Soft-Output Sphere Decoding with Modified Repeated Tree Search
3.2 Simulation Results
We now compare our modified RTS with the STS and the SOCA.
Consider an MIMO system transmitted over Rayleigh fading channels with possibly spatial or temporal correlation. Fast fading and slow fading scenarios as specified in [6]
are both considered, where in fast fading scenario, the channel realizations change per MIMO transmission, while in slow fading scenario, the channel realizations remain the same throughout an entire (turbo) transmission block but vary across (turbo) transmis-sion blocks. We assume that all channel matrix realizations can be perfectly estimated at the receiver. Four transmit antennas and four receive antennas, (i.e. NT = NR = 4) and 16-QAM constellation are adopted.
Two kinds of channel coding schemes are tested. The first one is a 3GPP-specified punctured turbo code of code rate R = 1/2 with codeword length 2000 bits [10]. After passing the code through a 40 × 50 block interleaver, 500 16-QAM symbols are formed and transmitted. At the receiver, the 8-iteration Max-Log-MAP decoder is used for turbo decoding.
The second channel coding scheme used in our simulation is a 3GPP-specified (2, 1, 8) convolutional code of code rate R = 1/2. The codeword length is 720 bits. After convolutional encoding, 180 16-QAM symbols are fed into a 15 × 48 block interleaver before they are sent. At the receiver, the Viterbi decoder is used for the decoding of this convolutional code.
The upper complexity limit T1 for the first stage in our modified RTS is set to 30, and the set of the restriction vectors b examined in our simulation contains b = [4444], [4442], [4422], [4222], and [2222], which respectively result in T2 = 40, 38, 34, 28, and 20.
For the SOCA, the tested numbers of paths extended in the first level, i.e., b1, include 6, 14, 12, 10, 8, 6, and 4, which results complexities 88, 80, 72, 64, 56, 48, and 40, respectively. Note that for the SOCA, bi = 1 for every i > 1.
It should be mentioned that the channel regularization algorithm [5] is used for all three algorithms, i.e., the STS, the SOCA, and our proposed modified RTS when performing the QR-decomposition step. In addition, the SQRD [13] is employed as the
sorting algorithm in QR decompositoin for both the STS and our proposed modified RTS, while the SOQR in [6] is implemented for the SOCA.
The performance index that we adopt in this thesis is the minimum SNR required to achieve a block error rate of 10−2 after channel decoding. The complexity measure is the number of visited nodes during the tree search. This complexity measure is widely adopted for one-node-per-cycle hardware implementation architecture [11]. We are now ready to present the simulation results.
We first examine what should e the proper selected value for Lmax. As previously mentioned, the value of Lmax chosen will affect performance and complexity of the STS, the SOCA and our modified RTS. As for the STS (as well as other algorithms), a larger Lmax implies a better performance but a larger complexity. We then show the the performance-complexity tradeoff for the STS in Fig. 3.2. Later, an Lmax value will be chosen according to this figure.
For the SOCA and our modified RTS, the relationship between Lmaxand performance-complexity tradeoff is a little messy. Various test results regarding different Lmax values are summarized in Figs. 3.3, 3.5, 3.4, and 3.6.
Specifically, in Fig. 3.3, the range of the tested Lmax values is ranged from 0.15 to 0.55 for the modified RTS under fast fading channels. We can clearly see from this figure that Lmax = 0.25 has the best complexity-performance trade-off. However, the implication from Fig. 3.4 is a little different under slow fading scenario. By testing Lmax from 0.15 to 0.55, we observe from Fig. 3.4 that under a slow fading environment, the smaller the Lmax, the better the performance-complexity tradeoff. To have a good balance choice that fits both fast fading and slow fading scenarios, we set Lmax = 0.25 for our modified RTS.
Similar tests, for which Lmaxassumes values from 0.15 to 0.55, are performed for the SOCA. From Fig. 3.5, where the fast fading scenario is assumed, we observe that the performance-complexity tradeoff improves as Lmax increases. Notably, the simulation
13 13.2 13.4 13.6 13.8 14
Minimum required SNR [dB] for BLER = 0.01
Average complexity
Minimum required SNR [dB] for BLER = 0.01
Average complexity
STS
Figure 3.2: Impact of Lmaxon the STS with turbo coding under fast (left subfigure) and slow (right subfigure) Rayleigh fading channels
results for Lmax ranging from 0.35 to 0.55 are almost indistinguishable. Under the slow fading scenario, however, different trends can be observed from Fig. 3.6. We note from this figure that when Lmaxis larger than 0.2, the performance-complexity tradeoff begins to degrade. Again, to compromise between two different scenarios, we choose Lmax = 0.3 for the SOCA.
Figs. 3.7 and 3.8 illustrate how different T1 affect BLERs and complexities. In short, we can see from Fig. 3.7 that under fast fading, the curve corresponding to T1 = 20 has already approached the curve of T1 = ∞. In Fig. 3.8, we then see that there is no visible gap between curves of T1 = 20 and T1 = ∞. Nonetheless, we set T1 = 30 to secure the (near-)ML performance.
After the settlement of the parameters used, we are now ready to compare the STS and the SOCA with our modified-RTS algorithm. First, we remark on the simulations results regarding turbo coding and fast fading scenario. As observed in Fig. 3.9, the proposed modified RTS achieves the best performance-complexity tradeoff, when it is
13.3 13.4 13.5 13.6 13.7 13.8 13.9
Minimum required SNR [dB] for BLER = 0.01
Average complexity
Figure 3.3: Impact of Lmax on the modified RTS with turbo coding under fast Rayleigh fading channels, where b = [4444] and [2222] are employed.
compared with the STS and the SOCA. In order to examine the variation in complexity, we also record that the 99.9th percentile complexities of the STS and our modified RTS in this figure. Since the SOCA has a fixed decoding complexity, the average complexity of the SOCA is exactly the same as its 99.9th percentile complexity. It is shown in Fig. 3.9 that since the 99.9th percentile complexity of the STS is much higher than its average complexity, the STS may suffer with high variation of complexity and hence may become a challenge for hardware implementation. The high variation of complexity of the STS also makes varying its computational delay.
On the other hand, the gap between the average complexity and the 99.9th percentile complexity of the modified RTS is considerably much smaller that that of the STS. The 99.9th percentile complexity of the modified RTS is even just slightly higher than the (fixed) complexity of the SOCA. This indicates that the complexity upper limit (T1+T2) we set for the modified RTS does decrease the variation of the decoding complexity, and therefore makes the soft-output SD algorithm more easily hardware-implementable.
Next we remark on the simulation results in the slow fading scenario. We observe
17 17.2 17.4 17.6 17.8 18 18.2 18.4 18.6 18.8
Minimum required SNR [dB] for BLER = 0.01
Average complexity
Figure 3.4: Impact of Lmax on the modified RTS with turbo coding under slow Rayleigh fading channels, where b = [4444] and [2222] are employed.
from Fig. 3.10 that the STS achieves the best complexity-performance tradeoff in the sense of average complexity. However, nonetheless, the high complexity variation of the STS remains, which again challenges its hardware implementation. In particular, the 99.9th percentile complexity of the STS is six times larger than its average complexity.
Similar conclusion as the one in the fast fading scenario can be obtained about the SOCA and the modified RTS that these two are more appropriate for hardware implementation due to their prohibitively bounded complexity. When comparing the modified RTS with the SOCA, the former requires a higher 99th percentile complexity but has a seemingly less average complexity.
In order to examine the impact on the coding algorithm such as turbo and con-volutional codes, we re-do the previous simulations by replacing the turbo code with the convolutional code. The simulation results are summarized in Figs. 3.11 and 3.12.
The results are similar to what obtained using the turbo code. As a result, the SOCA and the proposed modified RTS remain to be more attractive solutions for hardware implementation, regardless of the channel coding scheme.
13 13.1 13.2 13.3 13.4 13.5 13.6 13.7
Minimum required SNR [dB] for BLER = 0.01
Average complexity
Figure 3.5: Impact of Lmax on the SOCA with turbo coding under fast Rayleigh fading channels, where b1 = 16 and 8 are employed.
In order to have more detailed insight on the complexities of the STS and our modified-RTS algorithm, we show the 50%-percentile, 90%-percentile, 99%-percentile, and 99.9%-percentile complexities in Figs. 3.13 and 3.14 in both fast and slow fading scenarios under turbo coding scheme. Evidently, the gaps among 50%-percentile, 90%-percentile, 99%-90%-percentile, and 99.9%-percentile complexities for the modified RTS are much smaller that those of the STS.
We further investigate the complexity distribution of the STS with Lmax = 0.2 and also the complexity distribution of the modified RTS with Lmax = 0.25 and b = [4444]
in Figs. 3.15 and 3.16, respectively. Note that to achieve a BLER of approximately 10−2, the minimum SNR required for the STS is 13.41 dB, while the minimum SNR required for the modified RTS is 13.40 dB; so they are approximately operated at the same SNR. From the two figures, we can clearly see a drawback of the STS is its high complexity variation. Although its average complexity is only 44.57, its largest decoding complexity can be as large as 900 after testing 2,500,000 simulation samples. Such a high complexity variation of the STS may become a challenge for hardware implementation.
16.85 16.9 16.95 17 17.05 17.1 17.15 17.2 17.25 55
60 65 70 75 80 85 90
Minimum required SNR [dB] for BLER = 0.01
Average complexity
SOCA, Lmax=0.15 SOCA, Lmax=0.2 SOCA, L
max=0.25 SOCA, L
max=0.3 SOCA, L
max=0.35 SOCA, L
max=0.4 SOCA, L
max=0.45 SOCA, L
max=0.5 SOCA, Lmax=0.55
Figure 3.6: Impact of Lmax on the SOCA with turbo coding under slow Rayleigh fading channels, where b1 = 16 and 8 are employed.
As a contrary, the average decoding complexity for our modified RTS is only 25.62 and its true complexity is upper-bounded by T1+T2 = 70 as shown in Fig. 3.16, where unlike the complexity distribution of the STS, the complexity distribution of the modified RTS does not have a long tail.
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Figure 3.7: Impact of T1on BLERs and complexities under fast Rayleigh fading channels.
16 16.5 17 17.5 18
Figure 3.8: Impact of T1 on BLERs and complexities under slow Rayleigh fading chan-nels.
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Minimum required SNR [dB] for BLER = 0.01
Average and 99.9%−percentile complexity
STS average SOCA
modified RTS average STS 99.9%
modified RTS 99.9%
Figure 3.9: Performance versus complexity for the STS, the SOCA, and the modified RTS in fast Rayleigh fading channels. The numbers beside the STS marks are the Lmax
values used. The numbers next to the SOCA curve correspond to b1. The number next to each modified RTS mark is T2.
Minimum required SNR [dB] for BLER = 0.01
Average and 99.9%−percentile complexity
STS average SOCA
modified RTS average STS 99.9%
modified RTS 99.9%
Figure 3.10: Performance versus complexity for the STS, the SOCA, and the modified RTS with turbo code in slow Rayleigh fading channels. The numbers beside the STS marks are the Lmax used. The numbers next to the SOCA curve correspond to b1. The number next to each modified RTS mark is T2.
13.9 14 14.1 14.2 14.3 14.4 14.5 14.6
Minimum required SNR [dB] for BLER = 0.01
Average number and 99.9% number of visited nodes
STS average SOCA
modified RTS average STS 99.9%
modified RTS 99.9%
Figure 3.11: Performance versus complexity for the STS, the SOCA, and the modified RTS with convolutional code in fast Rayleigh fading channels. The numbers beside the STS marks are the Lmax used. The numbers next to the SOCA curve correspond to b1. The number next to each modified RTS mark is T2.
16.5 17 17.5 18
100 101 102 103
Minimum required SNR [dB] for BLER = 0.01
Average and 99.9%−percentile complexity
0.200
Figure 3.12: Performance versus complexity for the STS, the SOCA, and the modified RTS with convolutional code in slow Rayleigh fading channels. The numbers besidethe STS marks are the Lmax used. The numbers next to the SOCA curve correspond to b1. The number next to each modified RTS mark is T2.
13.1 13.2 13.3 13.4 13.5 13.6 13.7
Minimum required SNR [dB] for BLER = 0.01
50%, 90%, 99% and 99.9%−percentile complexity
STS 50%
Figure 3.13: Performance versus different percentile-complexity for the STS and the modified RTS with turbo code in fast Rayleigh fading channels. The numbers beside the STS marks are the Lmax used. The number next to each modified RTS mark is T2.
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Minimum required SNR [dB] for BLER = 0.01
50%, 90%, 99% and 99.9%−percentile complexity
STS 50%
Figure 3.14: Performance versus different percentile-complexity for the STS and the modified RTS with turbo code in slow Rayleigh fading channels. The numbers beside the STS marks are the Lmax used. The number next to each modified RTS mark is T2.
0 20 40 60 80 100 120 140 160 180 200 0
0.004 0.008 0.012 0.016 0.02
Complexity
Probability
Figure 3.15: Complexity distribution of the STS with Lmax = 0.2. The maximum range of the complexity is 200, where the probability of complexity exceeding 200 is 0.0035.
0 10 20 30 40 50 60 70
0 0.008 0.016 0.024 0.032 0.040 0.048
Complexity
Probability
Figure 3.16: Complexity distribution of the modified RTS with Lmax = 0.25 and b = [4444].