Branch and bound methods based tone injection schemes for PAPR reduction of DCO-OFDM visible light communications

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1 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 595 Branc and bound metods based tone injection scemes for PAPR reduction of DCO-OFDM visible ligt communications YONGQIANG HEI,,JIAO LIU, WENTAO LI, XIAOCHUAN XU,3 AND RAY T. CHEN3 State Key Laboratory of Integrated Service Networ, Department of Communication Engineering, Xidian University, Xi an 7007, Cina Science and Tecnology on Antenna and Microwave Laboratory, Department of Electronic Engineering, Xidian University, Xi an 7007, Cina 3 Microelectronics Researc Center, Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX USA yqei@utexas.edu Abstract: In tis paper, te pea-to-average power ratio (PAPR) reduction problem wit tone injection (TI) in te direct current-biased optical ortogonal frequency division multiplexing (DCO-OFDM) system is investigated, wic is formulated as a toug integer combinatorial optimization problem. Since it is a callenging tas to find te global optimal solution, te branc-and-bound metod (BBM), wic is extensively employed to deal wit te NP-ard problem, is introduced to solve tis problem. By splitting te superior brances and pruning te inferior brances, te close optimal solution is obtained. Simulation results reveal tat te proposed BBM-based TI metod as superior PAPR reduction and bit error rate (BER) performance compared wit some existing algoritms. Te proposed algoritm ensures a relatively low pea value, and tus provides an important bencmar in performance evaluations relative to oter existing algoritms targeting at te same problem. 07 Optical Society of America OCIS codes: ( ) Fiber optics and optical communications; ( ) Optical communications. References and lins. S. Rajagopal, R. Roberts, and S. K. Lim, IEEE visible ligt communication: Modulation scemes and dimming support, IEEE Commun. Mag. 50(3), 7 8 (0).. S. Hranilovic, On te design of bandwidt efficient signaling for indoor wireless optical cannels, Int. J. Commun. Syst. 8(3), 05 8 (005). 3. J. Armstrong, OFDM for optical communications, J. Ligtwave Tecnol. 7(3), (009). 4. Z. Yu, R. J. Baxley, and G. T. Zou, EVM and acievable data rate analysis of clipped OFDM signals in visible ligt communication, EURASIP J. Wirel. Commun. Netw. 0(), 6 (03). 5. H. Elgala, R. Mesle, and H. Haas, A study of LED nonlinearity effectson optical wireless transmission using OFDM, in Proc. Wireless Opt. Commun. Netw. (WOCN), 5(009). 6. Z. H. Yu, R. J. Baxley, and G. T. Zou, Iterative clipping for PAPR reduction in visible ligt OFDM communications, Military Communications Conference (IEEE,04), pp K. Bandara, P. Niroopan, and Y.-H. Cung, PAPR reduced OFDM visible ligt communication using exponential nonlinear companding, International Conference on Microwaves, Communications, Antennas and Electronic Systems (IEEE, 03), pp W. O. Popoola, Z. Gassemlooy, and B. G. Stewart, Pilot-assisted PAPR reduction tecnique for optical OFDM communication systems, J. Ligtwave Tecnol. 3(7), (04). 9. H. Zang, Y. Yuan, and W. Xu, PAPR reduction for DCO-OFDM visible ligt communications via semidefinite relaxation, IEEE Potonics Tecnol. Lett. 6(7), 78 7 (04). 0. N. Jaclin and Z. Ding, A linear programming based tone injection algoritm for PAPR reduction of OFDM and linearly precoded systems, IEEE Trans. Circ. Syst. 60(7), (03).. K. Cumanan, R. Krisna, L. Musavian, and S. Lambotaran, Joint beamforming and user maximization tecniques for cognitive radio networs based on branc and bound metod, IEEE Trans. Wirel. Commun. 9(0), (00).. T. Ding, R. Bo, F. Li, and H. Sun, A bi-level branc and bound metod for economic dispatc wit disjoint proibited zones considering networ losses, IEEE Trans. Power Syst. 30(6), (05). #79859 Journal 07 ttp://dx.doi.org/0.364/oe Received Nov 06; revised 30 Dec 06; accepted 3 Dec 06; publised 9 Jan 07

2 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS T. Jiang and Y. Y. Wu, An overview: pea-to-average power ratio reduction tecniques for OFDM signals, IEEE Trans. Broadcast 54(), (008). 4. J. C. Cen and C. K. Wen, PAPR reduction of OFDM Signal using cross-entropy-based tone injection scemes, IEEE Signal Process. Lett. 7(8), (00). 5. I. W. H. Tsang and J. T. Y. Kwo, Efficient yperernel learning using second-order cone programming, IEEE Trans. Neural Netw. 7(), (006). 6. Y. Nesterov and A. Nemirovsii, Interior-point polynomial algoritms in convex programming, Soc. Indust. Appl. Mat. Rev. 36(4), (994). 7. S. Boyd and L. Vandenberge, Convex Optimization (Cambridge, U.K.: Cambridge Univ. Press, 004).. Introduction Owing to te advantages of uge bandwidt, unlicensed frequency, immune to electromagnetic interference and ig security, visible ligt communications (VLC) system as attracted considerable attention recently. Unlie radio frequency (RF) communications, VLC wit wite ligt emitting diodes (LEDs) can provide bot illumination and communication simultaneously, and as te merit of ig efficiency, long life, low cost and power consumption. Hence, it is considered as one promising enabling tecnology for future wireless communications. VLC can be implemented by using on-off Keying (OOK), variable pulse position modulation (VPPM) or pulse amplitude modulation (PAM) []. However, tese modulation tecniques can be easily affected by te inter symbol interference (ISI). Nowadays ortogonal frequency division multiplexing (OFDM) is more favored in VLC [ 4] system because it cannot only combat te ISI induced by multi-pat, but also enance te spectrum efficiency. One possible metod to introduce OFDM into te VLC system is te DC-biased optical OFDM (DCO-OFDM) system, wic guarantees te positive real inputs by adding te DC bias. However, one intrinsic issue in te DCO-OFDM system is its ig pea-to-average power ratio (PAPR). Te signals wit ig PAPR passing troug power amplifier and ligt emitting diode (LED) [5] emitter can result in a nonlinear distortion. Furtermore, te ig signal pea value requires a large DC bias in DCO-OFDM system, wic causes serious degradation of system power efficiency. Tus, PAPR reduction in VLC OFDM systems is very crucial. Due to te difference in te transmitted signals, traditional PAPR reduction metods in RF-OFDM systems cannot be directly applied in VLC OFDM systems. In [6], an iterative clipping metod ensuring no in-band distortion is proposed to reduce PAPR in VLC OFDM systems. An exponential nonlinear companding tecnique is proposed in [7] for VLC OFDM systems, wic reduces te PAPR by compressing large signals and expanding te small signals simultaneously. Neverteless, bot metods mentioned above can result in te BER performance degradation caused by te nonlinear distortion of signals. Additionally, a pilot-assisted tecnique for PAPR reduction based on te selected mapping (SLM) algoritm is presented for an optical OFDM system in [8], wic is demonstrated to be better tan conventional SLM metods in terms of PAPR reduction for ig order constellations. Unfortunately, te pilot-assisted metod probably results in data rate loss because of te expense of te pilot. In RF-OFDM systems, te tone injection (TI) tecnique is regarded as a distortionless tecnique wic can reduce PAPR significantly witout transmission of any side information and data rate loss. An optimized TI strategy wit semidefinite relaxation (SR-TI) is proposed in [9] to reduce PAPR for DCO-OFDM system, in wic te final solution for te relaxed PAPR reduction optimization problem is generated by randomization metod based on te optimized positive semi-definite matrix. However, te optimization model after relaxation becomes a ig complexity second-order cone programing(socp). In order to furter reduce its complexity, te linear programming based TI algoritm (LP-TI) [0] is proposed for PAPR reduction. In tis paper, te PAPR reduction problem based on TI in DCO-OFDM systems is formulated as a combinatorial optimization problem. Ten, a branc-and-bound metod (BBM), wic is widely used to solve te NP-ard problem in wireless communications []

3 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 597 and acieve te close global optimal solution in power systems [], is employed to find te close optimal solution. Te proposed sceme starts wit getting a feasible solution, and ten splits some brances wit iger probability to obtain better solution and prunes te oter brances wit lower probability to obtain better solutions. Te iterative algoritm eeps splitting and pruning until te termination condition is satisfied. Finally, te close best solution is acieved and it is taen as te final optimal solution. Simulation results validate tat te proposed BBM sceme can acieve a superior PAPR reduction performance compared wit te existing TI scemes.. System model and problem formulation Consider a DCO-OFDM VLC system as sown in Fig.. Since te VLC system is based on intensity modulation and direct detection tecniques, te OFDM signal to modulate te LED must be real and unipolar. Tese requirements can be satisfied by te Hermitian symmetry property in te frequency domain and adding direct current (DC) bias to te time domain signal. At te transmitter, by using te inverse Fast Fourier Transform (IFFT), a set of multiplexed subcarriers in te frequency domain is transformed to its time-domain equivalent signal. Ten, a cyclic prefix (CP) is added and te resulting signal is digital to analog converted (DAC) to obtain a bipolar time-domain real-valued OFDM signal. After adding a DC bias to te time domain signal, te unipolar signal drives te LED to converts te electrical signals to te optical signals. At receiver, te optical signals are first converted to te electrical signals by potodiode. Ten a reverse process can be implemented to demodulate te data. Fig.. Te scematic diagram of te DCO-OFDM VLC system. N Denote { } / X 0 Ten te discrete-frequency signals [ ] constraint: as te original input data, were N is te number of te subcarriers. = X= X0, X,, XN sould satisfy te following X X =,,, N =, XN = N +,, N were X0 = X N = 0 and () represents te conjugation operation. Ten, te equivalent time-domain OFDM baseband signal over-sampled L-times is expressed as: N jπ n NL xn = Xe, n = 0,,,, LN. () N = 0 Te PAPR of te OFDM signal is given by: ()

4 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 598 max xn 0 n LN PAPR = 0 log 0, E xn were E[ ] denotes te expected value operation. Te basic concept of te tone injection (TI) in reducing te PAPR is to expand te original M-ary quadrature amplitude modulation (QAM) constellation by mapping eac point into several different constellation points [3]. For convenience, te expanded constellation in [4] is adopted in tis paper and it maps one constellation point to te outer ring of te original constellation. For better understanding, one example of te extended 6-QAM constellation is sown in Fig., in wic te constellations inside te dased box are te original ones, wile te constellations outside te dased box are te extension ones of te original constellations. Due to te extended constellation points do not overlap wit te original positions, tis TI sceme does not require any side information. Hence, te received symbol Y ˆ after fast Fourier Transform (FFT) can be directly decoded in accordance wit te extended constellation. After applying TI in te VLC OFDM system, te signal can be furter written as: X = ( ω ) X + ω C, (4) (3) C '' ' ' ( d ) a j( d ) M, ( a > M, b = M ) '' ' ' ( d ) a + j( d ) M, ( a < M, b = M ) '' ' ' ' ( d ) M j( d ) b,( a = M, M < b < M ) =, '' ' ' ' ( d ) M j( d ) b,( a = M, M < b < M ) '' ' ' ( d ) a j( d ) M, ( a = M, b = M ) '' ' ' ( d ) a + j( d ) M, ( a = M, b = M ) (5) were d is te minimum distance between two adjacent points, a and b are integer multiples of d taing values from te set { ±, ± 3,, ± M - }, i.e., ' '' X = ( d ) a + j( d ) b, M = M and M = M +. Wit te conjugate symmetry operation, te symbols assigned to OFDM subcarriers are expressed as: { X0} R, = 0 N X, =,, Y = N, I { X0}, = N XN, = +,, N were R {} and I represent {} te real part and te imaginary part respectively. (6)

5 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 599 Fig.. Te extended 6-QAM constellation diagram. Furtermore, by applying inverse fast Fourier transform (IFFT) to { Y }, te discrete-time signals over-sampled L-times can be written as: N jπ n LN yn = Ye, n = 0,,,, LN. (7) N = 0 Since tere only exists one equivalent constellation for eac original point, te PAPR optimization problem can be expressed as: were {0,} N min C ( ω) C ω sel sel ( ω) = max y ( ω) subject to n N ω {0,}, stands for a N -dimensional binary sequence. Te problem in Eq. (8) is a toug N integer optimization tas wit te complexity of O ( ). 3. Te BBM-based TI sceme for PAPR reduction In tis section, we focus on ow to address te PAPR reduction problem in Eq. (8) by BBM. Te specific details are provided below. 3. Branc To clarify te BBM metod, we first define some symbols first. Let Q j denote te j -t region generated by rounding off non-integral elements in ω to 0 or. LB( Q j ) represents te lower bound of te optimization problem. Initially, i.e. t = 0, Q0 = ω ω [ 0, ],. By solving { } (8) min ( ω) ω C sel st.. Q. 0 (9)

6 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 600 Te lower bound LB( Q 0 ) can be obtained wit interior-metod. Ten, Q0 is split into two regions Q and Q according to a non-integral element in ω (say, ω l ) at step t =. Correspondingly, two new sub-problems generated by Q and Q are expressed as follows: min C ( ω) ω sel [ ] st.. Q : ω 0,, l, ω = 0, l (0) min C ( ω) ω sel [ ] st.. Q : ω 0,, l, ω =. By addressing Eq. (0) and Eq. () wit interior-point metod, two new lower bounds LB( Q ) and LB( Q ) are obtained respectively. Ten, we select one of Q and Q as lower bound to split along an undetermined element ω ( l, te element ω l is a fixed integer and te index l is employed) for te next step. Tere are only non-integer variables after addressing te two new sub-problems. Tis spiting process will not stop until te termination criterion is satisfied. In BBM, tere are two ey issues required to be defined. One is te branc wit te smaller lower bound to split at eac step, from wic te better solution wit te igest probability can be found. Te oter is te rule to select wic element to be split at te -t step. In tis paper, it is defined as follows: = arg min ω. () {,, N} Te rule of Eq. () can be explained as follows, for eac branc, te region is split along te value wic is te closest to (most uncertain point in [0,]). 3. Bound In eac iteration, BBM requires to compute te lower bound LB( Q j ) for eac branc. From te previous subsection, we now tat LB( Q j ) can be obtained by directly solving te corresponding optimization problem wit relaxed Q j. For instance, LB( Q0 ) is obtained by relaxing Q 0 to { 0 ω, } in Eq. (9) and LB( Q ) is obtained by relaxing Q to{ 0 ω, l, ωl = 0} in Eq. (0). Besides, te feasibility of solutions to Eq. (8) can be determined by te criterion tat eac element of ω is an integer. After obtaining a feasible solution, we will prune te branc according to te following rules. Set te initial upper boundu =+. Suppose a feasible solution Q wic is obtained after te T -t iteration; If eac element of ω is an integer and LB( Q ) < U, Q becomes a candidate solution of Eq. (8) and set U = LB( Q ); If ω contains at least one non-integral element and LB( Q ) < U, it implies tat te smaller solution may be contained in tis branc. Ten find te feasible solution LB( Q m ) of tis branc; If LB( Q ) > U, prune tis branc. Tis is because tere is low probability to obtain any oter better solution included in tis branc tan te current one. l ()

7 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 60 Jointly considering branc and bound structure in BBM, te detailed procedures to optimize te PAPR problem in DCO-OFDM systems are provided by Algoritm below. Algoritm : BBM-based TI metod for PAPR reduction. Step Initialization: Set te initial upper bound U =+ and iteration indext = ; Set te initial region of brances required to be split as Q 0. If all elements of te solution in ω are integers, stop and tae ω as te final solution. Oterwise, split along te undetermined elements until a feasible solution is obtained according to te brancing operation in subsection 3.; Set U = LB( Q ),were Q represents te region of feasible solution. Step Wen te node witout branc exists, Prune te branc Q wic satisfies LB( Q) > U; Select te region wit index wic as te non-integral elements and te best lower bound to split; Determine te split index based on Eq. () for te selected region Q ; Split Q into two regions Q and T Q + based on ω. Continue to split one of Q and T Q T + T wit lower bound until a feasible solution ω is obtained. Te region of feasible solution ω is denoted as Q ; If LB( Q ) original branc; T = T +. > U, prune te branc corresponding to Q. Oterwise, set U = LB( Q ) and prune te Step 3 Tae ω as te final optimal solution corresponding to te pea valueu. 4. Simulation results To evaluate te proposed BBM-based TI sceme in terms of PAPR reduction in VLC OFDM systems, te complementary cumulative distribution function (CCDF) is used. In te simulation, we assume tat DCO-OFDM systems ave N = 8 subcarriers, in wic te transmit signals are modulated by 6-QAM and oversampled factor is set as L = 4. Classical metods, suc as te semidefinite relaxation (SR)-TI metod in [9] and te linear programming (LP)-TI metod in [0] for PAPR reduction are also simulated in te VLC OFDM system for comparison. Bot te SR-TI metod [9] and te LP-TI metod [0] metod use te extended 6-QAM constellation wit iger degrees of freedom tan tat of te BBM-based TI metod in order to testify its and effectiveness and efficiency. Additionally, te iteration number in SR-TI and LP-TI is selected as 8 since no obvious performance improvement can be observed after 8 iterations. Fig. 3. PAPR reduction comparison of different TI metods wit N = 8 and 6-QAM.

8 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 60 In Fig. 3, simulation results in terms of te PAPR CCDF for different TI metods are provided. At 0, observing tat BBM-based TI obtains about 5 db PAPR reduction over te original OFDM system. Besides, it performs better tan te SR-TI and te LP-TI metods in terms of PAPR reduction. Compared wit 4 equivalent constellation points in SR-TI and 9 equivalent constellation points in LP-TI for PAPR reduction, BBM-based TI as less tan one equivalent constellation point on average. Hence, te good PAPR reduction performance indicates te promising ability of te proposed BBM-based TI metod in dealing wit te PAPR optimization problem. To furter evaluate te BBM-based TI metod on te PAPR reduction in VLC OFDM systems, over 5000 OFDM signals are randomly generated to compare te pea power and te mean power acieved by BBM-based TI, LP-TI and SR-TI. Te corresponding results are plotted in Fig. 4 and Fig. 5, respectively. For reference, te pea and te mean power of te original OFDM signals are also provided. It is noticed tat te pea power by BBM-based TI metod is muc lower. From Fig. 5, we now tat te BBM-based TI metod consumes smaller energy compared wit te LP-TI metod and te SR-TI metod, wic terefore proves te effectiveness of te proposed BBM-based TI metod. Fig. 4. Pea power comparison of different TI metods. Fig. 5. Mean power comparison of different TI metods. In tis paper, a simplified LED nonlinear model is employed in te DCO-OFDM system wit te linear LED range of [ 0, A m ]. To assure a steady ligt intensity and maximize te modulation dept, te DC bias is set to A m. Ten te DCO-OFDM signals are expressed as:

9 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 603 Am, yn > Am yndco, = yn,0 yn Am. (3) 0, yn 0 Essentially, LED will degrade te BER performance of OFDM system due to te non-linear distortion. Hence, PAPR reduction tecnique is employed to mitigate te effect of non-linear distortion caused by LED. Figure 6 sows BER performance comparison among different TI scemes under te additive wite Gaussian noise (AWGN) cannel wit LED. Te simulation results indicate tat te BBM-based TI sceme not only obtains significant PAPR reduction, but also effectively reduces te nonlinear distortion and ence as better BER performance over LP-TI and SR-TI in AWGN cannel. Fig. 6. BER comparison of different TI metods wit N = 8 and 6-QAM. Complexity evaluation: wen taing te practical implementation into account, one important factor tat as to be considered is te complexity of te TI sceme, wic directly determines weter tis sceme can be applied or not. According to [0], te complexity of 3 linear programming problem is ONRL ( ), were N is te number of subcarriers, R denotes te total number of rotations and L is over-sampling size. In [9], te PAPR problem is formulated as a semi-definite programming (SDP) problem and ten cast as a second-order cone programing (SOCP) problem, wic is because SOCP runs far more efficiently tan SDP and as a muc better worst-case complexity. According to [5], te SOCP problem solved by 7 te interior-point metod as a complexity of O(( N + ) ) per iteration. However, if using te primal-dual metod to deal wit te SOCP problem, te accuracy of a given solution can be 3 improved by an absolute constant factor in O(( N + ) ) iterations [6]. In tat case, te total 8.5 complexity of te SOCP problem is O(( N + ) ). Hence, te semidefinite relaxation (SR) algoritm [9] is muc more complex tan te linear programming (LP) metod [0]. For te proposed BBM metod, it can be efficiently solved by te interior-point metod, wic as a 3 complexity of OC ( max(( N -),( N -) ( N-), F)) in solving te PAPR optimization problem (8), were C denotes te number of steps, N -represents te dimension of te variable ω, N- is te number of constraints and F is te cost in evaluating te first and te second derivatives of te objective and constraint functions [7]. Since (8) as only one objective function tat can be translated into real variable for eac element of ω, F can be ignored. Hence, te wole complexity of te BBM metod by using te interior-point metod 3 is OCN ( 4) per iteration. Terefore, te BBM metod owns a lower complexity tan te SR algoritm [9] per iteration. Since te interior-point metod in BBM-TI is responsible for

10 Vol. 5, No. 3 Jan 07 OPTICS EXPRESS 604 finding te local optimal solution of te PAPR optimization problem (8), te number of steps C can be muc smaller wen compared wit te case of finding a global optimization solution. More importantly, te interior-point metod in te BBM-TI metod can be replaced by oter lower complexity metods, provided tat tey can find te same local optimal solution. Tat is to say, te complexity of te proposed BBM-based TI metod can be furter reduced. In conclusion, te proposed BBM-TI metod as better PAPR reduction performance tan te SR-TI algoritm [9] and te LP-TI algoritm [0], and lower computational complexity wen compared wit te SR-TI algoritm [9]. 5. Conclusion In tis paper, an efficient BBM-based TI metod proposed for PAPR reduction in VLC DCO-OFDM systems is presented. Te PAPR reduction problem is formulated as a toug integer combinatorial optimization problem, and a widely used metod to solve NP-ard problem, i.e., branc-and-bound metod (BBM) is introduced to deal wit tis problem. Simulation results sowed tat te proposed sceme can acieve significant PAPR reduction performance and better BER performance for VLC system compared wit te existing TI scemes.