Wireless Information and Energy Transfer for Decode-and-Forward Relaying MIMO-OFDM Networks
aa r X i v : . [ c s . I T ] J a n Volume , Number , Xxxx XXXX pp. – Guanyao Du , Zhilong Dong , Ke Xiong and Zhengding Qiu School of Computer and Information Technology, Beijing Jiaotong UniversityBeijing 100044, P. R. China { } @bjtu.edu.cn State Key Lab. of Scientific and Engineering Computing Academy of Mathematics and Systems Science,Chinese Academy of SciencesBeijing 100190, P. R. [email protected] Author: Ke Xiong
Abstract.
This paper investigates the system achievable rate and optimization forthe multiple-input multiple-output (MIMO)-orthogonal frequency division multiplexing(OFDM) system with an energy harvesting (EH) relay. Firstly we propose a time switching-based relaying (TSR) protocol to enable the simultaneous information processing andenergy harvesting at the relay. Then, we discuss its achievable rate performance theoret-ically and formulated an optimization problem to maximize the system achievable rate.As the problem is difficult to solve, we design an Augmented Lagrangian Penalty Func-tion (ALPF) method for it. Extensive simulation results are provided to demonstrate theaccuracy of the analytical results and the effectiveness of the ALPF method.
Keywords:
Energy harvesting, MIMO-OFDM, Decode-and-forward (DF), AugmentedLagrangian Penalty Function (ALPF). Introduction.
Energy harvesting (EH) has emerged as a promising approach to over-come the limited energy budget of wireless networks [1]-[7] in recent years. Comparedwith conventional EH sources (e.g., solar, wind, thermoelectric effects or other physicalphenomena [1]-[2]), one prospective way is to harvest energy from the ambient radio-frequency (RF) signals [3]-[7], which is referred to as simultaneous wireless informationand energy transfer (SWIET).The idea of SWIET was first proposed in [3], where the performance tradeoff betweenthe energy and information rate was studied. Later, it was widely investigated in variousmodels [4]-[7]. Specifically, in [4], the multi-user orthogonal frequency division multiplex-ing (OFDM) system was considered, where the optimal design of SWIET was obtained.In [5], a three node multiple-input multiple-output (MIMO) broadcasting system wasconsidered, where the rate-energy bound and region were studied.Recently, efforts have been made to apply MIMO and OFDM technologies to wirelesscommunication system in order to support high data rates and provide high spectral ef-ficiency. However, only a few works have investigated the MIMO-OFDM system withSWIET technology [6]-[7]. Specifically, in [6] and [7], a two-hop MIMO-OFDM relayingsystem was considered, where the relay employed an amplify-and-forward (AF) coopera-tive scheme, and the optimum performance boundaries and the rate-energy region wereinvestigated.In this paper, we also focus on the SWIET for a two-hop MIMO-OFDM relaying system,where a source transmits its information to the destination with the help of an energy-constrained relay, as the relay network has great penitential by employing some advancedtechnologies, see e.g., [8, 9]. The main contributions of this paper can be summarized asfollows. Firstly, by applying the time-switching receiver architecture proposed in [10], we
Energy Harvesting at on each subcarrier over MIMO channel Information Transmission on each sub-carrier over MIMO channel T (1 ) / 2 T ! R R S Information Transmission on each sub-carrier over MIMO channel DR (1 ) / 2 T ! Figure 1.
System model and parametersdesign a transmission protocol to enable the simultaneous information processing and EHat the decode-and-forward (DF) relay. Secondely, we discuss the system achievable rateperformance theoretically and formulated an optimization problem to explore the systemperformance limit. Thirdly, as the problem is difficult to solve, we design an AugmentedLagrangian Penalty Function (ALPF) method for it. Finally, extensive simulation resultsare provided to demonstrate the accuracy of the analytical results and the effectivenessof the ALPF method.2.
System Model and Protocol Description.
Assumptions and Notations.
We consider a half-duplex two-hop DF relayingsystem which consists of a source S, a destination D, and an energy-constrained relay R.All nodes are equipped with multiple antennas, and the number of antennas at S, R, andD are denoted by N S , N R and N D , respectively.S has fixed energy supplying and wants to transmit information to D. We assumethat the direct link between S and D is unavailable. A relay R is used to assist theinformation forwarding from S to D. R is energy constrained since it has no internalenergy source. Thus, it relies on external charging. Specifically, R harvests energy fromthe received RF signals transmitted from D, and uses all the harvested energy to assistthe information relaying. We also assume that all nodes have perfect knowledge of thechannels of both hops. Broadband communication OFDM is considered in this model, andthe frequency-selective channel with total frequency band B is divided into K frequency-flat sub-channels. Moreover, we consider the block fading channel, where the channel gainof each sub-channels remains constant during each round of relaying transmission.2.2. The Proposed Time Switching-based Relaying (TSR) Protocol.
Fig. 1 de-picts the main transmission process in the proposed TSR protocol. By considering thetime switching receiver architecture described in [10], the proposed TSR protocol consistsof three phases: the energy transfer phase, the information transmission from S phase andthe information relaying from R phase, as shown in Fig. 1. The time durations assignedto each phase are αT , (1 − α ) T / − α ) T /
2, respectively, where 0 ≤ α ≤ y (EH)R ,k = H ,k x k + n R ,k , where H ,k ∈ C N R × N S denotes the channelmatrix from S to R at hop 1 over the k -th subcarrier, x k denotes the transmitted signalvector for energy transfer over subcarrier k , and n R ,k ∼ CN (0 , σ I N R ) is a N R × k is E R ,k = αη k H ,k x k k , where 0 < η ≤ K subcarrierscan be given by E = P Kk =1 E R ,k , and note that, the total transferred energy is limitedby the available power at S, i.e. P Kk =1 tr( x k x Hk ) ≤ P s . Since all the harvested energy inthe first phase is used to relay the information in the third phase, the available transmitpower at R in the information relaying phase is given by P R = E (1 − α ) T / αη − α K X k =1 k H ,k x k k (1) In the second phase, i.e. the information transmission from S phase, S delivers the signalvector s k ∈ C N S × to R, and the received signal at R over the k -th subcarrier can berepresented as y R ,k = W R ,k H ,k F S ,k s k + n R ,k (2)where E[ s k s Hk ] = I N S , F S ,k ∈ C N S × N S denotes the precoding matrix at S, and W R ,k ∈ C N R × N R is the receiver filter deployed at R to detect the relaying signal. Then, theachievable rate at R is given by R R ,k = log (cid:12)(cid:12) I + W R ,k H ,k F S ,k F H S ,k H H ,k W H R ,k σ − (cid:12)(cid:12) (3)In the third phase, i.e. the information relaying from R phase, R decodes the signal s k from (2) and forwards it to D by multiplying a forwarding matrix F R ,m ∈ C N R × N R . Byconsidering the subcarrier pairing, we assume that the k -th subcarrier over hop-1 in thesecond phase is paired with the m -th subcarrier over hop-2 in the third phase, and callthem subcarrier pair (SP) ( k, m ). Thus, the received signal at D over the SP ( k, m ) canbe expressed as y D ,m = W D ,m H ,m F R ,m s k + n D ,m (4)where H ,m ∈ C N D × N R denotes the channel matrix from R to D at hop 2 over the m -thsubcarrier, and W D ,m ∈ C N D × N D is the receiver filter deployed at D to detect the relayingsignal. n D ,m ∼ CN (0 , σ I N D ) is the N D × R D ,m = log (cid:12)(cid:12) I + W D ,m H ,m F R ,m F H R ,m H H ,m W H D ,m σ − (cid:12)(cid:12) (5)Since the achievable rate for the two-hop relaying system is bounded by the minimumof (3) and (5), the achievable rate over the SP ( k, m ) is given by R k,m = B K · − α R R ,k , R D ,m ) (6)where B denotes the total bandwidth of the OFDM system and (1 − α ) / Achievable Rate Analysis and Optimization Problem Formulation.
By per-forming the singular value decomposition (SVD) on H ,k and H ,m , the MIMO channelsof the two-hop relaying system can be decomposed into multiple parallel independentsubchannels with different gain. Specifically, the SVD of the channel matrices is givenby H i,q = U i,q Λ i,q V Hi,q , where q = k for i = 1, and q = m for i = 2. Both U i,q and V Hi,q are unitary, and Λ i,q ∈ C Rank( H i,q ) × Rank( H i,q ) is a diagonal matrix whose diagonal elements { p λ i,l } Rank( H i,q ) l =1 are nonzero singular values of H i,q in descending order.Due to the full channel state information (CSI) at all the nodes, we can use the SVDof the channel matrices to determine the precoding matrix and receiver filter matricesat the transmitter and receiver. Specifically, we choose the precoding matrix at S, theforwarding matrix at R and the receiver filters deployed at R and D as F S ,k = p P S ,k V ,k , F R ,m = p P R ,m V ,m , W R ,k = U H ,k and W D ,m = U H ,m , respectively, where P S ,k and P R ,m denote the available transmit power at S and R, respectively.Substituting (3) and above designed matrices into (6), the achievable rate over the SP( k, m ) can be rewritten as R k,m = (1 − α ) B K min(log (cid:12)(cid:12) I + P S ,k Λ ,k Λ H ,k σ − (cid:12)(cid:12) , log (cid:12)(cid:12) I + P R ,m Λ ,m Λ H ,m σ − (cid:12)(cid:12) ) (7)Though the above mentioned operations, each SP ( k, m ) is divided into N availableend-to-end (E2E) subchannels, where N denotes the number of available spatial sub-channels per OFDM subcarrier over the two hops, which is bounded to the minimumnumber of spatial subchannels of each hop, i.e., N = min { Rank( H , k ) , Rank( H , m ) } = GUANYAO DU, ZHILONG DONG, KE XIONG AND ZHENGDING QIU min { N S , N R , N D } . Since there are K subcarriers, the total number of effective E2E sub-channels in the MIMO-OFDM system is KN . We introduce the subscript n ∆ = ( k − K + l and n ′ ∆ = ( m − K + l ′ to simplify the notation, where 1 ≤ l, l ′ ≤ N . As a result,1 ≤ n, n ′ ≤ KN , and the above mentioned SP ( k, m ) can be rewritten as SP ( n, n ′ ) whichmeans that the n -th subchannel over hop 1 is paired with the n ′ -th subchannel over hop2. Further, we define P S ,n ∆ = P S µ n , P R ,n ′ ∆ = P R µ n ′ , where µ n and µ n ′ denote the powerallocating factor at S for subchannel n over hop-1 and the power allocating factor at Rfor subchannel n ′ over hop-2, respectively. Consequently, the achievable rate R n,n ′ overSP ( n, n ′ ) can be expressed as R n,n ′ = (1 − α ) B K min(log (1 + P S µ n λ ,n σ ) , log (1 + P R µ n ′ λ ,n ′ σ )) (8)Thus, the achievable rate of the TSR protocol in the DF MIMO-OFDM relaying systemis given by C (TSR) = P KNn =1 P KNn ′ =1 θ n,n ′ R n,n ′ , where θ n,n ′ ∈ { , } denotes the subchannel-paring, and the optimization problem of maximizing the achievable rate for a DF MIMO-OFDM relaying system can be formulated as follows max X S ,µ n ,µ n ′ ,θ n,n ′ ,α C (TSR) s.t. KN X n =1 µ n ≤ , KN X n ′ =1 µ n ′ ≤ , µ n ≥ , µ n ′ ≥ K X k =1 tr( x k x Hk ) ≤ P S , X i (cid:23) KN X n =1 θ n,n ′ = 1 , KN X n ′ =1 θ n,n ′ = 1 , θ n,n ′ ∈ { , } , ≤ α ≤ Specifically, θ n,n ′ = 1 means that the n -th subchannel over hop-1 is paired with the n ′ -thsubchannel over hop-2. Otherwise, θ n,n ′ = 0. Let X k = E { x k x Hk } denote the covariancematrix of x k , X S = { X , X , ..., X k } indicates the energy transfer pattern at S.4. Achievable Rate Optimization.
The main ideas to solve (9) are as follows: Firstly,only energy is delivered in the first phase, which means that only X S needs to be optimizedand it is independent with other variables. Thus, we could design X S independently whichwill not affect the global optimality. Secondly, according to the separation principledesigned in [11], the joint channel pairing and power allocation optimization problemcan be decoupled into two separate sub-problems. So, we can optimize θ independentlywithout considering other variables. Thirdly, based on the optimal X ♯ S and θ ♯ , we proposean Augmented Lagrangian Penalty Function (ALPF) method to jointly optimize µ n , µ n ′ and α to maximize C (TSR) .4.1. Optimal X ♯ S and optimal θ ♯ for TSR. To achieve the maximum energy transfer,all power at S for energy delivery should be allocated to the subcarrier with the maximum k e h ( i )S , k , where i ∈ { , , ..., K } , and e h ( i )S , denotes the first column of matric H S ,i V S ,i . Theorem 4.1.
The optimal subchannel pairing θ ♯ is performed in the order of sortedchannel gain which means that the subchannel with i -th largest channel gain (normalizedagainst the noise power) over hop-1 should be paired with the subchannel with i -th largestchannel gain (also normalized against the noise power) over hop-2. Proof:
According to the separation principle in [11], the joint channel pairing and powerallocation optimization problem can be operated in a separated manner, and the optimalchannel pairing is performed individually at the relay in the order of sorted channel gain.
Joint optimal µ n , µ n ′ and α for TSR. According to the max-flow min-cut theo-rem, the achievable rate of each subchannel is limited by the minimal rate over the twohops. Thus, the rates of two hops are equal to each other when C (TSR) is maximized.Further, by substituting the optimal X ♯ S and θ ♯ obtained from subsection 4.1 into (9), theoriginal optimization problem can be rewritten as follows min µ n ,µ n ′ ,α ( α − B K KN X n =1 log (1 + P S µ n λ ,n σ ) s.t. KN X n =1 µ n ≤ , KN X n ′ =1 µ n ′ ≤ , µ n ≥ , µ n ′ ≥ P S µ n λ ,n σ = P R µ n ′ λ ,n ′ σ , for θ n,n ′ = 1 , ≤ α ≤ It can be observed that the problem in (10) is still a nonlinear and non-convex optimiza-tion problem which is difficult to solve. Here, we introduce an Augmented LagrangianPenalty Function (ALPF) method [12]-[13] to find the joint optimal µ n , µ n ′ and α .The unconstrained augmented Lagrangian penalty function of (10) can be written as P ( x , ν, σ ) = ( α − B K KN X n =1 log (1 + A n µ n ) − ν ( KN X n=1 µ n + S − − ν ( KN X n ′ =1 µ n ′ + S − − X ( n,n ′ ) ν n,n ′ ( A n µ n − α − α B n ′ µ n ′ ) + 12 σ ( KN X n =1 µ n + S − + 12 σ ( KN X n ′ =1 µ n ′ + S − + X ( n,n ′ ) σ n,n ′ ( A n µ n − α − α B n ′ µ n ′ ) (11) where A n = P S λ ,n σ , B n ′ = P R λ ,n ′ σ , x = ( α, µ n , µ n ′ , S , S ), S and S are positive slackvariables, ν and σ denote the Lagrangian multipliers and the penalty parameter, respec-tively. We also define the constraint violation function as C ( − ) ( x ) = ( P KNn =1 µ n + S − , P KNn ′ =1 µ n ′ + S − , A n µ n − α − α B n ′ µ n ′ )In the k -th iteration, with ν ( k ) i and σ ( k ) i , x ( k +1) can be obtained by x ( k +1) = arg min P ( x ( k ) , ν ( k ) , σ ( k ) ) (12)Then the lagrange multipliers can be updated as ν ( k +1)1 = ν ( k )1 − σ ( k )1 ( P KNn =1 µ ( k +1) n + S ( k +1)1 − ν ( k +1)2 = ν ( k )2 − σ ( k )2 ( P KNn ′ =1 µ ( k +1) n ′ + S ( k +1)2 −
1) and ν ( k +1) n,n ′ = ν ( k ) n,n ′ − σ ( k ) n,n ′ ( A n µ ( k +1) n − α ( k +1) − α ( k +1) B ( k +1) n ′ µ ( k +1) n ′ ). And the penalty parameters can be updated by σ ( k +1)1 = ( σ ( k )1 , if | P KNn =1 µ ( k +1) n + S ( k +1)1 − | ≤ | P KNn =1 µ ( k ) n + S ( k )1 − | , max { σ ( k )1 , k } , otherwise. (13) σ ( k +1)2 = ( σ ( k )2 , if | P µ ( k +1) n ′ + S ( k +1)2 − | ≤ | P µ ( k ) n ′ + S ( k )2 − | , max { σ ( k )2 , k } , otherwise. (14) σ ( k +1) l,l ′ = ( σ ( k ) n,n ′ , if | ( A n µ ( k +1) n − α ( k +1) − α ( k +1) B ( k +1) n ′ µ ( k +1) n ′ ) | ≤ | ( A n µ ( k ) n − α ( k ) − α ( k ) B ( k ) n ′ µ ( k ) n ′ ) | , max { σ ( k ) n,n ′ , k } , otherwise. (15) We present the main steps of ALPF method as shown in Algorithm 1.
GUANYAO DU, ZHILONG DONG, KE XIONG AND ZHENGDING QIU φ A c h i e v ab l e R a t e ( bp s ) TS numericalTS simulationnon−optimizedPs=70dBmPs=50dBmPs=30dBm (a) φ O p t i m a l α TS P s =30dBmTS P s =50dBmTS P s =70dBm (b) Figure 2. (a) Optimal system achievable rate: numerical vs simulation(b) Optimal α vs φ Algorithm 1
ALPF Algorithm Initialize x (0) , ν (0) i and σ (0) i . k = 0 denotes the number of iterations. Solve (12) through the projected-gradient method to obtain x ( k +1) .If k C ( − ) ( x ( k +1) ) k ∞ ≤ ǫ , algorithm ends. Otherwise, go to Step 3. Update the penalty parameters in terms of (13)-(15). Update the Lagrange multipliers. k = k + 1, and return to step 2.5. Numerical Example.
In this section, we provide some numerical results. The dis-tance between S and D, which is denoted as d SD , is used to be the reference distance, andthe path loss factor is set to be 4. The variable φ ∈ (0 ,
1) denotes the ratio of the distancebetween S and R, i.e. d SR = φd SD . Unless specifically stated, we set η = 1, and the totalsystem bandwith is set to be B = 1kHz, so that each subcarrier is allocated with 1 /K kHz.The total receiving noise at R and D over the total bandwidth is set to be 10 − W, so thatthe noise over each subchannel is set to be σ = σ = − K W. In Algorithm 1, the initialparameters are set as x (0) = (0 . , /KN, /KN, . , . ν (0) i = 0 and σ (0) i = 1.To show the performance gain of the optimized TSR, we also show the results of anon-optimized scheme as a benchmark. In the non-optimized scheme, α is set to be 0.5, µ n and µ n ′ are set as µ n = λ ,n P KNn =1 λ ,n and µ n ′ = λ ,n ′ P KNn ′ =1 λ ,n ′ , which is proportional to thesingular value of each sub-channel.Fig. 2(a) verifies our theoretical analysis and the proposed ALPF algorithm. It canbe seen that, for different available power P S at S, the simulation results closely matchwith the numerical results. Moreover, the system achievable rate achieves higher with theincrement of P S due to the fact that high P S leads to high SNR for the system. Fig. 2(a)also shows the effect of relay location on the system achievable rate. Specifically, as φ increases, the achievable rates first decrease and then increase, and achieve the minimumwhen the relay is deployed in the middle of S and D. Fig. 2(b) shows the optimal α versus φ . It can be observed that, as φ increases, the optimal α first increases and then decreases.This is due to the fact that, when R is far away from S, the energy harvesting efficiencybecomes lower, R needs higher α to collect enough energy to decode the information fromS. And when R is close to D, the channel quality of R-D link gets better, R needs lessenergy to relay the information for D, which makes α get lower.Fig. 3 shows the effect of the number of antenna N on the system achievable rate. Inthe simulations, K is set to be 2, and N increases from 2 to 6. It can be observed that N O p t i m a l A c h i e v ab l e R a t e ( bp s ) TSnon−optimized
Figure 3.
System performance vs the number of antennas N the system achievable rate increases as the number of antennas increases. This is due tothe fact that more antennas yields more spacial subchannels, thus higher multiplex gainover subchannels can be achieved.6. Conclusions.
In this paper, we investigated the DF MIMO-OFDM relaying systemwith an EH relay. We first proposed a TSR protocol for the MIMO-OFDM relayingsystem. Then, in order to explore the system performance limit, we formulated an op-timization problem to maximize the system achievable rate, and we also proposed anALPF method to solve it. Moreover, the effects of the relay location and the number ofantennas on the system performance were also discussed. Numerical results verify ourtheoretical analysis on the system achievable rate and the effectiveness of our proposedALPF method. In the future, the performance of SWIET in the DF MIMO-OFDMrelaying system will be discussed for the case that the CSI is imperfect.
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