Spectral- and Energy-Efficient Transmission Over Frequency-Orthogonal Channels

Transcription

1 Spectral- and Energy-Efficient Transmission Over Frequency-Orthogonal Channels Liang Dong Department of Electrical and Computer Engineering Baylor University Waco, Texas 76798, USA liang November 21, 2018 IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

2 Introduction Green Communications Systems and Networks Spectral efficiency allows the network to maximize utilization of assigned frequencies to provide better services for more users. It is measured as the communication data rate per unit bandwidth used. Energy Efficiency Spectral Efficiency Energy efficiency allows the network to minimize energy consumption for transferring a certain amount of information. It is measured as the communication data rate per unit of power. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

3 Introduction 1. We are interested in frequency-orthogonal channels, e.g, orthogonal frequency division multiplex access (OFDMA) networks which is desirable to simplify the receiver design. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

4 Introduction 1. We are interested in frequency-orthogonal channels, e.g, orthogonal frequency division multiplex access (OFDMA) networks which is desirable to simplify the receiver design. 2. In general, parallel Gaussian broadcast channels. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

5 Introduction 1. We are interested in frequency-orthogonal channels, e.g, orthogonal frequency division multiplex access (OFDMA) networks which is desirable to simplify the receiver design. 2. In general, parallel Gaussian broadcast channels. 3. The maximum communication data rate is characterized by the sum capacity of channels from the transmitter to the multiple receivers. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

6 Introduction 1. We are interested in frequency-orthogonal channels, e.g, orthogonal frequency division multiplex access (OFDMA) networks which is desirable to simplify the receiver design. 2. In general, parallel Gaussian broadcast channels. 3. The maximum communication data rate is characterized by the sum capacity of channels from the transmitter to the multiple receivers. 4. The amount of total bandwidth and transmit power needs to be managed to maximize the spectral efficiency and the energy efficiency. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

7 Introduction 1. We are interested in frequency-orthogonal channels, e.g, orthogonal frequency division multiplex access (OFDMA) networks which is desirable to simplify the receiver design. 2. In general, parallel Gaussian broadcast channels. 3. The maximum communication data rate is characterized by the sum capacity of channels from the transmitter to the multiple receivers. 4. The amount of total bandwidth and transmit power needs to be managed to maximize the spectral efficiency and the energy efficiency. 5. There is a minimum rate requirement for each user. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

8 Introduction Background: Xiong et al. (2011) established a spectral efficiency energy efficiency tradeoff framework in downlink OFDMA networks. Deng et al. (2013) formulated the spectral efficiency energy efficiency optimization problem as a multi-objective optimization problem and then converted it into a single-objective optimization problem. Tang et al. (2014) proposed resource efficiency as a combined metric of spectral efficiency and energy efficiency in an OFDMA network with different transmission-bandwidth requirements. Tsilimantos et al. (2016) studied the spectral efficiency-energy efficiency tradeoff in the cellular network downlink over orthogonal channels. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

9 Introduction What is missing in the current research work is either a complete closed-form solution or a simple algorithm to find bandwidth and transmit power for maximum efficiency. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

10 System Model Frequency-orthogonal parallel broadcast: Base Station User Equipment 1 r 1 r K User Equipment K r 2 User Equipment 2 Maximum achievable data rate ( k K): ( r k = w k log p k h k 2 ) ( = w k log w k N 0 + I p ) kg k k w k where g k is the gain-to-interference-plus-noise-density ratio. g k is the channel quality that is known. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

11 System Model Frequency-orthogonal parallel broadcast: Base Station User Equipment 1 r 1 r K User Equipment K r 2 User Equipment 2 The kth active UE has a minimum rate requirement R k, such that r k R k, k K. İEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

12 System Model Spectral Efficiency Energy Efficiency Γ SE R W Γ EE R P R = k K r k W = k K w k W M W is the total assigned bandwidth in the cell. W M is the maximum allowed bandwidth for BS transmission. P = k K p k P M P is the total transmit power. P M is the maximum transmit power that the BS can deliver. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

13 Problem Formulation Problem of spectral- and energy-efficient transmission P 1 : maximize Γ W,P,{w k },{p k } SE, Γ EE ) subject to w k log 2 (1 + p kg k w k R k, k K k K w k = W W M k K p k = P P M. A multi-objective optimization problem. 1. Optimization of bandwidth assignment with a fixed power allocation. 2. Optimization of transmit power allocation with a fixed bandwidth assignment. 3. Joint bandwidth assignment and transmit power allocation. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

14 1a. Optimal Bandwidth Assignment With Fixed Transmit Power Allocation Problem of optimal bandwidth assignment P 2 : ) maximize Γ {w k } SE (W ) = 1 W k K w k log 2 (1 + p kg k ) w k subject to w k log 2 (1 + p kg k w k R k, k K k K w k = W W M. Optimal Solution: ( ŵ k = ˇw k + q k ρ ˇw ) + k, k K q k = p k g k q k ρ w 1/q 1 w 3/q 3 w 4/q 4 ˇw k minimum required bandwidth q 1 q 2 q 3 q 4 IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

15 1a. Optimal Bandwidth Assignment Define Set I The set of UEs that are assigned with their minimum required bandwidths, i.e., ŵ i = ˇw i, i I K. The maximum spectral efficiency is ˆΓ SE (W ) = { R0 /W 0 1 W [ i I R i + (W i I ˇw i) log 2 ( ρ )], W = W 0, W > W 0 where W 0 = k K ˇw k R 0 = k K R k ρ = W i I ˇw i j K\I q j IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

16 1a. Optimal Bandwidth Assignment Given a total bandwidth W, the optimal bandwidth assignment can be calculated as ŵ i = ˇw i, i I K ( q j ŵ j = j K\I q W j i I ) ˇw i, j K \ I. ρ w 1/q 1 w 3/q 3 w 4/q 4 q 1 q 2 q 3 q 4 IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

17 1a. Optimal Bandwidth Assignment 16 R (W ) R(W ) I Γ SE (W ) is continuously differentiable, and it is either strictly decreasing or strictly quasiconcave in W W0. I The maximum achievable sum rate R (W ) = W Γ SE (W ) is continuously differentiable, strictly increasing and concave in W W Γ SE (W ) ΓSE (W ) W Figure: Sum rate and spectral efficiency with various bandwidth assignments and a fixed transmit power allocation. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

18 1b. Find Total Bandwidth for Best Spectral Efficiency As ˆΓ SE (W ) is continuously differentiable, its first derivative is given by ( ) dˆγ SE (W ) d ˆR(W ) = = d ˆR(W ) 1 dw dw W dw W ˆR(W ) W W 0 ln 2 Φ(ρ 0) 1 R W 2 0, W = W 0 [ 0 ) 1 = W 2 i ˇw j i log 2 (1 + q j W i ˇw i ] i R W j i q j ( j q j+w i ˇw, W > W i) ln 2 0 where Φ(x) = ln(1 + 1/x) 1/(1 + x) ρ 0 = min k K ( ˇw k/q k ). IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

19 1b. Find Total Bandwidth for Best Spectral Efficiency Water-level Analysis Define ρ k = ˇw k /q k ( k K) as the water-level marks. Let {ρ k } K k=1 be sorted in ascending order and denoted as ρ 1 ρ 2 ρ K, where ρ k corresponds to ˇw k, q k and R k, i.e., ρ k = ˇw k /q k. The total bandwidth starts at the minimum level of W 0 = K i=1 ˇw i. According to the bandwidth water-filling solution, the critical levels of W are the ones at which the water level reaches the water-level marks. In an ascending order, the critical levels are given by W J 1 = K i=j+1 J ˇw i + ρ J q j, J = 1, 2,..., K. j=1 When W J 1 W < W J, the K J UEs that correspond to ρ J+1, ρ J+2,..., ρ K are in Set I. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

20 1b. Find Total Bandwidth for Best Spectral Efficiency At the critical bandwidth levels W J 1, J = 1, 2,..., K, the maximum spectral efficiency and its derivative are given by ˆΓ SE (W J 1 ) = 1 K Jj=1 R i q j + R J W J 1 dˆγ SE (W ) dw = 1 W 2 J 1 W =WJ 1 ( K i=j+1 i=j+1 ˇw i ˇw R J J K i=j+1 R i q J W J 1 (1+ρ J ) ln 2 ). IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

21 1b. Find Total Bandwidth for Best Spectral Efficiency When dˆγ SE (W )/dw W =WJ 1 0 and dˆγ SE (W )/dw W =WJ < 0, the optimal total bandwidth is in interval [W J 1, W J ). With the bisection method, the optimal total bandwidth W opt can be found as the root of Θ(W ) ( ) Θ(W ) = ˇW Q W Q I ln 1 + W ˇW R I ln 2 I Q + W ˇW I where ˇW I = R I = K ˇw i = J ˇw i, Q = q j = q j, i=j+1 i I j=1 j K\I K R i = R i. i=j+1 i I IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

22 Spectral-Efficient Transmission Algorithm Find optimal total bandwidth and its assignment that maximize Γ SE 0. If there is no minimum rate requirement of any UE, i.e., ˇw k = 0, k K, the problem is invalid because ˆΓ SE (W ) is strictly decreasing in W. Otherwise, there are some non-zero { ˇw k }. Calculate ˇw k from R k, k K; 1. Calculate ρ k = ˇw k /q k, where q k = p k g k ; 2. Sort {ρ k } in ascending order as {ρ k }K k=1 ; 3. Check the sign of dˆγ SE (W )/dw W =W0 : If ρ 1 = 0, the derivative is positive. Otherwise, the sign is Sgn[W 0 Φ(ρ 1 ) R 0 ln 2]. If dˆγ SE (W )/dw W =W0 0, W opt = W 0, go to Step 7; IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

23 Spectral-Efficient Transmission Algorithm Find optimal total bandwidth and its assignment that maximize Γ SE 4. Calculate the critical levels of total bandwidth W J 1, J = 1, 2,..., K. The bandwidth levels can be neglected if they are beyond W M ; 5. Calculate dˆγ SE (W )/dw W =WJ 1, starting with J = 2 and stopping when the derivative is negative; 6. When dˆγ SE (W )/dw W =WJ 1 0 and dˆγ SE (W )/dw W =WJ < 0, the optimal total bandwidth is in interval [W J 1, W J ). Establish Set I. Find W opt as the root of Θ(W ) using the bisection method; 7. With W opt, calculate the optimal bandwidth assignment and the maximum spectral efficiency. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

24 Simulation Results ˆΓSE(W) dˆγse(w)/dw Zero Crossing 0 W 0 W 1 W Figure: Search of the total bandwidth that maximizes the spectral efficiency. Fixed transmit power allocation. W IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

25 2a. Optimal Transmit Power Allocation With Fixed Bandwidth Assignment Problem of transmit power allocation P 3 : Optimal Solution: maximize Γ {p k } EE (P ) = 1 P k K w k logw 2 (1 + p kg k 1/q 1 ) subject to p k w R k q 2 k w g k (2 k 1, k K k K p k = P P M. h ) w 3/q 3 w k q 1 q 3 q 4 w 4/q 4 ˆp k = ˇp k + w k ( 1 µ α k) +, k K ) ˇp k = w R k k w g k (2 k 1 1/µ p 3/w 3 p 1/w 1 α k = 1 g k + ˇp k w k water-level marks w 1 w 2 w 3 w 4 IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34 1/g 1 1/g 2 1/g 3 p 4/w 4 1/g 4

26 2a. Optimal Transmit Power Allocation Define Set I The set that contains the UEs transmitting with their minimum required power, i.e., ˆp i = ˇp i, i I K. The maximum energy efficiency with a total transmit power P is given by ˆΓ EE (P ) = 1 P R i + i I j K\I ( ) gj w j log 2 µ where 1 µ = P P 0 + P 0 = ˇp k. k K j K\I w jα j j K\I w j IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

27 2a. Optimal Transmit Power Allocation 70 R(P ) 60 R (P ) ΓEE (P ) 2 45 I Γ EE (P ) Γ EE (P ) is continuously differentiable, and it is either strictly decreasing or strictly quasiconcave in P P0. 45 P Figure: Sum rate and energy efficiency with various transmit power allocations and a fixed bandwidth assignment. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

28 2b. Find Total Transmit Power for Best Energy Efficiency Water-level Analysis Let {α k } K k=1 be sorted in the ascending order and denoted as α 1 α 2 α K, where α k corresponds to g k, ˇp k and w k, i.e., α k = 1/g k + ˇp k /w k. As P increases from P 0, according to the transmit power water-filling solution, the critical levels of P are the ones at which the water level 1/µ reaches α 1, α 2,..., α K in its order. In an ascending order, the critical levels are given by J 1 P J 1 = P 0 + (α J α i)w i, J = 1, 2,..., K. i=1 IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

29 2b. Find Total Transmit Power for Best Energy Efficiency At the critical levels of total transmit power P J 1, J = 1, 2,..., K, the maximum energy efficiency is given by ˆΓ EE (P J 1 ) = R 0 + J 1 i=1 w i log 2(α J /α i ) P 0 + J 1 i=1 (α J. α i )w i The derivative of ˆΓ EE (P ) is given by dˆγ EE (P ) dp = d dp ( ) ˆR(P ) P = d ˆR(P ) dp 1 P ˆR(P ) P 2. As P > 0, the sign of dˆγ EE (P )/dp is determined by the sign of Λ(P ) Λ(P ) = d ˆR(P ) dp ˆΓ EE (P ). IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

30 2b. Find Total Transmit Power for Best Energy Efficiency Let Λ J 1 denote Λ(P J 1 ) with critical power level P J 1, J = 1, 2,..., K. Λ J 1 = 1 ( ) 1 1 P J 1 ln 2 ˆΓ EE (P J 1 ). α J If Λ 0 0, the optimal total transmit power to maximize energy efficiency is P opt = P 0 with transmit power allocation ˆp k = ˇp k, k K. In the situation when Λ J 1 > 0 and Λ J 0, J UEs with smallest α s are assigned excess transmit power beyond their minimum required power to achieve maximum energy efficiency. The optimal total transit power is in the interval (P J 1, P J ]. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

31 2b. Find Total Transmit Power for Best Energy Efficiency The maximum energy efficiency can be written as ˆΓ EE (P ) = 1 P ( A + W ) J log 2 (P B), P (P J 1, P J ] where W J = J w i i=1 A = J R 0 w i log 2 α i W J log 2 WJ i=1 B = J P 0 w iα i. i=1 IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

32 2b. Find Total Transmit Power for Best Energy Efficiency The derivative of ˆΓ EE (P ) in this interval is given by dˆγ EE (P ) dp = W J P (P B) ln 2 A + W J log 2 (P B) P 2. The sign of the derivative is determined by the sign of Θ(P ) Θ(P ) = P W J P B A ln 2 W J ln(p B). As Θ(P J 1 ) > 0 and Θ(P J ) 0, the root of Θ(P ), P (P J 1, P J ] can be found using the bisection method. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

33 Energy-Efficient Transmission Algorithm Find optimal total transmit power and its allocation that maximize Γ EE 0. Calculate minimum transmit power ˇp k from minimum rate requirement R k, k K; 1. Calculate α k = 1 g k + ˇp k w k ; 2. Sort {α k } in ascending order as {α k }K k=1 ; 3. Calculate the critical levels of total transmit power P J 1, J = 1, 2,..., K. The power levels can be neglected if they are beyond P M ; IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

34 Energy-Efficient Transmission Algorithm Find optimal total transmit power and its allocation that maximize Γ EE 4. Calculate ˆΓ EE (P J 1 ) and Λ J 1, starting with J = 1 and stopping when Λ is negative; 5. If Λ 0 0, P opt = P 0, go to Step 7; 6. When Λ J 1 > 0 and Λ J 0, the optimal total transmit power is in interval (P J 1, P J ]. Establish Set I. Find P opt as the root of Θ(P ) using the bisection method; 7. With P opt, calculate the optimal transmit power allocation and the maximum energy efficiency. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

35 Simulation Results ˆΓEE(P) Θ(P) Zero Crossing 0-10 P 0 P 1 P P Figure: Search of the total transmit power that maximizes the energy efficiency. Fixed bandwidth assignment. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

36 3. Joint Bandwidth Assignment and Power Allocation The optimum sum rate ˆR(W, P ) is achieved as follows. 1. The K 1 UEs with channel qualities g i, i = 2, 3,..., K, are transmitted to at their corresponding minimum required rates R i, i = 2, 3,..., K. All of the remaining resources of the spectrum and the transmit power is used for transmission to the one UE with the best channel quality g The maximum combined efficiency is ( 1 P + γ W ) ˆR(W, P ) = ( 1 P + γ ) ( K R i + w ( 1 W ln 2 i=2 W 0 ( ψg1 1 e ) ) ) + 1 where W 0 ( ) is the principal branch of the Lambert W function. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

37 Simulation Results Optimal ˆΓEE 3 Maximum Combined Efficiency W P Figure: Maximum combined efficiency ( ˆΓ EE ) as γ = Solid red curve indicates the peak ˆΓ EE with optimal total P. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34

38 Conclusion Over frequency-orthogonal broadcast channels, the problem of spectral- and energy-efficient transmission is formulated with maximum bandwidth and transmit power constraints and minimum rate requirements of the individual users. With a fixed transmit power allocation or a fixed bandwidth assignment, the problem is separated into two convex optimization problems. For each problem, the optimal bandwidth assignment or the optimal transmit power allocation is given by a water-filling solution. Effective procedures are provided to find total bandwidth W opt that maximizes the spectral efficiency and total transmit power P opt that maximizes the energy efficiency. IEEE Online Green Communications Spectral- and Energy-Efficient Transmission November 21, / 34