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MS. THESIS

Joint User Scheduling and Power Allocation for Energy Efficient Millimeter Wave NOMA Systems

밀리미터파 비직교 다중접속 시스템에서 사용자 스케줄링과 전력 할당

BY

Sunyoung Lee

AUGUST 2018

DEPARTMENT OF ELECTRICAL AND

COMPUTER ENGINEERING

COLLEGE OF ENGINEERING

SEOUL NATIONAL UNIVERSITY

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i

Abstract

Joint User Scheduling and Power Allocation for Energy Efficient Millimeter Wave NOMA

Systems

Sunyoung Lee Department of Electrical and Computer Engineering The Graduate School Seoul National University

Non-orthogonal multiple access (NOMA) and millimeter wave (mmWave) communications are promising technologies for the fifth generation (5G) wireless communication systems. NOMA is able to serve multiple users in the same resource block by exploiting successive interference cancellation (SIC). MmWave communications can use wide bandwidth available in the mmWave frequency band.

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ii

In this thesis, we investigate the user scheduling and power allocation scheme for a mmWave NOMA system. To reduce the feedback overhead, random beamforming is adopted at a base station. The optimization problem is formulated to maximize the energy efficiency. To solve this problem, we first address the user scheduling problem and power allocation problem separately, then an iterative algorithm is proposed to jointly optimize the user scheduling and power allocation. Simulation results show that the proposed scheme achieves higher energy efficiency than the conventional scheme.

Keywords: NOMA, mmWave, random beamforming, resource allocation, power allocation, user scheduling, energy efficiency.

Student Number: 2016-24274.

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List of Figures

Figure 2.1 Downlink mmWave NOMA system. . . 27

Figure 3.1 A system diagram for the addressed mmWave NOMA

system. . . 16

Figure 4.1. Energy efficiency versus maximum transmission power P_max for proposed and conventional schemes. . . 31

Figure 4.2. Energy efficiency versus maximum transmission power P_max for different values of the number of beams M. . . 34

Figure 4.3. Energy efficiency versus maximum transmission power P_max for

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different values of the number of antennas N. . . 35

Figure 4.4. Energy efficiency versus maximum transmission power P_max for different values of the number of antennas N. . . 36

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1

Chapter 1

Introduction

Non-orthogonal multiple access (NOMA) has been recognized as a promising candidate for the fifth generation (5G) wireless communication systems. In NOMA, multiple users are served in the same resource block by applying power domain multiplexing at the transmitter and successive interference cancellation (SIC) at the receiver [1], [2]. Since the communication resources are shared by users, NOMA improves the spectral efficiency compared with orthogonal multiple access [3].

Recently, multiple input multiple output (MIMO) has been applied to

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NOMA systems to further increase spectral efficiency. In a MIMO-NOMA system, users are paired into clusters and users in each cluster share the same beamforming vector. The performance is enhanced when users with high channel correlation are paired into a cluster [4].

Millimeter wave (mmWave) communication is another promising technology for 5G wireless communication systems. MmWave communication operates in the band of 30-300 GHz, where the available bandwidths are much wider than the microwave bands used in current wireless communications [5]. However, mmWave signals suffer from severe path loss compared to microwave signals. To compensate the large path loss, proper beamforming schemes are needed [6].

The use of NOMA in mmWave communications is desirable due to the highly directional nature of mmWave propagation, which makes users' channel highly correlated [7]. Furthermore, due to the large bandwidth available at mmWave frequencies, mmWave NOMA system can achieve high capacity.

Most of previous works on the coexistance of NOMA and mmWave communications focus on the spectral efficiency [7]-[9]. In [7], the sum rate and outage probabilities were analyzed for mmWave NOMA systems when random beamforming is used at the BS. In [8], the capacity of mmWave massive MIMO NOMA systems was analyzed in the low signal-to-noise ratio (SNR) and high SNR regimes. In [9], user scheduling and power allocation

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schemes are proposed to maximize the spectral efficiency of mmWave NOMA systems.

As the energy consumption of wireless communications increases due to the explosive growth of data traffic [10], an energy efficient resource allocation is needed. There has been few works on the energy efficiency of mmWave NOMA systems. In [11], an energy efficient power allocation scheme is proposed for mmWave massive MIMO NOMA systems.

In this thesis, we investigate a joint user scheduling and power allocation for a mmWave NOMA system to maximize the energy efficiency. A base station (BS) transmits signals to users by NOMA and adopt random beamforming to reduce the channel feedback overhead [12]. We formulate the joint user scheduling and power allocation optimization problem with the objective of maximizing energy efficiency under the quality-of-service (QoS) constraints, SIC constraints, and the transmission power constraint. To solve this challenging problem, we first decouple the problem into two subproblems, the user scheduling problem and the power allocation problem. For the user scheduling problem, a suboptimal algorithm is proposed to reduce the complexity. For the power allocation problem, the problem is approximated and reformulated into a convex problem. Then an iterative algorithm is proposed to obtain the optimal solution. We jointly optimize the user scheduling and power allocation by solving the user scheduling and the power allocation subproblems iteratively.

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The rest of this thesis is organized as follows. In chapter 2, the system model and channel model are described. In chapter 3, the optimization problem is formulated and a joint user scheduling and power allocation algorithm is proposed. In chapter 4, simulation results are shown. Finally, conclusions are drawn in chapter 5.

Equation Section (Next)

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5

Chapter 2

System Model

Consider a downlink mmWave NOMA system which consists of one BS and K users u_k, k1, 2, ,K . The BS has a uniform linear array (ULA) with N antennas, and each user has a single antenna. Suppose that the BS forms M beams where MN and MK/ 2 . Suppose that a user is scheduled on at most one beam and a beam serves at most two users. When two users are scheduled on a beam, the users are served by NOMA.

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6

BS

Figure 2.1. Downlink mmWave NOMA system

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7

2.1 Channel Model

As discussed in [13], [14], the mmWave channel has a characteristic of limited scattering to have a few number of paths. We adopt a mmWave channel model with L paths including a line-of-sight (LOS) path [15]. The channel vector between the BS and u_k is given by

, ,

1 ,

( ),

L k l

k k l

l k l

N a

L g 





h a (2.1)

where l1 for the LoS path, l1 for non-line-of-sight (NLoS) paths, and

,

ak l is the small scale fading gain which is distributed according to (0,1).

,

gk l denotes the path loss, which is given by

, 10 log (10 ) [dB],

k l k

g    d  (2.2)

where d_k is the distance between the BS and u_k,  ~ (0,²), and ,

 ,  are parameters which depend on whether the path is LoS or NLoS.

(_{k l},)

a denotes the array response vector which is given by

, ( 1) ,

,

( _{k l}) 1 [1,e^j ^{k l}, ,e^{j N} ^{k l}] ,^T N

 

  ^

a (2.3)

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where _{k l}_,  [ 1,1] is the normalized angle of departure (AoD) for the l-th path of the channel between the BS and u_k.

2.2 Random Beamforming

Suppose that random beamforming is adopted at the BS that does not require full channel state information (CSI) of all users. The random beamforming vector at the BS is given by [7], [16]

2( 1)

, 1, , ,

m

m m M

 M^

 

    

w a (2.4)

where  is a random variable uniformly distributed over [-1,1]. For simplicity, let _m denote the direction of the m -th beam, i.e.,

2( 1)

m

   M^ .

Suppose that each user knows the beamforming vectors so that it feeds back the effective channel gains for all beams, {|h w^H_k _m| |² m1, 2, ,M}, instead of full CSI.

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2.3 Data Transmission Model

For two users scheduled on each beam, the user with larger and smaller effective channel gains are referred to as the strong user and the weak user, respectively. Let _{k m}^{( )}ⁱ_, , k 1, 2, ,K , i1, 2 , m1, 2, ,M , denote the scheduling indicators. The indicator _{k m}⁽¹⁾_, 1 if u_k is scheduled for the strong user of the m -th beam and _{k m}⁽¹⁾_, 0 otherwise. The indicator

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, 1

k m  if u_k is scheduled for the weak user of the m -th beam and

( 2)

, 0

k m  otherwise. The transmit signal at the BS is given by

2

( )

, ,

1 1 1

,

M K

i

n j n n i j

n j i

 P s

  





x w (2.5)

where s_k is the data symbol transmitted tou_k , P_m_,1 and P_m_,2 are the transmission power allocated to the strong user and the weak user of the m -th beam, respectively.

Suppose that u_k is scheduled on the m-th beam. The received signal at uk is given by

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10

2 2

( ) ( )

, , , ,

1 1 1

disred signal

intra-beam interference 2

( )

, ,

1 1 1

inter-beam interference

,

K

H i H i

k k m k m m i k k m j m m i j

i j i

j k

M K

H i

k n j n n i j k

n j i

n m

y P s P s

P s n

 



  



  







 

 



h w h w

h w

(2.6)

where n_k is an additive white Gaussian noise with zero-mean and variance

2 . When u_k is the strong user of the m -th beam, the signal-to- interference-plus-noise ratio (SINR) for u_k to decode the weak user's signal is given by

2 (2)

, ,2

1 2 1

, 2

2 (1) 2 ( ) 2

, ,1 , ,

1 1 1

| |

.

| | | |

K H

k m j m m

j j k

k m M K

H H i

k m k m m k n j n n i

n j i

n m

P

P P





  





  





 





h w

h w h w

(2.7)

The decoding rate for u_k to decode the weak user's signal is given by

2 1 2 1

, log (12 , ).

k m k m

R ^   ^ (2.8)

If R_{k m}²_,^¹ is higher than the target rate R_min , u_k decodes the weak user's signal successfully [17]. Removing the weak user's signal by successive interference cancellation (SIC), the SINR and data rate of u_k are given by

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11

2 (1)

, ,1

1

, 2

2 ( ) 2

, ,

1 1 1

| |

H

k m k m m

k m M K

H i

k n j n n i

n j i

n m

P P

 

 

  







 h w

h w

(2.9)

and

1 1

, log (12 , ),

k m k m

R   (2.10)

respectively.

When u_k is the weak user of the m-th beam, the SINR and data rate of uk are given by

2 (2)

, ,2

2

, 2

2 (1) 2 ( ) 2

, ,1 , ,

1 1 1 1

| |

| | | |

H

k m k m m

k m K M K

H H i

k m j m m k n j n n i

j n j i

j k n m

P

P P

 

  

   

 



 

 

h w

h w h w

(2.11)

and

2 2

, log (12 , )

k m k m

R   (2.12)

respectively.

The energy efficiency of the system is given by

2 ,

1 1 1

2 ,

1 1

( , ) ,

M K

i k m

m k i

M

c m i

m i

R

P P

 ^ ^ ^

 







ρ P



(2.13)

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12

where P_c is the circuit power consumption, ρ ^{K M}^{ }² is the user scheduling matrix whose ( , , )k m i -th element is _{k m}^{( )}ⁱ_, , and P ^M² is the power allocation matrix whose ( , )m i -th element is P_{m i}_, .

Equation Section (Next)

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13

Chapter 3

Energy Efficient User Scheduling and Power Allocation

For joint user scheduling and power allocation, the optimization problem to maximize the energy efficiency is formulated as follows.

max,

: ( , )

ρ P ρ

1 P

P (3.1)

( )

, , min

s.t. R_{k m}ⁱ _{k m}ⁱ R , k ,m ,i{1, 2}, (3.2)

2 1 (1)

, , min, , ,

k m k m

R ^  R k m (3.3)

2

, max

1 1

,

M

m i m i

P P

 



 ^(3.4)

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14

2 ( )

,

1 1

2, ,

K i k m k i

 m

 

 



^(3.5)

2 ( )

,

1 1

1, ,

M i k m m i

 k

 

 



^(3.6)

( )

, {0,1}, , ,

i

k m k m

    (3.7)

where {1, 2, K} , {1, 2, M} , and P_max is the maximum transmission power of the BS. (3.2) is the QoS requirement for users, (3.3) is the SIC constraint, and (3.4) is the total transmission power constraint.

(3.5) is a constraint that at most two users are scheduled on a beam, and (3.6) is a constraint that each user is scheduled on at most one beam.

The joint optimization problem P1 is a mixed-integer programming which is difficult to solve [18]. To obtain a solution for this problem, we decouple the problem into two subproblems: a user scheduling problem for the given power allocation and a power allocation problem for the given user scheduling. We first address two subproblems separately, then propose an algorithm in which user scheduling and power allocation are performed iteratively to obtain a solution for the joint optimization problem.

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15

3.1 User Scheduling

Obtaining an optimal solution of user scheduling problem by exhaustive search requires high computational complexity [19]. To reduce the complexity, we propose a novel suboptimal user scheduling algorithm, as shown in Algorithm 1.

The first step is to find out the set of candidate users, _m. Define _m as a set of users whose AoD of the LoS path, _k_,1 , is in the range of [_m  ,

m ]

   , i.e., C_m{u_k |_k_,1_m| } , where  is the maximum angle difference. Due to the directional nature of mmWave channel, the users in _m can have large beamforming gain for the m-th beam [7]. At most two users among _m will be scheduled on the m-th beam.

The next step is to select one beam and two users iteratively. In each iteration, one beam and two users are selected to maximize the energy efficiency with all the chosen pairs in all the previous iterations. Let denote the set of indices of beams on which no user is scheduled. Initially, set

{1, ,M}

 . Let J_{k m i}_{, ,}  ^{K M}^{ }² denote the single entry matrix,

, , {1, 2},

k m i whose ( , , )k m i -th element is one and the other elements are zero [20]. Schedule *

k1

u and *

k2

u on the m^*-th beam which

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16

Figure 3.1. A system diagram for the addressed mmWave NOMA system.

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17 satisfy

1 2

* * *

1 2 , ,1 , ,2

, ,

( , , ) arg max ( , )

k k m

k m k m

m u u C

m k k 

 

 ρ J J P . Then remove

m* from , and remove *

k1

u and *

k2

u from _m ,  m . The above procedure is repeated until all beams are scheduled, i.e.,   or less than two users remains in _m for all m , i.e., | _m | 1 , m . In the case of   and  m , | _m | 1 , the remaining user in _m , m , is scheduled on the m-th beam.

Algorithm 1 Proposed User Scheduling

1: Initialize ρ 0 , C_m{u_k |_k_,1_m| } for m and {1, ,M}

2: repeat 3:

1 2

* * *

1 2 , ,1 , ,2

, ,

( , , ) arg max ( , )

k k m

k m k m

m u u

m k k 

 

 ρ J J P .

4: * * * *

1, ,1 2, ,2

k m k m

  

ρ ρ J J .

5:  \m^*. 6: Remove *

k1

u and *

k2

u from _m,  m . 7: until   or | _m| 1 ,  m

8: if   and  m , | _m| 1

9: Schedule u_k _m on the m-th beam.

10: end if

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18

3.2 Power Allocation

For a given user scheduling matrix ρ, the power allocation problem to maximize the energy efficiency is formulated as follows.

( )

: max  ,

P ρ

P2 P (3.8)

( )

, , min

s.t. R_{k m}ⁱ _{k m}ⁱ R , k ,m ,i{1, 2}, (3.9)

2 1 (1)

, , min, , ,

k m k m

R ^  R k m (3.10)

2

, max

1 1

.

M

m i m i

P P

 



 ^(3.11)

To solve this non-convex problem, we propose a power allocation algorithm.

First, we employ the successive convex approximation technique to sequentially approximate constraints by using the following inequality [21]:

2 , , 2 , ,

log (1_{k m}ⁱ )a_{k m}ⁱ log _{k m}ⁱ b_{k m}ⁱ , (3.12) where

, ,

1 , i i k m

k m i

k m

a 

 

 (3.13)

and

,

, 2 , 2 ,

,

log (1 ) log ,

1

i

i i k m i

k m k m i k m

k m

b 

 

   

 (3.14)

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19

for a given _{k m}ⁱ_, . The equality in (3.12) is satisfied when _{k m}ⁱ_, _{k m}ⁱ_, . Using the lower bound of (3.12), R¹_{k m}_, and R_{k m}²_, are approximated to

1 1 2 1 1 1

, , 2 , ,1 , 2 ,

log (| ^H | ) log (1)

k m k m k m k m m k m m k m

R a h w a P a  b (3.15)

and

2 2 2 2 2 2

, , 2 ,

(2

,2 , 2 ,

log (| ^H | ) log ) ,

k m k m k m k m m k m m k m

R a h w a P a  b (3.16)

respectively, where P_{m i}_, log₂P_{m i}_, ,

,

2

(1) 2 ( ) 2

,

1 1 1

| | 2 ^{n i} ,

M K

H i P

m k n j n

n j i

n m

 

  



 



^{h w}  ^(3.17)

and

,1 ,

2

(2) 2 (1) 2 ( ) 2

, ,

1 1 1 1

| | 2^m | | 2^{n i} .

K M K

P P

H H i

m k m j m k n j n

j n j i

j k n m

  

   

 

  ^{h w}







^{h w}  ^(3.18)

Similarly, R_{k m}²_,^¹ is approximated to

2 1 3 2 3 (2)

(

, , 2 , , ,2

1

3 3

,

3

2 ,

)

log (| | )

log ,

K H

k m k m k m k m j

m

m m j

j k

k m k m

R a a P

a b







 

  



h w

(3.19)

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20 where

,1 ,

2

(3) 2 (1) 2 ( ) 2

, ,

1 1 1

| | 2^m | | 2^{n i} ,

M K

P P

H H i

m k m k m k n j n

n i j

n m

  

  



  ^{h w} 



^{h w}  (3.20)

2 1 3 ,

, 2 1

,

1 ,

k m k m

k m

a 





 

 (3.21)

and

2 1

3 2 1 , 2 1

, 2 , 2 1 2 ,

,

log (1 ) log

1 ^{k m} ,

k m k m k m

k m

b 

 





 

   

 (3.22)

for a given _{k m}²_,^¹ . Note that (3.15), (3.16) and (3.19) are concave with respect to P_{m i}_, , since a log-sum-exponential function is a convex function.

From (3.15), (3.16) and (3.19), the problem P2 is approximated to

,

2 ,

1 1 1

2

1 1

ma

: x

2 ^{m i}

M K

i k m

m k i

M P c

m i

R P

  

 







P

P3 (3.23)

( )

, , min

s.t. R_{k m}ⁱ _{k m}ⁱ R , k ,m ,i{1, 2}, (3.24)

2 1 (1)

, , min, , ,

k m k m

R ^  R k m (3.25)

,

2

max

1 1

, 2^{m i}

M P m i

P

 



 ^(3.26)

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21

where P is a M2 matrix whose ( , )m i -th element is P_{m i}_, . However, the objective function (3.23) is still non-concave function. We first introduce slack variables _{k m}ⁱ_, , k , m , i{1, 2}, so that the problem P3 is reformulated as

,

2 ,

1 1 1

, 2

1 1

max

2 :

m i

M K

i k m

m k i

M P c

m i

P



  

 







P ξ

P4 (3.27)

( )

, , min

s.t. _{k m}ⁱ _{k m}ⁱ R , k ,m ,i{1, 2}, (3.28)

( )

, , , , , , {1, 2},

i i i

k m k mRk m k m i

     (3.29)

(3.25), (3.26).

Note that _{k m}ⁱ_, 0 if _{k m}^{( )}ⁱ_, 0 . The problem P4 is equivalent to the following problem.

,

2 2

, ,

1 1 1 1 1

2 2

mi

: n log 2^{m i} l go

M M K

P i

c k m

m i m k i

P 

    

   

    





 





P5 P ξ (3.30)

s.t. (3.25), (3.26), (3.28), (3.29).

Since the objective function (3.30) is a convex function, the problem P4 is a convex problem. Next, we apply Lagrange dual method [22] to solve it. The Lagrangian of the problem P5 is given by

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22

,

2 2

2 2 ,

1 1 1 1 1

2 2

( )

, min max

1 1 1 1 1

2

( ) 3

, , min ,

1 1 1

, ,

, , ,

( , , , , )

log 2 log

( ) 2

( ) (

m i

k m k m

k m k m k m

M M K

P i

c k m

m i m k i

M K M

i i i p

k m

m k i m i

M K

i i i i

k m k m k

m k i

L

P

R P

R R





   

    

    

  

 

 

 

   

 

     

   

 

 

 



P ξ λ μ

(1) 2 1

,

1 1

),

K M

m k m

k m

R ^

 



(3.31)

where _{k m}ⁱ_, , _{k m}ⁱ_, ,  are non-negative Lagrange multipliers, and λ , μ are collections of _{k m}ⁱ_, , _{k m}ⁱ_, , respectively. The Lagrange dual function is given by

( , , ) min, ( , , , , ).

g   L 

λ μ P ξ P ξ λ μ (3.32) The Lagrange dual problem is formulated as

max, , ( , , )

: g

 

P6 λ μ λ μ (3.33)

, 0,

s.t. _{k m}ⁱ  k ,m ,i{1, 2}, (3.34)

, 0, , , {1, 2, 3},

i

k m k m i

     (3.35)

 0. (3.36)

The subgradient method is utilized to solve the problem P6 [23]. The Lagrange multipliers in the t-th iteration are given by

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23

,

2

max

1 1

( 1) ( ) ( ) 2 ^{m i} ,

M P

m i

t t t P

  



 

  

    

 

 



 ^(3.37)

 

, min ,

( ) , ( ) , ,

( ) ( ) , 1,

( 1)

0, 0,

i i

k m k m

i k

i

k m

k m i

m

t t R

t   



 

    

 

  

  (3.38)

and

 

3

, mi

2 1 (1)

, ,

( ) n

, 3

,

( ) ( ) , 1,

( 1)

0, 0,

k m k m

i m

m k

k k m

t t R

t  R

 



 ^ ^

    

 

  

  (3.39)

where ( )t is the positive step size and [ ]a ^ max{ , 0}a . The Karush- Kuhn-Tucker (KKT) conditions result in

, ,

2 ,

,

1 1 1

1 0

ln 2

i i

k m k m

M K

i k m i

k m

m k i

L  

 

  

     





^(3.40)

for ( , , )k m i that satisfies _{k m}^{( )}ⁱ_, 1. From the above equation, we obtain

( ) 2 ,

, ,

1 1 1

( ) , ,

1 l

( 1) , 1,

( 1)

0 0

n

. 2

i M K k m

i k m

m k i

i k i

k m

i k m

m

t

 t









  

   

  

 





^(3.41)

For fixed λ , μ , and  , the optimal solutions are obtained by KKT conditions, which lead to

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24

,1

(1) 1

, , ,

,1 1

2

(1) 1 1

, , , (1)

1 3 2

( )

, , , ( )

1 1 1

2 1

ln 2 2

| | 2

0

m

P K

P

k m k m k m m tot k

H P K

k m

k m k m k m

k m

H P

M K

i i i k m

k m k m k m i

m k i m

L a

P P

a

  

 



  

   

    



 

 







h w

(3.42)

and

 

,2

(2) 2 2 (3) 3 3

, , , , , ,

,2 1

3 2 ( )

, , , ( )

1 1 1

2 ln 2 2

| | 2

0,

m

P K

P

k m k m k m k m k m k m

m tot k

H P

M K

i i i k m

k m k m k m i

m k i m

m m

L a a

P P

a

    

 



  

   



     



 





 

^{h w} (3.43)

where ² ^,

1 1

2^{m i}

M P tot

m i

P

 





^and ^^{k m}⁽³⁾^, ^^^{k m}⁽¹⁾^, . From (3.42) and (3.43), the power allocation coefficients are given by

(1) 1

,1 , , , ,1

1

1 /

K

m k m k m k m m

k

P   a A





 

 



 ^(3.44)

and



⁽²⁾ ² ² ⁽³⁾ ³ ³



,2 , , , , , , ,2

1

/ ,

K

m k m k m k m k m k m k m m

k

P   a   a A





 





  ^(3.45)

where

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25

2

(1) 1 1

,1 , , , (1)

1 3 2

( )

, , , ( )

1 1 1

| |

ln 2

| |

1 ^K ^H_k _m

m k m k m k m

tot k m

M K H

i i i k m

k m k m k m i

m k i m

A a

P

a

  

 



  

   

  



 





h w

h w (3.46)

and

2 ,2

3 ( )

, , , ( )

1 1

,2

1

| | 2

1 ln 2 .

Pm

M K H

i i i k m

k m k m k m i

m k i

tot m

m m

Am a

P   _ _ _

   



 



 

^{h w} (3.47)

The power allocation algorithm is summarized in Algorithm 2.

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26 Algorithm 2 Proposed Power Allocation

1: Set the initial point P⁽⁰⁾, the maximum error tolerance , the maximum number of iterations L_max, and the outer iteration counter l1.

2: Calculate ( ,ρ P⁽⁰⁾), _{k m}ⁱ⁽⁰⁾_, , and _{k m}²_,^¹⁽⁰⁾ based on P⁽⁰⁾. 3: repeat

4: Set _{k m}ⁱ_, _{k m}^{i l}⁽_,^¹⁾, _{k m}²_,^¹ _{k m}²_,^{ }¹⁽^l ¹⁾.

5: Calculate a_{k m}ⁱ_, and b_{k m}ⁱ_, according to (3.13), (3.14), (3.21), and (3.22).

6: Set the inner iteration counter t1 and the inner iteration initial point P(0)P⁽^l¹⁾.

7: repeat

8: Obtain ( )t , _{k m}ⁱ_, ( )t , and _{k m}ⁱ_, ( )t , according to (3.37)-(3.39), and (3.41) based on P(t1).

9: Calculate ^{( )}_mⁱ according to (3.17), (3.18), and (3.20).

10: Calculate P_tot.

11: Obtain P( )t from (3.44) and (3.45) based on ( )t , _{k m}ⁱ_, ( )t ,

, ( )

i k m t

 , ^{( )}_mⁱ , and P_tot. 12: t t 1.

13: until convergence 14: Set P^{( )}^l P(t1).

15: Calculate ( ,ρ P^{( )}^l ), _{k m}^{i l}^{( )}_, , and _{k m}²_,^^{1( )}^l based on P^{( )}^l . 16: l l 1.

17: until | ( , ρ P^{( )}^l )( ,ρ P⁽^l^¹⁾) | or lL_max

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27

3.3 Joint User Scheduling and Power Allocation

To solve the original joint user scheduling and power allocation problem P1, we perform Algorithm 1 and Algorithm 2 iteratively. Setting the power allocation coefficients P_m_,1 P_max/ (3M) and P_m_,2 2P_max/ (3M) initially in Algorithm 3, find user scheduling indicators by Algorithm 1. Then, given the user scheduling, find power allocation coefficients by Algorithm 2. This procedure is repeated until the energy efficiency converges or the maximum number of iterations is reached.