Although the optimal ˆΩ can be obtained by the exhaustive search, we still prefer to develop an efficient algorithm to obtain ˆΩ to reduce search space. In (C.4) of (4.9), the power allocation is across the subcarrier. Thus, the optimal beamforming weight in each subcarrier is correlated with others. A simplified formulation is first considered as:
where (C.2) represents equal power allocation over all subcarriers. Then, the power allocation, beamforming and scheduling in each subcarrier is independent with other subcarriers. That is, Ωj = [Ω1,j,· · · , ΩK,j]T can change without impacting other subcarrier. Therefore, an iterative algorithm can be developed for the single car-rier transmission, and may apply to multicarcar-rier transmission by utilizing to each subcarrier.
The following formulations are expressed as single carrier transmission. First, we define:
R2(W, Ω) =
Then, the optimization problem is formulated about Ω as follows:
R1(Ω) = max
W∈Θ(W)R2(W, Ω) (4.14)
where Θ(W) is the feasible set with respect to (4.9). Note that the size of set Θ depends on the value of Ω. In (C.4) of (4.9), if Ω becomes smaller, the set of available beamforming weight W would become smaller. Consider two vector Ω(1) = [Ω(1)1 ,· · · , Ω(1)K ] and Ω(2) = [Ω(2)1 ,· · · , Ω(2)K ], corresponding to two non-empty set Θ(1)
The first observation shows that the system sum rate can be improved by decreasing Ω, but the second observation seems to reveal opposite trend as Ω decrease. There-fore, we take a simple example to investigate the relationship between sum rate and Ω.
29
10−5
100
105 10−5
100
105 5
6 7 8
Ω1 Ω2
Sum rate [bps/Hz]
Figure 4.1: Sum-rate versus Ω with three transmit antennas and two secondary users
Assumes that the number of transmit antennas is three, and there are two secondary users. As shown in Fig. 4.1, the secondary system sum rate increases significantly by decreasing Ω = [Ω1, Ω2] when both Ω1 and Ω2 are large. However, the optimal value does not occurs at the boundary, which can be seen more clearly in Fig. 4.2 and Fig4.3. If Ω decreases without restrictions, the sum rate can always be not improved.
Note that if Ω is set to be zero, the solution will become zero-forcing beamforming, which is usually not an optimal solution. To summarize, the system sum rate can be improved by decreasing the intra-user interference power constraint for those users in serving set Ωk, k∈ S, but Ωk can not be too small.
Therefore, an iterative algorithm can be developed to maximize the sum rate
10−6 10−4 10−2 100 102 104 7.2
7.25 7.3 7.35 7.4 7.45 7.5 7.55
Sum Rate [bps/Hz]
Ω1
Figure 4.2: Sum-rate versus Ω with three transmit antennas and two secondary users, and fixed Ω2.
of the secondary system. The entire procedures of the iterative algorithm are shown in Fig. 4.4. First, we initialize i = 1 and k = 1, which corresponds iterative loop and user index, respectively. In order to not restrict the intra-user interference to those non-served users, the feasible set Ω(0) is set to a sufficiently large non-negative real value. Then, we solve (4.10), and check whether initial Ω(0) is feasible. Second, we choose a constant step size δ, where 0 < δ < 1, and update Ω(i−1) to Ω(i) by
Ω(i) = Ω(i−1)− (1 − δ)Ω(i)k ek . (4.15) where ek is the kth column of a K× K identity matrix IK. Then, we solve (4.10) for new Ω(i) and check whether the system sum rate improvement R1(Ω(i))− R1(Ω(i−1))
31
10−6 10−4 10−2 100 102 104 5
5.5 6 6.5 7 7.5
Ω2
Sum Rate [bps/Hz]
Figure 4.3: Sum-rate versus Ω with three transmit antennas and two secondary users, and fixed Ω1.
is larger than a pre-defined threshold ∆r ≥ 0, which is a small constant value. If so, set i = i + 1 and continue to decrease Ωk until the capacity improve insignificantly.
Otherwise, set Ω(i) = Ω(i−1), R1(Ω(i)) = R1(Ω(i−1)) and ˆW(i) = ˆW(i−1), and decrease the intra-user interference to the next secondary user until k = K. Now, we have the optimal beamforming ˆW = [ ˆW1,· · · , ˆWK], which includes rank one and zero matrix. The serving set ˆS includes the users with rank 1 solution, where the size of S is smaller than or equal to than the number of transmit antennas M . Although an iterative algorithm is utilized to find the optimal Ω, there is still an error compared with the optimal solution. The margin of this error depends on the step size δ and the number of secondary users. This error may increase as the number of secondary
users K increase. Therefore, the users, who are not in the serving set S, can be dropped and set K = |S|, where |·| denotes the size of a set. Again, we run the iterative algorithm for users in S. Note that it may still have a zero matrix solution for some users, which means the number of served users may be smaller than|S|. In the next chapter, numerical results with the single and multiple carrier transmissions are shown. The results of multicarrier transmissions are obtained by executing the iterative algorithm shown in Fig. 4.4 for each subcarrier.
33
Set (0) to a large
Find the serving set S, which is secondary users
with largest rate
35
CHAPTER 5
Numerical Results and Discussion
5.1 Single Carrier Case
5.1.1 Simulation Assumptions
In this subsection, we describe the simulation environments for evaluating the per-formance of the proposed iterative algorithm. The distribution of the primary and secondary systems are shown in Fig. 5.1. Assume that the primary and secondary BS are located at (0 m, 300 m) and (400 m, 0 m), respectively. The radius of the secondary system is 300 m. In a primary user with fixed location (0 m, 200 m), K secondary users are uniformly distributed at the secondary cell edge. The secondary BS equipped with M transmit antennas which are equal spacing with half a wave-length. Assume that the CSI is perfectly known at the secondary BS. The transmit power of the primary user and secondary BS are 20 dBm and 26 dBm, respectively.
The noise power is −104 dBm. The pathloss exponent is four and the standard de-viation of shadowing is 8 dB in both primary and secondary system. Simulation parameters are listed in Table 5.1.
There are two schemes for comparison: the first scheme is zero forcing beam-forming between primary BS and M−1 secondary users with the exhausted scheduling and the optimal power allocation, which is regarded as a lower bound. Another scheme is singular value decomposition (SVD) to the secondary users, and does not control
Primary BS Primary user Secondary BS Secondary user
(400m,0m) Secondary BS Primary BS
(0m,300m)
D (0m,200m)
Figure 5.1: Distribution of primary and secondary systems.
the interference to the primary BS, but the secondary users are still interfered by the primary user. The SVD includes the transmitter and receiver beamforming, and is the optimal beamforming for the downlink MIMO system. The receiver beamform-ing in the hierarchical network architecture is completed by the cooperation between secondary users. However, secondary users do not cooperate with each others in the proposed algorithm. Thus, the SVD scheme can be regards as an upper bound.
5.1.2 Effects of Number of Transmit Antennas
Fig. 5.2 shows that the sum rate versus the number of the secondary BS transmit antennas for three beamforming and scheduling schemes. We can observe that the proposed scheduling scheme has significant improvement compared with the zero-forcing scheme. The gain comes from allowing some interference power to the primary
Table 5.1: Simulation Parameters for Single Carrier Transmissions Position of primary BS (0 m, 300 m)
Position of primary user (0 m, 200 m) Position of secondary BS (400 m, 0 m)
Position of secondary users Uniformly distribution at cell edge
Cell radius 300 m
Antenna spacing equal spacing with λ2 Transmit power of primary user 20 dBm Transmit power of secondary BS 26 dBm
Noise power −104 dBm
Pathloss exponent 4
Standard deviation of shadowing 8 dB
Channel type Rayleigh fading
37
3 4 5 4
5 6 7 8 9 10 11 12 13
Number of transmit antennas
Spectral efficiency (bps/Hz)
Proposed algorithm ZF with optimal scheduling SVD with optimal scheduling
Figure 5.2: Sum rate for various numbers of transmit antennas, where the number of the secondary users is 10, and Imax = 10 dBm.
BS in our algorithm. However, interference power to the primary BS is zero. In addition, the proposed algorithm may serve M users in each transmission, but the zeor-forcing scheme served M−1 users fixedly. Shannon capacity formula is logarithm function, and the low SINR region can be regarded as the linear region. Thus, one more user can be served may lead to significant improvement.
5.1.3 Effects of Maximal Allowable Interference to Macro Cell
Fig. 5.3 shows that the sum rate versus the maximal allowable interference to pri-mary BS Imax for the three considered beamforming and scheduling schemes. The SVD scheme does not consider the interference to primary BS, and the zero-forcing
always forces the interference to zero. Thus, the sum rate of these two schemes will not change as Imax changes. However, our algorithm increase significantly as Imax increases. It means that our proposed scheme is very flexible for different interfer-ence constraints Imax. As Imax increases, the sum rate of the proposed algorithm will gradually approach to the SVD scheme. Nevertheless, they will not have the same performance even if Imax goes infinity because of the received beamforming of SVD scheme. As Imax decreases, the sum rate of the proposed algorithm will grad-ually approach to the zero-forcing scheme. In fact, the proposed algorithm and the zero-forcing scheme will have the same performance as Imax approach to the negative infinity. However, the proposed algorithm can improve capacity significantly when Imax is large. Note that Imax will affect the coverage and transmit power of the secondary BS. Thus, the adaption of Imax is very important for the underlying CR system.
5.1.4 Effects of Number of Cognitive Radio Users
Fig. 5.4 shows the sum rate versus the number of the secondary users for three beam-forming and scheduling schemes. The slope of the proposed algorithm is same as the SVD scheme, and larger than zero-forcing scheme. Thus, the user diversity of the proposed algorithm is same as the exhausted scheduling.