In this chapter, we compare the effective throughput retrieved by using different power allocation policies. Several convolutional codes [2, 3] are adopted. The parameters in the sense of the approximation given by (3.4) for these codes are listed in Table 5.1.
5.1 Throughput-Oriented Water-Filling: Noise Vari-ance in Each Channel is Known
In this section, three situations of parallel Gaussian channels with K = 3 are examined.
They are respectively referred to as Cases I, II and III.
In Case I, the noise variances for the three channels are σ12 = 1, σ22 = 3.5 and σ32 = 6, respectively. Here, codes with higher code rates are naturally assigned to less noisy channels;
hence we have R1 = 1/2, R2 = 1/3 and R3 = 1/4. The frame sizes for the three channels are N1 = 2(1000 + 6), N2 = 3(1000 + 6) and N3 = 4(1000 + 6), respectively. In Figure. 5.1, we depict the effective throughputs for the seven possible choices of the active channel set O. The figure indicates that all the power should be allocated to channel 1 if Pt < 5.14, and both channels 1 and 2 should be active when 5.14 < Pt < 10.05. Beyond the point Pt= 10.05, all three channels should be made active.
Table 5.1: The information of the used codes in the simulation.
adjusted adjusted codeword length generator polynomial code dfree Adfree
d A N (octal)
10.63 1478.07 2(500+6) (2, 1, 6) 10 11
11.02 4750.45 2(1000+6) [133 171]
15.79 593.83 3(500+6) (3, 1, 6) 14 1
16.12 1449.97 3(1000+6) [133 171 145]
22.42 962.51 4(500+6) (4, 1, 6) 20 2
22.13 1401.29 4(1000+6) [117 127 155 171]
(2, 1, 2) 5 1 5.31 111.56 2(500+2) [5 7]
(3, 1, 11) 24 13 29.04 41373.67 3(500+11) [5475 6471 7553]
(4, 1, 10) 29 3 35.54 11266.62 4(500+10) [2565 2747 3311 3723]
In Figure. 5.2, we compare the optimal effective throughput obtained from exhaustive search with that obtained from our throughput-oriented water-filling based on the FER approximation and from the capacity-achieving water-filling policy. We remark that our throughput-oriented water-filling can achieve a near-optimal effective throughput as an-ticipated. We also observe that the capacity-achieving water-filling policy yields a good throughput only when all the power is allocated to a single channel (which is the optimal choice only for small values of Pt).
In Figure. 5.3, we plot the optimal power ratio P2∗/Pt with respect to different power allocation policies. We note that a sudden increase for this ratio occurs in the exhaustive search curve at Pt = 4.98 which is exactly the instance the active channel set O changes from {1} to {1, 2} as shown in Figure. 5.4. This jump occurs when the total power is a little bit larger than the total power corresponding to ν = νmin = logdσ22
This is because Channel 2 can provide a solid contribution to the system effective throughput only when P2 is adequately larger than Pth,2. Figure 5.3 also indicates that the predicted jump point from the throughput-oriented water-filling based on the FER approximation,
i.e., Pt = 5.14, is very close to the true jump point, Pt = 4.98, while the capacity-achieving water-filling policy always suggests a continuous increase in the power ratio.
0 2 4 6 8 10 12 14 16
0 0.2 0.4 0.6 0.8 1 1.2 1.4
total power
effective throughput
case 1: Ch1 only case 2: Ch2 only case 3: Ch3 only case 4: Ch1 & Ch2 case 5: Ch1 & Ch3 case 6: Ch2 & Ch3 case 7: Ch1 & Ch2 & Ch3
Figure 5.1: Case I: Effective throughputs for the seven choices of the active channel set O.
For Case II, we exchange the codes used in Channels 1 and 3 in Case I. Hence, R1 = 1/4 and R3 = 1/2. The results are summarized in Figures. 5.5, 5.6 and 5.7. These figures point out that using a lower code rate for a less noisy channel will yield a better throughput only when the total power is very small. For moderate to high total power, exchanging the codes between channels 1 and 3 never results in a better effective throughput. This confirms the common intuition that when a channel is less noisy, a code with a higher rate should be used.
A side observation is that when assigning a code with lower rate to a less noisy channel, the set of active channels changes more often with respect to Pt. In particular, Channel 2 will
0 2 4 6 8 10 12 14 16 0
0.2 0.4 0.6 0.8 1 1.2 1.4
total power
effective throughput
exhaustive search throughput−oriented I capacity−optimizing
Figure 5.2: Case I: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achieving water filling policy.
have two cut-off regions given by Pt< 3.76 and 11.42 < Pt < 14.65 as shown in Figure. 5.5.
In addition, Figure. 5.5 shows that adopting a wrong O will noticeably degrade the effective throughput. Hence, exchanging the codes between Channels 1 and 3 will make complicated the optimization of the throughput.
Finally for Case III, the codes used for three channels are the same as those used in Case I, but the frame sizes are changed to N1 = 2(500 + 6), N2 = 3(500 + 6) and N3 = 4(500 + 6).
Thus di and Ai are changed simultaneously. Besides, the noise variances are changed to σ12 = 2, σ22 = 8, σ32 = 9. Similar behaviors can be observed from Figure 5.8 except that the capacity-achieving water-filling policy gives an effective throughput closer to the optimal
0 2 4 6 8 10 12 14 16 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
total power
power ratio in ch2
exhaustive search throughput−oriented I capacity−optimizing
Figure 5.3: Case I: Optimal power ratio for channel 2.
one for high values of the total power. This can be somehow anticipated from the discussion following (4.5) as when the noise variances of the active channels have larger gaps (between Channel 1 and Channels 2 or 3), the capacity-achieving water-filling policy will yield a power allocation closer to the throughput-oriented water-filling.
º º
Pth;1 Pth;1
CH1
CH1 CH2CH2 CH3CH3
Pth;1 Pth;1
Pth;2 Pth;2
CH1
CH1 CH2CH2 CH3CH3
Figure 5.4: Case I: Illustration of the optimal active set O changing from {1} to {1, 2}.
0 5 10 15 20 0
0.2 0.4 0.6 0.8 1 1.2 1.4
total power
effective throughput
case 1: Ch1 only case 2: Ch2 only case 3: Ch3 only case 4: Ch1 & Ch2 case 5: Ch1 & Ch3 case 6: Ch2 & Ch3 case 7: Ch1 & Ch2 & Ch3
Figure 5.5: Case II: Effective throughputs for the seven choices of active channel set O.
0 5 10 15 20 0
0.2 0.4 0.6 0.8 1 1.2 1.4
total power
effective throughput
exhaustive search throughput−oriented I capacity−optimizing
Figure 5.6: Case II: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achieving water filling policy.
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
total power
power ratio in ch2
exhaustive search throughput−oriented I capacity−optimizing
Figure 5.7: Case II: Optimal power ratio for Channel 2.
0 5 10 15 20 25 30 0
0.2 0.4 0.6 0.8 1 1.2 1.4
total power
effective throughput
exhaustive search throughput−oriented I capacity−optimizing
Figure 5.8: Case III: Optimal effective throughputs obtained from exhaustive search, the throughput-oriented water-filling based on the FER approximation, and the capacity-achiving water-filling policy.
5.2 Throughput-Oriented Water-Filling: Only Total Noise Variance is Available
In this section, three situations of parallel Gaussian channels are examined. They are re-spectively referred to as Cases I, II and III.
In Case I, we consider K = 3 and use the (2, 1, 6), (3, 1, 6) and (4, 1, 6) convolutional code in three channels, respectively. The frame size of the codes are N1 = 2(500 + 6), N2 = 3(500 + 6) and N3 = 4(500 + 6). The total noise variance σ2t is set to 10. For convenience, we will plot the ratios of the effective throughput against the maximum rate, which is the sum of the rates of the three channels in the following figures.
In Figure. 5.9, we compare the ratios of the effective throughputs against the maxi-mum rate, obtained from the throughput-oriented water-filling in (4.11) and the traditional worst-case capacity-achieving equal power allocation. The γ†th, at which value our proposed power allocation becomes optimal, is 4.72 dB. We can see that almost all of the power al-location methods achieve the maximum rate when system SNR is above γth† . Although we cannot guarantee the optimality of using the throughput-oriented water-filling for system SNR smaller than γ†th, we can still see that it has around 1.4 dB gain over the equal power allocation when the effective throughput of the system is required to achieve 85% of the maximum rate.
We also observe the distribution of worst-case noise variances for system SNR varying from 2 dB to 6 dB when using throughput-oriented water-filling as the power allocation method. The result shows that we should always give total noise power to Channel 2. This confirms our claim that the worst-case effective throughput is achieved by giving total noise power to only one channel for system SNR greater than γth† .
In Case II, the (2, 1, 2), (3, 1, 11) and (4, 1, 10) convolutional codes are used in three
2 2.5 3 3.5 4 4.5 5 5.5 6 40
50 60 70 80 90 100 110 120
SNR(Pt/St in dB)
effective throughput/maximum rate(%)
throughtpu−oriented II
worst−case capacity optimizing(equal power) 85% maximum rate
Figure 5.9: Case I: The worst-case effective throughputs obtained from throughput-oriented water-filling based on the FER approximation, and the worst-case-capacity-achieving equal power allocation.
channels, respectively. Compared with the codes used in Case I, the codes used in Case II has larger gaps in di, where d1 = 5.31, d2 = 29.40 and d3 = 35.54. It is anticipated that by using this set of codes, throughput-oriented water-filling should yield a greater gain than equal power allocation, when being compared with Case I. A simple way to prove this anticipation is by looking at the situation when system SNR is large. The proposed power allocation policy suggests that Pi should be allocated inversely proportional to di. For larger difference in the amount of di’s, the proposed power allocation deviates greatly from the equal power allocation, and thus yields better gain. Figure 5.10 confirms our deduction. We see that throughput-oriented water-filling yields around 2 dB gain when the effective throughput
achieves 85% of the maximum rate. Besides, when we look at the situation when system SNR is equal to γth† = 5.19 dB, throughput-oriented water-filling almost achieves the maximum rate while the equal power allocation achieves only 83% of the maximum rate.
3 3.5 4 4.5 5 5.5 6
40 50 60 70 80 90 100 110 120
SNR(P
t/S
t in dB)
effective throughput/maximum rate(%)
throughtpu−oriented II
worst−case capacity optimizing(equal power) 85% maximum rate
Figure 5.10: Case II: The worst-case effective throughputs obtained from throughput-oriented water-filling based on the FER approximation, and the the worst-case-capacity-achieving equal power allocation.
For Case III, we increase the number of channels to be K = 4. We use the (2, 1, 6), (3, 1, 6) and (4, 1, 6) convolutional codes in the first three channels as in Case I. Two different codes are chosen to be used in Channel 4 for comparison.
Firstly, we use the (2, 1, 6) in Channel 4, which is the same code as that used in channel 1. We yield only 0.89 dB gain when the effective throughput is required to achieve 85% of the maximum rate (See Figure 5.11), which is less than the gain in Case I. Secondly, we use
the (3, 2, 6) punctured convolutional code in Channel 4. It is punctured from (2, 1, 6) code with puncture pattern
1 1 1 0
.
The adjusted parameters for the punctured (3, 2, 6) code is d4 = 7.89 and A4 = 6469.15, where d4 is much less than the di of other used codes. From Figure 5.12, we could see that the gain enlarges to 1.89 dB when the effective throughput achieves 85% of the maximum rate, which is greater than the gain obtained in Case I. The result in this case confirms the anticipation that the throughput-oriented water-filling yields a larger gain from traditional equal power allocation when the characteristics of the used codes deviate largely from each other .
3 3.5 4 4.5 5 5.5 6 6.5 7 60
65 70 75 80 85 90 95 100 105 110 115
SNR(P
t/S
t in dB)
effective throughput
throughput−oriented II
worst−case capacity optimizing(equal power) 85% maximum rate
Figure 5.11: Case III: The worst-case effective throughput of using throughput-oriented water-filling and equal power allocation. K = 4. (2, 1, 6), (3, 1, 6) and (4, 1, 6) codes are used in the first three channels, and (2, 1, 6) code is used again in the fourth channel.
4 4.5 5 5.5 6 6.5 7 7.5 8 50
60 70 80 90 100 110
SNR(Pt/St in dB)
effective throughput
throughput−oriented II
worst−case capacity optimizing(equal power) 85% maximum rate
Figure 5.12: Case III: The worst-case effective throughput of using the throughput-oriented water-filling and equal power allocation. K = 4. (2, 1, 6), (3, 1, 6) and (4, 1, 6) code sare used in the first three channels, and (3, 2, 6) code is used in the fourth channel.
Chapter 6 Conclusion
In this paper, two power allocation policies are respectively proposed for the two situations:
(i) noise variance is known to each channel and (ii) only total noise variance is known. We aim to maximize the effective throughput and the so-defined worst-case effective throughput of the K coded parallel AWGN channels, subject to practical finite-length and fixed-rate coding constraints. These policies preserve the notion of the water-filling principle by additionally taking into consideration the code characteristics. Simulation and numerical results show that the proposed policy for the situation that noise variance is known to each channel can achieve a near-optimal effective throughput for all values of the total power. When only the total noise variance is known, the proposed policy can also achieve a near-optimal effective throughput for system SNR greater than a certain threshold.
In practice, standards usually provide a list of optional codes for each channel. For the case where noise variance in each channel is known, a natural future work is thus to provide a quick determination of the optimal active channel set O (instead of examining all (2K− 1) possibilities) such that our policy can readily determine the suitable code to be used in each channel. For the case where only total noise variance is available, the future work is to find the optimal power allocation policy for system SNR below the threshold system SNR.
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