Energy-Efficient Precoding for Multi-User Visible Light Communication with Confidential Messages
aa r X i v : . [ c s . I T ] F e b Energy-Efficient Precoding for Multi-User VisibleLight Communication with Confidential Messages
Son T. Duong ∗ , Thanh V. Pham † , Chuyen T. Nguyen ∗ , and Anh T. Pham † ∗ Hanoi University of Science and Technology, Vietnam. † Computer Communications Lab., The University of Aizu, Japan.Emails: ∗ [email protected], ∗ [email protected], † { tvpham, pham } @u-aizu.ac.jp Abstract —In this paper, an energy-efficient precoding schemeis designed for multi-user visible light communication (VLC)systems in the context of physical layer security, where users’messages are kept mutually confidential. The design problemis shown to be non-convex fractional programming, thereforeDinkelbach algorithm and convex-concave procedure (CCCP)based on the first-order Taylor approximation are utilized totackle the problem. Numerical results are performed to showthe convergence behaviors and the performance of the proposedsolution for different parameter settings.
Index Terms —Multi-user VLC, energy efficiency, physicallayer security, precoding.
I. I
NTRODUCTION
Visible light communication (VLC) has been becoming anattractive wireless solution to complement existing technolo-gies due to its high-capacity data transmission with license-free spectrum. Aside from this, the technology also takesadvantage of the widespread deployment of light-emittingdiodes (LEDs). This naturally enables VLC to fit into thefuture ubiquitous networks.However, multiple challenges still remain to make VLCmore viable, among which security in terms of informationprivacy and confidentiality (especially in public areas) are ofthe most important issues [1]. In this respect, physical layersecurity (PLS) has emerged as novel paradigm to enhancesecure communication by exploiting the randomness of thewireless channels, noise, and interference. The most promisingimplication of PLS is that a perfect secured communicationcan be achieved from information theoretic point of view. Thesecrecy of PLS is quantized by the secrecy rate that defines themaximum transmission rate at which unauthorized users areunable to extract any information from the received signalsregardless of their computational capability. While PLS hasbeen a well investigated topic in radio frequency (RF) com-munications, it has only been receiving considerable attentionin the past few years in the case of VLC. In practical VLCsystems, multiple LED luminaries should usually be deployedto provide a sufficient illumination. As such, multiple-inputsingle-output (MISO) channels are more prevalent. In suchscenarios, the degrees of freedom introduced by multiple LEDtransmitters enables the use of precoding as a means of secrecyenhancement [2]–[9].Aside from the security, energy efficiency is another es-sential criterion in designing a communications system. This increasing attention to energy consumption comes from thecurrent global effort to reduce the carbon footprint. In regardto PLS in VLC, there has been a few studies dealing withprecoding design from the energy consumption perspective. Inthe case of one legitimate user and multiple eavesdroppers, theauthors in [4] and [10] considered the problem of minimizingthe power of the information-bearing signal while ensuringa minimum achievable secrecy rate. With the same configu-ration, [11], [12] focused on designing artificial noise-aidedprecoding. The total power consumption of the information-bearing signal and the artificial noise is then minimizedtaking into account predefined thresholds on the signal-to-interference-plus-noise ratios (SINRs) of the legitimate userand eavesdroppers.In general, the above mentioned works concerned withthe issue of minimizing the consumed power given that acertain secrecy requirement is fulfilled. This design approach,however, is not necessarily optimal from the perspective ofenergy efficiency, which is defined as the number of bitsthat can be transmitted per Joule. In other words, it is theratio of the achievable (secrecy) rate to the total powerconsumption. Considering that an energy-efficient precodingdesign for PLS in multi-user (MU) VLC systems has notbeen studied, the objective of this paper is to fill this gap.Particularly, we examine an MU-MISO VLC system whereeach user treats others as eavesdroppers, hence its intendedmessage must be kept confidential. The energy efficiency inthe context of PLS of such system is then formulated usinga lower bound on the achievable secrecy sum-rate. Regardingthe power consumption, in addition to the power consumedfor the information-bearing signal and circuitry operations(e.g., as in radio frequency (RF) communications), the lightingfunction of VLC requires an additional power for illumination,which also impacts the achievable secrecy rate. The optimalprecoding design problem to maximize the energy efficiencyis shown to be non-convex fractional programming, whichrenders finding an exact solution difficult. Hence, our approachis to find a sub-optimal yet computational efficient solution byemploying the Dinkelbach algorithm and convex-concave pro-cedure (CCCP). Simulations under different system parametersare conducted to verify the efficiency of the proposed solution.The rest of the paper is structured as follows. The systemmodel together with a formulation of the energy efficiencyare described in Section II. Section III focuses on solving therecoding design to maximize the energy efficiency. Numericalresults and related discussions are given in Section IV. Finally,Section V concludes the paper.Notation: R m × n denotes the m × n real-valued matrices.The uppercase bold symbols, e.g. M , denote matrices, whilethe lowercase ones such as v represent column vectors. Thetranspose of M is written as M T . The k -th column vector of M is denoted as m k , while its ( i , j )-th element is written as m i , j . Additionally, | · | , k·k and | · | are L norm, the Euclideannorm and absolute value operator, respectively.II. S YSTEM M ODEL
Fig. 1: A simple example of the considered MU-MISO VLCsystem with N T = K = N T LED luminaries and K decentralizedusers, where each user is equipped with a photodiode (PD).It is reminded that the transmission is considered to beconfidential if each user is not able to decode any informationintended to the others. A. Signal Model
Let d = h d d · · · d K i T ∈ R K × be the vector of datasymbols for all users. Assume that the symbols are drawnfrom an M -ary pulse amplitude modulation ( M -PAM) and aremodeled as a random variable (RV) d following a certaindistribution over [ − , ] with zero-mean and variance σ d . Aninformation-bearing signal s n for the n -th LED transmitter isgenerated from a linear combination of the data vector and aprecoder v n = h w n , w n , · · · w n , K i ∈ R × K as s n = v n d . (1)For illumination, an DC bias I DC n should be added to s n tocreate a non-negative drive current x n for the LED. The drivecurrent, in addition, needs to be constrained to a maximumthreshold, i.e., I max , to ensure that LEDs operate normally.Therefore, 0 ≤ x n = s n + I DC n ≤ I max . (2) The emitted optical power of each LED luminary is given by P sn = η (cid:16) s n + I DC n (cid:17) , (3)where η is the LED conversion factor. Denote h k =[ h , k h , k · · · h N T , k ] T is the k -th user’s channel matrix, where h n , k is the line-of-sight (LoS) channel coefficient between the n -th LED array and the user. Details on the VLC channelmodel can be found in [1] and references therein. The electri-cal signal at the PD output is then given by y k = γ h Tk h P s P s · · · P sN T , k i + n k = γη h Tk w k d k + h k K ∑ i = , i = k w i d i + h k I DC + n k , (4)where w k = h w , k w , k · · · w N T , k i T is the k − th user’s precoderand I DC = h I DC1 I DC2 · · · I DC N T i T . It is noted that since | d i | ≤ −k v k k ≤ s n ≤ k v k k . (5)To ensure both (2) and (5), the following constraint should besatisfied K ∑ k = | w n , k | ≤ min (cid:16) I DC n , I max − I DC n (cid:17) . (6)The receiver noise n k in (4) can be modeled as a real-valuedzero-mean Gaussian RV whose variance is given by σ k = γ eP rk B + π eA r γχ amb ( − cos ( Ψ )) B + i B , (7)where P rk = η h Tk I DC is the average of received power at the k -th user, A r is the area of the PD, e is the elementary charge, Ψ is the optical field of view (FoV) ofthe PD, B is the modulationbandwidth, χ amb is the ambient light photo-current, and i amp is the pre-amplifier noise current density. B. Power Consumption
We now analyze the total consumed power at LED trans-mitters, which can be expressed by P total = P DC + P AC , (8)where P DC and P AC are the powers of the DC and ACcurrents, respectively. The DC current power includes thepower used for illumination by LEDs denoted as P DC, LEDs andthat used by other circuit components denoted as P DC, circuitry .While P DC, circuitry can be considered to be fixed, the powerconsumption of LEDs can be adjusted depending on therequired dimming level. However, under a specific usage whenthe illumination level is not changed, it is reasonable to assumethat P DC, LEDs is fixed as well. The DC power consumption isthen given as P DC = P DC, LEDs + P DC,circuitry , (9)with P DC, LEDs being calculated as P DC, LEDs = N T ∑ n = U LED I DC n , (10)here U LED being the forward voltage of the LEDs.The AC currents comes from the output current (or voltage)from the precoders of LED drivers, therefore can be calculatedas P AC = r K ∑ k = σ d k w k k , (11)where r is the equivalent resistance of the AC circuit. Withoutloss of generality, we denote ξ = r σ d as equivalent resistance,then rewrite the total power consumption as P total = P DC + ξ K ∑ k = k w k k . (12) C. Energy Efficiency
For demodulation, the DC component in the received signalin (4) is removed, yielding y k = h Tk w k d k + h k K ∑ i = , i = k w i d i + n k , (13)where n k = n k γη . According to [7], a lower bound of achievableconfidential secrecy rate of the k − th user is given as R s , k ( W ) =
12 log + ∑ Ki = a k (cid:0) h Tk w i (cid:1) + ∑ Ki = , i = k b k (cid:0) h Tk w i (cid:1) −
12 log + K ∑ i = , i = k b i (cid:16) h Ti w k (cid:17) , where W = h w w · · · w K i , a k = exp ( h d ) π e σ k , and b k = σ d σ k with σ k = σ k ( γη ) and h d being the differential entropy of d . Theenergy efficiency with respect to the achievable secrecy sum-rate of the considered system is therefore given by Φ ( W ) = ∑ Kk = R s , k ( W ) P DC + ξ Tr (cid:0) WW T (cid:1) . (14)III. S UB - OPTIMAL P RECODING D ESIGN
Our design objective is to maximize the energy efficiencywhile a minimum achievable secrecy rate for each user isguaranteed. Hence, the design problem can be formulated asfollows max W Φ ( W ) , (15a)s.t. R s , k ( W ) ≥ λ k , (15b) K ∑ k = | w n , k | ≤ min (cid:16) I DC n , I max − I DC n (cid:17) , (15c)where λ k is the threshold for the secrecy rate of the k -thuser. It is seen from (14) that the objective function is anon-concave fractional function with W . This requires the useof Dinkelbach algorithm [13], which is efficient in solvingfractional programming. Let N ( W ) = ∑ Kk = R s , k ( W ) , D ( W ) = P DC + ξ Tr (cid:0) WW T (cid:1) . Then for some µ ≥
0, the Dinkelbachalgorithm involves solving the followingmax W N ( W ) − µ D ( W ) , (16a) s.t. R s , k ( W ) ≥ λ k , (16b) K ∑ k = | w n , k | ≤ min (cid:16) I DC n , I max − I DC n (cid:17) . (16c)to obtain a precoder W ′ . The optimal value µ ∗ = N ( W ∗ ) D ( W ∗ ) isachieved when N ( W ′ ) − µ D ( W ′ ) =
0. This results in theDinkelbach algorithm described as follows
Algorithm 1:
Dinkelbach-type algorithm for solving(16).Choose the maximum number of iterations L max , andthe error tolerance ε .Initialize µ > l = while convergence == False and l ≤ L max , do For a given µ , solve (16) to get W ( l ) . if N (cid:16) W ( l ) (cid:17) − µ D (cid:16) W ( l ) (cid:17) ≤ ε then convergence == True ; W ∗ = W ( l ) ; µ ∗ = N (cid:16) W ( l ) (cid:17) D ( W ( l ) ) ; else convergence == False l = l + µ = N (cid:16) W ( l ) (cid:17) D ( W ( l ) ) ; endend Return the optimal W ∗ and µ ∗ .Nevertheless, it should be noted that (16) is not a convexoptimization problem due to the non-concave objective func-tion and the non-convex constraint in (16b). To overcome thisproblem, we make use of the CCCP approach based on thefirst-order Taylor approximation to approximate the originalproblem to a convex one. Specifically, we first introduce thefollowing slack variables r , k ∆ =
12 log + K ∑ i = a k (cid:16) h Tk w i (cid:17) ! , (17a) p , k ∆ = K ∑ i = a k (cid:16) h Tk w i (cid:17) , (17b) r , k ∆ =
12 log + K ∑ i = , i = k b k (cid:16) h Tk w i (cid:17) , (17c) p , k ∆ = K ∑ i = , i = k b k (cid:16) h Tk w i (cid:17) , (17d) r , k ∆ =
12 log + K ∑ i = , i = k b i (cid:16) h Ti w k (cid:17) , (17e) p , k ∆ = K ∑ i = , i = k b i (cid:16) h Ti w k (cid:17) , (17f)hen, the objective function ∑ Kk = (cid:0) r , k − r , k − r , k (cid:1) − µ (cid:16) P DC + ξ Tr (cid:0) WW T (cid:1)(cid:17) is a concave function with respect to W , r , k , r , k , and r , k . Also, (16) can be rewritten asmax W , r , k , r , k , r , k p , k , p , k , p , k K ∑ k = (cid:0) r , k − r , k − r , k (cid:1) − µ (cid:18) P DC + ξ Tr (cid:16) WW T (cid:17)(cid:19) , (18a)s.t. r , k ≤
12 log (cid:0) + p , k (cid:1) , (18b) p , k ≤ K ∑ i = a k (cid:16) h Tk w i (cid:17) , (18c) r , k ≥
12 log (cid:0) + p , k (cid:1) , (18d) p , k ≥ K ∑ i = , i = k b k (cid:16) h Tk w i (cid:17) , (18e) r , k ≥
12 log (cid:0) + p , k (cid:1) , (18f) p , k ≥ K ∑ i = , i = k b i (cid:16) h Ti w k (cid:17) , (18g) r , k − r , k − r , k ≥ λ k , (18h) K ∑ k = | w n , k | ≤ min (cid:16) I DC n , I max − I DC n (cid:17) . (18i)It is seen that constraints (18c), (18d), and (18f) are still notconvex. To cope with this, the first-order Taylor approximationis employed to approximate these non-convex constraints. TheCCCP is then used to iteratively solve a sequence of ap-proximating convex problems until a predefined convergencecriterion is met [14]. Specifically, at the m -th iteration, thefollowing problem is solvedmax W , r , k , r , k , r , k p , k , p , k , p , k K ∑ k = (cid:0) r , k − r , k − r , k (cid:1) − µ (cid:18) P DC + ξ Tr (cid:16) WW T (cid:17)(cid:19) , (19a)s.t. p , k ≤ K ∑ i = a k (cid:18)(cid:16) h Tk w ( m − ) i (cid:17) + (cid:16) w ( m − ) i (cid:17) T h k h Tk (cid:16) w ( m ) i − w ( m − ) i (cid:17)(cid:19) , (19b) r , k ≥
12 log (cid:16) + p ( m − ) , k (cid:17) + (cid:16) p ( m ) , k − p ( m − ) , k (cid:17) (cid:16) + p ( m − ) , k (cid:17) , (19c) r , k ≥
12 log (cid:16) + p ( m − ) , k (cid:17) + (cid:16) p ( m ) , k − p ( m − ) , k (cid:17) (cid:16) + p ( m − ) , k (cid:17) , (19d)(18b) , (18e) , (18g) − (18i) . where w ( m − ) i , p ( m − ) , k and p ( m − ) , k are the solutions obtainedfrom the previous iteration. Problem (19) is convex, thus canbe solved efficiently by using standard optimization softwares , such as CVX [15]. Finally, (18) can be solved by the proposed Algorithm 2 using CCCP.
Algorithm 2:
CCCP-type algorithm for solving (19)Choose the maximum number of iteration L max , andthe error tolerance ε > W ( ) , p ( ) , k , p ( ) , k to (19).Set m = while convergence == False and m ≤ L max , do m : = m + W ( m ) , p ( m ) , k , p ( m ) , k using W ( m − ) , p ( m − ) , k , p ( m − ) , k obtained from the previousiteration. if (cid:13)(cid:13)(cid:13) W ( m ) − W ( m − ) (cid:13)(cid:13)(cid:13) k W ( m ) k ≤ ε then convergence = TrueW ∗ = W ( m ) else convergence = False . endend Return the optimal value W ∗ .IV. S IMULATION R ESULTS AND D ISCUSSIONS
In this section, the performance convergence behaviors ofthe proposed solution are evaluated. Numerical results areobtained through averaging 10 ,
000 different users’ channelrealizations. Unless otherwise noted, the following parametersare used for simulations. LED bandwidth B =
20 MHz,beam angle φ = ◦ , LED conversion factor η = A r = , responsivity γ = . Ψ = ◦ , optical filter gain T s ( ψ ) =
1, refractiveindex of the concentrator κ = .
5, ambient light photocur-rent χ amp = . / ( m · Sr), and preamplifier noise currentdensity, i amp = / Hz − / . Moreover, the data symbols areassumed to be uniformly distributed over [ − , ] . (a) 2 × (b) 2 × (c) 3 × Fig. 2: Different layouts of LED transmitters.Figure 3 illustrates the convergence of the proposed solu-tion. For this purpose, three different scenarios of the numbersof LED transmitters and users, namely ( N T , K ) = ( , ) , ( , ) and ( , ) , are taken into consideration. The configurationsof LED transmitter are illustrated in Fig. 2. Additionally,the average emitted optical power of each LED luminary P sn = η I DC n =
30 dBm, U LED = . P DC,circuitry = ξ = Ω ,and λ k ’s = . W ( ) in Algorithm 2 is chosenrandomly. It is shown that considerably large numbers ofiterations are required for the normalized energy efficiency toconverge (i.e., 60, 75, and 90 iterations for ( N T , K ) = ( , ) , ( , ) and ( , ) , respectively). As we observe that the optimalsolution is usually a near zero-forcing (ZF) precoder, one canchoose the initial point as the ZF precoder to speed up theconvergence. Indeed, simulation results revealed significantimprovements in terms of the required number of iterations.Specifically, using ZF precoder as the initial point, roughly 8,9, and 20 iterations are needed when ( N T , K ) = ( , ) , ( , ) and ( , ) , respectively. Fig. 3: Convergence behaviors of the proposed solution.In Fig. 4, we assess the energy efficiency with respect toeach luminary’s average emitted optical power P sn for differentvalues of circuitry power consumption where ( N T , K ) = ( , ) .It is observed that the energy efficiency first increases withan increase in P sn until its maximum value. It then startsdecreasing as P sn continues to increase. This phenomenoncan be explained as follows. At its low value region, theincrease in the achievable secrecy sum-rate due to increasing P sn is dominant in improving the energy efficiency. When P sn further increases, it becomes the dominant factor that reducesthe energy efficiency as the achievable secrecy sum-rate onlylogarithmically increases with P sn . We also notice that theoptimal point of P sn increases in accordance with P DC, circuitry .V. C
ONCLUSIONS
In this paper, we have studied an energy-efficient precodingscheme for PLS in MU-MISO VLC systems. Due to the non-convex nature of the design problem, Dinkelbach and CCCPalgorithms were employed to find a sub-optimal solution withlower complexity. Numerical results shown that by choosingZF precoders as the initial points for the CCCP algorithm,the rate of convergence of the proposed solution could besignificantly improved. It was also demonstrated that thereexist an optimal value of the average emitted optical powerwhere the energy efficiency achieves its maximum value.
20 25 30 3500.20.40.60.811.2
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