針對下鏈路LTE-A網路所設計且以提昇傳輸速率為目標之資源管理技術

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針對下鏈路

LTE-A

網路所設計且以提昇

傳輸速率為目標之資源管理技術

研究生

:

何唯慶

指導教授

:

方凱田

國立交通大學電信工程研究所碩士班

摘

要

由於頻譜與能量資源的有限且珍貴，資源分配問題在近幾年越來

越重要。然而，不恰當的資源分配會導致強烈的細胞間干擾，此干擾

會嚴重的影響整體系統效能。因此在這篇論文中，我們將整個資源分

配問題轉化為數學最佳化問題，且根據此最佳化問題提出解決策略。

在聯合載波選擇與功率分配

JCP

策略中，通道選擇與功率分配將會

在幾何程序

(Geometric Programming)

的轉化與干擾未知的假設下被聯

合求解。然而，顧慮到

JCP

策略的高複雜度，提出另一個簡化策略

(JCP-S)

。此策略將在干擾固定假設的最佳化問題中求解，期望能達

到降低複雜度的效果。此外，儘管已經做出簡化，聯合問題依然是複

雜的。因此，提出更簡化的

HCP

策略以及

HCP-S

策略。其概念是將

整個聯合問題分解為兩個子問題，也就是說通道選擇和功率分配將根

據所提之演算法被依序分別求解。最後，模擬結果顯示

JCP

策略的

效能高於其他所提若干之簡化策略，且因為效能與複雜度的考量下，

兩者間存在著權衡概念。

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Rate-enhanced Resource Management for Downlink

LTE-A Networks

Student : Wei-Ching Ho Advisor : Kai-Ten Feng

Institute of Communications Engineering

National Chiao Tung University

Abstract

Because of the scarcity of spectrum and energy resource, the problems about allocation of these resources become more and more important in re-cent years. However, inappropriate resource allocation may bring about high inter-cell interference which has a great effect on the performance of the entire system. Thus, in this thesis, we provide the optimal formulation of the resource allocation problem and proposed several schemes to solve this problem. In joint component carrier selection and power allocation (JCP) scheme, the channel and power resource are jointly solved based on geo-metric programming and the consideration of undeterministic interference term. Pondering on the high complexity and computational cost in JCP scheme, another simplified scheme, called JCP-S scheme, is proposed where the interference term is assume to be fixed with expectation to lower the complexity. Besides, on account of that the complexity is still high to solve problem from joint view, heuristic scheme, called HCP scheme and HCP-S scheme, are then proposed and try to separate the joint problem into two parts. In other words, channel selection and power allocation problems are solved according to the corresponding algorithms successively. Simulation results demonstrate that the data rate performance of JCP scheme is better than that of the other simplified schemes. However, there exists a trade-off between JCP scheme and the other simplified schemes considering the performance and the complexity.

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誌

謝

時光飛逝，一轉眼碩士生活即將結束，這兩年來付出的努力以及師長們的指導，讓我的研究成果得以順利地在此篇論文呈現。我要感謝指導教授方凱田老師，帶領我在最熱門的 LTE 系統上從事資源分配的研究，也讓我在短短兩年內，從只會念書的大學生，變成具有研究能力的碩士生。在這段期間，老師總是用輕鬆地方式和我討論問題，遇到研究上有不順利的時候，老師也總以鼓勵代替責罵，引領我去克服困難。而從大四開始，我便是 MINT 實驗室的一員，在實驗室裡總能時時感受到歡樂的氣氛，讓我不會完全被研究的壓力所壟罩。感謝已經畢業的佳士學長帶我認識自己的研究領域，在我有問題的時候不吝情地給予幫助。感謝瑞鴻學長，總是在我徬徨不知道該怎麼辦的時候，總能安撫我緊張的情緒，且給我最有幫助的靈感，更是我閒暇運動健身的好夥伴。感謝博後及博班的學長們添壽、建華及子皓，在實驗室裡接受你們的照顧及分享的研究心得。感謝裕平、智偉、修銘和景維學長，你們的優秀表現是我最好的學習方向。感謝 98 級的學長宥儒，謝謝你的經驗讓懵懂的我可以上手。而同屆的治緯、可婷和之後國外求學的君容則是最好的戰友，在這兩年也接受了你們很多的幫助，不管未來是要回到家鄉，還是進入職場，希望大家都會成為人生的勝利組。而我也祝福實驗室的學長學姊學弟妹們瑞鴻、雯琪、培榮、培軒、秉正、宜修、群杰的研究能夠順利。除了實驗室的成員，我也感謝在交大六年來的好朋友們詠閎、勁廷、則亦、宣佑及其他電信的同學，讓我在遇到瓶頸的時候，可以找你們聊天或運動放鬆心情。感謝童博士，陪著我做研究，並給予我許多幫助。而最感激的，還是支持我完成碩士學位的爸爸、媽媽、姊姊以及我最愛的阿公，沒有你們就沒有這份榮耀，一路走來始終有你們的陪伴，你們是我在這裡努力的最大原動力。最後，此篇論文謹獻給所有幫助過我的家人、朋友以及師長。何唯慶謹誌于國立交通大學新竹中華民國一Ｏ二年七月

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List of Figures

2.1 Downlink LTE-A system. . . 9 2.2 Resource structure for downlink LTE-A system. . . 9 2.3 Mapping of PCFICH, PHICH, PDCCH, PDSCH in one PRB

pair. . . 10

3.1 Example of sequence order and allocation for control resource. 24 3.2 Whole process for JCP scheme. . . 26

5.1 Total data rate of JCP, JCP-S, HCP, HCP-S, and HCP-E schemes under diﬀerent total number of CC, which is changed from 1 to 5, in high SINR situation: total number of eNBs

M = 4, total number of UEs per eNB K = 3, code eﬃciency ν = 0.1523. . . 39 5.2 Total data rate of JCP, JCP-S, HCP, HCP-S, and HCP-E

schemes under diﬀerent total number of CC, which is changed from 1 to 5, in low SINR situation: total number of eNBs

M = 4, total number of UEs per eNB K = 3, code eﬃciency ν = 0.1523. . . 40 5.3 Total data rate of JCP scheme under diﬀerent number of

OFDM symbols in control region, which is changed from 1 to 7: total number of eNBs M = 4, total number of CCs J = 5, total number of UEs per eNB K = 3. . . 42

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5.4 Total data rate of JCP, JCP-S, and HCP schemes under dif-ferent code eﬃciency: total number of eNBs M = 4, total

number of CCs J = 5, total number of UEs per eNB K = 3,

number of OFDM symbol in control region Nc= 1. . . 44

5.5 Total data rate of JCP, JCP-S, and HCP schemes under dif-ferent number of UEs per eNB: total number of eNBs M = 4,

total number of CCs J = 5, number of OFDM symbol in

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Chapter 1 Introduction

The long term evolution (LTE) and its advanced version LTE-A system, developed by 3rd generation partnership project (3GPP) as a mobile commu-nication standard from the former 3G systems, has been proven that it can provide high data rate, high resource allocation efficiency, and larger trans-mission coverage. However, the first release of LTE, which is being described as 3.9G, cannot meet the requirements for 4G, such as peak data rates up to 1 Gbps in nomadic speed, defined by the International Telecommunication Union. For the purpose of pursuing higher data rate and spectrum efficiency in LTE-A system, the issues about allocating the two main kinds of radio resource, spectrum resource and power resource respectively, have become more and more important.

Orthogonal frequency division multiple access (OFDMA) technique, which is a multi-user version of orthogonal frequency division multiplexing (OFDM), has been well studied [1–3] and utilized in LTE downlink transmission. In the OFDMA structure, component carrier (CC) is the spectrum resource that can be allocated for data transmission in LTE-A network. The component carriers utilized in the LTE-A systems can be divided into two categories as follows. One is the primary component carrier (PCC) through which the user equipment (UE) handles the network entry process in the control chan-nel. The PCC also contains the data channel which is used for data signal

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transmission. Each UE only has one CC as PCC which is chosen by its eNB, i.e. eNodeB which represents the base station in LTE system. The other one is secondary component carrier (SCC), which is mainly responsible for data transmission. Thus, SCC only includes data channel and eNB can choose many CCs as SCCs for its own UEs if necessary. It is allowed for eNB to select the CCs for transmission even though these CCs have been chosen by neighboring eNBs. This way would bring about a higher spectrum efficiency, but also cause enormous inter-cell interference which has a great influence on transmission data rate if the component carriers are not well-allocated. The 3GPP proposes two methods, inter-cell interference coordination (ICIC) [4] for LTE and its enhanced version eICIC [5] for LTE-A respectively, to reduce inter-cell interference by adaptive channel selection and power con-trol. In [6, 7], component carriers are selected according to the background interference matrix (BIM), which records outgoing and incoming interference information to reduce inter-cell interference, and has not bad performance comparing to the methods proposed by 3GPP. Although these works intro-duce intuitive methods which effectively reintro-duce the background interference on each component carrier, they aren’t joint optimization considering chan-nel selection and power allocation. Furthermore, in previous research works, the design rational of PCCs and SCCs selection algorithms are from eNB’s viewpoint, i.e., all UEs in one eNB will be allocated with the same set of CCs as PCC and SCCs. Due to the lack of diversity in UEs, it is intuitive that this type of eNB-oriented CC selection will result in lower spectrum efficiency.

In order to meet the broadband requirement in 4G, carrier aggregation (CA) technique has been proposed in [8] to aggregate two or more component carriers (CCs), even if these CCs are not continuous in frequency domain, to support high data rate transmission. Referring to speciﬁcation [9], each aggregated carrier is seen as a CC and can have a bandwidth of 1.4, 3, 5, 10, 15 or 20 MHz. Maximum of ﬁve component carriers can be aggregated,

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hence the maximum aggregated bandwidth is 100 MHz. However, compared to the situation that only one CC can be used for data transmission, the enlarged available bandwidth bring about a larger number of data channel allocative information transmitted in control channel. In other words, with limited resources in control channel, the maximum total number of data channels for data transmission one UE can use is constrained to the total resources it has in control region for transmitting data channels’ allocative information. Therefore, how to allocate the resources in control channel to get higher transmission throughput in data channel is a new issue that should be considered in CC allocation problem since wide bandwidth transmission becomes available owing to CA technique.

Regarding power control, two common power control problems, sum rate maximization and sum power minimization, would be solved optimally when the optimization model has convex property. Unfortunately, the power op-timization problem would become a non-convex problem while considering inter-cell interference. In [10], the required transmission power for UEs lo-cated at the cell center would ﬁrstly be determined by their proposed power allocation approach. Besides, using scheduling strategy, cell-edge UEs would only mutually interfered with the center UEs of neighboring eNB. Thus, the optimal power allocation problem for the cell-edge UEs would become a convex problem, which can be solved by the Lagrangian method. However, spectrum eﬃciency can be degraded by stipulating that only cell-edge UEs and theirs neibhboring eNBs’ center UEs could select resource at the same time according to scheduling strategy. In addition, in [11], the resource al-location problem is divided into two parts, one is channel alal-location and the other is power allocation. The power allocation is executed with channel al-location being determined. And in these two schemes, the interference term is considered to be deterministic in power optimization. In [12], even though the optimization is not concave, the optimal solution for power allocation is proven to be the same as the solution for the corresponding dual optimization

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problem when the problem satisﬁes time-sharing property. However, it only considers the problem for power allocation in the optimization. Furthermore, with no CA technique, all the schemes mentioned before don’t consider the required transmission rate on PCC’s control channel, who would be a con-straint on the total data channels’ rate as mentioned before.

In this thesis, optimization is formulated to maximize the total channel capacity on all data channels under the constraints of maximum transmis-sion power and maximum number of data channels which is related to the resource allocation in control region. Firstly, a resource allocation scheme is proposed, called joint component carrier selection and power allocation (JCP) scheme which prefers to jointly solve the CC selection and power allo-cation. Secondly, in order to lower the high complexity in JCP scheme which consider underterministic interference, a simplified version of JCP (JCP-S) is proposed. Besides, considering the high complexity of the joint view, a heuristic component carrier selection and power allocation (HCP) shceme is proposed which divides the original optimization problem into two sub-problems, which are channel selection and power allocation. In the first stage of HCP, a channel selection method is proposed to not only reduce the inter-cell interference but also to enhance the total UEs’ data transmis-sion rates. In the second stage of HCP, a methods is proposed, considering underterministic interference as in JCP, to solve power allocation with the channel allocation being known from the first stage. Moreover, seeing that the high computation cost when considering underterministic interference, another power allocation method is proposed with the concept of derter-ministic interference. And call the scheme a simplified version of HCP, i.e. HCP-S. At final, simulations will be performed to compare their performance under different environments and validate that the scheme in joint view can achieve better performance than that in divided view even though the high complexity in joint view. The comparisons between proposed schemes and other schemes mentioned before are shown in Table 1.1.

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The rest of this thesis is organized as follows. Chapter 2 introduce the resource structure in LTE-A systems and formulates the complete resource allocation problem in optimization view. The proposed joint component carrier selection and power allocation (JCP) scheme and simpliﬁed version of JCP (JCP-S) are presented in Chapter 3. Next, the heuristic scheme (HCP) will be presented in Chapter 4. The simulation results and performance comparisons will be demonstrated in Chapter 5. Finally, Chapter 6 draws conclusion.

Table 1.1: Schemes comparison

Allocation type Channel allocation Power allocation Orientation ICIC Separated Heuristic Heuristic Interference reduced ACCS Separated Heuristic Heuristic Interference reduced Scheme in [10] Separated Heuristic Optimization Rate maximized Scheme in [11] Separated Heuristic Optimization Rate maximized Scheme in [12] Only power No allocation Optimization Rate maximized Proposed JCP Joint Optimization Optimization Rate maximized Proposed HCP Separated Heuristic Optimization Rate maximized

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Chapter 2 System Model and Problem

Formulation

2.1 Downlink Scenario for LTE-A System

As shown in Fig. 2.1, a multi-cell downlink homogeneous LTE-A system is considered and all eNBs are equipped with omnidirectional antennas. The cell deployment follows the wrap around topology [13] and each cell contains a centering eNB with a number of serving static user equipments (UEs). Under the consideration of interference from other eNBs, called inter-cell interference and represented as dotted line in Fig. 2.1, each eNB should determine allocation in control and data region for each integrated resource unit on each CC, called physical resource block pair (PRBP) in LTE, to its own UEs and decide how much power will be transmitted on the data region for each PRBP if used. Besides, there are no specific rules in specification that regulate all UEs belonging to the same serving eNB should select the same resource for transmission. For more flexibility and higher spectrum efficiency, assume that the UEs belonging to the same serving eNB can select different CCs as their PCC and SCCs. Moreover, in order to avoid the intra-cell interference which would greatly reduce the channel capacity if occurred,

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the speciﬁcation regulates that, not only in control region but also in data region for one PRBP, the UEs belonging to the same serving eNB cannot select the same resource for transmission. In other words, in both region for one PRBP and for one eNB, a resource can be at most allocated to one UE simultaneously.

In order to resist the outside interference, the coding and modulation techniques have been proven that it can effectively reduce the bit error rate (BER). According to different conditions of channel quality, UEs can decide the best channel quality indication (CQI) index so that the transport error probability not exceeding 10 percentage. In the specification, the modulation and coding scheme (MCS) is developed for the purpose of selecting appropri-ate code rappropri-ate and modulation to transmit data according to the CQI index. More explicitly, when encountering bad channel quality, it is prefer to select the scheme with low code rate, i.e. add more redundant bits to resist the strong interference, and noise-resisted modulation (e.g. QPSK) to transmit data. On the other hand, since guaranteeing the BER is not hard to achieve under condition of good channel quality, the scheme with high code rate and high-rank modulation (e.g. 16-QAM, 64-QAM) can be used to increase the transmission rate.

As shown in Fig. 2.2, in time domain, a downlink transmission frame is equally divided into 10 subframes and each subframe contains two time slots. Each slot has Ns= 7 OFDM symbols. In frequency domain, the total

band-width of a component carrier is equally divided into numerous subcarriers (SCs) and each subcarrier has 15 (kHz). According to the LTE-A speciﬁca-tion [14], the size of least resource unit, called resource element (RE), is 1 subcarrier and 1 OFDM symbol. Further, the size of physical resource block (PRB) is V = 12 subcarriers and 7 OFDM symbols. The integrated resource

unit is a pair of PRB, i.e. PRBP. Assume that the ﬁrstNcOFDM symbols in

a PRBP are responsible for control signal transmission, and the rest 2Ns−Nc

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the size of basic resource unit for allocation in control region is V = 12

sub-carriers and Nc OFDM symbols, and call it control resource unit. The size

of basic resource unit for allocation in data region is V = 12 subcarriers and

2Ns− Nc OFDM symbols, and call it data resource unit. As a result,

con-trol channel transmission and data channel transmission can be separated by time division duplex (TDD) technique. Then, in frequency domain, as-sume that there are total J component carriers in the system and each CC

has R PRBPs in one subframe. Note that since the information

transmit-ted in control channel is so important that it inﬂuences whether the whole system functions properly or not, these information should be transmitted carefully even without loss. As shown in Fig. 2.3, there are several types of control channels in the downlink control region, including physical down-link control channel (PDCCH), physical HARQ indicator channel (PHICH), physical control format indicator channel (PCFICH), etc. Each type of con-trol channel has its own responsible functions. For instance, the PDCCH is responsible for transmitting the allocative information of physical downlink shared channel (PDSCH), which is the main type of data channel in data region. In other words, the information records that which PDSCH should be assigned to which UE is all transmitted in the PDCCH. Besides, there are several types of physical signal transmitted on some control channels, like reference signal (RS) which is mainly responsible for channel estimation.

2.2 Problem Formulation for Resource

Allo-cation in LTE-A System

As mentioned before, the allocative information of PDSCH is transmitted by PDCCH. Assume that r is the ratio of resources which are responsible for

transmitting allocative information of PDSCH in control region. Considering the importance of control signal and in order to guarantee the control sig-nal can be transmitted to received termisig-nals with low error probability, the

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eNB 3 eNB 7 eNB 2 UE 2 eNB 1 UE 1 eNB 4 eNB 5 eNB 6

Figure 2.1: Downlink LTE-A system.

frame R Bs #0 #1 #2 #3 #4 #5 #7 #8 #9 #0 #1 #2 #3 #4 #5 #6 #7 #8 #9 #6 #0 #1 #2 #3 #4 #5 #6 #7 #8 #9 1 frame (10ms) SLOT 1 2 1 2 Su b ca rri e rs (1 8 0 kH Z ) 7 OFDM symbols (0.5 ms) 1 RE 1 PRB 1 RE

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Control Region Data Region RS PHICH PCFICH PDCCH PDSCH 14 OFDM Symbols (1 ms) 12 Su bcarri ers (180 kHZ)

Figure 2.3: Mapping of PCFICH, PHICH, PDCCH, PDSCH in one PRB pair.

scheme with low code rate and noise-resisted modulation is more preferable. Assume that the QPSK modulation, which can transmit at most A = 2 bits

on each resource element, and code rate κ is used in the control region, so

that the code eﬃciency is ν = ₁₀₂₄2·κ . And the maximum total bit rate that a

PRBP can provide for allocative information of PDSCH in one subframe can be represented as ζ = A· V · Nc· r (Kbps). That is to say, what modulation

and code eﬃciency is used inﬂuences the capability of one PRBP to transmit allocative information in control region for data channel.

A bitmap, one scenario for resource allocation in data region, is used to indicate the resource block groups (RBGs) where a RBG is a set of con-secutive data resource units. Assume a resource block group contains G

consecutive data resource units and then the total number of RBGs in the whole component carriers can be represented as ϕ =⌊J_G·R⌋ + 1. To make an

associative connection to the PDCCH in control region, the RBG is allocated to the UE if the corresponding bit value in the PDCCH is 1. For instance, if an UE wants to take all of the RBGs as PDSCH to transmit data, the UE needs φ = ϕ·_ν1 bits resource space in PDCCH to transmit PDSCH allocative

information in one subframe, i.e. needs rate φ (Kbps) in PDCCH. In other

words, allocating one RBG as PDSCH to an UE at least needs 1

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resource in PDCCH. And the maximum number of RBGs that an UE can use for PDSCH is determined by how much resource the UE gets in the control region, i.e resource in control region of one PRBP can provide ζ (Kbps) for

transmitting allocative information of data channel. Thus, the resource al-location in control region has great eﬀects on the resource alal-location in data region and the ﬁnal performance of system data rate.

The main goal is to maximize the total channel capacity in all data chan-nels under the constraint of maximum transmission power and the constraint of maximum data channel’s number each UE can get in data region accord-ing to the resource allocation in control region. In other words, the target in our optimization problem is to acquire the channel allocation in both regions and the power allocation on each channel in order to maximize the total data transmission rate on all data channels. Denote_C¯i,(D)

j,s,k as the channel capacity

of kth UE in eNB i on the subchannel s of the CC j in data region. It is assumed that the total numbers of eNBs in the LTE-A network are M and

the set of all eNBs is denoted as M. There are K UEs in each eNB and K

represents the set of all UEs. The sets of all CCs and all subchannels are denoted as J and S, respectively. For simplicity, assume that the RBG size

G equals to 1. Namely, one data resource unit is the allocative subchannel

unit in data region, and one control resource unit is the allocative subchannel unit in control region. Moreover, ρ¯is the channel selection indicator and is deﬁned as follows ¯ ρi,(C)_j,s,k =   

1, if UE k in eNB i selects subchannels of the CCj as CCH 0, otherwise (2.1) and ¯ ρi,(D)_j,s,k =   

1, if UE k in eNB i selects subchannels of the CCj as SCH 0, otherwise

(2.2) Note that ρ¯i,(C)_j,s,k ∈ ¯ρ(C) and ρ¯i,(D)_j,s,k ∈ ¯ρ(D),∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K.

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The bar means the indicator belongs to discrete domain. And deﬁne δi,(C)_j,k =    1, if ∑R_s=1ρ¯i,(C)_j,s,k > 0 0, otherwise (2.3)

From the viewpoint of CC’s categories, the component carrier j with δi,(C)_j,k = 1, the subchannels on this CC with ρ¯i,(D)_j,s,k = 1, and the subchannels

on this CC with ρ¯i,(C)_j,s,k = 1 can be view as the PCC, this PCC’s data channel, and this PCC’s control channel ofkth UE in eNBi, respectively. The remain

component carrier j′ with δ_ji,(C)′_,k = 0 can be view as the SCC of kth UE in

eNB i if there exist any subchannel s with ρ¯i,(D)_j′_,s,k = 1. With the notations

deﬁned above, the optimization problem can be formulated as

max ¯ ρ(C)_{, ¯}_ρ(D)_,P M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 ¯ C_j,s,ki,(D) (2.4) subject to: ¯

ρi,(C)_j,s,k ∈ {0, 1} , ¯ρi,(D)_j,s,k ∈ {0, 1} , ∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K (2.5)

J ∑ j=1 δi,(C)_j,k = 1, ∀i ∈ M, ∀k ∈ K (2.6) K ∑ k=1 ¯ ρi,(C)_j,s,k ≤ 1, K ∑ k=1 ¯ ρi,(D)_j,s,k ≤ 1 ∀i ∈ M, ∀j ∈ J, ∀s ∈ S (2.7) J ∑ j=1 R ∑ s=1 K ∑ k=1 P_j,s,ki ≤ Pmax, ∀i ∈ M (2.8) J ∑ j=1 R ∑ s=1 ¯ ρi,(D)_j,s,k ≤ ζ ν · J ∑ j=1 R ∑ s=1 ¯ ρi,(C)_j,s,k, ∀i ∈ M, ∀k ∈ K (2.9)

The main goal is to ﬁnd the power setP, channel selection set for control region ρ¯(C), and channel selection set for data region ρ¯(D) respectively that

maximize the total channel capacity. Equation (2.6) means that each UE can select only one CC as its control CC, i.e. PCC. In order to avoid the

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oc-currence of intra-cell interference, the constraint in equation (2.7) represents that one subchannel can be allocated to at most one UE in an eNB. TheP_j,s,ki

in equation (2.8) represents the transmission power of kth UE in eNB i on

the subchannelsof the CCjwhich belongs to the setPandPmax denotes the

maximum allowed transmission power for each eNB. The term ζ

ν in equation

(2.9) represents the equivalent total subchannels’ amount in data region that one subchannel in control region can guarantee and provide for transmitting allocative information. This constraint means that the total amount of sub-channels, which are selected by an UE for data sub-channels, should not exceed the maximum number which is guaranteed by the total resource the UE gets in control channel. In addition, observing the optimization problem, the way to measure the rate in control channel is from the perspective of practical data rate regardless of the channel capacity. As mentioned before, the reason is that, in general, the most important thing concerned in control channel is how to transmit the control signal with low error probability instead of how to increase the transmitting data rate. So the MCS with low error probabil-ity is assumed to be used in control channel and bring about the viewpoint of practical data rate. Besides, according to Shannon capacity theory, the transmission rates on data channels in equation (2.4) can be derived as

¯

C_j,s,ki,(D)= 2Ns− Nc

2Ns · B · log2

(1 + S_j,s,ki,(D)) (bits/s) (2.10)

where the term 2Ns−Nc

2Ns specify that the capacity belongs to data region. B

in (2.10) is the bandwidth of one subchannel, i.e. 180kHZ as shown in Fig. 2.2. The kth UE’s signal to interference plus noise ratio (SINR) in eNB i on

the subchannel s of the CC j in data region is denoted asS_j,s,ki,(D), and can be

written as follow S_j,s,ki,(D)= ρ¯ i,(D) j,s,k · Pj,s,ki · g i,i j,s,k Γ· (N0+ I_j,s,ki,(D)) , I_j,s,ki,(D)=∑ q̸=i K ∑ z=1 ¯ ρq,(D)_j,s,z · P_j,s,kq · gq,i_j,s,k (2.11)

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Theρ¯i,(D)_j,s,k indicate that whether the subchannel is selected for data trans-mission or not, i.e. the capacity on this subchannel equals to zero if not. According to channel model in [15] which considers the pathloss, shadowing, and fading eﬀects, the parameter g_j,s,ki,i′ in (2.11) is deﬁned to represent the

channel gain from eNB itokth UE of eNB i′ on the subchannel sof the CC j, and the channel gains of all the communication links are assume to remain

constant in one downlink transmission frame. Assume g represent the set of

all channel gain’s information. The parameter Γ =− ln(5BER)/1.5 in (2.11) can be obtained from [16] given M-ary quadrature amplitude modulation and target bit error rate (BER). N0 is the background channel noise, which

can be calculated by multiplying the bandwidthB and the background noise

spectrum density n0. I_j,s,ki,(D) represents the interference term of kth UE in

eNB i on the subchannel s of the CC j in data region and can be written

as the right hand side of (2.11), which consists of the numerous inter-cell interferences resulting from other eNBs.

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Chapter 3 Proposed Joint Component

Carrier Selection and Power

Allocation (JCP) and

Simpliﬁed JCP (JCP-S)

Schemes

3.1 Geometric Programming and Problem

Re-formulation

As presented in equations (2.4)-(2.9), exhaustively searching for every possible channel allocations and then finding the best power allocation among them is the most intuitive way to solve this problem, but almost cannot be realized because of unbelievably large computations. From the perspective of computational complexity theory, NP problem is defined as that it can be solved in non-deterministic polynomial time by using an infinite number of calculators. In other words, with finite number of calculators, a NP problem has much higher complexity than polynomial time and is difficult to solved.

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Observing our optimization problem and in order to ﬁnd the global optimum solution, the optimization problem is NP-hard whose complexity is at least as hard as the hardest problems in the NP problem.

With regard to the optimization problem, the problem contains two main kinds of variables to be solved. One is the power set P that belongs to continuous-type variable and the complete optimization theorem has been well-developed for this type of variable. The other one is the channel in-dicator sets that belongs to discrete-type variable, including ρ¯(C) and ρ¯(D)

which represent the indicator sets for control region and data region respec-tively. In the integer programming theory, there exists a series of methods to solve the discrete-type optimization problem. However, when considering the problem including both types of variables, this joint problem will become so complicated and difficult to solved. As a result, a modification to the original optimization problem is proposed owing to the complexity. The modification is that making releases of the discrete-type variables to continuous domain. In other words, the concept is trying to solve the joint problem with all of the variables being on continuous domain, then recover the solution, which are originally discrete-type, back to the discrete domain according to some designed algorithms.

Furthermore, on the aspect of channel allocation in the control region, modulation and coding scheme with noise-resisted modulation like QPSK and lower code eﬃciencyνis preferable and assumed to be used in this region.

As mentioned before, how much resource that one subchannel in the control region can provide for transmitting allocative information of data channels is related to what MCS is used, instead of where the subchannel is located in. That is to say, no matter which component carrier the subchannels are located in, one subchannel in the control region can guarantee to transmit allocative information of ζ

ν subchannels in the data region. On account of

this reason, another modiﬁcation is proposed which implies that the amount ∑J

j=1

∑R s=1ρ¯

i,(C)

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allocated for the control resource of kth UE in eNB i, is the term truely

cared about in the allocation problem.

With the modiﬁcations mentioned above, the optimization problem can be reformulated as follows max π,ρ(D)_,P M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 C_j,s,ki,(D) (3.1) subject to: πi_k∈ [0, 1] , ρi,(D)_j,s,k ∈ [0, 1] , ∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K (3.2) K ∑ k=1 π_ki ≤ J, ∀i ∈ M (3.3) K ∑ k=1 ρi,(D)_j,s,k ≤ 1, ∀i ∈ M, ∀j ∈ J, ∀s ∈ S (3.4) J ∑ j=1 R ∑ s=1 K ∑ k=1 P_j,s,ki ≤ Pmax, ∀i ∈ M (3.5) J ∑ j=1 R ∑ s=1 ρi,(D)_j,s,k ≤ ζ· R ν · π i k, ∀i ∈ M, ∀k ∈ K (3.6)

Diﬀerent to the original optimization problem in equations (2.4)-(2.9), the new indicator set for the resource allocation in control region are represented as π and the indicator πi_k ∈ π, ∀i ∈ M, ∀k ∈ K represents the ratio of the

total subchannels’ number allocated for the control channels of the kth UE

in eNB ito the total subchannels’ number in one component carrier, i.e. the

new indicator πi_k represent the term

∑J j=1 ∑R s=1ρ i,(C) j,s,k R ,∀i ∈ M, ∀k ∈ K, where

ρ(C)∈ [0, 1] represents the continuous allocation set for control resource.

Moreover, the constraints π_ki ∈ [0, 1] and ρi,(D)_j,s,k ∈ [0, 1] in equation (3.2)

represent the releases of discrete-type sets ρ¯(C) and ρ¯(D) to the continuous

domain respectively, which are both in the closed interval between 0 and 1. The constraint in equation (3.3) represents that the total resources selected

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for control channels by all UEs in one eNB cannot exceed the maximum resources that all J CCs can provide. In addition, there exists some same

concepts in the constraints between the origin and the reformulated problems. For example, responding to the constraint in equation (2.6),πi_k∈ [0, 1]implies

that each UE cannot select more than one CC as its control CC, i.e. PCC. Likewise, the concepts of the constraints in equations (3.4) and (3.6) also respond to the constraints of equations (2.7) and (2.9) respectively. Similar to the original optimization problem, the transmission rates on data channel

C_j,s,ki,(D) in equation (3.1) and the SINR term can both be written as the same

forms in equation (2.10) and (2.11) with continuous variables, respectively. As known in the optimization theory, if a function’s Hessian matrix, whose elements are constructed by the second partial derivatives of the func-tion with respect to two variables, is a positive semi-deﬁnite matrix, then the function is proven to have convex property. And if an optimization problem has convex property, the global optimal solution can be obtained by using some well-developed methods, like Lagrange method. Unfortunately, ob-serving the reformulated problem, the optimization problem has no convex property mainly due to the consideration of inter-cell interferences. Thus, the geometric programming [17] is utilized whose concept is to transform an original non-convex problem into a convex formulation by introducing some alternative variables and approximations. In our problem, the following lower bound,

µlog S0+ λ≤ log (1 + S0) (3.7)

which is tight at S0 when the approximation parameters are chosen as

µ = S0 1 + S0 (3.8) λ =log(1 + S0)− S0 1 + S0 log S0 (3.9)

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the equation (2.10) can be reformulated as ˆ C_j,s,ki,(D)= 2Ns− Nc 2Ns · B · ( µi_j,s,klog₂S_j,s,ki,(D)+ λi_j,s,k ) (3.10)

where µi_j,s,k and λi_j,s,k are ﬁxed parameters. Deﬁne µi_j,s,klog₂S_j,s,ki,(D)+ λi_j,s,k in equation (3.10) as the lower bound term, denoted as Li,(D)_j,s,k. Since Cˆ_j,s,ki,(D)

can be viewed as the lower bound of C_j,s,ki,(D) in equation (2.10), the original

optimization problem is then transformed to maximize the lower bound of total transmission rates on all data channels, in other words, try to solve the optimization problem from the viewpoint of lower bound. However, (3.10) is still non-convex which requires additional processing to such that it can be transformed into a convex function. Lemma 1 is presented as follows to conduct this transformation.

Lemma 1: The lower bound transmission rate (3.10) can be concaviﬁed by the variable transformations: _Pˆi

j,s,k=ln(Pj,s,ki ) and ρˆ i,(D) j,s,k =ln(ρ

i,(D) j,s,k).

Proof: With P_j,s,ki = exp ( ˆP_j,s,ki ) and ρi,(D)_j,s,k = exp (ˆρi,(D)_j,s,k), the lower bound term Li,(D)_j,s,k in (3.10) can be rewritten as

ˆ Li,(D)_j,s,k = µ i j,s,k ln 2 [

ϵ + ˆρi,(D)_j,s,k + ˆP_j,s,ki − ln (ˆI_j,s,ki,(D)+ N0)

]

+ λi_j,s,k (3.11)

where ϵ =ln g_j,s,ki,i _{− ln Γ}.

And the interference term_Iˆi,(D) j,s,k = ∑ q̸=i ∑K z=1e( ˆρ q,(D) j,s,z+ ˆP q j,s,k)· gq,i j,s,k.

Observ-ing (3.11), we can ﬁnd that this function is constructed by a linear function plus a log-sum-exp function ln ( ˆI_j,s,ki,(D)+ N0), which is proven to be convex in

[18]. Thus, the reformulated lower bound term is a concave function.

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func-tion of the GP optimizafunc-tion problem can be written as max π, ˆρ(D)_{, ˆ}_P M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 ˇ C_j,s,ki,(D) (3.12) where ˇ C_j,s,ki,(D)= ˆC_j,s,ki,(D)(ePˆ, eρˆ(D)) (3.13) And the new constraints can all be obtained from the constraints in equation (3.2)-(3.6) with ρ(D) and Preplaced by eρˆ(D) and ePˆ respectively.

3.2 Algorithms for JCP Scheme

After the GP transformation, ﬁnd that the objective function of the op-timization problem is a concave function and all of the constraints are linear functions. Therefore, the global optimal solutions of the control channel’s allocation π, the data channel’s allocation ρˆ(D), and the power allocation Pˆ

can all be jointly solved in continuous domain by the well-developed numer-ical analysis’s optimization method. The detailed processes to jointly solve these three kinds of variables in the JCP scheme are given in Algorithm 1. In the iterative progress of this algorithm, once the optimal solutions are acquired, the solutions that had been transformed into new domains should be transferred back to the original domain. For example, the solutions of the power in new domain _Pˆ _{should be transformed back to the original domain}

P byP_j,s,ki =exp ( ˆP_j,s,ki ).

The three kinds of solutions can be obtained from the output of Algo-rithm 1. However, observing these solutions, all of them are continuous and some of them are not matched to the original domain deﬁned in section 2.2 since the releases of variables from discrete domain to the continuous domain. For instance, consider the constraints in equations (2.5)-(2.7) and the deﬁnition of variable set π. Each continuous control channel’s allocation

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Algorithm 1: JCP scheme for joint channel and power allocation Input: g

Output: P , π, ρ(D)

1: Initialize all µi_j,s,k,(0) and λi_j,s,k,(0) according to (3.8)-(3.9) given a initial

SINR S0

2: Initialize counter of iteration n = 0

3: repeat

4: Maximize: after lower bound and variable transformation, solve the

optimization problem (3.12) and transform the solutions back to the original formulation with Pi

j,s,k,(n) = exp ( ˆP i

j,s,k,(n)) and ρi,(D)_j,s,k,(n) = exp (ˆρi,(D)_j,s,k,(n)),∀i, ∀j, ∀s, ∀k

5: Tighten : update µi

j,s,k,(n+1),λij,s,k,(n+1),∀i, ∀j, ∀s, ∀k according to the

new SINR Sn+1 calculated from P(n) and ρ (D)

(n), and then increase n

by one

6: until µi_j,s,k,(n+1),λi_j,s,k,(n+1),∀i, ∀j, ∀s, ∀k converge

πi_k is expected to be an integral multiple of _R1 because the basic allocative

unit in the control region is one subchannel. And the subchannels selected as control channels for an UE can only be located in the same component carrier owing to the constraint in equation (2.6) that regulates an UE can select only one CC as its control CC, i.e. PCC. Moreover, the constraint in equation (2.7) also stipulates that one subchannel can be allocated to one UE for one eNB simultaneously.

For the constraints mentioned above, a heuristic method, called control resource quantization (CRQ) , is proposed to perform the progress about recovering the solution set π, which is the global optimal solution from the

released viewpoint of continuous domain, back to the originally discrete do-main set π¯, and the concept of this method is trying to recover with least

shift. In addition, ifkth UE in eNBiis allocated with much control resource,

i.e. higher πi

k, it implies that this UE may have better average channel

qual-ity so that the performance of total data rate can be enhanced more when allocating more control resource to this UE, where the quantity of control resource inﬂuences the maximum number of data channels an UE can use as

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mentioned before. Consequently, in CRQ method, each eNB has a sequence, which determines the order of selecting control resource for all UEs belong-ing to this eNB, and the sequence order of kth UE in eNB i is decided by

the values of π_ki. In other words, comparing to the other UEs in the same

eNB i, the higher the kth UE’s πi_k is, the earlier this UE can select control

resource. After deciding the selecting sequence of each eNB, all UEs of each eNB will follow the sequence to select control resource and recover π to

dis-crete domain with least shift. For example, assume the total number of CCs is J = 3, the total number of subchannels in one CC is R = 5, the total

number of UEs in eNB iis K = 5, and the allocation for control resource,π_ki

obtained from the output of Algorithm 1 for eachkth UE in eNB i, is given

by the table listed in Fig. 3.1. For the ﬁrst three UEs in the sequence, owing to the existence of unoccupied CCs, each of these UEs can directly trans-form the π_ki to discrete domain with least shift and use the corresponding

amount of subchannels for control resource in the unoccupied CC. Taking 2th UE in eNB i as an instance, this UE transforms theπi₂ = 0.74to discrete ¯

πi₂ = _R4 = 0.8 and uses corresponding 4 subchannels (SCs) in unoccupied CC2, not to 3

R = 0.6 because 0.74 is nearer to 0.8 than to 0.6. Then, for the

last two UEs, the concept of least shift is still worked. Taking1th UE in eNB i as the example, there are two CCs, which are CC1 and CC2 respectively, with one SC available, and CC3 has 2SCs available. This UE transforms the

πi₁ = 0.48 to discrete π¯i₁ = _R2 = 0.4 with least shift. In other words, this UE uses 2SCs in CC3 and not use CC1 or CC2 because 0.48 is nearer to 0.4 than to 1

R = 0.2. Finally, after ﬁnishing the allocating process according to

selecting sequence, there are probably some SCs not being allocated. In this case, this kind of SCs in each CC is preferable to allocated to the UE, which selects the corresponding CC as control CC and has the highest value of π¯.

After the determination of discrete allocation for control resource in CRQ, the continuous allocation for data channel ρ(D) obtained fromAlgorithm 1

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value 0 or 1. However, on account of that the optimal solution of ρ(D)

and P are jointly calculated with the released allocation π, the solution of

allocation for data channel and power are supposed to be calculated again in the optimization problem where the allocation for control resource is known as constant and discrete value π¯ that had been decided in CRQ. In other

words, the more precise ρ˜(D) and P˜ are expected to be solved in the GP

optimization problem with π¯ being known. Once the reﬁned solutions ρ˜(D)

and _P˜ _{are calculated, another method, called data resource quantization}

(DRQ), is proposed to recover the allocation for data channel, ρ˜(D) , to

discrete domain.

With reference to the ρ˜(D) and P˜, all UEs in each eNB can calculate

their own equivalent capacity _C˜ _{on each subchannel and the proposed DRQ}

method can utilize this equivalent capacity to allocate data resource. The equivalent capacity of the kth UE in eNB i on subcarriers s in CC j can be

calculated as follows ˜ C_j,s,ki,(D)= 2Ns− Nc 2Ns ·B ·log2 (1 + ρ˜ i,(D) j,s,k · ˜Pj,s,ki · g i,i j,s,k Γ· (N0+ ∑ q̸=i ∑K z=1ρ˜ q,(D) j,s,z · ˜P q j,s,k· g q,i j,s,k) ) (3.14)

Considering the constraint that one subchannel can allocate to at most one UE for an eNB for the sake of avoiding intra-cell interference, each eNB is preferable to allocate resource to the UE, who has the highest equivalent capacity. Besides, in the course of allocation, each eNB should keep an eye on the total amount of control resource that each UE obtains. In other words, eNBiwould allocate the subchannelsin CCj to itskth UE, who has

the highest equivalent capacity comparing with the other UEs and still has available control resource which is used to transmit the additional allocative information of subchannel s. The detailed processes of transforming

contin-uous solutions to discrete domains in JCP scheme are given in Algorithm 2

and Algorithm 3 respectively.

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so-CC1 CC2 CC3

SC1

SC2

SC3

SC4

SC5

UE4

UE2

UE3

UE1

UE5

UE4

UE2 UE3

UE1

UE5

0.52

0.74 0.48 0.41

0.85

1.0

0.8

0.6

0.4

0.2

Figure 3.1: Example of sequence order and allocation for control resource.

lution have been solved, which are the allocation for control resource, π¯,

and for data resource, ρ¯(D), respectively. In the ﬁnal step of JCP scheme,

the ﬁnal power allocation can be calculated in the GP optimization problem with all allocations for channel resource being known. Summarily, the whole process for JCP scheme can be shown as Fig. 3.2.

3.3 Problem Reformulation in JCP-S Scheme

Although the existence of undeterministic inter-cell interference is a real-istic consideration in the optimization problem formed in previous chapters, the computational complexity and diﬃculty in mathematical analysis would also get much higher under this consideration. Consequently, an assumption is introduced to JCP-S scheme, which assumes that inter-cell interference term is a constant value in the optimization process. The reformulated opti-mization problem in JCP-S is shown as follows

max ρ(D)_,P M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 C_j,s,ki,(D) (3.15)

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Algorithm 2: Control resource quantization (CRQ) Input: π, g

Output: ¯π

begin

1. Sort π to form the selecting sequence for each eNB:

Initial πi 0 =−1, ∀i ∈ M and S(1 : N, 1 : K) = 0 for n = 1 to N do for k = 1 to K do for o = 1 to K do if πn k > πS(n,o)n then ot= o break Spre = S and S(n, ot) = k for ˆo = ot to K do if Spre(n, ô)̸=0 then S(n, ô + 1) = Spre(n, ô)

2. Recover continuous π to discrete ¯π:

Initial ¯π =−1 and A(1 : N, 1 : J) = 1

for n = 1 to N do for o = 1 to K do for j = 1 to J do if A(n, j) _{≥ π}n S(n,o) then u∗ = arg min(_Ru − πn S(n,o))2, u∈ {0, 1, .., R} ¯ πn S(n,o) = u∗ R , A(n, j) = A(n, j)− ¯π n S(n,o) break if ¯πn S(n,o) =−1 then

j∗ = arg min(πn_S(n,o)− A(n, j))2, j ∈ {0, 1, .., J}

¯

π_S(n,o)n = A(n, j∗) , A(n, j∗) = 0 break

for n = 1 to N do if J _{≤ K then} for j = 1 to J do ¯ π_S(n,j)n = ¯π_S(n,j)n + A(n, j) else for j = 1 to K do ¯ πn S(n,j) = 1

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Algorithm 3: Data resource quantization (DRQ) Input: ¯π, ˜ρ(D)_{, ˜}_{P , g}

Output: ¯ρ(D)

begin

1. Recover continuous ˜ρ(D) to discrete ¯ρ(D):

Initial Ki ∈ the set of all UEs in eNB i, ∀i ∈ M

Initial ¯ρ(D)= 0 for j = 1 to J do for s = 1 to R do for n = 1 to N do for k = 1 to K do ˜ C_j,s,kn,(D)=equation (3.14)

u∗ = arg max ˜C_j,s,un,(D), u_{∈ K}n

¯

ρn,(D)_j,s,u∗ = 1 and ¯ρn,(D)_j,s,u = 0, u /∈ u∗

if ∑J ˆ j=1 ∑R ˆ s=1ρ¯ n,(D) ˆ j,ˆs,u∗ =⌊ ζ·R ν · ¯π n u∗⌋ then Kn = Kn− {u∗} subject to: ρi,(D)_j,s,k ∈ [0, 1] , ∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K (3.16) K ∑ k=1 ρi,(D)_j,s,k ≤ 1, ∀i ∈ M, ∀j ∈ J, ∀s ∈ S (3.17) K ∑ k=1 J ∑ j=1 R ∑ s=1 P_j,s,ki ≤ Pmax, ∀i ∈ M (3.18) where C_j,s,ki,(D)= 2Ns− Nc 2Ns ·B·ρ i,(D) j,s,k·log2(1+ P_j,s,ki · gi,i_j,s,k Γ· (N0+ Ic) ) = 2Ns− Nc 2Ns ·B·ρ i,(D) j,s,k·f(P i j,s,k) (3.19) The Icis the constant interference term. Observing from the problem above,

the allocation for control resource π isn’t considered in the optimization

of JCP-S scheme and is supposed to be considered later. However, even though the logarithm function f (P_j,s,ki ) is concave in P_j,s,ki , the integrated

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function ρi,(D)_j,s,k · f(P_j,s,ki ) is not concave in (ρi,(D)_j,s,k, P_j,s,ki ). Hence, a variable transformation εi_j,s,k = P_j,s,ki · ρi,(D)_j,s,k is also utilized to concavify the problem and rewrite the integrated function as ρi,(D)_j,s,k · f(ε

i j,s,k

ρi,(D)_j,s,k), which is proven to be

a concave function in (εi_j,s,k, ρi,(D)_j,s,k). And the new variable set ε can be view

as the eﬀective transmission power. With the variable transformation, the channel capacity (3.19) can be rewritten as

C_j,s,ki,(D)= 2Ns− Nc 2Ns · B · ρ i,(D) j,s,k · log2(1 + εi_j,s,k· g_j,s,ki,i ρi,(D)_j,s,k · Γ · (N0+ Ic) ) (3.20)

and the constraint in equation (3.18) can also be rewritten as

K ∑ k=1 J ∑ j=1 R ∑ s=1 εi_j,s,k≤ Pmax, ∀i ∈ M (3.21)

Then, the optimization problem is reformulated as a concave maximization problem. Besides, in the process of concavifying, the value 1 in the formula of channel capacity, log₂(1 + S), doesn’t require to be canceled, which has been canceled in JCP scheme in order to concavify the optimization problem under the consideration of undeterministic interference. Note that if the number 1 has been canceled, the logarithmic function will equal to a negative value with the SINRS < 1and to a much negative value with the SINRS≪ 1. For

this reason, except for the determinism for interference term, JCP-S scheme has good approximation in low SINR environment.

3.4 Proposed JCP-S Scheme

In this section, the proposed JCP-S scheme will be designed to allocate subchannel in both regions and transmission power in order to maximize the total data rate of the entire system. According to optimization theorem, let

β, η, Φ and θ be the Lagrangian multiplier sets for the constraints in (3.16),

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Lagrangian multipliers. Then, the Lagrangian function L(ρ(D),P, Λ) of the previous reformulated concave optimization problem can be shown as follows

L(ρ(D),P, Λ) = M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 C_j,s,ki,(D)+ M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 β_j,s,ki (ρi,(D)_j,s,k) − M ∑ i=1 K ∑ k=1 J ∑ j=1 R ∑ s=1 ηi_j,s,k(ρi,(D)_j,s,k − 1) − M ∑ i=1 J ∑ j=1 R ∑ s=1 Φi_j,s( K ∑ k=1 ρi,(D)_j,s,k − 1) − M ∑ i=1 θi( K ∑ k=1 J ∑ j=1 R ∑ s=1 εi_j,s,k− Pmax) (3.22)

After that, the Karush-Kuhn-Tucker (KKT) conditions for deriving the op-timal solution are

∂L(ρ(D),P, Λ) ∂εi_j,s,k    ≥ 0, if εi j,s,k= 0 = 0, if εi j,s,k> 0 (3.23) ∂L(ρ(D),P, Λ) ∂ρi,(D)_j,s,k    ≥ 0, if ρi,(D)_j,s,k = 0 = 0, if ρi,(D)_j,s,k > 0 (3.24) The partial derivative of Lagrangian function with respect toεi_j,s,kcan further

be expressed as ∂L(ρ(D),P, Λ) ∂εi_j,s,k = Bρi,(D)_j,s,kg_j,s,ki,i ln 2[ρi,(D)_j,s,k(N0+ Ic) + εi_j,s,kg_j,s,ki,i ] (3.25)

Consequently, according to equations (3.23) and (3.25), the eﬀective trans-mission power εi_j,s,k can be derived as

εi_j,s,k = ρi,(D)_j,s,k[ B

θiln 2 −

(N0+ Ic)

gi,i_j,s,k ]

+ _(3.26)

where the expression [z]+ in equation (3.26) indicates that [z]+ = z if z ≥

0 and [z]+ = 0 if z < 0. The term _θB

iln 2 can be viewed as the concept

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function with respect to ρi,(D)_j,s,k can be derived as ∂L(ρ(D),P, Λ) ∂ρi,(D)_j,s,k = − ηi j,s,k+ βj,s,ki − Φij,s+ B ln 2[ln(1 + εi_j,s,kg_j,s,ki,i ρi,(D)_j,s,k(N0+ Ic) )− ε i j,s,kg i,i j,s,k ρi,(D)_j,s,k(N0+ Ic) + εi_j,s,kgi,i_j,s,k ] (3.27) By replacing the result in equation (3.26) into equation (3.27), the function

Ri

j,s,k can be deﬁned as eﬀect capacity and written as follows

Ri_j,s,k = B ln 2[ln(1 + εi_j,s,kgi,i_j,s,k ρi,(D)_j,s,k(N0+ Ic) )− ε i j,s,kg i,i j,s,k ρi,(D)_j,s,k(N0+ Ic) + εij,s,kg i,i j,s,k ] = B ln 2{[ln( Bgi,i_j,s,k θiln 2(N0+ Ic) )]+− [1 − (N0+ Ic)θiln 2 Bg_j,s,ki,i − θiln 2(N0+ Ic− 1) ]+} (3.28) And from equation (3.24), the result below can be inferred

Ri_j,s,k    ≤ ηi j,s,k− βj,s,ki + Φij,s, if ρ i,(D) j,s,k = 0 = ηi_j,s,k− β_j,s,ki + Φi_j,s, if ρi,(D)_j,s,k > 0 (3.29)

Thus, solve the simultaneous equations containing equations (3.26) and (3.29). We can get the continuous solution sets ρ(D) and ε.

Furthermore, for the sake of obtaining the continuous solution sets ρ(D)

andεfrom the simultaneous equations containing equations (3.26) and (3.29),

the values of Lagrangian multipliers are required to be solved. Another it-erative approach that use the subgradient method as in [19] is utilized to update the value of Lagrangian multipliers. Thus, the Lagrangian multiplier

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sets β, η, Φand θ can be calculated by the following updated equations.

β_j,s,ki,(n+1)= [β_j,s,ki,(n)+ s(n)(ρi,(D)_j,s,k)]+

η_j,s,ki,(n+1)= [η_j,s,ki,(n)− s(n)(ρi,(D)_j,s,k − 1)]+ Φi,(n+1)_j,s = [Φi,(n)_j,s − s(n)( K ∑ k=1 ρi,(D)_j,s,k − 1)]+ θ_i(n+1)= [θ(n)_i − s(n)( K ∑ k=1 J ∑ j=1 R ∑ s=1 εi_j,s,k− Pmax)]+ (3.30)

whereβ_j,s,ki,(n), η_j,s,ki,(n), Φi,(n)_j,s andθ_i(n)represent thenth iteration of the Lagrangian

multipliers β_j,s,ki , ηi_j,s,k, Φi_j,s and θi respectively. Beside, s(n) = √χ_n is the step

size and χ is a tunable constant.

However, considering the solution set ρ(D) belonging to continuous

do-main and the constraint that each subchannel can be allocated to at most one UE for an eNB, the following proposition can be proposed

Proposition 1. (Necessary condition for exclusively optimal channel assignment): Assuming ρi,(D)_j,s,k,∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K is the optimal subchannel

allocation for problem in equations (3.15)-(3.17) and (3.21), if subchan-nel ˆs on CC ˆj is exclusively allocated to ˆkth UE in eNB ˆn, i.e. ρn,(D)_ˆˆ

j,ˆs,ˆk = 1

and ρn,(D)_ˆˆ

j,ˆs,k = 0,∀k ̸= ˆk, then it should satisfy:

Rn_ˆ_j,ˆˆ_s,ˆ_k> R_ˆˆn_j,ˆ_s,k , ∀k ̸= ˆk (3.31)

and implies that:

ˆ k =arg max k R ˆ n ˆ

j,ˆs,k , ∀k ∈ the set of all UEs in eNB ˆn (3.32)

Proof: The Karush-Kuhn-Tucker conditions are:

(1) β_j,s,ki (ρi,(D)_j,s,k) = 0,∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K ⇒ so if ρi,(D) j,s,k = 0, then βj,s,ki > 0, else βj,s,ki = 0 (3.33) (2) ηi_j,s,k(ρi,(D)_j,s,k − 1) = 0, ∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K ⇒ so if ρi,(D) j,s,k = 1, then ηj,s,ki > 0, else ηij,s,k= 0 (3.34)

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subchannel ˆs on CC ˆj is exclusively allocated to ˆkth UE in eNB ˆn, i.e. ρˆn,(D)_ˆ j,ˆs,ˆk = 1 and ρ ˆ n,(D) ˆ_j,ˆ_s,k = 0,∀k ̸= ˆk, then R_ˆnˆ j,ˆs,ˆk− R ˆ n ˆ_j,ˆ_s,k ≥ (ηˆ_j,ˆnˆ_s,ˆ_k+ Φ ˆ n ˆ_j,ˆ_s)− (−βˆ_j,ˆnˆ_s,k+ Φˆ_j,ˆnˆ_s) = ηnˆ_j,ˆˆ_s,ˆ_k+ β ˆ n ˆ_j,ˆ_s,k > 0 (3.35)

In other words, the JCP-S scheme would allocate subchannel s on CC j to

the kth UE in eNB n who has the largest Rn_j,s,k comparing with the other

UEs in the same eNB n. However, as mentioned before, the allocation for

data channel is released from the original discrete set, i.e. ρ¯(D)∈ {0, 1}, into

the continuous set, i.e. ρ(D)∈ [0, 1]. As a consequence, the result of optimal

solution can happen to be situated at the interval [0, 1], i.e. not exclusive concept. In such case, the discrete solution set of allocation for data re-sourceρˆ(D)_{, which is obtained according to}_{Proposition 1}_{, is suboptimal not}

optimal unless the continuous solution set ρ(D) _{belongs to 0 or 1 originally.}

With the solutions of each Lagrangian multiplier which is the convergent result in equations (3.30), the suboptimal discrete solution set ρˆ(D) can be

determined byProposition 1. However, this result doesn’t consider the con-straint of allocation for control resource. Therefore, an instinct method is proposed to make the allocation be constrained by control resource. In this method, there is a selecting sequence for all UEs in each eNB. And the con-cept of this sequence is that the more subchannels an UE in one eNB gets, the earlier this UE can occupy control resource. And if the total amount of sub-channelsn′ that one UE gets for data channels exceeds the maximum amount n′′ that the remaining control resource can guarantee, the total amount of

subchannels of this UE is adjusted to this maximum amount n′ → n′′ and

select the top n′′ subchannel which is big in value of eﬀective capacity as

its updated subchannels for data resource. However, after completing the selecting sequence of one eNB, there are probably subchannels that are not allocated to any UE owing to the prior adjustment. In such case, it prefers to allocate to the UE with largest eﬀective capacity on this subchannel who still has available and remaining control resource can use. After that, the

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Algorithm 4: Detailed steps for JCP-S scheme Input: g

Output: ¯P , ¯ρ(D)

1: Initialize β_j,s,k(0) , η_j,s,k(0) , Φ(0)_j,s and θ_(i)(0),∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K 2: Initialize counter of iteration n = 0

3: repeat

4: Calculate : With the Lagrangian multipliers obtained in iteration n,

solve the simultaneous equations containing equations (3.26) and (3.29). And get ρi,(D),(n)_j,s,k and εi,(n)_j,s,k,∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K 5: Update : Obtain β_j,s,k(n+1), η(n+1)_j,s,k , Φ(n+1)_j,s and

θ(n+1)_(i) ,∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K according to the updated

equations (3.30), and then increase n by one

6: until all Lagrangian multipliers converge

7: Obtain the suboptimal allocation for data channel ˆρ(D) _{according to}

Proposition 1

8: Execute the method that makes the allocation ˆρ(D) _{be constrained by}

control resource, and get ﬁnal solution for ¯ρ(D)

9: Obtain ﬁnal power ¯P in the optimization problem where the allocation

for data channel is known as ¯ρ(D) _{and the interference term is similarly}

viewed as constant value

ﬁnal allocation ρ¯(D) _{for data resource can be obtained.}

Like the JCP scheme, the ﬁnal power _P¯ _{will be calculated again in the}

optimization problem where the allocation for data channel is known as ρ¯(D)

and the interference term is similarly viewed as constant value. The Algo-rithm 4 shows the detailed steps to get the solution in JCP-S scheme.

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