5.4 A Low-complexity Power-Minimization Solution
5.4.1 Correlation Based User Selection Method via Dual Decomposition 71
First, we show a simply user selection without considering the orthogonality of users’
channels in space domain. In addition, we choose the number of users at each subcarrier to meet the maximal number of users ˆK = NT/nr. Hence, the ˆK users whose efficiency is larger than other K − ˆK users will be selected at each iteration. After each decision is completed, the channel GNR of the selected users will be modified based on block diago-nalization approach and the unselected users will cant’t occupy this subcarrier assigned at this iteration. Then, run K OMPA algorithm again and go into next iteration.
However, such user selection scheme don’t take the orthogonality into account. Al-though the selected ˆK users have larger efficiency computed by dual decomposition under perfect interference free communication. However, it is possible the channel of these ˆK users may be high-correlated, this case will make the channel quality decrease much through block diagonalization in order to fit the interference free communication criteria. So, we propose another user selection scheme which jointly consider the users’
channel correlation and the users’ channel efficiency.
We propose a second user selection scheme based on the semi-orthogonal user selec-tion (SUS) and all users’ spectrum efficiency within each iteraselec-tion.
Specifically, the transmitter selects the first user from initial user set A0 = {1, 2, · · · , K}
over the mth subcarrier as
νk,m =
η
X
l=1
k,m,l (5.22)
π(1) = arg max
k∈A0
νk,m. (5.23)
If the maximal number of users can simultaneously over the same subcarrier is equal to one for SISO system discussed in chapter 3, the user selection is completed now.
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For MIMO system there are more than one user can transmit over the same subcarrier.
Hence, after select p users, if i < ˆK, the p + 1th user is selected within the user set Ap = {1 ≤ k ≤ K : g(hk, hπ(j)) ≤ T , 1 ≤ j ≤ p} (5.24) as
π(p + 1) = arg max
k∈Ap
νk,m. (5.25)
where T is a design parameter that indicates the maximum spatial correlation allowed between users’ channels. In this way, the transmitter will choose users that have high efficiency over current discussed subcarrier and mutually semi-orthogonal. g(x, y) de-notes spatial correlation value. For the case x, y are vector in MISO system, g(x, y) can be given by
g(x, y) = |xyH|
|x||y| (5.26)
For MIMO system, x, y are matrices, g(x, y) is computed by g(x, y) = |ˆxHy|ˆ
|ˆx||ˆy| (5.27)
where the columns of ˆx represent the right singular vectors of x. We summarize the proposed joint subcarrier (channel) and user assignment algorithm below.
5.5 Numerical Results and Discussion
Some simulated performance of the proposed resource allocation algorithms are pre-sented in this section. We assume the channel matrix with i.i.d zero-mean, unit-variance complex Gaussian entries. For simplicity, we assume that all users are with the same required data rate and BER, i.e. Rj = R, ∀j. The same required data rate is possible decided after some scheduling in MAC layer. In Fig. 5.2, we compare the fixed subcar-rier assigned and our proposed algorithms in MISO communication with base station has 4 antennas and all mobile terminals are equipped with single antenna. The proposed
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Step 1: (Initial channel allocation)
Set C(j) = {i | 1 ≤ i ≤ N }, for 1 ≤ j ≤ K.
set S = {1, 2, · · · , N } and t = 0 Step 2: (channel selection )
each user individually invokes the OMPA algorithm based on its SCS C(j) if t < N
i∗ = argi∈Sνi according to (5.21)
Set S = S\{i∗}, t = t + 1, then goto Step 3.
else goto Step 6.
end
Step 3: (initial user election )decide the first user in the i∗ subcarrier Set A0 = {1, 2, · · · , K} and
νk,i∗ =Pη
l=1k,i∗,l
π(1) = arg maxk∈A0νk,i∗. Set p = 1 goto Step 4 Step 4: (user election )
Set Ap = {1 ≤ k ≤ K : g(hk, hπ(j)) ≤ T , 1 ≤ j ≤ p}
if Ap = ∅ then t = t + 1 goto Step 5.
else if p = ˆK then t = t + 1 goto Step 5.
else
π(p + 1) = arg maxk∈Apνk,m
Set p = p + 1 and goto Step 4.
end
Step 5: (modify GNR via block diagonalization and channel set C(j) for 1 ≤ j ≤ K.) for j = 1 : K
ifj /∈ {π} then C(j)\{i∗} end end
modify GNR via block diagonaization for selected users’ indexes {π}
goto Step 2
Step 6: (Output) The final channel allocation and user election
Table 5.1: A joint channel assignment and user selection algorithm
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algorithm 2 has be described in Table 5.1. In addition, we show the performance of the proposed algorithm1 in order to see what importance the user selection. The algorithm1 is done by always selecting the ˆK users which have better efficient values, where ˆK is equal to NT/nr = 4. The proposed algorithm 2 with user selection is superior to the algorithm 1 without user selection. The reason is that even if the number of eigen-channels in the same subcarrier in algorithm 1 than in algorithm 2, but the users which have good efficient values may be correlated each other. It produces that after block diagonlization the original good channel condition of these users will decrease very much.
Simultaneously, we can find the difference among these three schemes increase when the required rate increase. It presents that subcarrier assignment play an important role when the traffic load of system increases and the wireless resource is still limited.
5 10 15 20 25 30 35
10−5 10−4 10−3 10−2 10−1
BER
Average Power(dB) fixed
algorithm1 algorithm2 fixed algorithm1 algorithm2
R=10 R=20
Figure 5.2: Average power per user for the case of 8 subcarriers, 8 users; GNR=0dB, NT = 4, nr = 1, and R = 10, 20 respectively.
Fig. 5.4 plots the effect of the transmit antenna numbers on the system performance.
Several performance trends are observed. First, the more the transmit antennas the more
74
important the user selection from the comparison between algorithm 1 and algorithm 2.
The difference between fixed assignment and the algorithm 1 decreases when transmit antenna number increasing. In other words, the traditional fixed assignment can improve its poor performance by adding antennas. Finally, this figure shows that the proposed algorithm 2 at the NT = 8 case outperform the traditional fixed assignment at NT = 16 and is close to the performance of the algorithm 2 at NT = 16. For the same performance constraint, we can reduce the transmit antennas by the joint user selection and subcarrier assignment proposed algorithm described in Table 5.1.
−5 0 5 10 15 20
10−5 10−4 10−3 10−2 10−1
BER
Average Power (dB) fixed(8x2)
algorithm 1(8x2) algorithm 2(8x2) fixed (16x2) algorithm 1(16x2) algorithm 2(8x2)
Figure 5.3: Average power per user for K = 32, N = 64, R = 20, nr = 2, GNR=0dB, and NT = 8, 16 respectively.
Next, we discuss the case GNR=-10dB and the other parameters are the same in Fig.
5.4. We can observe that the more transmit antennas are robust to the poor channel conditions. In the same way, the user selection still play an important role in resource allocation.
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5 10 15 20 25 30 10−4
10−3 10−2 10−1
BER
Average Power (dB) fixed(8x2)
algorithm 1(8x2) algorithm 1(8x2) fixed(16x2) algorithm 1(16x2) algorithm 2(16x2)
Figure 5.4: Average power per user for K = 32, N = 64, R = 20, nr= 2, GNR=-10dB, and NT = 8, 16 respectively.
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Chapter 6 Conclusion
OFDMA is an effective multiple access scheme for wideband wireless mobile net-works. Besides its anti-fading capability, an OFDMA system can achieve high spectral efficiency in a multiuser environment by taking advantage of the multiuser diversity and adaptively allocating subcarriers and time slots to the the most suitable users with the minimum required transmit power. Alternatively, one can also maximize the total (weighted) rate or product rate, if fairness is of concern, with power and/or some QoS constraints. Efficient dynamic RA algorithms to solve the above constrained optimiza-tion problems in real time is thus crucial for realizing this potential advantage.
Based on the principles of dynamic programming and branch-and-bound, we propose two algorithms–the DPRA and BBRA algorithms–which give either near-optimal or op-timal solution. In contrast to the existing algorithms, which suffer from the shortcomings of requiring high complexity and/or unsatisfactory performance, the DPRA algorithm renders near-optimal performance with relative low complexity. Since the existing effi-cient algorithms are designed with a discrete-rate constraint and use some suboptimal water-filling solution, we make some modifications for fair comparisons. As expected, the resulting ACG and RCG based DPRA algorithms are shown to provide less satis-factory performance with reduced complexities. With proper reuse of the water-filling solution obtained in earlier stages, the average DPRA complexity can be further reduced and is insensitive to d, N and the required sum rate. The average complexity of the
77
BBRA algorithm, on the other hand, is at least an order higher than that of the DPRA algorithm when the number of users is greater than 10 but is still much less than the known algorithms for obtaining the optimal solution.
The DP-based weighted sum rate or product rate maximization algorithms are as efficient as their counterpart for power minimization. The product rate maximization enjoys the benefit of being highly fair. We also apply the dual decomposition approach to develop low-complexity algorithms for solving the weighted sum rate and product rate maximization problems. Channel ordering is not needed for these cases and the number of calls to single user optimization process (water-filling or OMPA) is at most equal to the product of the subcarrier and user numbers. Finally, we extend our work to MIMO system and propose a low-complexity high performance suboptimal algorithm for MIMO-OFDMA networks.
Our numerical experiment in both independent and correlated fading environments have demonstrated that the near-optimal power-minimization DPRA algorithm is suit-able for real-time resource allocation application. In fact, when operating in a more practical correlated fading environment, the outstanding features of the proposed sub-optimal algorithms become even more obvious: the performance gains of our algorithms are much higher than those in the ideal i.i.d. fading environments.
On the other hand, the optimal BBRA algorithm is practical only if the user number is small, e.g., d ≤ 5. Nevertheless, the latter algorithm offers the optimal solution and performance for large N and d with reasonable complexity, which has never been achieved before and is needed for benchmarking and comparison purposes.
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Appendix A
An optimal mono-rate power allocation algorithm
Let us redefine the normalized channel capacity ri by
ri = log2
1 + |hi|2pi σ2i
≡ log2(1 + piai) (A.1)
where the subscript i denotes the ith channel and |hi|2, pi, σi2 are the corresponding channel gain, transmitted power, and noise power, respectively. In addition, the N orthogonal channels are sorted according to descending channel gain-to-noise ratio, e.g., a1 > a2 > · · · , aN, ai ≡ |hi|2/σ2i. Note that because of (A.1), power and rate allocations are equivalent provided that ai is known.
For the mono-rate case, (4) becomes
minP N
X
i=1
pi , s.t.
N
X
i=1
ri ≥ R, pi ≥ 0, (A.2)
The water-filling solution implies that only the strong channels (those whose reciprocal channel gains are below the water-filling level) will be used. Hence we assume that only the strongest x channels are used so that the power and rate for the weakest N − x channels are identically zero, i.e., pi(x) = ri(x) = 0, x < i ≤ N , where pi(x), ri(x) denote the power and rate of the ith channel when only the first x channels are activated. The optimization problem (A.2) then become that determining the optimal x. Define the
79
Lagrange dual function as
and omit the constraints 0 ≤ pi for the moment. Taking derivative with respect to ri
for i = 1, 2, · · · , x we obtain λ = eR/xˆa(x)ln 2, where ˆa(x) =
Obviously, it is possible ri(x) < 0 as the constraint pi ≥ 0 has been removed.
Note that
To find the constrained solution we need the following definition.
Definition 4. An unconstrained solution r(x, N ) = (r1(x), r2(x), .., rx(x), 01×(N −x)) is said to be admissible if the least rate rx(x) > 0. The admissible active channel number sets for the problem defined by (A.2) is defined by F = {x|rx(x) > 0, 1 ≤ x ≤ N }, where rx(x) is given by (A.4).
Lemma 2. The total transmitted power associated with the admissible unconstrained optimal rate assignment (A.4) is a decreasing function of the number of channels used.
In other words, N1 < N2 =⇒PN1
The minimum required power for the case x = m is given by expressed as a function of rm+1.
P˜m+1 = P˜m0 + pm+1 =
For 0 ≤ rm+1(m + 1) < R, the second derivative of g(rm+1(m + 1))
g(2)(rm+1(m + 1)) = −1 ˆ
a(m)am+1
ham+1
m e(R−rm+1(m+1))/m+ ˆa(m)erm+1(m+1)i
(A.12)
is always negative. Since g0(r∗m+1(m + 1)) = 0, g0(rm+1(m + 1)) > 0, for 0 ≤ rm+1(m + 1) < rm+1∗ (m + 1), the fact that g(0) = 0 then lead to the desired conclusion that g(rm+1∗ (m+1)) > 0. In other words, the minimum power for the case x = m is larger than that for the case x = m + 1 which can be achieved with rm+1(m + 1) = r∗m+1(m + 1).
The above two Lemmas suggest that the solution to the constrained optimization problem (A.2) can be found by repeatedly calculating the unconstrained solution (A.5) for x = N, N − 1, N − 2, · · · until the constraints pi ≥ 0, ∀ 1 ≤ i ≤ x are satisfied. A similar but less efficient solution was proposed by Fischer and Huber [2] who iteratively recompute (A.5) by excluding all negative-rate channels and setting x ← x − l, where l is the number of negative-rate channels. Such an approach does not necessarily give the optimal solution and the issue of optimality was not addressed in [2]. Instead of sequentially decreasing x with a decrement of 1, we accelerate the process of locating the optimal x through a bisection search so that the optimal power allocation can be found in Table V.
The resulting algorithm will be referred to as the optimal mono-rate power allocation (OMPA) algorithm henceforth. Note the OMPA algorithm can easily be modified to solve the maximum sum-rate problem
maxX
i
ri, s.t. X
i
pi ≤ P, pi ≥ 0, (A.13)
Table V: An Optimal Mono-rate Power Allocation Algorithm
82
Step 1: (Initialization) Given ai, 1 ≤ i ≤ N ,and R, set upbound = N , lowbound = 1,
and x∗ = [(upbound + lowbound)/2].
Step 2: (Update the lowest rate) rx∗(x∗) = x∗R + log2h
ax∗
ˆ a(x∗)
i , where ˆa(x∗) =
Qx∗ j=1aj
1/x∗
, the number of iterations.
Step 3: If rx∗(x∗) ≥ 0, lowbound ← x∗, else upbound ← x∗
Step 4: If lowbound < upbound − 1, x∗ ← [(upbound + lowbound)/2], go to Step 2; else x∗ ← lowbound,
ri(x∗) ← 0, for i > x∗ and compute ri(x∗), for 1 ≤ i < x∗.
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