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), (3.17), and (3.21) respectively. In addition, Λ is defined as the set of all

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) =

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

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 = Bρ^i,(D)_j,s,kg_j,s,k^i,i

ln 2[ρ^i,(D)_j,s,k(N0+ Ic) + εⁱ_j,s,kg_j,s,k^i,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 = ρ^i,(D)_j,s,k[ B

θiln 2 −(N₀+ I_c)

g^i,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 _θ^B

iln 2 can be viewed as the concept of conventional water level. Similarly, the partial derivative of Lagrangian

function with respect to ρ^i,(D)_j,s,k can be derived as 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 And from equation (3.24), the result below can be inferred

Rⁱ_j,s,k

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

sets β, η, Φand θ can be calculated by the following updated equations. 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 ρ^i,(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. ρ^n,(D)_ˆ^ˆ Proof: The Karush-Kuhn-Tucker conditions are:

(1) β_j,s,kⁱ (ρ^i,(D)_j,s,k) = 0,∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K

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

⇒ so if ρ^i,(D)_j,s,k = 1, then η_j,s,kⁱ > 0, else ηⁱ_j,s,k= 0 (3.34)

According to the result in equation (3.29) and the conditions above, if

subchannel ˆs on CC ˆj is exclusively allocated to ˆkth UE in eNB ˆn, i.e.

ρ^ˆn,(D)_ˆ

j,ˆs,ˆk = 1 and ρ_ˆ^ˆn,(D)

j,ˆs,k = 0,∀k ̸= ˆk, then R_ˆ^nˆ

j,ˆs,ˆk− Rˆ^ˆnj,ˆs,k ≥ (η_ˆj,ˆ^nˆ_s,ˆk+ Φ_ˆ^ˆn_j,ˆs)− (−βˆj,ˆ^nˆs,k+ Φ_ˆ^ˆn_j,ˆs) = ηⁿ_ˆ^ˆ

j,ˆs,ˆk+ β_ˆⁿ_j,ˆ^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 toProposition 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

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 ρ^i,(D),(n)_j,s,k and ε^i,(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.

Chapter 4

Proposed Heuristic Component Carrier Selection and Power

Allocation (HCP) and

Simplified HCP (HCP-S) Scheme

4.1 Proposed HCP and HCP-S Schemes

Pondering on the high complexity of the joint problem which considers two kinds of resource allocation at the same time, the HCP and HCP-S scheme prefer to separate this joint problem into two subproblems, which are channel allocation and power allocation respectively. That is to say, the concept of HCP and HCP-S schemes would decide how to allocate data and control channel resource to UEs in the first step. Then, secondly determine how much power should be transmitted on the channels whose allocation have been decided in the first step.

Assume that equal power is used in the first step of HCP and HCP-S

schemes. In the first step, each eNB will allocate each subchannel to the adaptive UE for data transmission. In other words, for each subchannel on each CC, each eNB will select one of its UEs, who is the most suitable UE to occupy this subchannel for data channel with the least total interference comparing among the UEs who satisfies two conditions, i.e. Condition 1 and Condition 2 respectively. The first condition is that the CC, where this UE’s PCC is located in, still has remaining unoccupied control resources for transmitting the allocative information of this additional subchannel, or this UE had never selected PCC. The second condition is that the total transmis-sion rate of the entire system will increase after allocating this subchannel to this UE. The reason of the first condition is to consider the relation between control and data resource as mention before. The reason of the second con-dition is that if the total data rate of the entire system will not increase after allocating this subchannel to this UE, it means that, even though this UE has the least total interference comparing to the other UEs, this subchannel might be in a saturated state where the negative effect of total interference on total data rate is more than the positive effect of rate improvement. In other words, subchannel resource will be truly allocated to the UE with least interference and these conditions. In addition, if UE had never select PCC for control channel, it will select the most unoccupied CC as PCC owing to having more chance to get subchannel for data transmission. Note that the total interference of one UE contains two sorts. One is the outgoing inter-ference and the other is the incoming interinter-ference, which represent the total interference to and from other eNBs respectively. And the background in-terference matrix (BIM) is to record the outgoing and incoming inin-terference for each UE.

In the second step, the power allocation will be calculated from the op-timization problem where the allocation for data channel is known in the first step. Besides, the optimization problem for power allocation can be for-mulated to GP-form which consider the undeterministic interference term as

mentioned in Section 3.1, and call it as HCP scheme if the power allocation is obtained from this way. Or can also be formulated to the form where the interference is view as deterministic term as mentioned in Section 3.3, and call it as HCP-S scheme. Finally, the detailed steps for HCP and HCP-S schemes are shown in Algorithm 5

Algorithm 5: Detailed steps for HCP and HCP-S schemes Input: g

Output: ¯ρ^(D), ¯P^(D) begin

1. Obtain the allocation for data channel ¯ρ^(D): Initial P_j,s,kⁱ = P_equal,∀i ∈ M, ∀j ∈ J, ∀s ∈ S, ∀k ∈ K Initial ¯ρ^(D)= 0

for j = 1 to J do for s = 1 to R do

for n = 1 to N do 1. Initial ˆk = 0

2. Find the UE ˆk with least total interference among the UEs served by eNB n that satisfy Condition 1 and Condition 2

3. if ˆk ̸= 0 then

if (UE ˆk has no PCC) then

UE ˆk selects CC which has the most unoccupied subchannels as PCC

¯ ρ^n,(D)

j,s,ˆk = 1

2. Calculate power allocation ¯P^(D) with fixed ¯ρ^(D):

With the allocation for data channel ¯ρ^(D) being known, the power allocation ¯P^(D) can be calculated from optimization problem as the form in Section 3.1 or in Section 4.1

Chapter 5

Performance Evaluation

In addition to the proposed JCP, JCP-S, HCP, and HCP-S schemes, the heuristic scheme mentioned above with equal power allocation, which is called HCP-E scheme, is also considered in this thesis for comparison pur-poses. That is, the power in HCP scheme is allocated equally across all used channels. For simplicity, assume that all UEs are stationary and each UE chose the eNB as serving base station (BS) with highest SINR according to reference signal (RS) in control region. In the simulations, the cell deploy-ments follow the wrap around topology [13] and each cell contains a centering eNB with a number of UEs, who chose this eNB as serving BS, uniformly distributed in the cell coverage.

In this section, the simulations are presented to demonstrate the perfor-mance of proposed JCP, JCP-S, HCP, HCP-S, and HCP-E schemes from the perspective of total transmission rate in the data region. The simulation is conducted via MATLAB and utilizes CVX [20] as the tool to solve optimiza-tion problem. Moreover, the results are all averaged from 100 simulaoptimiza-tion runs and the related simulation parameters are listed in Table 5.1.

In order to demonstrate the total data rate of JCP, JCP-S, HCP, HCP-S, and HCP-E schemes under different environments where the total number of CC is changed from 1 to 5, the Fig. 5.1 and Fig. 5.2 show the results in high and low SINR situations respectively. In other words, UEs are intentionally

Table 5.1: Simulation Parameters Bandwidth of each CC for downlink 1 [MHz]

Modulation parameter (Γ) − ln (5 · 0.01)/1.5 Noise spectrum density (n₀) −174 [dBm]

Maximum Power (P_max) 46 [dBm]

Total subchannels’ number in one CC (R) 5

Carrier Frequency 700 [MHz] - 2.5 [GHz]

Path loss model from eNB to UE 128.1 + 37.6 log₁₀[dB]

Shadowing standard deviation 10 [dB]

ISD 500 [m]

located in the central area of each cell in Fig. 5.1, which implies that the average SINR is high in general. On the contrary, UEs are intentionally located in the edge of each cell in Fig. 5.2, which implies that the average SINR is low.

Observing these two figures, the performance of JCP scheme is always best comparing to the other schemes. However, JCP scheme has the highest complexity where the joint problem and undeterministic interference term are both considered in JCP scheme. Moreover, the total data rate of HCP scheme is higher than the total data rate of JCP-S scheme in situation of low SINR, but lower in situation of high SINR. This result implies that the concept of interference reduction in the HCP scheme is dominant enough to the performance in situation of low SINR, and it also makes sense that the effect of interference has a great influence on the throughput of the entire system in low SINR condition. Besides, we can see that the performance of HCP-E scheme, which uses simply equal power allocation for power control, is not much worse than the other schemes in situation of high SINR. This result implies that the effect of power allocation doesn’t has great influences on the throughput because it doesn’t cause much interference no matter how the power allocation is in situation of high SINR.

According to the specification in [21], there exist a modulation and coding scheme (MCS) to stipulate that what modulation and code efficiency should be used in different conditions of channel quality, i.e. channel quality index

1 2 3 4 5 0

10 20 30 40 50 60 70

Total number of CC

Total data rate (Mbps)

JCP JCP−S HCP HCP−S HCP−E

Figure 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.

1 2 3 4 5 5

10 15 20 25 30 35 40 45 50 55

Total number of CC

Total data rate (Mbps)

JCP HCP JCP−S HCP−S HCP−E

Figure 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.

(CQI) in MCS. As mentioned before, what modulation and code efficiency is used influences the capability of one resource block in control region to transmit allocative information for data channel. Assume low-rank modu-lation QPSK is used. Furthermore, in order to observe system performance under the condition of insufficient control resource, in Fig. 5.3, Fig. 5.4, and Fig. 5.5, assume that only one CC can be used as control resource for all UEs in each eNB even though there are J = 5 CCs in this simulation.

Fig. 5.3 demonstrate total data rate of JCP scheme under different num-ber of OFDM symbols in control region N_c, which is changed from 1 to 7.

Observing Fig. 5.3, we can see that there exist the optimal number of OFDM symbols in control region under each code efficiency. For example, when code efficiency equals to 0.1523, the optimal number of OFDM symbols in control region is 4, which result in the maximum value of total data rate. When number of OFDM symbols equals to 1 to3, the capability of one control CC can’t carry the allocative information of all data channels. In other words, in these cases, not all data channels in all CCs can be used for data trans-mission. Thus, the total data rate will not reach the maximum value until number of OFDM symbols in control region equals to 4. After the optimal value, the system throughput will decrease. This is because control resource is so enough to carry whole allocative information of all data channels that the increase in number of OFDM symbols in control region will be useless and will decrease the transmission time of data region 2Ns− Nc, which leads to the decrement in data rate. Besides, we can see that the optimal number of OFDM symbols in control region will be smaller when the code efficiency gets higher. This is because if the code efficiency gets higher, it requires less control resource to carry the allocative information of all data channels and will have more transmission time in data region.

The Fig. 5.4 shows total data rate of JCP, JCP-S, and HCP schemes un-der different code efficiency. Observing from this figure, the total data rate of three schemes increase in the first four efficiency on account of that the

1 2 3 4 5 6 7 20

25 30 35 40 45 50 55 60 65

Total number of OFDM symbol(s) in control region

Total data rate (Mbps)

Code efficiency=0.6016 Code efficiency=0.3770 Code efficiency=0.2344 Code efficiency=0.1523

Figure 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.

maximum amount of data channel, which can be used for data transmission at the same time, also increase when the value of code efficiency increases.

However, we can see that there is not much increment in the total data rate when code efficiency raises from 0.6016 to 0.8770. This is because the maxi-mum amount of data resource has reached the value of total data resource, i.e. N_total= R· J = 25. As mentioned before, note that if code efficiency gets higher, it means that less amount of data channels can be used for data trans-mission. Comparing the detailed value of performance between JCP scheme and HCP scheme, we can observe that the difference in percentage between the performance of these two schemes raises when the code efficiency reduce.

This atmosphere implies that JCP scheme can make much better selection for channel resource than HCP scheme especially in the situation of lower code efficiency.

In Fig. 5.5, it shows that the total data rate of JCP, JCP-S, and HCP schemes under different number of UEs per eNB. Observing this figure, we can see that when there is one UE in each eNB, the performance of HCP scheme is near to the performance of JCP scheme. It is because that in this special case, the channel allocation in both schemes are just the same, where the channel resource will be all allocated to this only UE no matter what the channel quality this UE has. Besides, we can see that the performance of JCP scheme gradually grows when there are more and more UEs in each eNB. This might because that the probability of having the kind of UEs, which has better channel quality, also grows when there are more and more UEs in each eNB. And in JCP scheme, the channel resource will prefer to be allocated to this kind of UE and result in better data rate. However, the HCP scheme has the opposite tendency because that the HCP scheme, whose target is to reduce total interference, will not indeed allocate channel resource to this kind of UE.

0.152310 0.2344 0.3770 0.6016 0.8770 20

30 40 50 60 70

Code efficiency

Total data rate (Mbps)

JCP JCP−S HCP

Figure 5.4: Total data rate of JCP, JCP-S, and HCP schemes under different 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 N_c= 1.

1 2 3 4 5 38

39 40 41 42 43 44 45 46

Number of UE(s) per eNB

Total data rate (Mbps)

JCP JCP−S HCP

Figure 5.5: Total data rate of JCP, JCP-S, and HCP schemes under different number of UEs per eNB: total number of eNBs M = 4, total number of CCs J = 5, number of OFDM symbol in control region N_c= 1.

Chapter 6 Conclusion

In this thesis, the joint component carrier selection and power allocation (JCP) scheme based on geometric programming which transforms non-convex problem to convex problem is proposed to determined the allocation for both kinds of resource, channel and power respectively. And JCP scheme has great performance in the viewpoint of total data rate. However, JCP scheme has high complexity and computational cost since the consideration of the unde-terministic interference term even though the problem is convex. Therefore, JCP-S scheme, a simplified version of JCP scheme, is then proposed where

In this thesis, the joint component carrier selection and power allocation (JCP) scheme based on geometric programming which transforms non-convex problem to convex problem is proposed to determined the allocation for both kinds of resource, channel and power respectively. And JCP scheme has great performance in the viewpoint of total data rate. However, JCP scheme has high complexity and computational cost since the consideration of the unde-terministic interference term even though the problem is convex. Therefore, JCP-S scheme, a simplified version of JCP scheme, is then proposed where

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