Problem Formulation

In this thesis, we tried to solve the interference control problem at secondary BS. The secondary BS should schedule secondary users to make the interference to primary BS as small as possible. When receiving the data from served users, the secondary BS uses beamforming technique to cancel the interference from concurrent primary users. Based on the signal model and performance metric described in Section 3.1 and 3.2, the single carrier transmission can be regarded as a special case in multicar-rier transmissions. Therefore, the following problems are formulated in multicarmulticar-rier transmissions. Before data transmission, the secondary BS should know the CSI from scheduled primary users and all the secondary users to the primary BS in each uplink resource block. Then, the secondary BS schedules users whose MIMO channel vector is near orthogonal to the scheduled primary users’ receive beamforming weights.

S = {j| ˜w_i,nsc⊥˜gj,nsc, i∈ Sp, n_sc ∈ [1, Nsc], j = 1, ..., K}, |S| ≤ M . (3.8) where N_sc is the amount of subcarriers in one resource block, K is the amount of secondary users requiring for service, and M is the number of receive antennas on secondary BS. The symbol ”⊥” represents near-orthogonal here. After knowing the scheduled users in the hierarchical CR system, the sum rate maximization beamform-ing problem for the secondary network is formulated as follows:

C_sum = max

wi,1,...,w_i,Nsc N∑sc

nsc=1

∑

i∈Slog₂(1 + γ_i,nsc) , (3.9) s.t.∥wi,nsc∥ ≤ 1 .

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22

CHAPTER 4

Quasi-Convex Beamformer Design

In this chapter, we present a joint beamforming and scheduling design to make the MIMO-OFDM hierarchical CR system have good performance. A receive beamform-ing problem is proposed to maximize the sum-rate of the secondary system when knowing the users served in each system. It is transferred to a quasi-convex problem.

An iterative algorithm is proposed to choose the secondary users who have better channel condition and relatively cause less interference to the primary system.

4.1 Sum-Rate Maximization Receive Beamform-ing

In this thesis, the goal of using MIMO beamforming technique at secondary base station is to maximize the sum-rate shown in (3.3). As we can see from (3.3), the objective function is the sum of capacity of every scheduled users on a designated group of subcarriers. Since a receive beamforming problem is considered here, the beamforming weights for a user don’t cause any effect on the other users’ signal, which is not like transmit beamforming. Due to this feature, the original optimization problem can be decomposed into many sub-problems to maximize the capacity of each

user as problem (4.1).

Moreover, the multi-carrier signal can be processed in frequency domain and the subcarrier are orthogonal, so the beamforming weights for each user on each subcarrier can be determined by solving the optimization problem in (4.2).

max The objective function in (4.2) is log₂(1 + γ_i,nsc). Because log function is mono-tonic increasing, which means the output value is maximized when the input variable reaches its maximum value, the capacity maximization problem can be simplified fur-ther into an SINR maximization problem, i.e.maximizing the variable γ_i,j by deciding the beamforming weights for each subcarrier of every users.

w^⋆_i,nsc = arg max γ_i,nsc,

The above optimization problem is not convex since the variable wi,j is on the nom-inator and denomnom-inator of the objective function simultaneously. It will be showed that this problem can be transformed into a quasi-convex problem in the following subsection.

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4.2 Convexity of Beamforming Problem

Before transform the original problem into a equivalent solvable form, some definitions of the reformulation technique we used are given.

Epigraph reformulation: An epigraph of a function f :Rⁿ→ R is a set of points in Rⁿ⁺¹ domain lying on or above its graph. Every optimization problem has a general epigraph form by introducing a new variable t representing the output value of the objective function. The epigraph form of a typical optimization problem is defined in problem (4.4). A simple diagram expressing the meaning of epigraph reformulation is shown in Fig. 4.1, where the objective function f is R → R.

min f₀(x)

s.t. f_i(x)≤ 0, i = 1, ..., m

h_i(x) = 0, i = 1, ..., p

≡ min t (4.4)

s.t. f₀(x)≤ t

f_i(x)≤ 0, i = 1, ..., m

h_i(x) = 0, i = 1, ..., p.

Second order cone programming: There are many function or sets of vectors are known for its convexity in their domain of definition. Here, we introduce a convex set inRⁿ⁺¹ called second order cone. It has a specific form: K ={(x, t) ∈ Rⁿ⁺¹| ∥x∥2 ≤ t}. Its shape in R³ space is shown in Fig. 4.2. Because of the convexity of a second order cone, a specific form of optimization problem called second order cone program-ming comes up as problem (4.5), which is a convex optimization problem.

Figure 4.1: Epi-graph of a function f (x) :R → R

min c^Tx (4.5)

s.t. ∥Aix + b_i∥2 ≤ fiTx + d_i, i = 1, ..., m (4.6)

Fx = g.

where Ai ∈ R^ni×n, bi ∈ R^ni×1, fi ∈ R^n×1. The feature of SOCP is that each inequality constraints involves a generalized inequality defined by a second order cone K_i ={(x, t) ∈ Rⁿⁱ⁺¹| ∥x∥2 ≤ t}, i.e. (Aix + b_i, f_i^Tx + d_i)∈ Ki. Since x in constraint set i are mapped onto a convex set K_i through affine mapping, constraint set i is proven to be convex. The objective function and the equality constraints are just linear combinations of the elements in x which are convex obviously. Thus, if an optimization problem can be transformed to a SOCP, it is solvable and a global

25

Figure 4.2: A typical second-order cone in R³

optimal is guaranteed. So far, the discussions in this section focus on functions in domain of real numbers, but the definitions of SOCP and epigraph reformulation is same the domain of problem is expanded to complex numbers (i.e. replace R by C). The above definitions for problem reformulation are used for our beamforming problem which is in the domain of complex numbers.

Quasi-convex functions: A real valued function f (x) is quasi-convex if each of sub-level set S_α(f ) = {x|f(x) ≤ α} is a convex set for every α. The negative of a quasi-convex function is said to be quasi-concave. f (x) is quasi-concave if S_α(−f) = {x|f(x) ≥ α} is convex for every α. The objective function of beamforming problem in (4.3) is quasi-concave, which can be proved by the definition of quasi-concave functions:

S_α(−f) = {wi,nsc|f(wi,nsc)≥ α} (4.7)

where f (wi,nsc) =

(4.8) will be proved to be a second order cone in C^n+m+2 from (4.9) to (4.11):

Q_sw^†i,nsch_i,nsc² ≥ α side by Qsti,nsc. After a transposition process, the inequality becomes (4.10). Finally, the original inequality constraint is transformed into a form, which is similar to second order cone, by taking the square root on both side and introducing a channel matrix A_i,nsc and a noise vector b. As shown in (4.11), for every α, the sub-level set is like (4.6) which is a second order cone constraint and is convex. So, this objective function is a concave function. The minimum value of a quasi-convex function can be converged by iterative bisection algorithm (if one exists). On the other hand, the maximum value of a quasi-concave function can be converged too.

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4.3 Beamforming Algorithm

Now, we begin the transformation of (4.3). The optimal beamforming weight for subcarrier n_sc of SU i can be obtained by solving problem (4.12) and its epigraph form is written as problem (4.13). Next, we should change the inequality constraint in (4.13) into a convex one.

max

Note that the constraint in (4.13) is same as (4.9) by just changing α into t_i,nsc. Therefore, it can be reformulated into a convex one by taking the same procedure from (4.9) to (4.11): For fixed t_i,nsc, the above constraint of the optimization problem is a second-order cone constraint which is convex. The optimization problem can be solved by solving a series

of SOCP using bisection method. The original SINR maximization beamforming

Bisection method is a way to search for the root of a continuous function by dichotomy.

The bisection method is used to find the optimal value of the beamforming problem (4.17) and the algorithm is as follows:

1. Set the expected SINR upper bound u and lower bound l: u := 10⁴ and l := 0.

Set the convergence tolerance: ϵ := 1.

2. For i = 1 :|S|, for nsc = 1 : N_sc; while u− l ≥ ϵ :

(a) t_i,nsc := (l + u)/2.

(b) Solve the convex problem (4.17).

(c) if the problem is solvable, l := t_i,nsc; otherwise u := t_i,nsc.

When we set a small number of error tolerance ϵ, and set the bounds properly, i.e.

l ≤ t^⋆i,nsc ≤ u, the bisection algorithm can find a solution close to the global optimal enough. One thing needs to be mentioned is that the convex problem (4.17) is solved by the CVX Matlab toolbox [1].

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30

CHAPTER 5

Channel Dependent Multi-User Scheduling

5.1 Proposed Channel Dependent Scheduling Al-gorithm

So far, the secondary base station is able to have the best signal quality for every scheduled UE using MIMO beamforming technology. The next challenge is how to select a set of UEs which have good channel condition and cause less interference to the primary base station. In order to minimize the interference power to the primary BS, we should focus on the inter-cell interference term of the signal model for primary receive signal. The inter-cell interference power on i-th primary user’s n_sc-th subcarrier is formulated in subsection 3.2.2, equation (3.6):

P_ICI(i, n_sc) = Q_s∑

j∈S

 ˜w^†_i,nscg˜_j,nsc². (5.1) where ˜w^†_i,n

sc represents primary user i’s receive beamforming weights on subcarrier n_sc. ˜g_j,nsc denotes the MIMO channel vector of j-th SU to PBS link on subcarrier n_sc. S are the set of scheduled PUs and SUs. In order to avoid serious interference to primary system, the secondary BS should choose a set of SUs so that PICI is smaller than a power constraint ρ for each PU i and subcarrier n_sc. In reality, the

beam-forming vector, ˜w^†_i,nsc, for i-th scheduled primary user’s n_sc-th subcarrier is unknown for secondary BS. But the secondary BS can assume that the beamforming weight is determined by ideal zero-forcing beamforming (ZFBF) method which takes the chan-nel vectors of PUs in S_p into account. The orthogonality of channel vectors plays an important role in MIMO beamforming. For the beamforming vector of PU i ( ˜w_i,nsc), it will have the following property: ˜w_i,nsc ∥ ˜hi,nsc and ˜w_i,nsc ⊥ ˜hj,nsc,∀j ̸= i, j ∈ Sp. If the scheduled SU’s channel vector is nearly parallel to the beamforming vector of i-th primary user, ˜w_i,n^† _sc is unable to null the interference signal, which will cause great interference. Therefore, we should choose SUs whose channel vectors to the pri-mary BS is nearly orthogonal to scheduled PUs’ beamforming vectors. The channel correlation metric for PU i and SU j on subcarrier n_sc is defined as:

Ωⁿ_i,j^sc = | ˜w^†_i,nscg˜_j,nsc|

∥ ˜w_i,nsc∥∥˜gi,nsc∥. (5.2) The value of this metric is between 0 to 1. The smaller the value is, the two vectors are more orthogonal. Besides the orthogonality of vectors, the pathloss effect also affect the value of inner product: ˜w_i,nscg˜_j,nsc =∥ ˜w_i,nsc∥∥˜gj,nsc∥ cos θ. For instance, although SU j has a highly correlative channel with PU i, on the condition that its location is far away from the primary BS, its signal also causes little interference. This scenario can’t be discovered by the ordinary channel correlation metric in (5.2) because the channel gain is divided by the denominator. We introduce a upper bound of channel gain ”g_d” here, which will be used in the scheduling algorithm. g_drepresents the gain of pathloss effect for SU-PBS link. It can be set as ρ−Qsin dB, where ρ in dBm is the ICI power constraint for each subcarrier and Q_s in dBm is secondary users’ transmit power on each subcarrier. If the channel gain of a SU is less than gd, it is expected to cause ICI power less than ρ dBm to the primary system on each subcarrier. Assume there are K SUs requiring for service. In a specific resource block, a set of PUs called S_p is about to transmit data concurrently. User scheduling in the resource block

31

with N_sc subcarriers is considered here. Before introducing the scheduling algorithm, we define two functions which will be used afterwards. Firstly, the average channel correlation metric for SU i and SU j on a resource block:

ω_i,j = 1 where h represents the channel vector of SU-SBS link. Secondly, the average channel gain in step (3) is:

The following is the proposed user schedule algorithm:

1. Set an orthogonality tolerance δ := 0.1 and ICI power threshold ρ :=−120 dBm.

2. ∀j ∈ [1, K] :

If the channel gain∥˜gj,nsc∥² ≤ gd,∀nsc, SU j is put into a candidate set S^′, else:

(a) Calculate the channel correlation metric between SU j and every PU i∈ Sp on all the subcarriers in the resource block.

(b) Find the maximum value of all the correlation metrics in one resource block:

Ω^max_j = max

i,nsc

Ωⁿ_i,j^sc

(c) Put SU i, whose Ω^max_j ≤ δ, into a set S^′ Finally the candidate set becomes:

S^′ ={j|∥˜gj,nsc∥² ≤ gd,∀nsc} ∪ {j|Ω^maxj ≤ δ}.

3. Sort the element in S^′ by the average channel gain G(i) in descent order i.e.

G(S^′(1))≥ G(S^′(2))≥ ... ≥ G(S^′(|S^′|)).

4. Initialize S = S^′(1) and set nsc = 1.

5. While |S| < M ∧ PICI(k, n_sc) < ρ,∀k ∈ Sp:

In the above algorithm, we set δ := 0.1 to ensure the selected SUs whose channel is near-orthogonal to the PUs. After choosing the qualified SUs who will cause less interference to the primary system, we select the secondary user with best CSI into set S first. Then, we select the one who have better channel gain, and average channel correlations ω between it and those in S must be small relatively. Before adding a qualified SU, we should check if the inter-cell interference (ICI) power exceed the threshold ρ. This threshold is about 10 dB to the noise power (-132 dBm) on the bandwidth of one subcarrier. The value of ρ is acceptable for our system scenario since SNR for primary users are 40 dB on average in our pre-simulation.

In order to promote the overall spectrum efficiency in CR systems, the sacrifice of primary system’s performance is inevitable. The secondary system is responsible to control the interference and ensure the QoS of primary users. The basic idea of proposed scheduling scheme is to control the interference power under a threshold.

The effectiveness of our scheduling scheme will be proved by comparing with some optimal scheduling schemes which are very complicated. The performance of proposed scheme is close to the optimal scheme in simulation results.

5.2 Random and Optimal Scheduling Scheme

Since the proposed scheduling algorithm is suboptimal. We need to evaluate the effec-tiveness of the proposed scheme. In this section, we introduce three other scheduling schemes and the performance of all the scheduling schemes will be compare in chap-ter 6. The first scheme is selecting M SUs randomly from the candidate SUs. This scheme is the simplest and is esteemed as a lower bound in performance

compari-33

son. Secondly, an optimal scheduling scheme, which is dedicated to maximizing the throughput of primary system, is taken into account. The optimal solution is gotten by solving problem (5.5). The optimal solution guarantees that the scheduled SUs cause least affection to primary system in a resource block. Because problem (5.5) is a 0-1 integer programming problem which is NP hard, we can solve the problem by exhaustive search. This ICI-minimizing scheduling scheme will be combined with our quasi-convex beamforming algorithm and the throughput performance will come up in chapter 6.

The beamforming weights for PUs considered in problem (5.5) are decided by ZFBF technique previously. If (5.5) is unsolvable, which means the optimal user set of M SUs cause ICI power greater than the constraint, we should reduce the number of served SUs (i.e. M = M − 1) and solve (5.5) again. Finally, we also discuss a scheduling scheme for sum rate maximization. This scheduling scheme takes the beamforming technique of each cell into consideration and select the SUs to maximize the system sum rate. Since this scheme guarantees the maximum sum rate, it is esteemed the upper bound for all the scheduling scheme. The optimization problem is shown in problem (5.6). The optimal solution is obtained by exhaustive search.

S,wmax_j,nsc

where ˜γ_i,nsc =

Q_p ˜w^†_i,n

sc

h˜_i,nsc² Qp

∑

k∈Sp,k̸=i

 ˜w^†_i,nsch˜k,nsc

²+ Qs

∑

j∈S

 ˜w^†_i,nscg˜j,nsc

²+ ˜σ²_N

γ_j,nsc =

Q_swj,n^† sch_j,nsc² Q_s ∑

k∈S,k̸=j

w^†j,nsch_k,nsc²+ Q_p∑

i∈S^p

wj,n^† scg_i,nsc²+ σ²_N .

˜

γ_i,nsc is the throughput of primary user i on n_sc-th subcarrier and γ_j,nsc is the through-put for secondary user j. Note that the optimal beamforming weights for SUs should be decided by quasi-convex algorithm every time when the different set of scheduled SUs is considered.

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36

CHAPTER 6

Simulation Results

6.1 Assumptions

In this subsection, we describe the simulation environments for evaluating the perfor-mance of proposed beamforming and scheduling algorithm. We establish a hexagon primary cell with l kilo-meter radius and a smaller secondary cell of 500 meter ra-dius overlapping with the primary cell. The position relationship of the two cells is illustrated in Fig.6.1. For approaching real circumstance, we use the MIMO spa-tial channel model (SCM) for urban macro cell, specified by 3GPP work group TR 25.996 [18]. The pathloss, shadowing effect, and the subcarrier spacing are set accord-ing to 3GPP TR 36.814 [19] for 2 GHz carrier frequency and 10MHz bandwidth. For simulation and evaluation convenience, we only allocate the first resource block for uplink signalling and perform the signal processing. Fairness is not considered here.

The primary BS select 4 users with best channel gain simultaneously and implement zero-forcing beamforming. The secondary BS executes the proposed scheduling and beamforming technique. The detail simulation parameters are listed in Table 6.1.

Table 6.1: Simulation Parameters for Uplink Transmissions

Primary cell radius 1 Km

Secondary cell radius 500 m

Position of users random distribution for each cell.

Number of antennas 1 Tx for UE. 4 Rx for BS.

Beamforming method

primary: ZFBF

secondary: optimal beamformer

Scheduling method

primary: greedy for best CSI secondary: proposed algorithm Transmit power of primary user 23 dBm

Transmit power of secondary user 17 dBm

Noise power −174 dBm/Hz

Bandwidth of subcarrier 15 KHz

Number of subcarriers 12 (i.e. 1 RB)

Pathloss model L = 128.1 + 37.6log₁₀(R), R in Km Standard deviation of shadowing 8 dB

Channel model 3GPP SCM MIMO channel model

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Figure 6.1: Simulation environment of proposed hierarchical cognitive radio network

Table 6.2: FDD Uplink Peak Spectrum Efficiency Simulator 3GPP rel.8 ITU-R

1× 4 SIMO reqirement Spectrum efficiency 16.2 (bps/Hz) 16.8 (bps/Hz) 6.75 (bps/Hz)

6.2 Results

6.2.1 Spectrum Efficiency Performance

The system performance of our proposed CR system is shown in this subsection. We assume that there are N UEs requiring for uplink service in each cell. Although the number of users may be different in different cell and at different time in reality, we set N = 50 in each cell at each time slot, to see the general performance of the proposed algorithms. For the secondary system, N = 50 is large enough to schedule M (= 4) SUs simultaneously. In each simulation process, we spread N UEs randomly in the domain of each cell. Then, we perform the beamforming and and scheduling

0 5 10 15 20 25

Figure 6.2: CDF plot for UEs’ spectrum efficiency of the CR system, where proposed BF and scheduling algorithm is adopted.

algorithm in Table 6.1 and get the capacity for each scheduled user. Since fairness is not considered here, the PUs with best CSI is chosen in each simulation round, which means that the outcome represents the peak spectrum efficiency of the system.

The peak spectrum efficiency of the ordinary primary cell, which doesn’t coexist with secondary cell, is compared with the 3GPP calibration result in Table 6.2. The value is very similar and achieves the ITU-R requirement. This shows the validity of the simulator. We obtain the experimental CDF plot of UE’s throughput distribution in Fig. 6.2 after 200 independent rounds. From Fig. 6.2, it is found that the addition of secondary cell causes little performance degradation of primary system, but the additional throughput of secondary system is much more than the decreased quantity of primary system’s. In order to see the gain of spectrum efficiency brought from

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0 5 10 15 20 25 30

Figure 6.3: CDF for spectrum efficiency of the CR system, where the secondary BS schedules users randomly/sub-optimally while using ZFBF/quasi-convex technique.

secondary system, the spectrum efficiency of each PU in each simulation round is added by one corresponding SU. The spectrum efficiency of PUs and SUs are arranged in decreasing order, respectively, in advance. Those with the same order are summed up. This value stands for the spectrum efficiency which the CR system can achieved while a primary user is served.

For the purpose of performance comparison between beamforming schemes, we establish a scenario where secondary BS selects users by our scheduling algorithm and perform ZFBF on receive signal of scheduled users. The other more simplified

For the purpose of performance comparison between beamforming schemes, we establish a scenario where secondary BS selects users by our scheduling algorithm and perform ZFBF on receive signal of scheduled users. The other more simplified

在文檔中 階層式感知無線電使用多輸入多輸出正交分頻多工之最佳上傳模式設計 (頁 33-63)