð24Þ
8 cðr2 þr2 Þk 9M jdðnÞj2 > N cðr2 þr2 Þ > S d 1 S cÞ ¼ 1 e > k! ðk 1Þ! > : n¼0 k¼1 ;
Pr ðPAR > cÞ ¼ 1 e
ð27Þ
1 Pe0 ¼ Pðn < bÞ ¼ pffiffiffiffiffiffiffi 2prn
According to the incomplete gamma function defined below as Eq. (24), the CCDF of PAR is:
Mc 1b
nbit N bit
Pð0Þ þ Pð1Þ ¼ 1
1 X ðg=2Þ2nþk n¼0
BER is an important index to measure the validity and reliability of the communication, which is defined as:
ð18Þ
n¼0
f R ½r ¼ 1 e
4.4. Theoretical analysis of BER
n0 Pð1Þ 1 ln Pð0Þ 2 2
Z
Ts
Z
b
e
x2 2r2 n
dx
ð33Þ
1
½s1 ðtÞ s0 ðtÞ2 dt
ð34Þ
0
Therefore, Eq. (29) can be converted into Eq. (35):
1 Pe ¼ Pð0Þ pffiffiffiffiffiffiffi 2prn
Z
b
1
2
x2 2r
e
n
1 dx þ Pð1Þ pffiffiffiffiffiffiffi 2prn
Z
a
2
x2 2r
e
n
dx
ð35Þ
1
With given noise intensity, the BER is determined by the distinction among signal codes. The correlation coefficient q of signal codes is:
R Ts s1 ðtÞs0 ðtÞdt 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r q¼ n onR o R Ts Ts ½s0 ðtÞ2 dt ½s1 ðtÞ2 dt 0 0
ð36Þ
RT RT Here, E0 ¼ 0 s ½s0 ðtÞ2 dt , E1 ¼ 0 s ½s1 ðtÞ2 dt are the powers of signal codes s0 ðtÞ, s1 ðtÞ, respectively and 1 6 q 6 1. When the powers are the same, E0 ¼ E1 ¼ Eb :
R Ts
q¼
0
s1 ðtÞs0 ðtÞdt Eb
ð37Þ
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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^ is written as: C
^ ¼ 1 C 2
R Ts 0
2
½s1 ðtÞ s0 ðtÞ dt
ð38Þ
¼ Eb ð1 qÞ
When Pð0Þ ¼ Pð1Þ ¼ 12, substitute Eq. (38) into Eq. (35), then BER can be simplified to Eq. (39): 1 Pe ¼ pffiffiffiffi 2pr
¼
R C^ n
1 pffiffiffiffi 2prn
2
x2 2r
e 1
n
R Eb ð1qÞ 1
dx ð39Þ
2
x2 2r
e
n
dx
pffiffiffi pffiffiffi If z ¼ x= 2rn , its differential is dz ¼ dx = 2rn , and the BER is shown as Eq. (40): 1 Pe ¼ pffiffiffiffi 2pr
R Eb ð1qÞ=pffiffi2rn n
¼ p1ffiffipffi ¼
1 p2ffiffiffi ½ 2 p
pffiffiffi 2rn dz
ez
1
ez dz
R Eb ð1qÞ=pffiffi2rn
¼ p1ffiffipffi
2
1
R1
pffiffi Eb ð1qÞ= 2rn
2
2
ez dz
R1
pffiffi Eb ð1qÞ= 2rn
ð40Þ
z2
e dz n h io E ð1 ¼ 12 1 erf bpffiffi2r qÞ n
Rx 2 where error function is erfðxÞ ¼ p2ffiffipffi 0 ez dz , and its complementary error function is erfcðxÞ ¼ 1 erfðxÞ. Replace rn in Eq. (32) with n0 , and the BER is expressed as Eq. (41):
n hqffiffiffiffiffiffiffiffiffiffiffiffiio Eb ð1qÞ pffiffi Pe ¼ 12 1 erf 2n0 hqffiffiffiffiffiffiffiffiffiffiffiffii Eb ð1qÞ 1 pffiffi ¼ 2 erfc
ð41Þ
2n0
As a result, BER is related to the Signal to Noise Ratio (SNR) and the correlation coefficient q. 5. The analysis of simulation results 5.1. The simulation of PAPR To prove the effectiveness of the two schemes in OFDM–ROF systems, we will simulate Proposed ED-SLM and Proposed SLM in this part. The simulations of the Proposed ED-SLM method is in Fig. 6 with the Proposed SLM method. Most importantly, the simulations of the Proposed SLM method will be compared with different schemes using Hadamard matrix in Refs. [11] and [12]. The
results are shown in Figs. 7–9. The number of sub-carriers is 16, 128. The signal sequence number is 1000. H-SLM in [11] introduces a new multicarrier system with low computational complexity transform that combines the Walsh–Hadamard transform (WHT) and the discrete Fourier transform (DFT) into a single fast orthonormal unitary transform. H-SLM in [12] designs a cubic constellation, called the Hadamard constellation, whose boundary is along the bases defined by the Hadamard matrix in the transform domain. Fig. 6 shows the PAPR curves of original, traditional SLM and the two improved SLMs (U = 4) with the QPSK modulation in OFDM– ROF systems. When CCDF = 103, in the same condition that the phase sequence number is U = 4, the PAPR of the Proposed EDSLM algorithm is 1.2 dB lower than that of the traditional one. The PAPR of the Proposed SLM algorithm is 2.5 dB lower than that of the traditional one and is 1.4 dB lower than that of the Proposed ED-SLM algorithm. Although Proposed ED-SLM can reduce the PAPR of the OFDM–ROF systems, the effect is not obvious, so the proposed SLM algorithm works better in reducing PAPR of the OFDM–ROF signal. The above comparison between different schemes using Hadamard matrix in [11] and [12] shows that the proposed SLM scheme has better performance in PAPR reduction with the same 16-QAM modulation in OFDM–ROF systems. And the H-SLM in [12] has the same performance to reduce PAPR, but it needs U times Hadamard transform while the proposed method only needs once. Different from the H-SLM in [11,12], the row vectors of Hadamard matrix are used as phase sequences without generating new phase sequences. From the Fig. 7, when CCDF = 103, in the same condition that the phase sequence number is U = 16, the PAPR of the proposed SLM algorithm is 0.6 dB lower than that of the H-SLM [11] algorithm, but has the almost same performance compared with the H-SLM [12] algorithm. When U = 128, the PAPR of the proposed SLM algorithm is 0.7 dB lower than that of the HSLM [11] algorithm, and is 0.2 dB lower than that of the H-SLM [12] algorithm. So the proposed SLM algorithm works better in reducing PAPR of the OFDM–ROF signal when the U is greater. Comparative analysis of PAPR performance demonstrates that the Proposed SLM scheme can reduce PAPR of OFDM–ROF signals more effectively than the traditional one with the QPSK modulation in OFDM–ROF systems. Fig. 8 shows that on the same condition that the phase sequences number is 128, and the CCDF = 103, the PAPR of the proposed SLM scheme is 1.1 dB lower than that of the traditional one. The PAPR of the Proposed SLM scheme with the phase sequences number 16 is mostly the same
0
Orignal OFDM traditional SLM(u=4) Proposed ED-SLM(u=4) Proposed SLM(u=4)
0
10
original traditional SLM,u=16 H-SLM in [11],u=16 H-SLM in [12],u=16 proposed SLM,u=16 traditional SLM,u=128 H-SLM in [11],u=128 H-SLM in [12],u=128 proposed SLM,u=128
-1
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CCDF (Pr[PAPR>PAPR0])
CCDF (Pr[PAPR>PAPR0])
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Fig. 6. The PAPR curves of original, traditional SLM and the two improved SLM (U = 4).
-3
10
3
4
5
6 7 PAPR0 [dB]
8
9
10
Fig. 7. CCDF of the PAPR comparisons between different schemes (U = 16, 128).
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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Orignal OFDM traditional SLM(u=128) Proposed SLM(u=128) traditional SLM(u=16) Proposed SLM(u=16)
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Fig. 8. The PAPR curves of original, SLM and proposed SLM (U = 128, 16).
Orignal(u=1) proposed SLM proposed SLM proposed SLM proposed SLM
Fig. 10. The BER curves of the two improved schemes in the OFDM–ROF systems.
L = 4, respectively in OFDM–ROF systems. For the purpose of comparison, we also give the BER performance obtained in original and traditional SLM. The side information is correctly received. As is shown in Fig. 10 that the BER performance of the Proposed EDSLM is almost the same as that of the original and traditional SLM. Proposed scheme can achieve lower BER than Proposed EDSLM, which means that side information can be correctly restored by the Proposed SLM scheme.
0
10
CCDF (Pr[PAPR>PAPR0])
Orignal OFDM traditional SLM Proposed ED-SLM Proposed SLM
10
BER
CCDF (Pr[PAPR>PAPR0])
10
(u=2) (u=4) (u=8) (u=16)
-1
10
6. Conclusion -2
10
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7 PAPR0 [dB]
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Fig. 9. The PAPR curves of proposed SLM with different phase sequences number U.
as that of the traditional one with the phase sequences number being 128. Therefore, the Proposed SLM method can efficiently reduce the PAPR of OFDM–ROF signals. To further verify the advantages of the Proposed SLM scheme, Fig. 9 presents different PAPR curves with different phase sequences number U in the condition of 4-QAM modulation. For the traditional SLM, the bandwidth of side information increases with the increase of phase sequence number U, so does the system complexity. But the row vectors of Hadamard matrix used as phase sequences retain low complexity. From Fig. 9, the Hadamard matrix illustrates its advantages of better performance with larger phase sequence number. In brief, the simulations show that the Proposed SLM scheme is effective in reducing the PAPR of the OFDM–ROF transmission systems.
5.2. The simulation of BER In the following simulations, we assume a zero-mean complex additive white Gaussian noise (AWGN) channel with various values of Eb/N0. Fig. 10 shows BER performance of the Proposed ED-SLM and Proposed SLM scheme employing QPSK and over-sample factor
In this paper, we propose two improved schemes without sending explicit side information to reduce PAPR and BER in OFDM–ROF systems. The Proposed ED-SLM and Proposed SLM schemes have better PAPR performance than the traditional ones in the PAPR reduction. By Proposed ED-SLM scheme, we put forward a algorithm to reduce PAPR with the initialized phase sequences and also a Euclidean phase distance detection method to detect the side information at the receiver in the OFDM–ROF systems. Proposed SLM does not need to generate new phase sequences thanks to the Hadamard matrix which will not transmit and restore the explicit side information about the phase rotation by using superimposed method. The simulation results prove that the Proposed ED-SLM and Proposed SLM schemes can reduce peak power of the OFDM–ROF systems, save the bandwidth and lower systems’ complexity. At the same time, the proposed SLM scheme can obtain better BER performance than both the Proposed ED-SLM and traditional SLM scheme in the OFDM–ROF systems. Finally, results of the computer simulation show that the proposed scheme achieves an excellent performance in terms of both BER and PAPR reduction. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (No. 61310306022 and No. 61072073), ‘‘1000-elite program’’ foundation of Sichuan Province and Signal Processing Scientific Research and Innovation Team in Southwest Petroleum University (No. 2013XJZT007), and Science and technology support project in Sichuan province (No.2012FZ0021). References [1] I.B. Djordjevic, B. Vasic, Orthogonal frequency division multiplexing for highspeed optical transmission, Opt. Express 14 (9) (2006) 3767–3775.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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[2] K. Xu, X. Sun, J. Yin, et al., Enabling ROF technologies and integration architectures for in-building optical–wireless access networks, IEEE Photonics J. 2 (2) (2010) 102–112. [3] H.T. Huang, C.T. Lin, C.H. Ho, et al., High spectral efficient W-band OFDM–ROF system with direct-detection by two cascaded single-drive MZMs, Opt. Express 21 (14) (2013) 16615–16620. [4] Y. Shao, N. Chi. PAPR reduction and computational complexity analysis of interleaved segmentation in 60 GHz OFDM–ROF system, in: Optical Fiber Communication Conference, Optical Society of America, (2013) 1–3. [5] F. Li, J. Yu, Z. Cao, et al., Reducing the peak-to-average power ratio with companding transform coding in 60 GHz OFDM–ROF systems, IEEE/OSA J. Opt. Commun. Networking 4 (3) (2012) 202–209. [6] X. Zhu, W. Pan, H. Li, et al., Simplified approach to optimized iterative clipping and filtering for PAPR reduction of OFDM signals, IEEE Trans. Commun. 61 (5) (2013) 1891–1901. [7] A.S. Alaraimi, T. Hashimoto. PAPR and OOBP of OFDM and their improvement by using self cancellation codings, in: Wireless Personal Multimedia Communications (WPMC), 2011 14th International Symposium on. IEEE, (2011) 1–5. [8] H.B. Jeon, J.S. No, D.J. Shin, A low-complexity SLM scheme using additive mapping sequences for PAPR reduction of OFDM signals, IEEE Trans. Broadcast. 57 (4) (2011) 866–875.
[9] S.S. Jeng, J.M. Chen, Efficient PAPR reduction in OFDM systems based on a companding technique with trapezium distribution, IEEE Trans. Broadcast. 57 (2) (2011) 291–298. [10] T. Jiang, C. Ni, L. Guan, A novel phase offset SLM scheme for PAPR reduction in Alamouti MIMO-OFDM systems without side information, IEEE Signal Process. Lett. 20 (2013) 383–386. [11] M.S. Ahmed, S. Boussakta, B.S. Sharif, et al., OFDM based on low complexity transform to increase multipath resilience and reduce PAPR, IEEE Trans. Signal Process. 59 (12) (2011) 5994–6007. [12] A. Mobasher, A.K. Khandani, Integer-based constellation-shaping method for PAPR reduction in OFDM systems, IEEE Trans. Commun. 54 (1) (2006) 119– 127. [13] L. Li, D. Qu, Joint decoding of LDPC code and phase factors for OFDM systems with PTS PAPR reduction, IEEE Trans. Veh. Technol. 62 (1) (2013) 444–449. [14] M. Niranjan, S. Srikanth. Adaptive active constellation extension for PAPR reduction in OFDM systems, in; Recent Trends in Information Technology (ICRTIT), 2011 International Conference on. IEEE, (2011) 1186–1189. [15] C.L. Wang, Y. Ouyang, Low-complexity selected mapping schemes for peak-toaverage power ratio reduction in OFDM systems, IEEE Trans. Signal Process. 53 (12) (2005) 4652–4660.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems Luo Renze a, Li Rui a,⇑, Dang Yupu b, Yang Jiao a, Liu Weihong c a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu 610500, China Department of Well Tech, China Oilfield Services Limited, Beijing 101149, China c Turpan Hami Oilfield Company, Xinjiang 838202, China b
a r t i c l e
i n f o
Article history: Received 16 April 2014 Revised 12 July 2014 Available online xxxx Keywords: Orthogonal frequency division multiplexing–radio over fiber (OFDM–ROF) Peak-to-average power ratio (PAPR) Selected mapping (SLM) Bit error ratio (BER)
a b s t r a c t OFDM–ROF (orthogonal frequency division multiplexing–radio over fiber) system has high spectral efficiency and high transmission rate. However, due to the non-linear power amplifier, the high peakto-average power ratio (PAPR) of the transmission signals is one of the major setbacks in the OFDM– ROF transmission systems. In order to reduce PAPR, this paper proposes two improved selected mapping (SLM) methods without explicit side information (SI): Proposed ED-SLM and Proposed SLM. In the Proposed ED-SLM method, signals will be transmitted without side information which is scrambled by special initial phase sequences and at the receiver an Euclidean phase distance detection (EPD) method is used to detect the SI and finishes the demodulation. In the proposed SLM method, the signals are transformed by Hadamard matrix to reduce PAPR. Then the row vectors of Hadamard matrix are used as phase sequences and superimposed on the data signals. The theory and simulation results show that the two improved SLM methods perform better in PAPR and BER reduction than the conventional SLM method in the OFDM–ROF transmission systems. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction With the construction of the experimental networks such as Long Term Evolution (LTE), and Worldwide Interoperability for Microwave Access (WiMAX), wireless communication has entered a high-speed data era. In order to solve the problem of insufficient spectrum resources, ROF wireless network and OFDM–ROF systems have been proposed [1]. Because of its low transmission loss and ultra-wide bandwidth, the ROF system proved to be a key technology for next generation network (NGN) access applications [2]. The benefits that OFDM brings to wireless communication systems include high transmission capacity, resistance to interference and robustness to multipath fading. Thus, the combination of orthogonal frequency division multiplexing and radio over fiber systems (OFDM–ROF) has attracted much attention for future gigabit broadband wireless communication [3]. However, due to the drawbacks of fiber-optic such as dispersion, nonlinearity, the PAPR of OFDM signals cannot be ignored when data are transmitted over the fiber. The drawbacks of high peak-average power ratio (PAPR) ⇑ Corresponding author. E-mail addresses: [email protected] (R. Luo), [email protected] (R. Li), [email protected] (Y. Dang), [email protected] (J. Yang), 153691280@qq. com (W. Liu).
may outweigh all the potential advantages of the OFDM–ROF transmission systems [4,5], so the system must overcome the defect. Various methods for PAPR reduction are proposed, including clipping [6], coding [7], selected mapping (SLM) [8–12], partial transmit sequence (PTS) [13], and active constellation (ACE) [14]. In these methods, selective mapping (SLM) is effective and promising to reduce PAPR. Ref. [8] introduces a modified SLM scheme which generates alternative signal sequences by adding mapping signal sequences to an OFDM signal sequence for the purpose of reducing computational complexity at the cost of PAPR performance. Ref. [10] explains a scheme to reduce PAPR without transmitting side information, and in the receiver a minimum Euclidian distance (MED) decoder is used to recover the side information. Nevertheless, the computational complexity seems larger and BER performance is not satisfactory. In Ref. [15], the inverse discrete Fourier transform (IDFT) is replaced by a conversion matrix in the time domain and it significantly reduces the computational complexity but bit error ratio (BER) degradation cannot be avoided. To remove a great deal of side information and further reduce PAPR in OFDM–ROF systems, this paper introduces two effective methods based on SLM algorithm: Proposed ED-SLM and Proposed SLM. In Proposed ED-SLM algorithm, signals will be transmitted without side information which is scrambled by special initial phase sequences and at the receiver an Euclidean phase distance
http://dx.doi.org/10.1016/j.yofte.2014.07.007 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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detection method is used to detect SI and finish demodulation. In order to have better PAPR and BER performance, another improved method is proposed. In the Proposed SLM method, signal is transformed by Hadamard matrix to reduce PAPR. Then the row vectors of Hadamard matrix are used as phase sequences and superimposed on data signals. After channel estimation, the superimposed Hadamard sequence on the signals will be removed at the receiver and the SI will be recovered at the same time. The method has a good BER performance after the OFDM–ROF demodulation. The paper is organized as follows: firstly, the OFDM–ROF transmission systems are introduced. Secondly, the Proposed ED-SLM and Proposed SLM schemes are presented. Thirdly, the performance analysis of Proposed ED-SLM and Proposed SLM methods are shown. Fourthly, the PAPR and BER simulation results are compared with different schemes. Conclusion is presented in the last part. 2. The OFDM–ROF transmission systems Fig. 1 shows the block diagram of the OFDM–ROF transmission system. ROF is a technology which takes optical fiber as transmission medium, light wave as carrier, and high-frequency microwave as modulation wave. As is shown in Fig. 1, ROF combines wireless communications and optical fiber transmission, where the wireless access system is composed of the central station (CS), optical fiber link, base stations (BS) and mobile terminals. Shortened as OFDM–ROF system, OFDM–ROF optical wireless communication system combines OFDM modulation and ROF technology. Central Station is mainly responsible for generation of the downlink radio-on-fiber OFDM wireless signals, and the inspection and receiving of the uplink data. Base station is mainly responsible for the detection of the downlink radio-on-fiber OFDM wireless signals and converting them into electricity OFDM wireless signals which will be transmitted through antennas. At the same time, it also receives uplink signals from the antennas and modulates the optical carrier signals to generate upward radio-on-fiber signals. Mobile terminal, called subscriber unit, receives wireless OFDM signals by antennas and demodulate to baseband signals, and at the same time, it launches uplink signals. In practical communication network, a central station is connected to multiple base stations, and a base station is connected to more than one subscriber units. In order to save costs and facilitate the management of the whole communication network, the processing of complex and expensive equipment is usually placed in the central station. 3. Proposed ED-SLM and Proposed SLM schemes
sub-carriers are used, and each sub-carrier is modulated using symbols, the OFDM symbol sðtÞ is expressed as N1 1 X n sðtÞ ¼ pffiffiffiffi Sn ej2pNt ; 0 6 t 6 N 1; N n¼0
ð1Þ
where Sn are the data symbols, and N is the number of sub-carriers. The PAPR of an OFDM signal is defined as
PAPRðSÞD
max06t6N1 jsðtÞj2 EfjsðtÞj2 g
ð2Þ
where E½ denotes the expected value. The OFDM–ROF systems have high PAPR, which requires linear, large-dynamic-range amplifiers that are inefficient and expensive. This paper focuses on improving traditional SLM scheme to reduce PAPR without distortion, and achieve good BER performance. In the conventional SLM scheme, the symbol sequences multiply the phase sequences to generate alternative symbol sequences. After the IFFT blocks, the OFDM–ROF signal with the lowest PAPR is selected to transmit but the side information is also transmitted. Therefore, the bandwidth occupied by the redundant side information is one of the main bottlenecks of the traditional SLM method. 3.1. Proposed ED-SLM The signals will be launched without side information in Proposed ED-SLM algorithm, and the receiver will take advantage of the Euclidean phase distance judgment method to detect side information. The ED-SLM algorithm can obtain certain improvement on PAPR performance and the BER isn’t worse than that of the traditional SLM method. Fig. 2 is the block diagram of the Proposed ED-SLM scheme. As shown in Fig. 2, the binary data bit is put into the transmitter through OFDM–ROF signal transmission systems, and frequency OFDM modulation signal can be obtained through the mapping and series-to-parallel conversion: Y n ¼ ½Y 1 ; Y 2 ; Y N , n 2 ½1; N. Then initialize the phase sequences according to Eq. (3):
u u u Lu ¼ diag ejhp0 ; ejhp1 ; ; ejhpN1
ð3Þ
u
Here, L represents the u-th phase sequence u 2 ð1; UÞ. The rotation phase h 2 ð0; pÞ is obtained randomly and the scrambler coefficient is pui 2 f1; 0; 1g, i 2 ð0; N 1Þ, j ¼ sqrtð1Þ. U initialization phase sequences will be used to scramble the transmitting signals, signals after IFFT can be expressed as:
yun ¼ IFFTðY n Lu Þ
ð4Þ ym n
In the OFDM–ROF transmission systems, signals are demodulated onto the carriers which are orthogonal to each other. If N
Then the signals with a minimum PAPR will be launched, and m 2 ð1; UÞ. Since the Euclidean phase distance judgment method will be used to detect phase sequence at the receiver, the transmit-
Fig. 1. Block diagram of the OFDM–ROF transmission systems.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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IFFT
IFFT Input data
S/P ...
...
Phase initiating Phase optimal
Select the sequence with lowest PAPR to transmit
IFFT
Receive data
Phase Detection select The min Euclidean phase distance
decison
Input data
Hadamard transform
S/P
wireless channel
Receive the Signals and have OFDM demodulat -ion
Fig. 2. The block diagram of Proposed ED-SLM scheme. Fig. 3. The block diagram of the Proposed SLM scheme.
ter will not transmit side information. Transmitting signals will be demodulated at receiver after they pass through wireless channels. The signals are descrambled with all phase sequences after FFT and ^ u , which is written as Eq. (5): restored to U demodulation signals Y n u Y^ un ¼ FFTðym n Þ ðL Þ
ð5Þ u
^ u can be The phase Dem phase of the demodulated signals Y n worked out. In the Euclidean phase distance judgment method, Eq. (6) will calculate the difference of two squares of the phase between receiver signal and initial phase:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N1 uX u DifðuÞ ¼ t jDem phase ðkÞ hj2
ð6Þ
k¼0
where DifðuÞ represents U differences of all the Euclidean phase distances. As is shown in Eq. (7), the indexing of the phase sequence with minimum difference is named temp:
temp ¼ minðDifðuÞÞ
ð7Þ
temp Signals ym n are descrambled with the conjugation sequence L of the indexed phase sequence to obtain output data bits.
3.2. Proposed SLM The Proposed SLM methods using Hadamard were introduced in [11, 12], but the new method presented in Fig. 3 has better performance on PAPR reduction than the schemes in [11, 12]. Previous researches show that if the signals are transformed by the Hadamard matrix before IFFT blocks, PAPR and BER of OFDM–ROF signals can be reduced. In the diagram shown in Fig. 3, the input OFDM–ROF data sequences are converted from series to parallel and the signals X n ¼ ½X 1 ; X 2 ; X N are generated. Then each signal X n is multiplied by an Hadamard matrix H, which is called Hadamard transform. The row vectors of Hadamard matrix Hu are set as phase sequences. To reduce bandwidth efficiency reduction, the phase sequences Hu ; u ¼ 1; 2; U are superimposed onto signal sequences X n . It is called the superimposed process. At the same time, the PAPR reduction should be guaranteed. So the phase sequences should be added as follows:
X un ¼
pffiffiffi u pffiffiffiffiffiffiffiffiffiffiffiffi bH þ 1 bX n
ð8Þ
Here b is a power allocation factor, 0 6 b 6 0:1. When the power ratio factor b is between 0 and 0.1, the systems have a good performance without hardware change at receiver [15]. The phase sequences, according to Eq. (8), can effectively lower the band radiation of the systems and omit the transmission of explicit side information. The channel estimation should be used to restore the transmitted signals at the receiver port. After the superimposed process, the processed signals multiply u the phase sequences h i H to produce alternative signal sequences u 1 2 U X n ¼ X n ; X n ; ; X n ; u ¼ 1; 2; U. xun ¼ x1n ; x2n ; ; xUn with the lowest PAPR Finally, the sequence ~ is transmitted after IFFT block. From what’s been discussed above, it is clear that the SLM scheme proposed improves the traditional one in reduction PAPR and side information.
4. The performance analysis of Proposed ED-SLM and Proposed SLM method 4.1. Proposed ED-SLM In order to have correct demodulation at the receiver, the Proposed ED-SLM algorithm will adopt Euclidean phase distance judgment method to test side information and complete the demodulation of data symbols. Euclidean distance is a common definition of the distance, and it is the real distance between two points in the n dimensional space. It denotes as Eq. (9):
d ¼ sqrt j Si1
X
j
j 2
ðSi1 Si2 Þ
ð9Þ j Si2
is the j dimensional coordinates for the first point, and is the j dimensional coordinates for the second point. Euclidean distance can be viewed as the degree of similarity among signals. The proposed ED-SLM algorithm analogy the distance of two points in the space for phase distance, so the Euclidean phase distance judgment method is put forward. According to the Euclidean phase distance judgment method, u the phases Dem phase of the descrambled signals ym n at the receiver have relation to initial phases at the transmitter as is shown in Fig. 4 below:
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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Nomalized autocorrelation function
1
Fig. 4. The deviation of the phases at the receiver.
Fig. 4 shows that the U phase vectors of the phase sequences distribute around the initialization phase, with deviation of distances, which is written as cu in Eq. (10):
cu ¼ jDem phaseu hj
Original QPSK sequence Hadamard transformed sequence
0.8
0.6
0.4
0.2
0
-0.2
ð10Þ
0
20
40
60
80 100 Subcarrier
120
140
160
The deviation of the phase factors in each phase sequence will be accumulated and the minimum deviation will be chosen to apply into Eq. (6) to compute indexed phase sequence.
Fig. 5. The normalized autocorrelation function of the Original QPSK sequence and Hadamard transformed sequence.
4.2. Proposed SLM
4.3. Theoretical analysis of PAPR
In order to further reduce the PAPR of the signal and achieve good BER performance, a Proposed SLM method with the superimposed training sequence is put forward, in which the superimposed training sequence is chosen from Hadamard matrix. A Hadamard matrix is the matrix of 1’s and 1’s, whose columns are orthogonal. So a Hadamard matrix H of order n satisfies the following equation:
In order to have a comprehensive description of algorithm performance, we will make a full derivation of the PAPR performance in the two algorithms above. The signal in the proposed ED-SLM scheme is YðkÞ, k 2 ½0; N 1. The discrete-time-domain OFDM– ROF signal sð nÞ is obtained by taking the normalized IDFT of SðkÞ:
0
HH ¼ nIn
ð11Þ 0
where In is the n n identity matrix and H is the transposition of H. The Walsh matrix [9] can be applied in the systems proposed. Then H H the partitioned matrix is a Hadamard matrix of order 2n. H H So there are:
H1 ¼ ½1; 1 1 H2 ¼ ; 1 1 .. . H2w1 H2w1 ¼ H2 H2w1 H2w ¼ H2w1 H2w1
N1 1 X n sðnÞ ¼ pffiffiffiffi Sð kÞej2pNk N k¼0
ð13Þ
The OFDM–ROF signals are subject to the distribution for v2 with mean value being zero and the degree of freedom two. The cumulative distribution function of v2 ð2Þ can be expressed as: N
Pr ðPAPR 6 cÞ ¼ ð1 ec Þ
ð14Þ
where, P r ðÞ is probability function. Conventionally, the probability function with PAPR over a certain threshold value-the complementary cumulative distribution function (Complementary CDF, CCDF) is adopted to represent the distribution of PAPR. It is written as:
Pr ðPAPR > cÞ ¼ 1 ð1 ec Þ ð12Þ
where denotes the Kronecker product. It is noted that Hadamard transform can be implemented by a butterfly structure as FFT, so the transform will not cause a sharp increase in the system complexity. For demodulation receiver, it is also very simple, with FFT data multiplied by the inverse Hadamard matrix. Because Hadamard matrix is orthogonal, it just need multiply the matrix transpose H0 . And PAPR values of OFDM signals relate to the autocorrelation of the input sequences, and the greater the input sequences’ autocorrelation is, the lower the sidelobe value of its spectrum function is. The proposed SLM uses Hadamard Matrix to lower the autocorrelation of the input signals in OFDM–ROF systems. Fig. 5 compares the normalized autocorrelation functions of the Original QPSK sequence and Hadamard transformed sequence. From Fig. 5, the maximum value is 1 in both sequences. But the Hadamard transformed sequence has lower sidelobe value than the Original QPSK sequence. It means that applying the Hadamard transformation to the IFFT input sequence can reduce PAPR values of OFDM–ROF signals.
N
ð15Þ
c is threshold. The peak-to-average power (PAR) of a symbol block sðnÞ should be defined as Eq. (16): PAR½sðnÞ ¼
maxo6n6N1 jsðnÞj2 PN1 2 1 n¼0 jsðnÞj N
ð16Þ
P 2 large N, the N1 N1 can be replaced by n¼0 jsðnÞj P N1 h i 2 E½jsðnÞj 2 n¼0 jsðnÞj E , so PAR of sðnÞ should be represented by N For
Eq. (17):
PAR½sðnÞ ¼
maxo6n6N1 jsðnÞj2 h i jsðnÞj2 E
ð17Þ
The signal in Proposed SLM scheme is XðkÞ ¼ SðkÞ, assume that the frequency-domain OFDM–ROF signal SðkÞ is i.i.d. and has the known constellation with variance r2s . According to the Central Limit Theorem, when the number of subcarriers N is large, the time-domain OFDM–ROF signal sðnÞ shown in (13) is approximately i.i.d. complex Gaussian distributing with zero-mean and average power r2s . For the Proposed SLM scheme with the superimposed training framework, we add row vectors of Hadamard
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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R. Luo et al. / Optical Fiber Technology xxx (2014) xxx–xxx
matrix denoted by dðnÞ onto sðnÞ to obtain SxðnÞ ¼ sðnÞ þ dðnÞ. Hence, SxðnÞ is independent complex Gaussian distributing with time-varying mean dðnÞ, the average power of pðnÞ is r2d .
Pr ðPAR > cÞ ¼ 1
N 1 Y
fR ½cðr2s þ r2d Þ
Pe ¼
where f R ðrÞ is the CDF of the noncentral follows:
"
jdðnÞj2 þr r2s
v distribution given as 2
pffiffiffi #
k 1 X jdðnÞj 2jdðnÞj r pffiffiffi Bk r2s r k¼0
ð19Þ
where Bk ðÞ is the kth-order modified Bessel function of the first kind, which can be represented by the infinite series, and g > 0:
Bk ðgÞ ¼
ð20Þ
n!ðn þ kÞ!
Because of the Hadamard matrices used as superimposed training sequence, it can be viewed as a special periodic dðnÞ with the period D ¼ N. Because the noncentrality parameter is jdðnÞj2 and dðnÞ is periodic, the CDF fR ðrÞ in (19) is also periodic in N. The CCDF of SxðnÞ in (18) can thus be simplified to Eq. (21): D1 Y
Pr ðPAR > cÞ ¼ 1
fR cðr2s þ r2d Þ
!M ð21Þ
n¼0
where M ¼ ND, and M is an integer. Add Eqs. (19) and (20) into Eq. (21), and the simplified expression for the CCDF is shown as Eq. (22):
8 k 0 l 19M jdðnÞj2 c > > N1 1 jdðnÞj2 X N1 < = Y X Nc 1b r2s B 2 C Pr ðPAR > cÞ ¼ 1 e1b @e rs A > > l! : n¼0 k¼1 k! ; l¼0 ð22Þ where b is power allocation factor defined below:
b¼
r2d r þ r2d
ð23Þ
2 S
Cðn; aÞ ¼ ðn 1Þ!ea
n1 l X a
l!
l¼0
where nbit represents error code number, N bit represents the total code number in the transmission. Suppose that in the wireless channels along which signals travel in OFDM–ROF systems, the noise is band-limited gaussian white noise with the mean value being zero, unilateral power spectral density n0 , binary code ‘‘1’’ and ‘‘0’’, and the prior probability Pð0Þ and Pð1Þ respectively, and we derive:
ð28Þ
where Pð0Þ represents the probability of transmission code ‘‘0’’ and Pð1Þ represents the probability of transmission code ‘‘1’’. Then the total error rate in Eq. (27) can be expressed as:
Pe ¼ Pð0ÞPe0 þ Pð1ÞP e1
ð29Þ
where Pe0 ¼ Pð1=0Þ, which is the conditional probability when ‘‘0’’ is sent and ‘‘1’’ is received;
1 Pe1 ¼ Pðn < aÞ ¼ pffiffiffiffiffiffiffi 2prn
a¼
Z
n0 Pð0Þ 1 ln Pð1Þ 2 2
r2n ¼ DðnÞ ¼
n0 2
Z
Ts
After simplification:
0 k 1M c b 1 C k; 1b c NM BX 1b C 1 e1b @ A k!ðk 1Þ! k¼1 ð26Þ
When b ¼ 0, the value of the signal power allocation factor is 0, which means no training sequence is superimposed onto signals. When pðnÞ ¼ 0, SxðnÞ ¼ sðnÞ, and the CCDF expressions of the PAR for OFDM–ROF signal sðnÞ will be shown as Eq. (15), which is the same as the PAPR expression in Proposed ED-SLM method.
2
a
e
x2 2r n
dx
ð30Þ
1
rf are shown as follows:
½s1 ðtÞ s0 ðtÞ2 dt
ð31Þ
0 Ts
½s1 ðtÞ s0 ðtÞ2 dt
ð32Þ
0
where, s0 ðtÞ is the signal waveform of the transmission code ‘‘0’’, s1 ðtÞ is the signal waveform of the transmission code ‘‘1’’. T s is the length of the code, r2n is the power of the noise. Similarly, we can get the conditions error probability when sending s0 ðtÞ and receiving s1 ðtÞ:
b¼
ð25Þ
Z
The parameters of a and
ð24Þ
8 cðr2 þr2 Þk 9M jdðnÞj2 > N cðr2 þr2 Þ > S d 1 S
Pr ðPAR > cÞ ¼ 1 e
ð27Þ
1 Pe0 ¼ Pðn < bÞ ¼ pffiffiffiffiffiffiffi 2prn
According to the incomplete gamma function defined below as Eq. (24), the CCDF of PAR is:
Mc 1b
nbit N bit
Pð0Þ þ Pð1Þ ¼ 1
1 X ðg=2Þ2nþk n¼0
BER is an important index to measure the validity and reliability of the communication, which is defined as:
ð18Þ
n¼0
f R ½r ¼ 1 e
4.4. Theoretical analysis of BER
n0 Pð1Þ 1 ln Pð0Þ 2 2
Z
Ts
Z
b
e
x2 2r2 n
dx
ð33Þ
1
½s1 ðtÞ s0 ðtÞ2 dt
ð34Þ
0
Therefore, Eq. (29) can be converted into Eq. (35):
1 Pe ¼ Pð0Þ pffiffiffiffiffiffiffi 2prn
Z
b
1
2
x2 2r
e
n
1 dx þ Pð1Þ pffiffiffiffiffiffiffi 2prn
Z
a
2
x2 2r
e
n
dx
ð35Þ
1
With given noise intensity, the BER is determined by the distinction among signal codes. The correlation coefficient q of signal codes is:
R Ts s1 ðtÞs0 ðtÞdt 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r q¼ n onR o R Ts Ts ½s0 ðtÞ2 dt ½s1 ðtÞ2 dt 0 0
ð36Þ
RT RT Here, E0 ¼ 0 s ½s0 ðtÞ2 dt , E1 ¼ 0 s ½s1 ðtÞ2 dt are the powers of signal codes s0 ðtÞ, s1 ðtÞ, respectively and 1 6 q 6 1. When the powers are the same, E0 ¼ E1 ¼ Eb :
R Ts
q¼
0
s1 ðtÞs0 ðtÞdt Eb
ð37Þ
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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R. Luo et al. / Optical Fiber Technology xxx (2014) xxx–xxx
^ is written as: C
^ ¼ 1 C 2
R Ts 0
2
½s1 ðtÞ s0 ðtÞ dt
ð38Þ
¼ Eb ð1 qÞ
When Pð0Þ ¼ Pð1Þ ¼ 12, substitute Eq. (38) into Eq. (35), then BER can be simplified to Eq. (39): 1 Pe ¼ pffiffiffiffi 2pr
¼
R C^ n
1 pffiffiffiffi 2prn
2
x2 2r
e 1
n
R Eb ð1qÞ 1
dx ð39Þ
2
x2 2r
e
n
dx
pffiffiffi pffiffiffi If z ¼ x= 2rn , its differential is dz ¼ dx = 2rn , and the BER is shown as Eq. (40): 1 Pe ¼ pffiffiffiffi 2pr
R Eb ð1qÞ=pffiffi2rn n
¼ p1ffiffipffi ¼
1 p2ffiffiffi ½ 2 p
pffiffiffi 2rn dz
ez
1
ez dz
R Eb ð1qÞ=pffiffi2rn
¼ p1ffiffipffi
2
1
R1
pffiffi Eb ð1qÞ= 2rn
2
2
ez dz
R1
pffiffi Eb ð1qÞ= 2rn
ð40Þ
z2
e dz n h io E ð1 ¼ 12 1 erf bpffiffi2r qÞ n
Rx 2 where error function is erfðxÞ ¼ p2ffiffipffi 0 ez dz , and its complementary error function is erfcðxÞ ¼ 1 erfðxÞ. Replace rn in Eq. (32) with n0 , and the BER is expressed as Eq. (41):
n hqffiffiffiffiffiffiffiffiffiffiffiffiio Eb ð1qÞ pffiffi Pe ¼ 12 1 erf 2n0 hqffiffiffiffiffiffiffiffiffiffiffiffii Eb ð1qÞ 1 pffiffi ¼ 2 erfc
ð41Þ
2n0
As a result, BER is related to the Signal to Noise Ratio (SNR) and the correlation coefficient q. 5. The analysis of simulation results 5.1. The simulation of PAPR To prove the effectiveness of the two schemes in OFDM–ROF systems, we will simulate Proposed ED-SLM and Proposed SLM in this part. The simulations of the Proposed ED-SLM method is in Fig. 6 with the Proposed SLM method. Most importantly, the simulations of the Proposed SLM method will be compared with different schemes using Hadamard matrix in Refs. [11] and [12]. The
results are shown in Figs. 7–9. The number of sub-carriers is 16, 128. The signal sequence number is 1000. H-SLM in [11] introduces a new multicarrier system with low computational complexity transform that combines the Walsh–Hadamard transform (WHT) and the discrete Fourier transform (DFT) into a single fast orthonormal unitary transform. H-SLM in [12] designs a cubic constellation, called the Hadamard constellation, whose boundary is along the bases defined by the Hadamard matrix in the transform domain. Fig. 6 shows the PAPR curves of original, traditional SLM and the two improved SLMs (U = 4) with the QPSK modulation in OFDM– ROF systems. When CCDF = 103, in the same condition that the phase sequence number is U = 4, the PAPR of the Proposed EDSLM algorithm is 1.2 dB lower than that of the traditional one. The PAPR of the Proposed SLM algorithm is 2.5 dB lower than that of the traditional one and is 1.4 dB lower than that of the Proposed ED-SLM algorithm. Although Proposed ED-SLM can reduce the PAPR of the OFDM–ROF systems, the effect is not obvious, so the proposed SLM algorithm works better in reducing PAPR of the OFDM–ROF signal. The above comparison between different schemes using Hadamard matrix in [11] and [12] shows that the proposed SLM scheme has better performance in PAPR reduction with the same 16-QAM modulation in OFDM–ROF systems. And the H-SLM in [12] has the same performance to reduce PAPR, but it needs U times Hadamard transform while the proposed method only needs once. Different from the H-SLM in [11,12], the row vectors of Hadamard matrix are used as phase sequences without generating new phase sequences. From the Fig. 7, when CCDF = 103, in the same condition that the phase sequence number is U = 16, the PAPR of the proposed SLM algorithm is 0.6 dB lower than that of the H-SLM [11] algorithm, but has the almost same performance compared with the H-SLM [12] algorithm. When U = 128, the PAPR of the proposed SLM algorithm is 0.7 dB lower than that of the HSLM [11] algorithm, and is 0.2 dB lower than that of the H-SLM [12] algorithm. So the proposed SLM algorithm works better in reducing PAPR of the OFDM–ROF signal when the U is greater. Comparative analysis of PAPR performance demonstrates that the Proposed SLM scheme can reduce PAPR of OFDM–ROF signals more effectively than the traditional one with the QPSK modulation in OFDM–ROF systems. Fig. 8 shows that on the same condition that the phase sequences number is 128, and the CCDF = 103, the PAPR of the proposed SLM scheme is 1.1 dB lower than that of the traditional one. The PAPR of the Proposed SLM scheme with the phase sequences number 16 is mostly the same
0
Orignal OFDM traditional SLM(u=4) Proposed ED-SLM(u=4) Proposed SLM(u=4)
0
10
original traditional SLM,u=16 H-SLM in [11],u=16 H-SLM in [12],u=16 proposed SLM,u=16 traditional SLM,u=128 H-SLM in [11],u=128 H-SLM in [12],u=128 proposed SLM,u=128
-1
10
CCDF (Pr[PAPR>PAPR0])
CCDF (Pr[PAPR>PAPR0])
10
-2
10
-1
10
-2
10
-3
10
3
4
5
6
7 8 PAPR0 [dB]
9
10
11
12
Fig. 6. The PAPR curves of original, traditional SLM and the two improved SLM (U = 4).
-3
10
3
4
5
6 7 PAPR0 [dB]
8
9
10
Fig. 7. CCDF of the PAPR comparisons between different schemes (U = 16, 128).
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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R. Luo et al. / Optical Fiber Technology xxx (2014) xxx–xxx 0
0
10
Orignal OFDM traditional SLM(u=128) Proposed SLM(u=128) traditional SLM(u=16) Proposed SLM(u=16)
-1
-1
10
-2
10
-2
10
-3
10
-4
-3
10
10
3
4
5
6
7 8 PAPR0 [dB]
9
10
11
12
2
4
6
8
10
12
14
SNR
Fig. 8. The PAPR curves of original, SLM and proposed SLM (U = 128, 16).
Orignal(u=1) proposed SLM proposed SLM proposed SLM proposed SLM
Fig. 10. The BER curves of the two improved schemes in the OFDM–ROF systems.
L = 4, respectively in OFDM–ROF systems. For the purpose of comparison, we also give the BER performance obtained in original and traditional SLM. The side information is correctly received. As is shown in Fig. 10 that the BER performance of the Proposed EDSLM is almost the same as that of the original and traditional SLM. Proposed scheme can achieve lower BER than Proposed EDSLM, which means that side information can be correctly restored by the Proposed SLM scheme.
0
10
CCDF (Pr[PAPR>PAPR0])
Orignal OFDM traditional SLM Proposed ED-SLM Proposed SLM
10
BER
CCDF (Pr[PAPR>PAPR0])
10
(u=2) (u=4) (u=8) (u=16)
-1
10
6. Conclusion -2
10
-3
10
3
4
5
6
7 PAPR0 [dB]
8
9
10
11
Fig. 9. The PAPR curves of proposed SLM with different phase sequences number U.
as that of the traditional one with the phase sequences number being 128. Therefore, the Proposed SLM method can efficiently reduce the PAPR of OFDM–ROF signals. To further verify the advantages of the Proposed SLM scheme, Fig. 9 presents different PAPR curves with different phase sequences number U in the condition of 4-QAM modulation. For the traditional SLM, the bandwidth of side information increases with the increase of phase sequence number U, so does the system complexity. But the row vectors of Hadamard matrix used as phase sequences retain low complexity. From Fig. 9, the Hadamard matrix illustrates its advantages of better performance with larger phase sequence number. In brief, the simulations show that the Proposed SLM scheme is effective in reducing the PAPR of the OFDM–ROF transmission systems.
5.2. The simulation of BER In the following simulations, we assume a zero-mean complex additive white Gaussian noise (AWGN) channel with various values of Eb/N0. Fig. 10 shows BER performance of the Proposed ED-SLM and Proposed SLM scheme employing QPSK and over-sample factor
In this paper, we propose two improved schemes without sending explicit side information to reduce PAPR and BER in OFDM–ROF systems. The Proposed ED-SLM and Proposed SLM schemes have better PAPR performance than the traditional ones in the PAPR reduction. By Proposed ED-SLM scheme, we put forward a algorithm to reduce PAPR with the initialized phase sequences and also a Euclidean phase distance detection method to detect the side information at the receiver in the OFDM–ROF systems. Proposed SLM does not need to generate new phase sequences thanks to the Hadamard matrix which will not transmit and restore the explicit side information about the phase rotation by using superimposed method. The simulation results prove that the Proposed ED-SLM and Proposed SLM schemes can reduce peak power of the OFDM–ROF systems, save the bandwidth and lower systems’ complexity. At the same time, the proposed SLM scheme can obtain better BER performance than both the Proposed ED-SLM and traditional SLM scheme in the OFDM–ROF systems. Finally, results of the computer simulation show that the proposed scheme achieves an excellent performance in terms of both BER and PAPR reduction. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (No. 61310306022 and No. 61072073), ‘‘1000-elite program’’ foundation of Sichuan Province and Signal Processing Scientific Research and Innovation Team in Southwest Petroleum University (No. 2013XJZT007), and Science and technology support project in Sichuan province (No.2012FZ0021). References [1] I.B. Djordjevic, B. Vasic, Orthogonal frequency division multiplexing for highspeed optical transmission, Opt. Express 14 (9) (2006) 3767–3775.
Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007
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Please cite this article in press as: R. Luo et al., Two improved SLM methods for PAPR and BER reduction in OFDM–ROF systems, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.07.007