(4.40) where
gR(n, k) = gRm(n, k)|m=mopt, mopt = arg max
m gRm(n, k) (4.41) After examining all relayed links and making necessary link switches, we compute the corresponding sum rate and fairness index. The resulting algorithms are summarized in Tables 4.1 and 4.2, respectively.
4.6 Numerical Results and Discussions
Numerical performance of the proposed algorithms is presented in this section. We consider a network with several MS nodes that are random distributed within a
120-degree section of the 600-meter radius circle centered at the BS. The relay stations are placed on a circle with a 200-meter radius with a equal angular spacing. The probability density function (pdf) of the MS locations is given by [11]
P = r40
3.3. Each transmitted signal experiences attenuation with a path loss exponent value of 3.5 and, in any direct or relay link, each subcarrier suffers from independent Rayleigh fading. For the convenience of comparison, we normalized the link gain with respect to the worst-case gain corresponding to the longest link distance. We set σ2 = 1. 4 × 105 simulation runs were carried out to estimate the performance. We compare the sum rate and fairness performance of our algorithms with that of the modified Awad-Shen (MAS) algorithm which is a modified version of the original AS algorithm given in [12].
Because the original AS algorithm considers amplify-and-forward cooperative relay and allow each source to use at most one relay node, we modify it so that the comparison with ours is as fair as possible. The MAS algorithm is listed in Table 4.3.
In Figs. 4.2–4.3, we compare the performance of Algorithm B when a source is allowed or forbidden to use the corresponding SD link in the second phase, i.e., whether a SD link’s second phase is idled or not. As expected, if the sources can send extra data packets via direct links (without relaying) in the second phase, the resulting sum rate performance is much improved. However, there exists minor loss of fairness if packets transmitted in the second phase can use the direct links only. In Fig. 4.4 and Fig.
4.5 we compare the performance of our algorithms with that of the algorithm which is designed to achieve the optimal sum rate without fairness consideration and the MAS algorithm. We consider the situation when the system has 2 MS users and 3 relay nodes with 8 subcarriers, 80 W total transmit power and a required BER of 10−3. We find that our algorithms achieve about 94% of the optimal sum rate but the corresponding fairness indices are significant better than that offered by the optimal sum rate algorithm.
0 10 20 30 40 50 60 70
rate constraint (bits/2 OFDM symbols)
sum rate (bits/2 OFDM symbols)
with idled slots without idled slots
Figure 4.2: The effect of SD link’s idled slots on Algorithm B’s sum rate performance;
4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
rate constraint (bits/2 OFDM symbols)
fairness index
with idled slots without idled slots
Figure 4.3: The effect of SD link’s idled slots on Algorithm B’s fairness performance; 4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
0 5 10 15 20
rate constraint (bits/2 OFDM symbols)
sum rate (bits/2 OFDM symbols)
optimal Algorithm A Algorithm B MAS Algorithm
Figure 4.4: Sum rate performance of the proposed and MAS algorithms; 2 MS users, 3 relay nodes, N = 8, PT = 80, BER = 0.001.
rate constraint (bits/2 OFDM symbols)
fairness index
optimal Algorithm A Algorithm B MAS Algorithm
Figure 4.5: Fairness performance of the proposed and MAS algorithms; 2 MS users, 3 relay nodes, N = 8, PT = 80, BER = 0.001.
0 5 10 15 20 90
91 92 93 94 95 96 97 98
rate constraint (bits/2 OFDM symbols) sum rate percentage % (normalize with optimal)
Algorithm A Algorithm B MAS Algorithm
Figure 4.6: Relative sum rate performance of the proposed and MAS algorithms; 2 MS users, 3 relay nodes, N = 8, PT = 80, BER = 0.001.
0 5 10 15 20
75 80 85 90 95 100
rate constraint (bits/2 OFDM symbols) fairness index percentage % (normalize with Algorithm B)
optimal Algorithm A MAS Algorithm
Figure 4.7: Relative fairness performance of the proposed and MAS algorithms; 2 MS users, 3 relay nodes, N = 8, PT = 80, BER = 0.001.
The sum rate of MAS algorithm is about 5% higher than that of our algorithms while our fairness index performance is also much improved. In Fig. 4.6, we normalize the sum rate with respect to the optimal sum rate. In Fig. 4.7, we normalize the fairness index with respect to the fairness index of Algorithm B. It is clear to compare our performance with other algorithms’. In Fig. 4.14 and Fig. 4.15, we consider another scenario in which there are 4 MS users and 3 relay nodes with 128 subcarriers. The total transmit power is 128 W while the required BER is again 10−3. The sum rate of the MAS algorithm [12] is also about 4% higher than that of our algorithms but their fairness index performance is inferior to ours by a margin of about 35%. These two figures indicate that both proposed algorithms give more robust and much better fairness index performance than the MAS algorithm can offer. Another advantage of our algorithms that was not shown in Fig. 4.14 is that when the minimum rate requirement is high, say > 80 (bits/2 OFDM symbols), our algorithms are capable of providing a solution that meet all MS rate requirements while the MAS algorithm fails. In Fig. 4.8, Fig. 4.9, Fig. 4.10 and Fig. 4.11, we consider another scenario in which there are 3 relay nodes with 128 subcarriers. The total transmit power is 128 W while the required BER is again 10−3. We find that the sum rate of the MAS algorithm is about 4%-8%
higher than that of our algorithms but their fairness performance degrades when the number of users increases. As far as fairness is concerned, our algorithms is very robust against the user number’s variation and outperform the MAS algorithm by 20%-40%.
Algorithm B outperforms Algorithm A since the latter, which uses the small signal approximation (18), suffers from performance loss in Step one. In Figs. 4.12-4.13, part of the performance loss is recovered by the additional step to fine-tune the designated link for each subcarrier. Algorithm B achieves a better performance at the expense of higher computation complexity in Step 1. However, both proposed algorithms offer satisfactory balance between maximizing the sum rate and the fairness performance.
2 3 4 5 6 7 8
sum rate (bits/2 OFDM symbols)
Algorithm A Algorithm B MAS Algorithm
Figure 4.8: Sum rate performance of the proposed algorithms and the MAS algorithm;
rate constraint 50, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
Figure 4.9: Fairness performance of the proposed algorithms and the MAS algorithm;
rate constraint 50, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
2 3 4 5 6 7 8 90
91 92 93 94 95 96 97
user number sum rate percentage % (normalize with MAS Algorithm)
Algorithm A Algorithm B
Figure 4.10: Relative sum rate performance of the proposed algorithms; rate constraint 50, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
2 3 4 5 6 7 8
60 65 70 75 80 85 90 95 100
user number fairness index percentage % (normalize with Algorithm B)
Algorithm A MAS Algorithm
Figure 4.11: Relative fairness performance of the proposed algorithm and the MAS algorithm; rate constraint 50, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
0 10 20 30 40 50 60 70 1000
1010 1020 1030 1040 1050 1060 1070 1080
rate constraint (bits/2 OFDM symbols)
sum rate (bits/2 OFDM symbols)
with step4 w.o. step4
Figure 4.12: Sum rate performance of the proposed Algorithm A; 4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
0 10 20 30 40 50 60 70
0.9 0.91 0.92 0.93 0.94 0.95 0.96
rate constraint (bits/2 OFDM symbols)
fairness index
with step4 w.o. step4
Figure 4.13: Fairness performance of the proposed Algorithm A; 4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
0 10 20 30 40 50 60 70 800
900 1000 1100 1200 1300
rate constraint (bits/2 OFDM symbols)
sum rate (bits/2 OFDM symbols)
Algorithm A Algorithm B MAS Algorithm
Figure 4.14: Sum rate performance of the proposed algorithms and the MAS algorithm;
4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
0 10 20 30 40 50 60 70
0.5 0.6 0.7 0.8 0.9 1
rate constraint (bits/2 OFDM symbols)
fairness index
Algorithm A Algorithm B MAS Algorithm
Figure 4.15: Fairness performance of the proposed algorithms and the MAS algorithm;
4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
Step 1: for n = 1: N for k = 1: K
if gRm(n, k) > gD(n, k) m = arg max`gRl(n, k) else
m = 0 end
Compute gELG(n, k) end
end
Step 2: Decide the assignment order n0 for n0 = 1: N
Compute ∆(n0, k) k∗ = arg maxk(∆(n0, k)) Nk∗ ← Nk∗∪ {n0} end
Step 3: for k = 1: K
while(Rk < Rk,min)
k∗ = arg maxk(Rk− Rk,min)
n0 = arg minngELG(n, k) , n ∈ Nk∗ Nk ← Nk∪ {n0} Nk∗ ← Nk∗\ {n0} end
end
Step 4: Check each relayed subcarrier.
Compute gELG(n, k) and make necessary link switches.
Calculate R and F .
Table 4.1: Maximization Sum Rate Algorithm A (MSRA): A subcarriers, power and relay assignment scheme for a multiple-relay cooperative communication network.
Step 1: for n = 1: N for k = 1: K
if Gm(n, k) > 4N ΓσPT 2
m = arg max`G`(n, k) else
m = 0 end
Compute gELG(n, k) end
end
Step 2: Decide the assignment order n0 for n0 = 1: N
Compute ∆(n0, k) k∗ = arg maxk(∆(n0, k)) Nk∗ ← Nk∗∪ {n0} end
Step 3: for k = 1: K
while(Rk < Rk,min)
k∗ = arg maxk(Rk− Rk,min)
n0 = arg minngELG(n, k) , n ∈ Nk∗ Nk← Nk∪ {n0} Nk∗ ← Nk∗\ {n0} end
end
Table 4.2: Maximization Sum Rate Algorithm B (MSRB): A subcarriers, power and relay assignment scheme for a multiple-relay cooperative communication network.
Satisfy sources’ rate requirements while K 6= ∅ do
n ← random(N ) k∗ = argkmax R(k, n)
Nk∗ ← Nk∗∪ {n} N ← N \ {n}
Rk∗ = Rk∗+ R(k∗, n) while Rk∗ < Rk,min do
n∗ = argnmax R(k∗, n)
Nk∗ ← Nk∗∪ {n∗} N ← N \ {n∗} Rk∗ = Rk∗+ R(k∗, n∗)
end while
N ← N \ Nk∗ K ← K \ {k∗} end while
Allocate remaining subcarrier while N 6= ∅ do
k∗ = argkmax R(k, n)
Nk∗ ← Nk∗∪ {n} N ← N \ {n}
end while
Table 4.3: The Modified Awad-Shen (MAS) Algorithm.
Chapter 5
Resource Allocation in AF Cooperative Networks
With minor modifications, our algorithms can be applied to other system setup. In this chapter, we consider amplify-and-forward cooperative relays instead of DF relays.
RA in networks with more elaborate cooperative scheme such as estimate-and-forward can also be solved by our proposals.
5.1 Signal-Channel Model and Relay Selection
For an AF-based cooperative network, the relay receives xk in the first time slot and transmits an amplified version of xk in the second time slot. The received samples in the destination and the mth relay satisfy the following relations.
ySD(n, k) = hSD(n, k)xk+ n(n, k) (5.1)
ySRm(n, k) = hSRm(n, k)xk+ n(n, k) (5.2) yRD(n, m) = βhRD(n, m)xk+ n(n, k) (5.3) where
β = s
PR(n, m)
PSRm(n, k)|hSRm(n, k)|2+ σ2 (5.4) is the power amplification factor at relay nodes. The AF scheme is a suitable choice when a relay node does not have a sufficiently large SNR to decode the transmitted
symbol. It, however, suffers from noise amplification. The achievable rate of the kth user on subcarrier n with the aid of relay m is given by
RRm(n, k) = log2 where α is the gain-to-noise ratios defined by (4.13).
Due to the presence of the item “1” in the denominator of (5.5), the optimal PR and PSRm do not vary linearly with P , which cannot reach an equivalent form of channel gain. Thus, an approximation is made by trying to maximize
RRm(n, k) = log2 and the link power gain (gRm(n, k)) of the composite link becomes
gRm(n, k) = |gRD(n, m)|2(g∗+ |gSD(n, k)|2)2
(g∗+ |gRD(n, m)|2)2 (5.7) where g∗ =p
|hSRm|2|hRmD|2 + |hRmD|2|hSD|2− |hSRm|2|hSD|2. The approximation and the corresponding equivalent channel gain can also be found in [14]. By applying La-grange multiplier method we obtain the optimal power ratio
PR(n, m)
PSRm(n, k) = |hSRm|2|hRmD|2− |hSRm|2|hSD|2
h|hRmD|2+ |hSD|2|hRmD|2 (5.8) For the conventional AF scheme, cooperative relay is beneficial if it offers a higher achievable rate with the same power or, equivalently, the composite link should require less power to obtain the same achievable rate. If multiple relay nodes are available, (4.2) and (5.6) imply that this happens iff
maxm gRmD(n, k) > gD(n, k) (5.9) The above conditions are necessary but not sufficient for the AF scheme under consid-eration. Assuming the optimal power ratio (5.8), we can show that a necessary and sufficient condition for a single-relay system is
gRm− gD
g2 > γ (5.10)
where γ = P (n,k)4Γσ2 and the link gains’ dependence on the pair (n, k) is omitted for the sake of brevity. For multiple-relay systems, (5.10) becomes
maxm
gRm− gD g2D
def= max
m Gm > γ (5.11)
We can find that the condition is the same as DF scheme so our proposed algorithms can be extended to AF networks.
5.2 Numerical Results and Discussions
Numerical performance of the proposed RA algorithms for AF-based cooperative networks is presented in this section. All the parameter values used in our simulation are the same as those used in simulating the performance of DF networks. In Figs.
5.1 and 5.2, we consider the scenario in which there are 4 MS users and 3 relay nodes with 128 subcarriers. The total transmit power is 128 W while the required BER is again 10−3. The sum rate of the MAS algorithm is about 5% higher than that of our algorithms but their fairness index performance is inferior to ours by a margin of about 35%. We can find that the performance of decode-and-forward cooperative relay networks is a little better than amplify-and-forward cooperative networks under the same parameters in our simulations. The reason is our relay nodes are located nearer to the source than the destination so decode-and-forward relaying has better performance.
Our algorithms are also have not only achieve near-optimal sum rate but also provide very robust fairness performance in AF schemes because the derived results has the same forms as DF schemes. If the cooperative scheme can be derived to find the effective link gain, our algorithms can be used under the same constraints. If the objective functions are also maximizing the sum rate and the fairness index with more constraints, we can adjust our algorithms to meet more constraints.
0 10 20 30 40 50 60 70
rate constraint (bits/2 OFDM symbols)
sum rate (bits/2 OFDM symbols)
Algorithm A Algorithm B MAS Algorithm
Figure 5.1: Sum rate performance of the proposed algorithms and the MAS algorithm;
4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
rate constraint (bits/2 OFDM symbols)
fairness index
Algorithm A Algorithm B MAS Algorithm
Figure 5.2: Fairness performance of the proposed algorithms and the MAS algorithm; 4 MS users, 3 relay nodes, N = 128, PT = 128, BER = 0.001.
Chapter 6 Conclusion
Cooperative relays provide additional transmission opportunities and offer the po-tential to improve overall system’s capacity, throughput and the coverage range. It is thus natural to regard relay stations as part of the network radio resource and their allocation should be considered in conjunction with other conventional radio resources to optimize the system performance.
We first propose two algorithms that minimize the total transmitted energy and simultaneously satisfy the individual user’s rate (bit) and BER constraints. Our algo-rithms achieve suboptimal performance with a computational complexity that is linear in NK. Numerical results indicate that our low-complexity algorithms do give significant reduction in energy consumption when compared with the performance of the no-relay network.
We have also proposed another two algorithms that maximize the sum rate and fairness index while meeting the individual user’s minimum rate and QoS (BER) re-quirements. Numerical results indicate that our low-complexity algorithms not only achieve 94% of the optimal sum rate but also provide very robust fairness performance with respect to the minimum rate constraint. Moreover, our algorithms can offer near-optimal allocation solution while meeting a large range of the minimum rate constraints.
No practical optimal solution to the problems discussed herein is known, the required computational complexities of our algorithms are only moderate but is far less than that
of the exhaustive search approach.
Several issues remain to be addressed and solved. First, the time-frequency resource unit considered in this thesis is perhaps the smallest one, i.e., a subcarrier in a time duration much less than the channel coherent time. Such an assumption is not very practical for it costs too much overhead in relaying the RA information to the users.
Second, the performance of the proposed algorithms presented in this thesis is estimated by Monte-Carlo simulations. It is desirable that analytic expressions be derived such that the impacts of system parameters (such as the numbers of users, relays, subcarriers) and those of the channel conditions (such as GNR) can be assessed. Third, the optimal RA solutions for the scenarios considered in this thesis are still lacking and need further efforts to develop and find them. Finally, fairness issue is often solved via scheduling over a longer period of time. Our solutions implicitly assume an uncorrelated and stationary environment. Extensions to non-stationary and/or Markovian channels are much needed.
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作 者 簡 歷
盧彥碩，高雄市小港區人，1984 年生
高雄市立高雄高級中學 2000.9 ~ 2003.6 國立交通大學電信工程學系 2003.9 ~ 2007.6 國立交通大學電信工程學系研究所系統組 2007.9 ~ 2009.6
Graduate Course:
1. Random Process
2. Digital Signal Processing 3. Digital Communications 4. Coding Theory
5. Detection and Estimation Theory 6. Adaptive Signal Processing
7. Computer Communication Networks 8. Embedded System Design