Fig. 5.9. BER simulation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with S-DF/RT relaying scheme.
For S-DF/Idle relaying scheme, the optimal 0's for both PA-ABER and PA-MGEC are plotted in Fig. 5.10 as well as that for PA-EG. As was predicted in Section 5.3.3, when
0 2
E Nb , PA-MGEC allocates all power to source, and so does PA-ABER. As the SNR in-creases, the source power of both methods decline and finally converge to about 0 0.715, as
0 16
E Nb . Similarly, PA-ABER tends to allocate slightly less power to source than PA-MGEC.
Compared with Fig. 5.8 for S-DF/RT, the 0's in Fig. 5.10 are higher than those in Fig. 5.8 for all E Nb 0 2. This is because, in S-DF/Idle, power allocated to relay will be wasted if the re-lay cannot decode correctly. Thus, power allocation tends to be more conservative and put more power on source. The corresponding BER is given in Fig. 5.11, wherein PA-ABER slightly out-performs PA-MGEC. This is intuitive because their power allocation results are rather close. Both the proposed method significantly outperform PA-EG with about 2 dB gain at BER of 105,
Fig. 5.10. Power allocation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with S-DF/Idle relaying scheme.
-4 -2 0 2 4 6 10-7
10-6 10-5 10-4 10-3 10-2 10-1 100
Eb/N 0
BER
PA-EG PA-ABER PA-MGEC
Fig. 5.11. BER simulation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with S-DF/Idle relaying scheme.
For S-DF/AF relaying scheme, the power allocation results are plotted in Fig. 5.12. As was predicted in Section 5.3.4, when E Nb 0 2, both method allocate all power to source. As the SNR increases, the source power declines and finally converge to about 0 0.607, as
0 16
E Nb . PA-ABER tends to allocate slightly less power to source than PA-MGEC. Compared with S-DF/RT and S-DF/Idle, the 0's in Fig. 5.12 are lower than those in Fig. 5.8 and Fig. 5.10.
This is because S-DF/AF is less sensitive to decoding failure at relay. Thus, power allocation tends to put more power on relay for transmitting the packet through the R-D link, which is better than the S-D link. The corresponding BER is given in Fig. 5.13, wherein PA-ABER outperforms PA-MGEC with 0.2 dB at BER of 105 and outperforms PA-EG with about 1.2 dB gain at BER of 105.
-5 0 5 10 15 20
Fig. 5.12. Power allocation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with S-DF/AF relaying scheme.
Fig. 5.13. BER simulation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with S-DF/AF relaying scheme.
Numerical results for 2 relays are also provided by considering Network-10 in Table IV, where relay 1 has the same S-R and R-D link is those in Network-9, but relay 2 has worse S-R link and worse R-D link than relay 1. For AF relaying scheme, there is no decoding issue. The power allocation result for PA-EC (which allocates 0 1 2 0.333) and PA-EG are plotted in Fig. 5.14 and the corresponding BER performance in Fig. 5.15. As is observed, at low E N , b 0 all power is allocated to source. After E Nb 0 8, 1 becomes non-zero, though the difference on BER performance between PAEC and PA-EG is trivial before E Nb 0 2. PA-EC starts to allocate power to relay 2 when E Nb 04. A gain of 1 dB is observed at BER of 105 for PA-EC over PA-EG.
For S-DF schemes, it is reasonable to predict that relay 1 is more likely to achieve correct decoding than relay 2. Therefore, we might start to allocate non-zero power to relay 1 at the SNR smaller than that for relay 2. Similar results are observed for S-DF/RT on Network-10.The power allocation results are provided in Fig. 5.16 and the corresponding BER performance in Fig. 5.17.
In spite of the observations mentioned in AF, in Fig. 5.16, the results of the two power allocation methods are quite similar, which leads to that, as is shown in Fig. 5.17, PA-ABER and PA-MGEC perform almost the same. They both have a more than 2 dB gain over PA-EG. Such a gain be-comes more significant for S-DF/Idle, whose corresponding results are provided in Fig. 5.18 and Fig. 5.19 for the power allocation and BER performance, respectively. In Fig. 5.19, an about 3.4 dB gain achieved for both methods over PA-EG. This is because, as mentioned before, our pro-posed power allocations tend to allocate more power to source (compared to PA-EG) so as to in-creases the probability of correct decoding at relay and to avoid power waste due to inactive re-lays. The results for S-DF/AF are provided in Fig. 5.20 and Fig. 5.21. Although PA-EG benefits significantly from S-DF/AF over S-DF/Idle because the relays now always use their power on forwarding, our proposed method still provide an almost 2 dB gain over PA-EG.
-10 -5 0 5 10 15
Fig. 5.14. Power allocation results for PA-EG and PA-EC on Network-10 with AF relaying scheme.
Fig. 5.15. BER simulation results for PA-EG and PA-EC on Network-10 with AF relaying scheme.
-5 0 5 10 15 20
Fig. 5.16. Power allocation results on Network-10 with S-DF/RT relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Fig. 5.17. BER simulation results on Network-10 with S-DF/RT relaying scheme.
-5 0 5 10 15 20
Fig. 5.18. Power allocation results on Network-10 with S-DF/Idle relaying scheme.
-2 -1 0 1 2 3 4 5 6 7 8
Fig. 5.19. BER simulation results on Network-10 with S-DF/Idle relaying scheme.
-5 0 5 10 15 20
Fig. 5.20. Power allocation results on Network-10 with S-DF/AF relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Fig. 5.21. BER simulation results on Network-10 with S-DF/AF relaying scheme.
Another interesting example is on Network-11, in which the S-R and R-D links of relay 1 remains the same, but the S-R link of relay 2 is better than that of relay 1. For AF relaying scheme, PA-EC still prefers to allocate non-zero power to relay 1 earlier than relay 2 as the SNR increases, as shown in Fig. 5.22 with the BER performance given in Fig. 5.23. But, for S-DF re-laying schemes, both PA-ABER and PA-MGEC prefer to allocate power to relay 2 earlier than relay 1, as the SNR increases. The power allocation results and BER performance are shown in Fig. 5.24 and Fig. 5.25 for S-DF/RT, while those for S-DF/Idle and S-DF/AF are similar. In Net-work-11, all methods allocate more power on relay 2 when SNR approaches infinity. The reason could be that, for AF, relay 2 provides a better equivalent channel than relay 1 and that, for S-DF, relay 2 is more likely to decode erroneously under the condition that both relay have the same R-D link quality.
-10 -5 0 5 10 15 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Eb/N0
j
PA-EG PA-EC,
0 PA-EC, 1 PA-EC, 2
Fig. 5.22. Power allocation results on Network-11 with AF relaying scheme.
-4 -2 0 2 4 6
Fig. 5.23. BER simulation results on Network-11 with AF relaying scheme.
-5 0 5 10 15 20
Fig. 5.24. Power allocation results on Network-11 with S-DF/RT relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 10-7
10-6 10-5 10-4 10-3 10-2 10-1 100
Eb/N0
BER
PA-EG PA-ABER PA-MGEC
Fig. 5.25. BER simulation results on Network-11 with S-DF/RT relaying scheme.
The proposed power allocation methods (PA-EC, PA-ABER and PA-MGEC) straightfor-wardly applied to Network with 3 or more relays. In fact, it is important to know that the gain of these methods over PA-EG generally increases with the number of relays. The reason is simply that as the relay increases, less power is expected to be allocated to source for PA-EG, e.g.,
1
PT R . Therefore, a high SNR is required for the source to activate relays. Oppositely, for PA-ABER and PA-MGEC, the source usually takes full power at low SNRs. This implies that PA-ABER and PA-MGEC can start to benefit from active relays at a lower SNR than PA-EG.
Thus, a gain can be expected. As an example, another example is conducted on Network-12 with 3 relays and the channel gains given in Table VI. The proposed power allocation results and the corresponding BER performance of AF are shown in Fig. 5.26 and Fig. 5.27, S-DF/RT in Fig.
5.28 and Fig. 5.29, S-DF/Idle in Fig. 5.30 and Fig. 5.31, as well as S-DF/AF in Fig. 5.32 and Fig.
5.33, respectively.
-10 -5 0 5 10 15
Fig. 5.26. Power allocation results on Network-12 with AF relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Fig. 5.27. BER simulation resultss on Network-12 with AF relaying scheme.
-5 0 5 10 15 20
Fig. 5.28. Power allocation results on Network-12 with S-DF/RT relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Fig. 5.29. BER simulation results on Network-12 with S-DF/RT relaying scheme.
-5 0 5 10 15 20
Fig. 5.30. Power allocation results on Network-12 with S-DF/Idle relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Fig. 5.31. BER simulation results on Network-12 with S-DF/Idle relaying scheme.
-5 0 5 10 15 20
Fig. 5.32. Power allocation results on Network-12 with S-DF/AF relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Fig. 5.33. BER simulation results on Network-12 with S-DF/AF relaying scheme.
In the last example of this chapter, we compare our proposed method with the power allo-cation in [84] which maximized the capacity of a 3-node cooperative repaying network (denoted by PA-MC). We consider Network-13 in Table IV with S-DF/Idle and plot the power allocation results of our methods and PA-MC in Fig. 5.34. As is seen in Fig. 5.34, the optimal 0 of PA-MC is invariant to the change E N (b 0 0 0.357). It can be expected that, at low SNRs, the insufficient source power is not able to active the relay so that the power allocated to the relay is wasted. In this case, the BER at destination will be high because destination could only decode the packet based on the signal transmitted from source with a low power ratio 0 0.357.At high SNRs, the power allocation of the three methods are very similar. The BER performance is plotted in Fig. 5.35. As is expected, PA-MC is outperformed by our methods with about 0.8 dB gain at BER of 105. Such a gap is expected to diminish, as the SNR further increases. However, the corresponding BER is too low to be obtained through computer simulation.
-5 0 5 10 15 20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Eb/N
0
0
PA-ABER PA-MGEC PA-MC
Fig. 5.34. Power allocation results on Network-13 with S-DF/Idle relaying scheme.
-5 -4 -3 -2 -1 0 1 2 3 4 10-7
10-6 10-5 10-4 10-3 10-2 10-1 100
Eb/N
0
BER
PA-ABER PA-MGEC PA-MC
Fig. 5.35. BER simulation results on Network-13 with S-DF/Idle relaying scheme.
5.5 Summary
Chapter 5 investigates the power allocation of the BICM-coded cooperative network. Four relay-ing schemes are considered: AF, S-DF/RT, S-DF/Idle and S-DF/AF with the general formulation in Section 2.4. For AF, the union bound of the BER proposed in [47] is simplified by considering only the worst-case error event and the shortest Euclidean between constellation points. An ap-proximate BER is obtained and is shown to be monotonically decreasing with an equivalent channel gain, which is then taken as the cost function of PA-EC. For S-DF relaying schemes, two power allocation methods, named PA-ABER and PA-MGEC, are proposed. In PA-ABER, by fol-lowing similar steps in deriving PA-EC, an approximate BER is obtained as the cost function and is shown to be a convex function for S-DF/RT, S-DF/Idle and S-DF/AF. Therefore, gradient method can be adopted to find the solution. Then, PA-MGEC transforms the approximate BER to a max-min problem, which can be optimized with even lower complexity, e.g., Simplex algorithm
can be used for S-DF/RT and S-DF/Idle. Examples are given to demonstrate how power is allo-cated as the SNR increases for PA-EC and PA-MGEC on the AF and S-DF relaying schemes, re-spectively. In general, our methods tend to allocate full power to source at low SNR to avoid power waste on inactive relays. Simulation results are provided to confirm that our proposed methods outperform PA-EG with large margins for different network setups.
Chapter 6
Power Allocation on
De-code-Remap-and-Forward
This chapter extends the results of the previous chapter to S-DRF, which allows active re-lays to change the constellation mapping used in modulation before forwarding. S-DRF has been proved to provide significant remapping gain over conventional S-DF [47]-[49]. In spite of it importance, this section shows how PA-ABER and PA-MGEC are applied work on S-DRF3.
For notation clarity, denote x j j
v the operation that a label v is mapped by a mapper j to a complex symbol x j , where j is the mapper used at relay j . (Note that 0
is used for the mapper at source.) With this notation, the general form representation in Sec-tion 2.4 is modified as
, 1 , 1 , 1, 1, 2, , 1
j R j R j j j R
y h P v j R , (6.1) where
3 In our previous work [50], the power allocation for S-DRF/Idle had been presented. In this dissertation, the derivation is provided in the general form so that S-DRF/Idle will also be covered.
on forwarding when decoding correctly. Upon decoding failure, 1) for S-DF/Idle, nothing is
going to be transmitted, 2) for S-DF/RT, the source remains its original mapper 0 (Or, in oth-er words, send x 0 again through the orthogonal-channel.4), and 3) for S-DF/AF, relay forwards without neither decoding nor re-mapping. Consequently, the LLR for the i -th bit of the label v at the destination is evaluated by
Although new notations are used, the BER analysis is not much different. In fact, one can follow (5.13)-(5.14) and modify (5.15) as (or refer to our work [50]) consider the re-mapping at source in this work, the corresponding power allocation can be easily extended based on our derivations.
Since the summation (6.6) contains only real exponential functions, at high SNRs, such a summation is dominated by the terms with the largest exponent, i.e.,
An alternative representation of (6.8) is
for the following setups.Setup-1: R1, 16QAM, and (0) (1) G (the Gray mapping) in Fig. 6.1. (Note that all elements in D are the same if ( )j (0) for all j .)
Setup-2: R1, 16QAM, (0) G and (1) A in Fig. 6.1.
Setup-3: R2, 16QAM, (0) G, (1) A and (2) B in Fig. 6.1.
Table V. for Setup-1, Setup-2 and Setup-3
[3;C;5;4] MBER mappings which maximize the minimum Euclidean distance between transmit symbols for the second, third and fourth transmissions of the hybrid automatic repeat-request system in listed for Setup-1, 2 and 3. After applying Lemma-3, the number of elements in is reduced dramatically from 9, 23 and 50 to 1, 2 and 10 in ˆ for Setup-1, 2 and 3, respectively.
Table VI. ˆ for Setup-1, Setup-2 and Setup-3
2
(for S-DRF/RT, S-DRF/Idle and S-DRF/AF). Here, we provide point out the differences during the proof.
a) S-DRF/RT and S-DRF/Idle: Firstly, following Appendix D, the problem turns to prove the concavity of Mˆ
α , which is (for S-DRF/RT)The summation in (6.18) is a linear function of α , which is both convex and concave. Since the minimum of concave functions is still concave [80], (6.18) is a concave function of α . With this property, we can continue the steps in Appendix D and finally prove the convexity of G α . ˆ
b) S-DRF/AF: In this case, Mˆ
α becomesSimilarly, the first summation is linear and the second is concave (as is proved in Appendix C).
Their sum is still concave, and (6.19), which is the minimum of concave functions, remains con-cave.
However, G α may not be differentiable because ˆ
Mˆ
α is the minimum of concave functions of α and may not be differentiable. Such a constrained convex optimization problem with a non-differentiable cost function can be solved by applying the projected sub-gradient method [83].6.2 PA-MGEC
PA-MGEC can also be applied for S-DRF. Following the same steps from (5.26) to (5.29) yields
Here, some detail about solving (6.20) is provided for different schemes.
a) S-DRF/RT: We have
Since (6.23) is also a linear programming problem which can be solved through Simplex algo-rithm. For the case of S-DRF/Idle, simply replacing the summation
Rj0
by
j
.b) S-DRF/AF: Bringing (6.19) into (6.20) yields
which is not linear and not differentiable. Fortunately, it is still concave so that (6.20) is convex, and its optimum can be obtained through the sub-gradient method.
6.3 Power Allocation Example
Similar to what in Section 5.3, this section provides an example of how power is allocated by PA-MGEC on S-DRF modes. Consider again Network-9 in Table IV. As will be also em-ployed in the simulation results, we use a half-rate convolutional codes CC(171,133) with
f 10
d and W dI
f 33. The length of the information sequence is K506 such that the length of the coded sequence is N1024. The mappers G and A are used on a 16-QAM constellation at source and relay, respectively. Note that the total transmit power is calculated asT b C
P E R l where E is the bit energy, and b RC 0.5 is the channel code rate.
6.3.1 PA-MGEC on S-DRF/RT
For S-DF/RT, Mˆ
α depends on the active relay set , which could be or {1} in this example. Specifically, (6.23) becomes
The problem becomes to determine the optimum of the minimum of three linear functions of 0. Define
cation. As the SNR further increases, the optimum point changes to the intersection between L 0 and L . Observing from Fig. 6.2, the optimal 2 0 decreases with SNR, starting from 1 to about 0.9 when E Nb 0 (or ˆ1 0). In summary, when SNR is low, all power is allocated to source. As the SNR increases, the source power decreases to a final point. This result is intuitive because when SNR is too low, more power should be allocated to source to increase the probabil-ity of correct decoding at the relay.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PA-MGEC on S-DRF/Idle is very similar to that on S-DRF/RT. The main different is that
Mˆ α now becomes
jD jj hj R, 12, which means that the orthogonal-channels of inac-tive relays will never contribute to Mˆ
α . Specifically, (6.23) becomes S-DF/Idle relaying scheme is adopted. Therefore, more power (compared to S-DRF/RT) should be allocated to source to avoid this power waste.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Define as those for S-DRF/RT and S-DRF/Idle. The main difference is that the curve of L now bends 0 to be concave, though it still shifts up as SNR increases (ˆ1 decreases).
To determine the optimal power, we need to maximize min
L L L . As is shown in Fig. 0, 1, 2
6.4, when the SNR is very low, e.g., E Nb 0 1.2, the whole curve of L is below both 0 L 1 and L , so that 2 min
L L L0, 1, 2
L0. The problem turns to optimize L alone. In this case, 0 L 0 is maximized at 0 1. (Note that the optimum could be somewhere else for different channel realizations.) As the SNR increases such that E Nb 0 1.2, L and 0 L begin to intersect. 1 This intersection is chosen as the optimal power allocation. As the SNR further increases, the op-timum moves to the intersection between L and 0 L . Observing from Fig. 6.4, the optimal 2 0 decreases with SNR, starting from 1 to about 0.8 when E Nb 0 (or 10).Not surprisingly, when SNR is low, all power is allocated to source. As the SNR increases, the source power decreases to a final point. This final point 0 0.8 is even lower than that for S-DRF/RT or S-DRF/Idle. This is because, in Network-9, the relay uses AF can provide a better equivalent channel than re-transmission through the S-D link.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
6.4 Numerical Results
This section verifies PA-ABER and PA-MGEC on S-DRF through BER simulation and
As first, the power allocation and simulation results for examples in Section 6.3 with Net-work-9 are provided. For S-DRF/RT, the optimal 0's for both PA-ABER and PA-MGEC are plotted in Fig. 6.5 as well as that for PA-EG. As was predicted in Section 6.3.1, when at low
SNRs, e.g., E Nb 0 1.5, PA-MGEC allocates all power to source, and PA-ABER does so when E Nb 0 5. As the SNR increases, the source power of both methods decline and finally converge to about 0 0.9, as E Nb 0 10. Generally, PA-ABER tends to allocate slightly less power to source. The corresponding BER is given in Fig. 6.6, wherein PA-ABER outperforms PA-MGEC by 0.4 dB at BER of 105 and PA-EG by 1.1 dB. At high SNRs, PA-ABER and PA-MGEC are expected to have the same performance, although the corresponding BER is too low to be simulated.
-5 0 5 10 15 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Eb/N0
0
PA-EG PA-ABER PA-MGEC
Fig. 6.5. Power allocation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with
Fig. 6.5. Power allocation results for PA-EG, PA-ABER and PA-MGEC on Network-9 with