Uncoordinated Multiple Access Schemes for Visible Light Communications and Positioning

Transcription

1 06 IEEE International Symposium on Information heory Uncoordinated Multiple Access Schemes for Visible Light Communications Positioning Siu-Wai Ho Institute for elecommunications Research University of South Australia Adelaide, SA 5095, Australia Chi Wan Sung Dept of Electronic Engineering City University of Hong Kong Hong Kong Abstract In visible light communication (VLC) systems, information are conveyed by visible light instead of radio-frequency electromagnetic waves Based on received signal strength, accurate indoor positioning systems can also be built Since a receiver obtains the superposition of signals from all light sources within line of sight together with ambient light, a multiple access scheme is necessary for the receiver to distinguish the received symbol signal strength from each light source his paper proposes two multiple access schemes for VLC he first scheme supports information broadcast positioning By using N timeslots, N transmitters transmit N symbols in total he second scheme supports positioning only but places emphasis on minimizing the required timeslots In each N timeslots for an odd integer N, the channel gains of 3N transmitters can be estimated I INRODUCION Visible light communications (VLC), which uses visible light instead of radio-frequency (RF) electromagnetic waves for communications, has attracted much attention in recent years Lighting devices are undergoing a revolution, from fluorescent lamps tubes to light emitting diodes (LEDs) Most smartphones tablets today contain built-in sensors such as photodiodes (PDs) VLC systems using LEDs as transmitters PDs as receivers are energy efficient have low deployment cost Since visible light does not interfere RF signals, VLC systems will be compatible with other RF systems, with the potential to form powerful hybrid systems Some of the promising applications of these systems include information broadcast indoor positioning For information broadcast using VLC, there are different techniques proposed in the literature including expurgated pulse position modulation [], optical code division multiple access (OCDMA) [][3] an adaptive modulation scheme based on power control [4] Precoding techniques for RF systems are adapted for VLC in [5][6] Note that all these techniques require a central unit to coordinate the transmissions among the LEDs through an extra infrastructure in a VLC system he disadvantages of using central unit will be explained after some positioning systems are introduced Visible light positioning systems can provide highly accurate indoor positioning [7][8] Experimental results show that positioning error can be less than 0 centimetres on average Since the positioning algorithms in these systems are based on the received light intensity or estimated channel gain from each LED, any interference among the signals from different LEDs would severely degrade the accuracy Furthermore, it is common to have ambient light sources which give extra energy in each measurement ime division multiple access (DMA) was used in [7], [8] o implement DMA, a central unit is again required to coordinate the transmissions among LEDs It is important to know a multiple access scheme for VLC systems without using central unit Such scheme allows us to build an information broadcast system with positioning capability by simply replacing the existing light sources by st-alone LED transmitters Without the need of extra infrastructure to control the LEDs, the installation cost is reduced o get rid of central unit, we need to solve a few technical problems Intensity modulation/direction detection (IM/DD) is used in VLC systems herefore, the transmitted signals are always positive real Multiple receivers may use the system at rom time at rom locations with arbitrary orientation in a 3-dimensional space As a result, the transmitted signals suffer rom channel gains which are unknown to the receiver Since the transmitters are also light sources, they may be controlled by different switches be switched on at different time So the transmitters the receiver are asynchronous It is also preferable to consider simple transmitter design that no communication link presents among transmitters We begin with the introduction of channel model notations in Section II In Section III, we will propose an efficient scheme in which a receiver obtains N symbols from N transmitters one estimate of ambient light intensity in N timeslots Section IV explores another scheme which can reduce the required timeslots by supporting only positioning II CHANNEL MODEL AND NOAIONS Suppose there are N transmitters in the system where a subset of them are active within the line-of-sight (LoS) of the receiver (see Fig ) Let S be this subset so that S,, N For i S, let a ik 0,,, M be the k-th message that ransmitter i wants to deliver since it is activated Let X i [n] 0 be the n-th symbol transmitted by ransmitter i Suppose ransmitter i has transmitted δ i symbols before the receiver starts to detect signals Let Φ be the background light intensity which is not negligible Receiver detects the superposition of the signals from the transmitters together with /6/$ IEEE 350

2 06 IEEE International Symposium on Information heory + Fig ransmitters want to broadcast different messages hey are suffering from rom delays due to the different activation time he superposition of ambient light the transmitted signals is received by the receiver the background light Let Y [n] be the n-th symbol received at the receiver So Y [n] = i S g i X i [n + δ i ] + Φ, () where g i is the channel gain which depends on the distance between ransmitter i the receiver, angle of irradiance of the LED, angle of incidence of the PD, etc Since these parameters depend on the receiver s location which is rom, we assume without lost of generality that the cumulative distribution function (cdf) of g i is continuous if i S We further assume that the cdf of g i given g k with i k is continuous Here, g i is considered as invariant over a short period of time like block fading channels his is justified by that it is common to achieve transmission rate over 0 6 samples per second in VLC [9] If the receiver s displacement is negligible within 0 3 seconds, the gain g i can be seen as invariant for more than 0 3 symbols In general, noise should be considered in () his paper considers the noiseless case as the starting point of this problem to illustrate what can be achieved Note that our proposed scheme has potential to be extended to noisy scenarios Notations: L(S, δ) means the cyclic shift of the sequence S to the left by δ steps L(S, δ)[n] means the n-th symbol after shift For example, if S = (s, s, s 3 ), then L(S, ) = (s, s 3, s ) L(S, )[3] = s he complex conjugate of W is denoted by W he imaginary number is j = III POSIIONING AND COMMUNICAIONS IN VLC Our goal of this section is to design a coding scheme such that if ransmitter i is within LoS, the receiver can determine i) the channel gain g i for positioning ii) the broadcasted messages a ik In this section, a ik is assumed to be independent uniformly distributed in 0,,, M Scheme for Positioning with Comms Capability: ) For i N, ransmitter i outputs i consecutive a ik followed by i consecutive 0 o be specific, for k 0, the n-th output from ransmitter i is a ik for + k i n < + (k + ) i X i [n] = 0 otherwise () ) Let a row vector S i ( ) be the first symbols from a periodic sequence with a repeating pattern of i consecutive s followed by i consecutive s So S i ( ) = [ ] (3) Let Y( ) = [Y [] Y [] Y [ ]] be a column vector 3) Let ˆδ i ( ) = argmax 0 δ< i L(S i ( ), δ) Y( ) 4) ransmitter i is considered as inactive if lim Y [n]l(s i ( ), ˆδ i )[n] = 0 (4) 5) According to (4), let Ŝ be the estimated set of active transmitters 6) Set Ŝ Ŝ Construct a (0, ) matrix A by concatenating the row vector ( + L(S i( ), ˆδ i )) for i Ŝ in an increasing order of i he positions of s in A represent the timeslots when ransmitter i sends positive X[n] 7) here must exist a zero column vector in A suppose it is the i-th column he estimated background intensity is ˆΦ = Y [i] Set Y [i] Y [i] ˆΦ for all i 8) Set i = maxk : k Ŝ 9) here must exist columns in A with the form [0 0 ] Let Ω be the position of those columns For n Ω, Y [n] = g i X i [n + ˆδ i ] which is kept in a set M i 0) For n Ω, let L n n be the set of consecutive s in the last row of A Update Y [l] Y [l] Y [n] for all n Ω l L n ) Remove the last row of A set Ŝ Ŝ \ i ) Go to Step 8 if Ŝ is non-empty 3) For i Ŝ, the receiver has collected a set M i = g i a ik where the range of k depends on δ i 4) If maxµ : µ M i minµ > 0 : µ M i < M (5) for any i, set go back Step 5) he estimated g i is given by ĝ i = minµ > 0 : µ M i herefore, a ik can be estimated from ĝ i M i he scheme ends here heorem : If a ik for all i k are independent uniformly distributed on 0,,, M, then lim Pr ˆδ i ( ) = δ i = (6) Proof: Note that L(S i ( ), δ i )[n] = if only if X i [n + δ i ] > 0 herefore, from () Y [n] = i S g i X i [n + δ i ]L(S i ( ), δ i )[n] + Φ (7) Now consider L(S i ( ), δ i )[n]l(s k ( ), δ)[n] for n Firstly, suppose i < k Consecutive s with length k repeatedly appear inside L(S k ( ), δ)[n] Let I + be the positions of those consecutive s so that L(S k ( ), δ)[n] = for n I + I + is an integer multiple of k On the other h, consecutive s with length i repeatedly appear inside L(S i ( ), δ i )[n] Since k is divisible by i, the number of consecutive s in L(S i ( ), δ i )[n] for n I + is 35

3 06 IEEE International Symposium on Information heory equal to I+ Similarly, let I be the positions of consecutive s with length k in L(S k ( ), δ) For n I, the number of consecutive s in L(S i ( ), δ i )[n] is equal to I Without lost of generality, we consider that is an integer multiple of k so that I + = I hus, for arbitrarily ɛ > 0, ( ) Pr g i X i [n + δ i ] g i X i [n + δ i ] > ɛ < ɛ n I + n I for sufficiently large ogether with lim g i X i [n + δ i ]L(S i ( ), δ i )[n]l(s k ( ), δ)[n] (9) k lim g i X i [n + δ i ] + g i X i [n + δ i ] (0) n I + g i X i [n + δ i ] + g i X i [n + δ i ] () n I n= k + ( ) = lim g i X i [n + δ i ] g i X i [n + δ i ], () n I + n I we have Pr g i X i [n + δ i ]L(S i ( ), δ i )[n]l(s k ( ), δ)[n] > ɛ < ɛ (3) for any ɛ > 0 sufficiently large Secondly, suppose i > k We can still prove (3) by swapping the roles of i k in the above argument So we do not repeat the argument here Finally, consider i = k It is easy to verify that g i X i [n + δ i ]L(S i ( ), δ i )[n]l(s i ( ), δ)[n] (4) < (8) g i X i [n + δ i ]L(S i ( ), δ i )[n]l(s i ( ), δ i )[n] (5) for δ δ i ogether with lim ΦL(S k ( ), δ)[n] = 0, (6) for finite Φ, (6) is verified heorem : If ransmitter k is inactive, the probability that the receiver considers it as active is arbitrarily small because Pr Y [n]l(s k ( ), ˆδ k )[n] > ɛ < ɛ (7) for any ɛ > 0 sufficiently large Proof: If ransmitter k is inactive, X k [n] = 0 for all n Since (3) holds for i k, the theorem is proved Lemma : For k, if ransmitter i is active for all i k, there exists k < k such that for all integer l 0, k X i [l k + k ] = 0 (8) i= Proof: We are going to prove the lemma by mathematical induction When k =, k i= X i[n] = X [n] By taking n = l + δ, (8) is verified to be true for k = Assume that (8) is true for k hen = k+ X i [l k+ + k+ ] (9) i= k X i [(l) k + k+ ] + X k+ [l k+ + k+ ] (0) i= If X k+ [ k ] = 0, then X k+ [l k+ + k ] = 0 for all l due to (), hence take k+ = k If X k+ [ k ] > 0, then X k+ [ k + k ] = 0 hence, X k+ [l k+ +( k + k )] = 0 for all l In this case, k+ = k + k herefore, the lemma is verified by mathematical induction heorem 3: If Ŝ = S ˆδ i = δ i for all i, then the column vectors described in Steps 7 9 must exist Proof: Let S be the set of active transmitters let k = maxi : i S Note that L(S i ( ), δ i )[n] = if only if X i [n + δ i ] > 0 By Lemma, the column vectors described in Step 6 exist Similarly, applying Lemma with k replaced by k shows that for all l, k X i [l k + k ] = 0 () i= Since either X k [l k + k ] > 0 or X k [(l + ) k + k ] > 0, the column vectors described in Step 9 exist Remarks: ) From heorems 3, we conclude that the receiver can successfully recover g i a ik for i S with arbitrarily small error probability when is sufficiently large ) When all transmitters are active within the LoS of the receiver, the receiver can obtain an estimate of Φ together with N N i = N () i= symbols in N timeslots his is better than [0] where only N symbols can be transmitted in N timeslots Example : Suppose N = 3 = 8 ables I III demonstrate some possible transmitted patterns Receiver receives the summation of each column together with Φ Assume that the receiver first correctly estimates δ i by considering >> 8 according to Step 3 he receiver can construct A in Step 6 which is similar to the tables except that all g i a ik s are replaced by In Step 7, the background light intensity is estimated by Y [8], Y [8], Y [6] in ables I III, respectively, according to the positions of zero column vectors Now, we focus on only able III After the effect due to Φ is removed from Y [n] in Step 7, g 3 a 3 is obtained from Y [] In Step 0, let L =, 3, 4, 5 set Y [l] Y [l] Y [] 35

4 06 IEEE International Symposium on Information heory ABLE I SCENARIO : δ = δ = δ 3 = 0 g a 0 g a 0 g a 3 0 g a 4 0 g a g a 0 0 g a g a 0 0 g 3 a 3 g 3 a 3 g 3 a 3 g 3 a ABLE II SCENARIO : δ = 0; δ = ; δ 3 = 0 g a 0 g a 0 g a 3 0 g a g a g a 0 0 g a g a 0 g 3 a 3 g 3 a 3 g 3 a 3 g 3 a ABLE III SCENARIO 3: δ = 0; δ = 3; δ 3 = g a 0 g a 0 g a 3 0 g a 4 0 g a g a g a 0 0 g a 0 g 3 a 3 g 3 a 3 g 3 a 3 g 3 a for all l L to remove the effect due to a 3 After removing the last row in A, we follow Step 9 find two positions (Ω = 4, 8) match the condition herefore, g a g a are obtained from Y [4] Y [8], respectively In Step 0, let L 4 = 4, 5 L 8 = 8 After setting Y [l] Y [l] Y [4] for all l L 4 Y [l] Y [l] Y [8] for all l L 8, the last row in A is removed go back to Step 8 We do not repeat the details here At the end, the receiver obtains g a, g a 3, g a 4, g a, g a g 3 a 3 If >>, these sets will be large contain both g i g i (M ) hen Step 5 is reached where g i a ik are determined IV VISIBLE LIGH POSIIONING his section investigates visible light positioning systems without communication capability Our focus is to minimise the number of symbols required for finding channel gain g i for i S Smaller number of symbols takes shorter transmission time; this improves the sensitivity to the changes in channel gains is important for delicate positioning systems Compared with wireless communications using radio frequency, one of the major differences in VLC is that the transmitted signals are required to be positive real Under this requirement, suppose ransmitter i outputs periodic signals X i [n] with period Here, if the transmitters output signals with different periods, we can take as the least common multiple of the periods Since X i [n] is positive real, the Fourier transform of X i denoted by W i must satisfy W i [n] = W i [ n] As a result, only half of the elements in the Fourier transform of the received signal Y [n] are useful In order to support N transmitters by using N timeslots, we can use the following simple scheme to decide the values of X i [n] Let C i [n] be the inverse Fourier transform of the i-th row in a N N matrix B = (3) For i N n N, let C i [n] = C i [n] + (4) Let X i [n + k N] = C i [n] for k 0 By considering the Fourier transform of the received N symbols at the receiver denoted by Ỹ [n] for 0 n N, the structure of B ensures: ) there is no interference among the transmitters in frequency domain ) C i [n] are positive real for all i n According to (), Ỹ [i] = g ie jiπ N δi where δ i causes a phase shift in frequency domain Although δ i is unknown, g i can be determined by Ỹ [i] he interesting question is: If the transmitted signal is required to be positive real the delay δ i is unknown, can we design a scheme which can support more than N transmitters by using N timeslots? We are going to reveal that it is possible by using the property that g i is positive real while Ỹ [n] is complex Definition : For an odd integer N, let C = C i, i 3N where C i is defined in (4) for i N For N + i 3N k = 0 or, let φ k (i) = (i N) + k, (5) ( Z i,k = (4 k) + exp jφ ) k(i)π, (6) N θ i,k = arg (Z i,k ), (7) C i = Z i,0 (4 C φ0(i) + L(C φ0(i), ))+ Z i, (3 C φ(i) + L(C φ(i), )) (8) Note that if C i is sent alone, Ỹ [φ k (i)] = Z i,k Z i,k for k = 0, N Lemma : Both θ i,k π θ i,0 θ i, N π are not integers he proof is omitted due to the limited space Lemma gives a property which will be frequently used in this section Instead of combining C φ0(i) C φ(i) as the way shown in (8), it is possible to use other weightings as long as Lemma still holds Hence, we cannot replace both 4 3 in (8) by Multiple Access Scheme for Visible Light Positioning: ) For i 3N, ransmitter i repeatedly sends C i ) Let Ỹ = [Ỹ (0), Ỹ (),, Ỹ (N )] be the discrete Fourier transform of Y which is a vector of N consecutive symbols received at the receiver 3) he estimate of g N is given by ĝ N = Ỹ [N] 4) For i > N, g φ0(i), g φ(i) g i are estimated together from (Ỹ [φ 0(i)], Ỹ [φ (i)]) as follows 5) For integers l, r 0,,, N k 0,, let 6) Suppose Ỹ [φ k(i)] = ρ x + jρ y Let µ l = θ i,k lπφ k(i) N, (9) ν r = rπφ k(i) N (30) γ x (i, k, r, l) = ρ y ρ x tan(µ l ) tan(ν r ) tan(µ l ) (3) 353

5 06 IEEE International Symposium on Information heory Fig he complex plane used in Lemma 3 (a) Step 6 finds (γ x, γ y) put the distances between (γ x, γ y) (ρ x, ρ y) into A(i, k) (b) he slopes of the green lines are µ l the slopes of the blue lines are ν r he line ξ connects the ending points of any two green lines with length g i Fig 3 he slopes of the short green line short red line are µ l the slopes of the long blue lines are ν r with k = in (9) (30) We first consider the probability of that two elements at positions (r, l) (r, l ) in A(i, k) are equal to g i his is equivalent to require that the triangle in Fig (a) is an isosceles triangle By writing the two angles in terms of µ l, µ l ν r, an isosceles triangle is formed only if π θ i,k = κ (37) N for certain integer κ that contradicts Lemma Now suppose the two elements in A(i, k) are the same at positions (r, l) (r, l ) where r r l l his is equivalent to require that the line ξ touches the axes Γ r Γ r in Fig (b) Consider a fixed g i hen the length the slope of ξ are fixed for given (l, l ) herefore, it can be verified that there are only a finite number of g φk (i) such that ξ touches the axes Γ r Γ r Since the cdf of g φk (i) given g i is continuous, the lemma is proved if g φk (i) > 0 Finally, if g φk (i) = 0, the lemma follows from that ξ cannot be parallel to any Γ r due to Lemma Lemma 4: For N + i 3N, if g i > 0, then A(i, 0) A(i, ) = almost surely Proof: Due to the limited space, the detail is omitted here γ y (i, k, r, l) = tan(ν r )γ x (i, k, r, l); (3) Note that tan(ν r ) because N is odd Define A(i, k) = ((γ x ρ x ) + (γ y ρ y ) ) r, l 7) For i > N, if A(i, 0) A(i, ) =, let (33) ĝ i = A(i, 0) A(i, ) (34) Let (r i,k, l i,k ) be the position of ĝ i in A(i, k) the estimate of g φk (i) is given by ĝ φk (i) = (γ x(i, k, r i,k, l i,k) + γ y(i, k, r i,k, l i,k)) (35) 8) For i > N, if A(i, 0) A(i, ) >, let ĝ i = 0 ĝ φk (i) = Ỹ [φ k(i)] he scheme ends here Now, we prove the robustness of the proposed scheme explain the rationale in the proofs Lemma 3: We can always find g i A(i, k) If g i > 0, then g i appears only once in A(i, k) almost surely Proof: Due to the definition of (ρ x, ρ y ), (4) (8), ρ x + jρ y = Ỹ [φ k(i)] = g φk (i) exp(jν r ) + g i exp(jµ l ), (36) where r = δ φk (i) l = δ i Due to Lemma, a nonorthogonal basis on the complex plane can be formed by using the arguments of exp(jν r ) exp(jµ l ) hus projecting Ỹ [φ k (i)] on the right basis with correct (r, l) gives g i By exhaustively trying all possible values of r l, g i must appear in A(i, k) heorem 4: he scheme generates ĝ i = g i almost surely for i 3N Proof: It is easy to verify the case that g i = 0 Suppose g i > 0 From Lemma 4, g i can be identified by considering A(i, 0) A(i, ) Only one pair (ri,k, l i,k ) can give g i due to Lemma 3 so that g φk (i) can be calculated from (35) herefore, we conclude that the receiver can estimate g i almost surely for i 3N by receiving N symbols REFERENCES [] M Noshad, M Brt-Pearce, Application of Expurgated PPM to Indoor Visible Light Communications-Part II: Access Networks, J of Lightwave echnology, vol3, no5, pp , March 04 [] Y Chen, Y Chang, Y seng Y Chen, A Framework for Simultaneous Message Broadcasting Using CDMA-Based Visible Light Communications, IEEE Sensors Journal, vol 5, no, Aug 05 [3] MH Shoreh, A Fallahpour JA Salehi, Design concepts performance analysis of multicarrier CDMA for indoor visible light communications, IEEE/OSA J of Opt Comm Netw, Jun 05 [4] A A Saed, S-W Ho C W Sung, Adaptive Modulation for wo users in VLC, in IEEE GlobeCom Workshop, Dec 05 [5] Y Hong, J Chen C Yu, Performance improvement of the precoded multi-user MIMO indoor visible light communication system, Int Symp on Comm Systems, Networks & Dig Signal Proc, Jul 04 [6] M Hao, L Lampe S Hranilovic, Coordinated Broadcasting for Multiuser Indoor Visible Light Communication Systems, IEEE ransactions on Communications, vol 63, no 9, pp , Sept 05 [7] Y Hou, S Xiao, H Zheng, W Hu, Multiple Access Scheme Based on Block Encoding ime Division Multiplexing in an Indoor Positioning System Using Visible Light, IEEE/OSA Journal of Optical Communications Networking,, vol 7, no 5, pp , May 05 [8] M Yasir, S-W Ho B N Vellambi, Indoor Position racking Using Multiple Optical Receivers, in IEEE/OSA Journal of Lightwave echnology, vol 34, no 4, pp 66-76, Feb 06 [9] S Rajagopal, RD Roberts, S-K Lim, IEEE 8057 visible light communication: modulation schemes dimming support, in IEEE Communications Magazine, vol50, no3, pp 7-8, Mar 0 [0] M Yasir, A A Saed S-W Ho, Uncoordinated multiple access scheme for VLC systems with positioning capability, 06 Australian Communications heory Workshop (AusCW), pp 8-3, Jan