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to satisfy the increasingly higher data rate demands , visible light communication ( vlc ) has drawn tremendous interest from both industry and academia as a promising complement to traditional radio frequency communication ( rfc ) that suffers from spectrum saturation @xcite . the maturing of led manufacturing techniques during the recent decade largely boosts the trend of replacing traditional lighting systems with led alternatives for both indoor and outdoor illumination purposes , and the resulting infrastructures are ready for deployment of vlc . it is a low - cost technology where one can use the simple intensity modulation and direct detection ( im / dd ) techniques . in addition , one can enjoy a bunch of additional advantages such as eye - safety , high security and causing no electromagnetic inference . vlc is both unique from and similar to rfc . with regard to the uniqueness of vlc , it only allows positive and real signals to drive the leds as intensities ( a non - informative dc - bias is typically used ) ; its channel especially for indoor environment is much more slower varying than a rfc counterpart ; baseband waveforms modulates the leds directly instead of being up - converted first , etc . as for the similarity between vlc and rfc , many existing rfc techniques can be applied , although possibly with non - straightforward modifications to vlc systems , e.g. optical multiple input multiple output ( o - mimo ) @xcite , optical orthogonal frequency division multiplexing ( o - ofdm ) @xcite , and other advanced signal processing techniques @xcite . these ( and relevant ) works nicely take advantage of various diversities a vlc system provides , such as spatial diversity , frequency diversity , color diversity and adaptive dc - bias , to improve system performance . it is worth noting that the color diversity and adaptive dc - bias configuration are specific to vlc . the motivation behind this work is to exploit the benefits of various diversities jointly for a very power efficient vlc , while the focus of this paper is on the problem of constellation design in high dimensional space . this space is formed by several dimensions of freedoms including adaptive dc - bias , baseband subcarriers and multiple wavelengths corresponding to r / g / b led lights . according to the fundamental idea that spheres ( i.e. , constellation points ) can pack more compactly in a higher dimensional space , a constellation with larger minimum euclidean distance ( med ) can be expected in a higher dimensional space .. this med maximization problem is formulated in a non - convex optimization form , and is then relaxed to a convex optimization problem by a linear approximation method . key practical lighting requirements are taken into account as constraints , e.g. , the optical power constraint , average color constraint , non - negative intensity constraint , color rendering index ( cri ) and luminous efficacy rate ( ler ) requirements @xcite .        for rfc , one well - known shortcoming with utilizing multiple subcarriers is the excessive peak - to - average power ratio ( papr ) problem introduced , which can cause severe nonlinear distortion to degrade system performance . plenty of methods have been proposed to reduce papr ( see @xcite and references therein ) . in fact , when using multiple subcarriers for vlc , high papr is also a very severe issue , due to the limited linear dynamic range of amplifiers and leds . this paper shows that such distortion can be avoided by formulating the dynamic range requirement as ( convex ) constraints of the optimization problem . in such way distortion control becomes a offline process or a by - product of constellation design . the remainder of this paper is organized as follows . in section ii , we first provide an overview of dc - informative modulation schemes for optical communications . dc - informative multicarrier modulation is introduced as a power efficient alternative to traditional non - dc - informative optical ofdm configurations . in section iii , we propose the dci - jcfm method for systems with rgb led . the key lighting constraints including dynamic range control are discussed . the cases of `` balanced '' , `` unbalanced '' and `` very unbalanced '' systems are introduced . in section iv , we discuss the pros and cons of using dynamic range constraint , short time papr constraint and long time papr constraint . in section v , we provide simulation results to verify the significant performance gains of the proposed method over a decoupled method for balance , unbalance , and very unbalanced systems . finally , section vi provides conclusions . optical communications based on im / dd has a unique feature of requiring all signals modulating the leds to be positive ( and real ) , so multiple schemes are proposed accordingly such as the well - know asymmetrically clipped optical ofdm ( aco - ofdm ) , dc - biased optical ofdm ( dco - ofdm ) and optical multisubcarrier modulation ( msm ) . these schemes all discard the dc - bias at the receiver , which causes significant power loss . the dc - informative modulation schemes were then proposed such that @xmath0 optical power is used for data transmission ( see @xcite for single carrier case and @xcite for multiple carrier selective fading case ) . to be specific , consider the channel model @xmath1 , \label{1}\ ] ] where @xmath2 is a symbol waveform mapped from @xmath3 that contains @xmath4 bits of information , @xmath5 denotes the convolution operator , @xmath6 is either flat - fading or selective - fading channel , @xmath7 denotes white noise , @xmath8 is the received signal , @xmath9 and @xmath10 are electrical - to - optical and optical - to - electrical conversion factors respectively , with out loss of generality ( w.o.l.g ) . ] and @xmath11 stands for the constellation size . the key feature of a dc - informative modulation is that the following basis are used _ jointly _ to carry information @xmath12 @xmath13 @xmath14 where both i and q channels are used . @xmath15 is the dc - bias basis , @xmath16 is the symbol interval , @xmath17 is the @xmath18-th subcarrier , @xmath19 is the total number of subcarriers , and a rectangular pulse - shaper is used @xmath20 the relationship between symbol waveforms @xmath2 and constellation points @xmath21 $ ] is @xmath22 and one of the reasonable goals is to minimize the ser subject to fixed electrical / optical power by properly design the constellation matrix @xmath23 where each column of @xmath24 is a constellation point , and the med of all columns should be maximized to reach the goal . we typically stack the columns into a single vector instead , i.e. @xmath25^t,\ ] ] for simplification of the formulation of the optimization problem discussed in section ii . based on the idea discussed in section ii , we propose two constellation design methods taking advantage of the informative dc - bias for a visible light communication system employing one rgb led as shown by fig.1 . if the inputs into r / g / b carry independent bit information as shown by fig.1(a ) , it is termed a decoupled scheme . in comparison , if the inputs into r / g / b modulators only carry information jointly , it is a joint scheme instead as shown by fig.1(b ) . in other words , although for a joint scheme still three modulators are used to create continuous domain waveforms to drive corresponding leds , information can not be estimated though recovery of a single ( or any pair ) of them . it is observed that the joint scheme may utilize four types of diversities during per channel use , including frequency diversity , color(wavelength ) diversity , adaptive dc , and spatial diversity . spatial diversity can be achieved by extending from employing only one rgb led to include @xmath26 ones , which is out of scope of this paper . we only emphasize the first three diversities here . this scheme is termed dc - informative joint color - frequency modulation ( dci - jcfm ) . the decoupled system shown in fig.1(a ) works as follows : at the transmitter - side three independent bit stream @xmath27~i\in[1,n_c]$ ] , of length @xmath28 are mapped to corresponding constellation points @xmath29 of size @xmath30 first , which are modulated separately to generate continuous symbol waveform ( current ) @xmath31 by for each tunnel . if cross - talks exist for any tunnel , a corresponding precoder @xmath32 needs to be applied before modulation . waveforms @xmath31 are then electrical - to - optical converted to intensity signals @xmath33 to drive the leds . at the receiver - side , photo detectors of each tunnel collect the waveforms ( convoluted with channel and corrupted by noise ) . the received signal is sent through red , green , and blue color filters respectively and after optical - to - electrical conversion waveforms @xmath34 are obtained . then @xmath35 matched filters are employed for each tunnel to demodulate @xmath34 to obtain signal vector @xmath36 . three symbol detectors follow to provide @xmath37 , estimates of the symbol vectors , which are de - mapped separately and the estimate of original bit sequences @xmath38 are finally obtained . if cross - talks exist , post - equalizers @xmath39 are applied before the symbol detectors . system using dci - jcfm as shown by fig.1(b ) works differently . a joint bit sequence @xmath40^t,\ ] ] is firstly mapped jointly to a constellation point @xmath41 of size @xmath42 . then @xmath41 is converted by a joint modulator to the continuous domain to obtain @xmath43 through @xmath44 if cross - talks exist , a joint precoder @xmath45 is applied before modulation . also we observe the expectation of @xmath46 as follows @xmath47&=\sum_{p=1}^3 s_{(p-1)(2k+1)+1,i}\phi_1(t),\end{aligned}\ ] ] since all non - dc basis has zero time averages . therefore , both the average optical power and average color of system are determined solely by these three adaptive dc - bias . while the dynamic range of waveform , instead , is influenced by all subcarriers of all leds .    for our design , we will demonstrate with a line - of - sight ( los ) scenario when channel has cross - talks , due to the imperfectness of receiver color filters . the discrete channel model can be written as follows @xcite @xmath48 where @xmath49 $ ] is termed the `` cross - talk index ( ci ) '' and @xmath50 .      with working snrs for vlc which are typically medium - to - high , the minimum euclidean distance between constellation pairs governs ser . therefore , we seek to minimize the system ser by maximizing the minimum euclidean distance , through carefully design the constellation vector @xmath51 subject to key lighting constraints . for a constellation containing @xmath52 points , distances of a total of @xmath53 pairs have to be constrained as follows @xcite @xmath54 where we define @xmath55 and @xmath56 where @xmath57 denotes kronecker product , @xmath58 is the @xmath59-th column of identity matrix @xmath60 , and @xmath61 where @xmath62 . the distance constraints are nonconvex in @xmath51 . we choose to use the follows linear approximation at point @xmath63 @xmath64      for our design problem , practical lighting issue considered include average optical power , average illumination color , ler and cri , non - negative intensity , and flickering - free requirements . the first three requirements can be constraint using only one equation written as follow @xmath65 where @xmath66 is the average optical power of a rgb led , @xmath67 is a selection matrix ( containing only ones and zeros ) adding up r / g / b components in @xmath51 respectively by a multiplication of each row with it , @xmath68^t$ ] is termed the average color ratio vector and the follow equation holds @xmath69 thus the optical power and illumination color requirements are constrained together . the luminous efficacy rate and color rendering index requirements can be satisfied by properly choosing @xmath70 .      since the linear dynamic range of leds are limited , the ranges of signal for each led have to be constrained to avoid nonlinear distortion , i.e. @xmath71 where @xmath72 is the highest current level and for simplicity we have assumed that red , green and blue leds have the same dynamic range . we propose to constrain dynamic range of a sequence of sampled signal @xmath73 and @xmath74 is picked as @xmath75 where @xmath76 is the oversampling rate , @xmath77 , and @xmath78 is the total number of sample points . it should be noted that although does not guarantee , which means the continuous signal waveforms designed subject to could result in negative amplitudes in between the sample instances , the negative peak is very small compare to the dynamic range of signal . we can compensate this effect by adding a small post dc - bias after obtaining an optimized constellation . therefore , we can formulate this point - wise dynamic range constraints as follows    @xmath79    @xmath80    where @xmath81 selects the @xmath82-th constellation point and @xmath83 selects the corresponding coefficients for color @xmath84 , @xmath85^t$ ] , @xmath86 , @xmath87 , and @xmath88 .      we first formulate the optimization problem when there is no cross - talks among different colored leds , i.e. @xmath89 , as follows    @xmath90    which is convex in @xmath51 and @xmath91 , and specialized solver such as cvx toolbox for matlab can be utilized @xcite . start from initial point @xmath63 , the scheme can iteratively converge to a local optima with each run . the best constellation is chosen from local optimal constellation obtained from multiple runs . when the channel suffers from cross - talks , we choose to deal with it by employing the well - known singular value decomposition ( svd ) based pre - equalizer @xmath92 and post - equalizer @xmath93 for our system , where @xmath94 . constellation is designed by an optimization with transformed constraints as follow    @xmath95    where @xmath96 is defined is the kronecker product . ] and apparently this optimization is convex as well .    for the decoupled system , three independent problems can be formulated to find the meds for each color , i.e.    @xmath97    where @xmath98 , @xmath99 , @xmath100 , @xmath101 , @xmath102 , and @xmath103 are defined in a similar manner with corresponding parameters in . for brevity we omit the explicit definitions . in fact , although a hard constraint on the dynamic range of symbol waveforms can help avoid non - linear distortion completely , it may bring with side effects such as excessive power efficiency decrease . this is particularly true if only one or few symbol waveforms have notably larger dynamic range than the majority . in such case , one can consider using certain papr constraint to replace the dynamic range constraint . in other words , there is a tradeoff between allowable papr and power efficiency . two types of papr constraints ( for each led light ) can be considered . one is the so - called long - term papr ( l - papr ) , i.e. the ratio of the peak power of all waveforms and the time average of them . assuming no cross - talks , the l - papr constraint for led @xmath84 can be written as @xmath104 ^ 2 } { \mathbf{s}_j^t\mathbf{s}_j / n_c}\leq \beta_x\notag\\ & = \frac{n_c[\max_{i , n}(\mathbf{u}_n^t\mathbf{k}_x\mathbf{j}_i\mathbf{s}_j)]^2 } { \mathbf{s}_j^t\mathbf{s}_j}\leq\beta_x,\label{27}\end{aligned}\ ] ] where @xmath105 is the required l - papr for led @xmath84 . thus , a set of constraints can be formulated as follows @xmath106 which is non - convex in @xmath51 . a way to deal with this is to use a similar linear approximation as in at the same initial point @xmath63 . the above constraints are thus transformed as follows @xmath107    the other is the individual papr ( i - papr ) , i.e. the ratio of the peak power of each waveform and the average power of it . the i - papr for @xmath82-th waveform for led @xmath84 can be written as @xmath108 ^ 2 } { \mathbf{s}_j^t\mathbf{j}_i^t\mathbf{j}_i\mathbf{s}_j}\leq \beta_{x , i}\end{aligned}\ ] ]    the corresponding constraint can be written as @xmath109 and a similar linear approximation process is applied to convert them to convex constraints . to the best of our knowledge , there is no comprehensive comparison on performance of systems applying those three constraints available so far . an interesting observation recently in @xcite shows that the non - linearity mitigation for an im / dd vlc is a more involved problem than expected . the reason is that the low part of the baseband frequencies are causing larger non - linear distortion than the higher part . this effect , if taken into account along with the three constraints discussed above , is expected to make the design problem even more worthwhile to look into . in this section , we compare the performances of the decoupled scheme and dci - jcfm by assessing the maximum med and bit error rate ( ber ) under different channel cross - talk and color illumination assumptions . each constellation point is assumed to have equal probability of transmission , and the union bound for ser of both the scheme can be written as ( * ? ? ? * eq.25 ) @xmath110 where @xmath111 is the number of neighbor constellation pairs @xcite and @xmath112 denotes the gaussian q - function , @xmath113}$ ] stands for the med of the dci - jcfm and meds for decoupled schemes . the bit error rate is thus calculated as @xmath114 where @xmath115 is the number of wrongly detected bits in each bit sequence , which can be minimized by employing the bsa mapper . we first compare the dci - jcfm and the decoupled scheme when channel cross - talks do not exist . to guarantee a fair comparison , the following system parameters are chosen : the number of monte - carlo runs for each scheme @xmath116 , the length of bit sequence for each channel use is @xmath117 for dci - jcfm and @xmath118 respectively for each tunnel of the decoupled system , the number of subcarriers for each led is @xmath119 , the average optical power @xmath120 , the symbol interval @xmath121 is used is chosen , since the design is rate independent . ] , the upper bound of waveform amplitude @xmath122 , the average color ratio vector for a balanced system @xmath123^t,\ ] ] for an unbalanced system @xmath124^t,\ ] ] and for a very unbalanced system @xmath125^t.\ ] ]    the meds of the two schemes for three systems obtained through picking the best constellatio from the @xmath126 local optimums are summarized by table i.    from table i , key observations include : a. with the dci - jcfm , the `` joint med '' is much larger than the r / g / b `` decoupled meds '' , except for the cases with very unbalanced illumination . while for the very unbalanced case the blue and green tunnels could suffer from severe performance loss with the small meds , and therefore the dci - jcfm is still expected to work better . b. larger meds are obtained with an increased number of subcarriers . we are only listing the cases when @xmath127 and @xmath128 for brevity , while we have observed through additional simulations that this gain continue to grow with @xmath19 . c. the more balanced a system is , the better performance is expected . .med comparison with no cross - talk , dci - jcfm ( row 1@xmath1292 ) vs decoupled ( row 3@xmath1294 ) . [ cols="^,^,^,^,^,^,^,^",options="header " , ]     from table ii , key observations include : a. with increased channel cross - talks , the performances of all system degrade monotonously . b. the performance of the balanced system remains the best with any level of channel cross - talks . c. the dci - jcfm is kind of robust with cross - talks , since with a severe cross - talk level , i.e. @xmath130 , the system performance is still comparable or even better than a decoupled counterpart . we pick the optimized constellation @xmath131 designed for unbalanced system as an example in this section . if cross - talks do not exist , the corresponding subcarrier symbol waveforms @xmath132 obtained with dci - jcfm are plotted in fig . 4 . with fixed optical power @xmath133 and varying noise power @xmath134 , fig . 5 includes bit error rate curves of the two scheme with different color illumination across selected working electrical snrs , which is defined as @xmath135 for the dci - jcfm and snr for the decoupled scheme is defined similarly . significant power gains of the dci - jcfm over the decoupled scheme are observed . also , the more unbalanced the system is , the worse performance is expected . if cross - talks exist and @xmath136 , the corresponding subcarrier symbol waveforms are plotted in fig . 6 - fig . 8 . with each figure , the symbol waveforms are differentiated by color . in practice , the sampled version of these waveforms can be pre - stored in the memory of a high speed waveform generator . , no cross - talks . ]    , no cross - talks . ]    , no cross - talks . ]        , @xmath136 . ]    , @xmath136 . ]    , @xmath136 . ] we have propose a joint constellation design scheme termed dci - jcfm taking advantage of the wavelength , frequency , and adaptive bias diversities at the same time for indoor visible light communication systems . by applying the dci - jcfm scheme , waveform symbols with a much larger med can be obtained than those from a decoupled scheme with or without channel cross - talks . future works will include a comprehensive comparison among three systems : one applying dynamic range , one with long - term papr , and one with instantaneous papr constraint respectively ; comparison of power efficiency of the dci - jcfm and the popular dco / aco - ofdm schemes for multi - carrier multi - color vlc systems ; and advanced precoder design to replace the svd - based pre and post - equalizers utilized in this paper . t. komine and m. nakagawa , `` fundamental analysis for visible light communication system using led lights , '' _ ieee trans . consumer electronics _ , vol . 100 - 107 , 2004 . s. rajagopal , r. roberts , and s. lim , `` ieee 802.15.7 visible light communication : modulation schemes and dimming support , '' _ optics express _ , vol . 21835 - 21842 , dec . h. elgala , r. mesleh , and h. haas , `` indoor optical wireless communication : potential and state - of - the - art , '' _ ieee commun . mag . 56 - 63 , 2011 . l. zeng , d. obrien , h. minh , g. faulkner , k. lee , d. jung , y. oh , and e. won , `` high data rate multiple input multiple output ( mimo ) optical wireless communications using white led lighting , '' journal on selected area in communications , vol . 9 , pp . 1654 - 1662 , 2009 . r. you and j. kahn , `` average power reduction techniques for multiple - subcarriers intensity - modulated optical signals , '' _ ieee trans . commun . _ , vol . 49 , no . 12 , pp . 2164 - 2170 , dec . s. teramoto and t. ohtsuki , `` multiple - subcarrier optical communication systems with subcarrier signal - point sequence , '' _ ieee trans . _ , vol . 53 , no . 10 , pp . 1738 - 1743 , oct . 2005 j. karout , e. agrell , k. szczerba , and m. karlsson , `` optimizing constellations for single - subcarrier intensity modulated optical systems , '' _ ieee trans . inf . theory _ , vol . 58 , no . 7 , pp . 4645 - 4659 , july 2012 . e. monteiro and s. hranilovic , `` constellation design for color - shift keying using interior point methods , '' _ ieee owc - ws _ , pp . 1224 - 1228 , dec . 2012 . q. gao , j.h . manton , g. chen , and y. hua , `` constellation design for a multicarrier optical wireless communication channel , '' _ ieee trans . _ , pp . 214 - 225 , jan . 2014 . g. cossu , a.m. khalid , p. choudhury , and e. ciaramella , `` 3.4 gbit / s visible optical wireless transmission based on rgb led , '' _ optics express _ , vol . b501 - b506 , 2012 . b. bai , q. he , z. xu , and y. fan , `` the color shift key modulation with non - uniform signaling for visible light communication , '' the 1st international workshop on optical wireless communications in china , pp . 37 - 42 , 2012 . j. vucic , c. kottke , k. habel , and k.d . langer , `` 803 mbit / s visible light wdm link based on dmt modulation of a single rgb led luminary , '' _ optical fiber communication conference _ , los angeles , mar . ieee 802.15.7 visible light communication task group . 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in this paper , we consider the problem of constellation design for a visible light communication ( vlc ) system using red / green / blue light - emitting diodes ( rgb led ) , and propose a method termed dc - informative joint color - frequency modulation ( dci - jcfm ) . 
 this method jointly utilizes available diversity resources including different optical wavelengths , multiple baseband subcarriers , and adaptive dc - bias . 
 constellation is designed in a high dimensional space , where the compact sphere packing advantage over lower dimensional counterparts is utilized . taking into account multiple practical illumination constraints , 
 a non - convex optimization problem is formulated , seeking the least error rate with a fixed spectral efficiency . 
 the proposed scheme is compared with a decoupled scheme , where constellation is designed separately for each led . 
 notable gains for dci - jcfm are observed through simulations where balanced , unbalanced and very unbalanced color illuminations are considered .    * constellation design , visible light communication , dc - informative , joint color - frequency , im / dd . *