Scientia Iranica D (2019) 26(3), 1617{1626
Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering http://scientiairanica.sharif.edu
A new modulation scheme in polymer optical ber communications using Jacket matrix spreading
P. Miaoa;b;, L. Wub, and L. Pengb a. School of Electronic and Information Engineering, Qingdao University, Ningxia Road, Qingdao, 266071, China. b. School of Information Science and Engineering, Southeast University, 2 Sipaolou, Nanjing, 210096, China.
Received 2 March 2015; received in revised form 7 May 2017; accepted 17 June 2017
KEYWORDS Abstract. In this study, a novel DMT scheme based on Jacket matrix Spreading (JS) is proposed to mitigate the high Peak-to-Average Power Ratio (PAPR) of Discrete Multi- Polymer optical ber; Tone (DMT) modulation in Polymer Optical Fibers (POF) which serve communication Peak to average power systems. Approximations of PAPR distribution for DMT signals are presented accurately. ratio; Single-band JS-DMT and Multi-Band (MB)-JS-DMT are introduced in POF transmission Modulation scheme; link for PAPR reduction. Oine processing and simulations are adopted to investigate Jacket matrix the performance of these new schemes with regard to the PAPR reduction, Power Spectral spreading. Density (PSD), data rate, and Bit Error Rate (BER). The results demonstrate that, by applying the JS methodology, up to 4 dB PAPR reduction is achieved. JS-DMT is sensitive to the bandwidth range, whereas MB-JS-DMT owns the robustness ability against ber nonlinearity. Compared with some well-known PAPR reduction techniques, the proposed schemes can be easily applied to the DMT modem and achieve the PAPR reduction eciently without degrading the BER, data rate, and PSD performance, demonstrating the feasibility and validity of these two methods in POF transmission. © 2019 Sharif University of Technology. All rights reserved.
1. Introduction fading, is regarded to be future-proof solutions for max- imizing the transmission rate in SI-POF system [8,9]. Large core diameter Step-Index Polymer Optical Fibers Therefore, employing the DMT with bit-loading tech- (SI-POFs) are proposed as a preferred alternative niques makes the Gigabit transmission over SI-POF transmission medium for high-speed short-range data available. However, there are still some challenging communications and have attracted signi cant research design issues. One of the major problems is high interest [1,2]. Most recent works have already shown Peak-to-Average Power Ratio (PAPR) of the DMT experimentally that adopting spectrally ecient mod- signals [10]. When the DMT signals with high PAPR ulation techniques within Intensity Modulation with are being transmitted over the POFs, distortion occurs Direct Detection (IM/DD) structures could overcome and nonlinear e ects increase, resulting in degradation the bandwidth limitation of the SI-POF and extend the of the overall performance. Therefore, adopting an system capacity [3]. With the low-cost consideration, appropriate method for reducing the PAPR should Discrete Multi-Tone (DMT) modulation [4-7], demon- receive the utmost degree of attention. strating strong ability of anti-multipath and anti- Seeking a solution to reduce the high PAPR, extensive researches have been carried out and various *. Corresponding author. Tel.: +86-182-6623-6416 PAPR reduction approaches have been proposed in E-mail address: [email protected] (P. Miao) wireless communication systems [11,12]. However, some of these common methods may not be directly doi: 10.24200/sci.2017.4382 used in POF systems. Clipping, ltering, and com- 1618 P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626 panding are types of famous signal distortion schemes 1) based on the Quadrature Amplitude Modulation that directly limit the peak envelope of the trans- (QAM) constellation, and then the DMT signals are mitting signal to a desired value [13-15]. However, illustrated as follows [4-7]: the signal processing leads to in-band distortions and 2N 1 out-of-band high frequency components, which will 1 X j2nk xk = p Cn exp ; (1) further degrade the Bit Error Rate (BER) performance 2N 2N n=0 and reduce the spectral eciency. Coding, selec- tive mapping (SLM), and Partial Transmit Sequences C2N u = Cu = Xu; (2) (PTS) are the typical signal scrambling schemes [16,17] that statistically improve the characteristics of the where xk(k = 0; 1; ; 2N 1) is a real value which PAPR distribution and provide pretty good PAPR requires 2N points of Inverse Fast Fourier Transform reduction performance. However, they have very high (IFFT) computation, Ck follows Hermitian symmetry computational complexity, and the side information, property given by Eq. (2), and C0 = CN = X0 = 0. which contains the optimal phase factors, must be We assume Xu to be statistically independent and transmitted to the receiver for the data recovery. identically distributed. As N is large enough, both Au 2 Besides, large amounts of hardware resources will be and Bu have a Gaussian distribution N (0; 2 ) due to consumed and the BER may be degraded if the SI the central limit theorem. After determining conjugate is not detected accurately. In addition, it is worth symmetry in Eq. (2), we can obtain EfCkg = 0 and 2 2 noting that the design of phase vectors for SLM and EfjCkj g = . Therefore, the real xk approaches PTS is also troublesome. Therefore, signal distortion N (0; 2). schemes and scrambling techniques are limited by the The PAPR of the transmitted signal is de ned BER performance and the computational complexity. as the ratio of the maximum peak power to the In this paper, the Jacket matrix [18-20] Spreading average power during a symbol period, expressed as (JS) in terms of JS-DMT and Multi-Band (MB)-JS- follows [21,22]: DMT is proposed in POF system. The main goal 2 2 is to focus on the application of JS technique in max jxkj max jxkj PAPR , 0k2N 1 = 0k2N 1 : DMT modem for PAPR reduction. Extensive results d 2 2 E fjxkj g and discussions show that these proposed schemes are (3) able to achieve a better compromise with regard to According to the central limit theorem, we can obtain: the PAPR reduction, BER, data rate and spectral 2 eciency. jxkj p xk p P r = P r r The remainder of this paper is organized as r 2 r follows. Section 2 outlines the PAPR of DMT and r investigates its statistical distributions. Section 3 de- r erf ; (4) scribes the correlation property between DMT symbols 2 and peak powers. In Section 4, two kinds of DMT p R where erf( ) = 2= e t2 dt, P () is the probabil- modem based on single-band and multi-band spreading 0 r are proposed in POF transmission. System perfor- ity, and r is the threshold. Generally, the Complemen- mance comparisons regarding the PAPR reduction, tary Cumulative Distribution Function (CCDF), which BER and Power Spectral Density (PSD) are illustrated denotes the probability of the PAPR that exceeds a and discussed in Section 5. Section 6 presents the given threshold within one symbol duration, is used to conclusion. evaluate the performance in PAPR reduction. There- Unless otherwise mentioned, lower and upper case fore, the CCDF can be calculated as follows: bold-face letters denote time-domain and frequency- r 2N r domain vectors, respectively, e.g., x and X. xn denotes CCDF = P fPAPR > rg = 1 erf : d r d 2 the (n+1)th element of column vector x. Let Efg,()T , (5) (), j j, Rfg, and Z() denote the ensemble average, transpose, complex conjugate, absolute, real compo- Continuous-time base-band DMT signals can be ap- nent, and symmetric conjugate operator, respectively. proximately represented by L0 times oversampled dis- Let N (; 2) be the Gaussian distribution with mean crete time DMT signals. Thus, the IFFT outputs are and variance 2. illustrated as follows: 1 2NLX0 1 j2nk x = p C0 exp 0 ; 2. PAPR problem and statistical distribution k0 2NL n 2NL 0 n=0 0 There are N sub-carriers for a DMT symbol; the bit k0 = 0; 1; ; 2NL0 1: (6) stream is mapped onto Xu = Au +jBu (u = 1; ;N P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626 1619
0 For 0 n (L0 1)N 1, Cn is presented as follows: ( 0 Xn; 0 n N 1 Cn = (7) 0;N n (L0 1)N 1
0 For (L0 1)N + 1 n 2L0N 1, Cn also follows the Hermitian symmetry property, and C0 = C0 = 0 (L0 1)N 0. Therefore, as N is suciently large, the real value ; k = 0; 1; ; 2NL 1g converges in distribution fxk0 0 0 to fx(t); 0 t T g, where T = f 1 is the symbol period and fs = 2NL0f is the sampling rate. Thus, the PAPR of x(t) can be written as follows:
max x2(t) max x2(t) 0tT 0tT PAPRc , = : (8) Efjx(t)j2g 2
x(t) Obviously, variable (t) = converges to stationary N (0; 1). According to the extreme value theory, the corresponding CCDF can be calculated by: 2N r CCDFc =PrfPAPRc > rg=1 exp p e 2 : 3 (9)
Therefore, stochastic characteristics of the PAPR for discrete-time xk and continuous-time x(t) can be esti- mated by Eqs. (5) and (9), respectively. It is shown in [11] that choosing L0 4 is sucient to approximate the peak value of x(t). There- fore, the sequences of xk and xk0 are obtained by 2N point and 16N point IFFT computations, respectively. Considering N = 64, 128, 256, and 512, the theoretical Figure 1. Comparison of approximations with approximations and computer simulations of PAPR in simulations of PAPR: (a) Discrete time and (b) continuous terms of CCDFd and CCDFc are shown in Figure 1, time. where the solid lines denote the simulation results and 2NP 1 q the dash lines represent the approximations. It is De ning C (q) = Cp+qCp , is the aperiodic clearly manifested that the approximations agree well p=0 with the simulations and can predict the distribution of autocorrelation function (ACF) of Ck. For an arbitrary the PAPR for discrete-time and continuous-time DMT complex Xk, the following inequality holds: RfXkg PN 1 PN 1 accurately. In addition, as N increases, the PAPR jXkj and j k=0 Xkj k=0 jXkj. Thus, the PAPR problem deteriorates; thus, it should receive greater with unit mean power of 2 = 1 satis es: more attention. max jx j2 k 1 1 2XN 1 PAPR = 0k2N 1 + j (q)j 3. Correlation property 2 2 N C q=1 Taking the example of the discrete-time DMT signal, 1 1 the power of x can be calculated by: = + ; (11) k 2 N C
2N 1 2N 1 P 1 X X j2(p q)k 1 where = 2N 1 j (q)j. When input symbols jx j2 = C C exp = C q=1 C k p q 2 2N 2N 2 Cn, are equal to each other, max fjxkj g within p=0 q=0 0k2N 1 one symbol duration could reach the maximum value. ( ) 2XN 1 2NX 1 q Therefore, there is a close relationship between the 1 j2qk + R exp Cp+qC : N 2N p PAPR and the correlation property of input Cn. If q=1 p=0 (10) an appropriate method is performed to make input Cn 1620 P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626
odr with lower C , then the transmitted signal with lower where [odr] = 2 denotes the matrix order. 2 max fjxkj g will be obtained, and the probability 0k2N 1 4.2. Single-band JS-DMT modem of the PAPR that exceeds a given threshold within one The block diagram describing the proposed single-band symbol duration is reduced, meaning that the PAPR JS-DMT transceiver is illustrated in Figure 2 where reduction can be achieved consequently. Based on this the input sequence is transformed by the Jacket matrix principle, the novel DMT scheme will be proposed in before the IFFT. Compared to the Conventional DMT the following to lower C , thus reducing the PAPR (C-DMT), an extra JS precoding stage and an extra indirectly. inverse JS module emerged at the transmitter and the receiver, respectively. 4. The proposed scheme Let X be the original input vector and C = [X; Z(X)]T follow Hermitian symmetry property. As- 4.1. Jacket matrix suming that J signi es an N N Jacket matrix, the Derived and extended from the weighted Hadamard new transformed vector is given by: matrix, Jacket matrix is a class of matrix that is simple to calculate, easily element-wise invertible, and size- S = JX: (16) exible [18-20]. It is widely used in data compression, images processing, and orthogonal space-time coding. T Cnew = [S; Z(S)] is fed into IFFT module to yield The entries of Hadamard matrix are +1 or 1, whereas the new transmitted signals. Therefore, the aperiodic those of Jacket matrix are 1 and i. Assuming that ACF of C and Cnew which satis es: J = (Jm;l)N signi es an N N Jacket matrix, the inverse matrix can be calculated by: Cnew < C: (17) 1 T J+ = J 1 ; (12) A detailed proof is given in the Appendix. According N m;l N to Eq. (11), we can nd that the PAPR of the new where Jm;l is an element of J in the mth row and the transmitted signals will be reduced eciently. lth column. For example: 2 3 4.3. Multi-band JS-DMT modem 1 1 1 1 In fact, if the used bandwidth of DMT exceeds the 61 i i 17 J = 6 7 ; (13) optimum range, the performance of JS-DMT is sensi- [2] 41 i i 15 tive to non-linearity distortion and, thus, requires a 1 1 1 1 robust equalizer at the receiver, which will increase + the complexity of the whole system. In addition, is a Jacket matrix; thus, J[2] can be easily calculated by: the PAPR may be rapidly modi ed so that the low 2 3 PAPR advantage at the transmitter will be negated and 1 1 1 1 the PAPR reduction might be ine ective. Therefore, 1 61 i i 17 J+ = 6 7 : (14) an improved MB-JS-DMT scheme with split symbol- [2] 4 41 i i 15 spread blocks is proposed to strengthen the robustness 1 1 1 1 against ber nonlinearity. Figure 3 demonstrates the Furthermore, according to the Sylvester construction in conceptual diagram of MB-JS-DMT modulator with terms of Kronecker product, the higher order of Jacket sub-band mapping. The entire spectrum of DMT is matrix can be recursively expressed by: split into L bands with M sub-carriers per sub-band, which can be assigned with di erent modulation orders J[odr] = J[odr 1] J[1]; (15) based on the sub-channel SNR.
Figure 2. The diagram of DMT modem based on Jacket matrix spreading. P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626 1621
Figure 3. Block diagram of an MB-JS-DMT modulator.
Assuming that there are N = M L sub-carriers, the time-domain MB-JS-DMT signals y, represented l Xm(l = 0; ;L 1; m = 0; ;M 1) denotes the as follows: (m + 1)th data symbol in the (l + 1)th band. Contrary 8 9 l > > to the single-band JS-DMT, original symbol Xm is < = precoded by an M M Jacket matrix rstly, and a y = IFFT C ; 0; ; 0 ; Z(C ) : (20) > MB | {z } MB > new symbols is generated by: : ; 2(L0 1)N MX 1 The demodulation process, designed as in Figure 4, is l l 0 Sk0 = XmJm;k0 ; k = 0; ;M 1: (18) the reverse of Eqs. (18)-(20). Additionally, a simple m=0 zero forcing equalizer and a channel estimator are also l adopted on the receiver side. Sk0 is mapped onto: 8 < 5. Results and discussions C = C0(0); ;C1(M); ; ; MB :| {z } | {z } To evaluate the performance of these proposed trans- 1th 2th mission schemes over IM/DD SI-POF link, extensive 9 theoretical simulations and experimental tests are con- = ducted. The setup diagram, based on oine processing CL 1((L 1)M); ; | {z }; in Matlab/Simulink, is depicted in Figure 5. The low- Lth cost commercially available components, such as Res- onant Cavity Light-Emitting Diodes (RC-LED) and l where C (k) 2 CMB is the complex value according PIN diode, with a Trans-Impedance Ampli er (TIA) to the sub-band constellation mapping, expressed as are used. A standard 50 m Polymethyl Metacrylate follows: (PMMA) SI-POF (150 dB/km@650 nm) is employed. ( DMT signals are pre-generated by computer and Sl ; k =k0 +M l Cl(k)= k0 k =0; ;N 1: stored in the Arbitrary Waveform Generator (AWG) 0; others (19) memory. The analogue electrical signal of the AWG output is then used to drive a 650 nm RC-LED with In addition, the input vector is obtained after Hermi- a 10 mA bias current. After standard 50 m SI-POF tian symmetry and fed to the IFFT block to generate transmission, the received optical signal is detected
Figure 4. Block diagram of an MB-JS-DMT receiver. 1622 P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626
Figure 5. The experimental setup with oine processing.
Figure 7. The CCDF comparisons among JS-DMT, C-DMT, and PTS.
Figure 6. Amplitude comparisons between JS-DMT and C-DMT. Figure 8. PDF comparisons between JS-DMT and Table 1. The modulation parameters. C-DMT, in terms of Cnew and C. DMT Parameters Sampling frequency (GHz) 1 tudes are compressed, resulting in a signi cant PAPR reduction. Figure 7 shows the CCDF performance of IFFT/FFT 2048 the JS-DMT. In addition, the well-known PTS is also Subcarrier spacing (MHz) 0.488 implemented [23]. The parameters of PTS are chosen Subcarrier number, N 1 1024 as follows: U = 16, V = 4, and W = 16 (U, V , and W Cyclic Pre x (CP) ratio 1/64 denote the number of modi ed candidate sequences, disjoint sub-vectors, and phase angles, respectively). by a PIN diode and captured by ADC components. Obviously, the probability of the PAPR in C-DMT, 4 Finally, the digital signal processing is implemented being larger than 14 dB, is nearly 10 . After JS oine in Matlab. The DMT modulation parameters precoding, the JS-DMT owns a much lower PAPR and are depicted in Table 1. According to the received almost 4 dB improvement is achieved. The PAPR probing symbols, the estimated channel information is performance of PTS is better than that of C-DMT, obtained and the normalized noise PSD is measured yet inferior to that of JS-DMT. However, the side around (106 114) dB/Hz. information of PTS must be accurately known by the receiver; otherwise, the overall BER performance will 5.1. Single-band DMT be degraded. Furthermore, searching for optimal phase To assess the PAPR reduction in a single-band JS- factors of PTS will cost more hardware resources and DMT modem, N = 512 in Table 1 is selected. In have higher computational complexity. our experiment, 10000 DMT symbols are randomly The ranges of [min C; max C] and [min Cnew ; generated by a computer. Figure 6 shows the com- max Cnew ] are divided into 99 equidistant intervals. parison result of amplitudes of JS-DMT and C-DMT. Then, Probability Density Functions (PDFs) of C- It is clearly manifested that the energy distribution of DMT and JS-DMT, with respect to C and Cnew , JS-DMT becomes more uniform and the large ampli- are calculated and exhibited in Figure 8. The curves' P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626 1623
Figure 9. The PSD comparisons between JS-DMT and C-DMT. Figure 10. The PAPR performance of di erent schemes. shapes of PDF are alike, and has an approximate g become Gaussian distribution. Cnew and EfCnew smaller, implying that more PAPR reduction in JS- DMT is achieved. Figure 9 depicts the comparison of the PSDs' performances. According to the gure, the rectangular-like and very sharp characteristics of DMT power spectrum remain retained and the sideband lev- els are also reduced, indicating that JS transformation does not damage the signal's PSD and the spectral eciency.
5.2. Multi-band JS-DMT Before investigating the MB-JS-DMT performance, sub-band number, L, which might a ect the trans- mission performance, should be considered rstly to adopt the optimal used bandwidth. The performances of these four schemes, such as C-DMT, JS-DMT, 4- Figure 11. Transmission rate of di erent schemes. band-JS-DMT (abbreviated as `4-band'), and 8-band- JS-DMT (abbreviated as `8-band'), are compared. For implying that the capacity of JS-DMT will be severely a fair comparison, all four schemes own the xed input degraded once the used bandwidth beyond this optimal power and the same modulation parameters, which are range. However, fortunately, based on the gure, depicted in Table 1. In addition, the BER target is the MB-JS-DMT scheme owns the robustness ability 3 set to 1 10 which is convenient for reception with against ber nonlinearity, and their optimal bandwidth Forward Error Correction (FEC). Sub-carrier number range is large. In addition, 4-band and 8-band achieve N varies from 1 to 1024 with the xed sub-carrier similar data rate and optimal used bandwidth. In order spacing f = 0:488 MHz. Besides, the simplest Chow to make a good trade-o between transmission rate and algorithm [24] is also employed due to its practicality PAPR performance, 4-band is usually chosen in the and e ectiveness in POF transmission. MB-JS-DMT since the ecient PAPR reduction can Figure 10 shows CCDF performances of these bene t from the SNR gains of the system. four schemes. The C-DMT owns the highest PAPR, whereas the JS-DMT obtains the lowest. Considering 5.3. BER performance MB-JS-DMT, the less the band split is, the lower the The BER performances of JS-DMT, MB-JS-DMT (L = PAPR will be. Figure 11 shows the total achievable 4), clipping, and PTS are investigated in this section. transmission rate of each system with respect to dif- The DMT signal of these four schemes owns the same ferent used bandwidths. Accordingly, each scheme can input power and the same modulation parameters, reach the maximal transmission rate with its optimal which are depicted in Table 1. The Chow algorithm [24] used bandwidth. However, the transmission rate of JS- is adopted with the objective BER of 1 10 3; in DMT is sensitive to the bandwidth and its optimal addition, the used sub-carrier of N = 512 and the range is limited only form 150 MHz to 200 MHz, desired transmission rate of 1.2 Gbps are selected. 1624 P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626
which is bene cial to high-speed POF transmissions based on the performance investigation of the CCDF, BER, data rate, and PSD.
6. Conclusions This paper discussed and investigated the PAPR prob- lem in DMT modulator of the SI-POF system. The Jacket matrix spreading technique was introduced, and the novel modulation schemes, in terms of JS-DMT and MB-JS-DMT, were proposed for the PAPR reduction. By means of the new methodologies, the DMT signal with lower PAPR was produced. Extensive results and discussions manifested that JS precoding could reduce the PAPR eciently while having no adverse impacts on PSD and BER performances. In addition, the MB- Figure 12. BER performance of di erent schemes. JS-DMT scheme was more ecient for non-linearity mitigation by investigating the relationship between After IFFT computations, the used bandwidths of the transmission rates and used bandwidth. Both DMT symbols are measured around 250 MHz. The theoretical and experimental results demonstrated that crest factor (CR) of clipping is chosen by the typical the proposed JS modem was able to o er a favorable value 2.5. Figure 12 manifests the measured noise trade-o among the PAPR reduction, BER, data rate, PSD against BER performance for various transmission and spectral eciency, presenting a fact that the novel schemes. transmission schemes are practical and e ective for The gure clearly indicates that JS-DMT o ers future POF communications. the worst BER performance than the other schemes, because the bandwidth of 250 MHz is beyond its optimum range. Clipping scheme is the simplest way Acknowledgments to reduce the PAPR, whereas the total performance is This work is supported in part by the Qingdao Post- limited by CR; besides, a higher PAPR reduction leads doctoral Applied Research Project. to a higher BER. PTS owns better BER performance The authors would like to express gratitude to than clipping; however, the overall BER is limited Prof. Wu, Dr. Peng, and Dr. Chen for their constructive by side information detection. In the presence of suggestions and assistance to this article. We thank Dr. side information detection failure, the demodulator Peng for the assistance with equipment. cannot recover the original signal correctly; then, the BER performance is degraded. Moreover, multiple IFFT computations are required, and many operations References of complex addition and multiplication have to be 1. Okonkwo, C.M., Tangdiongga, E., Yang, H., Visani, performed, which would cost much higher hardware D., Loquai, S., Kruglov, R., and Gaudino, R.C. \Re- resources. cent results from the EU POF-PLUS project: Multi- Comparing the clipping and PTS at BER of 10 4, gigabit transmission over 1 mm core diameter plastic nearly 6.6 dB and 4.7 dB of power improvements optical bers", J. Lightwave Technol., 29(2), pp. 186- are achieved by the proposed MB-JS-DMT schemes, 193 (2011). respectively. JS precoding is the linear transformation 2. Nespola, A., Abrate, S., Gaudino, R., Zerna, C., O en- without signal distortion at the transmitter, and the beck, B., and Weber, N. \High-speed communications better BER performance is then obtained. In addition, over polymer optical bers for in-building cabling and only a very simple inverse Jacket matrix is employed home networking", IEEE Photonics J., 2(3), pp. 347- for decoding and no side information is required in the 358 (2010). proposed MB-JS-DMT scheme. What's more, not only 3. Randel, S., Breyer, F., Lee, S.C., and Walewski, J.W. did they have much lower computational complexity, \Advanced modulation schemes for short-range optical but also they own the best BER performance and o er communications", IEEE J. Sel. Top. Quant, 16(5), pp. a better PAPR reduction, which is a very suitable 1280-1289 (2010). trade-o . 4. Peng, L., Helard,M., and Haese, S. \On bit-loading for In summary, the results demonstrate that MB-JS- discrete multi-tone transmission over short range POF DMT are kinds of potential solution for the PAPR re- systems", J. Lightwave Technol., 31(24), pp. 4155- duction and can be easily applied to the DMT modem, 4165 (2013). P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626 1625
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EfSug = EfJuXg = JuEfXg = 0; (A.1) 13. Nadal, L., Moreolo, M.S., Fabrega,J.M., and Junyent, G. \Low complexity PAPR reduction techniques for where Ju denotes the uth row of J. The variance of S clipping and quantization noise mitigation in direct- can be calculated as follows: detection O-OFDM systems", Optical Fiber Technol., 2 H H H 2 20(3), pp. 208-216 (2014). S = EfSS g = JEfXX gJ = IN; (A.2)
where IN is the identity matrix. S and X are random 14. Armstrong, J. \Peak-to-average power reduction for variables; thus, we have: OFDM by repeated clipping and frequency domain ( ) ltering", Electron. Lett., 38(5), pp. 246-247 (2002). NX 1 NX 1 EfSg = E jS(q)j = EfjS(q)jg; (A.3) 15. Jiang, T., Yao, W., Guo, P., Song, Y., and Qu, q=1 q=1 D. \Two novel nonlinear companding schemes with iterative receiver to reduce PAPR in multi-carrier NX 1 modulation systems", IEEE Trans. Broadcast., 52(2), EfXg = EfjX(q)jg: (A.4) pp. 268-273 (2006). q=1
16. Wang, C.L. and Ouyang, Y. \Low-complexity selected For q = N (1 N), we can calculate: mapping schemes for peak-to-average power ratio Efj (q)j2g reduction in OFDM systems", IEEE Trans. Signal S Proces., 53(12), pp. 4652-4660 (2005). ( ! ! ) X 1 X 1 = E S S S S : 17. Alsusa, E. and Yang, L. \Redundancy-free and BER- k+N k k+N k maintained selective mapping with partial phase- k=0 k=0 (A.5) 1626 P. Miao et al./Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1617{1626
Owing to SN 1; ;SN , and S0 ; ;S 1 are mu- EfSg < EfXg: (A.14) tually independent, and Eq. (A.5) can be simpli ed as and , respecting follows: Finally, the mean values of Cnew C C = [S; Z(S)]T and C = [X; Z(X)]T , have the 2 2 2 new E jS(q)j = E jSN S0 j + E jSN +1S1 j following relation: 2 + + E jSN 1S j : (A.6) 1 EfCnew g < EfCg: (A.15) 2 Accordingly, the similar expression of EfjX(q)j g can approaches an approximate As N is large enough, Cnew be derived as follows: Gaussian distribution. The reduction of the mean value 2 2 implies a lower C . Therefore, it is inferred that the E jX(q)j = E jXN X0 j new lower C is, the better the PAPR performance will new 2 be. +E jXN +1X1 j 2 + + E jXN 1X 1j : (A.7) Biographies For the special case = 1, on the one hand, we have: Pu Miao received his BS and MS degrees from Zhengzhou University and Henan University of Science EfjS(N 1)jg = E fjSN 1S0 jg and Technology in 2008 and 2011, respectively, and the PhD degree in Information and Communication =E fjSN 1j jS0jg=E fjJN 1Xj jJ0Xjg : (A.8) Engineering from School of Information Science and Engineering, Southeast University, Nanjing, China By the Cauchy-Schwartz inequality, Eq. (A.8) can be in 2015. He is currently an Assistant Professor in further arrived at: the School of Electronic and Information Engineering, Qingdao University, Qingdao, China. His current EfjJN 1Xj jJ0Xjg research interests include signal processing, coding, r n o channel estimation and multicarrier modulation tech- 2 2 < E fjJN 1Xj g E jJ0Xj niques applied to wireless communications and optical communications. q = J E fXXH g J H J E fXXH g J H N 1 N 1 0 0 Lenan Wu received the PhD degree in Signal and In- formation Processing from Southeast University, Nan- 2 = : (A.9) jing, China, in 1997. He is currently a Full Professor in Institute of Multimedia Technology, School of Infor- On the other hand, we have: mation Science and Engineering, Southeast University, E fj (N 1)jg = EfjX j jX jg = 2: (A.10) Nanjing, China. He has authored or coauthored over X N 1 0 100 technical papers in major journals and conferences Therefore, we can obtain: in the areas of signal processing and communica- tions. His current research interests include the areas E fjSN 1S0 jg < E fjXN 1X0 jg : (A.11) of wireless communications and corresponding signal processing, especially for high eciency modulation Similarly, for = 2, we will arrive at: technique. E fjS Sjg + E fjS Sjg N 2 0 N 1 1 Linning Peng received his BS and MS degrees in 2006 < E fjS Sjg and 2010, respectively, from the School of Information N 2 0 Science and Engineering, Southeast University, China, + E fjS Sjg : (A.12) and the PhD degree in 2014 from the Electronics N 1 1 and Telecommunications Institute of Rennes (IETR) For = 3; 4; ;N, the similar inequality relationships laboratory at the National Institute of Applied Sci- as Eqs. (A.11) and (A.12) are achieved. Thus, we can ences (INSA) of Rennes, France. In 2014, he joined deduce that: Southeast University where he is currently an Associate n o n o Professor in the School of Electronic and Information 2 2 E jS(q)j < E jX(q)j ; (A.13) Engineering. His current research interests focus on physical layer security, optical communications, and and: indoor optical networking.