Chapter 2 Efficient Search Algorithm for Over- determined
2.5 Computer Simulation and Discussions
This section simulates the symbol error rate (SER) performance and complexity of the proposed K-Best SDA, and compares it with the SE SDA and conventional K-Best SDA [17]. Although many variants of the K-Best SDA have been proposed, the
conventional one has the best decoding performance and is chosen here as a benchmark.
For a fair comparison in each simulation, the preprocessing technique mentioned in Section 4.1 is applied to all algorithms. Complexity is measured in terms of the average number of floating point operations (flops). All real additions, multiplications, memory read/write, and comparison are equally treated as flops. We set d as the distance ' between the Babai estimate and the received signal [10], and Eéêë
( )
d' 2ùúû is then obtained in advance for each SNR as the average from 100000 independent trials. In each simulation, we generate 100 noise realizations per channel realization, and at least 5000 channel realization for each SNR value. The SER is obtained as the average from 500000 independent trials.We first investigate the effectiveness of the proposed CML strategy by comparing the performance of the complex K-Best SDA, which is mentioned in Section 2, with various
configurations. An extreme value of K = 4 is chosen for the complex K-Best SDA incorporating CML. Note that K = 4 is in general the maximal acceptable value for a MIMO-OFDM system, where the ML solution needs to be obtained for each sub-carrier.
The normalized deviation of SER is set as d =15%, the threshold T is set as 0.42 r according to (2.19), and the corresponding probability of performing the ML search is 9.08%, according to (2.14)-(2.16). On the other hand, the K values of the conventional
complex K-Best SDA without CML are chosen as K = 4, 8, 12 for 4 4´ 16-QAM, and K = 4, 12, 24 for 64-QAM, respectively, to illustrate the performance difference. Figure 2-11(a), (b) show the 4 4´ 16-QAM and 64-QAM simulations of SER respectively.
From the results, the complex K-Best SDA incorporating CML can significantly improve the decoding performance with a small K value. The reason is that the proposed CML strategy keeps all possible candidates in the first search layer when the channel is in a
poor condition, significantly reducing the probability of the ML solution being dropped.
The conventional complex K-Best SDA needs to choose K = 12 and 24 respectively to achieve the similar performance. Such high K configurations would inevitably increase the computational complexity, decoding latency and infeasibility for practical MIMO-OFDM systems. For demonstration, we also include the cases of K = 8 and K = 12 for the complex K-Best SDA incorporating CML for 16-QAM and 64-QAM respectively. It is evident that both can achieve nearly the same performance as the ML detector.
Next, we evaluate the SER performance and complexity of the proposed complex K-Best SDA incorporating the CML strategy. Figure 2-12(a), (b) and Figure 2-13(a), (b), respectively, show the 4 4´ 16-QAM and 64-QAM simulations of SER and complexity with K = . The normalized deviation of SER is set as 8 d =5% and
d =10% respectively and the threshold T is set as 0.291 and 0.3532 respectively, r according to (2.19), and the corresponding probability of performing the proposed ML search is 4.26% and 5.62% respectively, according to (2.14)-(2.16). Comparing Figure 2-12(a) with Figure 2-13(a), the SER curves of the proposed K-Best SDA and the SE SDA are nearly the same. This shows that the threshold constraint significantly reduces the probability of performing the ML search and there is almost no performance degradation in the proposed K-Best SDA. In contrast, the SER of the conventional K-Best SDA tends to become saturated at high SNR. This is due to the fact that the conventional K-Best SDA with a smaller K drops the ML solution with a high probability when the channel is in poor conditions, which always occurs with a certain probability in practice. In the 64-QAM case, the proposed K-Best SDA achieves nearly a 3 dB gain over the conventional K-Best SDA at SER = 10-3. Note that this performance gap between the proposed K-Best SDA and a conventional one is larger than that of the
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Complex K−Best SDA without proposed CML (K=4) Complex K−Best SDA without proposed CML (K=8) Complex K−Best SDA without proposed CML (K=12) Complex K−Best SDA with proposed CML (K=4) Complex K−Best SDA with proposed CML (K=8) MLD (4x4)
Complex K−Best SDA without proposed CML (K=4) Complex K−Best SDA without proposed CML (K=12) Complex K−Best SDA without proposed CML (K=24) Complex K−Best SDA with proposed CML (K=4) Complex K−Best SDA with proposed CML (K=12) MLD (4x4)
(b)
Figure 2-11: Performance of complex K-Best SDA for 4x4 MIMO systems. (a) 16-QAM modulation. K = 4 and 8 for complex K-Best SDA incoporating proposed CML strategy; K = 4, 8, and
12 for regular complex K-Best SDAs. (b) 64-QAM modulation. K = 4 and 12 for complex K-Best SDA
incoporating proposed CML strategy; K = 4, 12, and 24 for regular complex K-Best SDAs.
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10−3 10−2 10−1 100
SNR (dB)
SER
Conventional K−Best SDA (K=8) (4x4) Proposed K−Best SDA (K=8) (4x4) SE SDA (4x4)
(a)
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103 104
SNR (dB)
Complexity
SE SDA (4x4)
Conventional K−Best SDA (K=8) (4x4) Proposed K−Best SDA (k=8) (4x4)
(b)
Figure 2-12: Performance and complexity of SDA for 4x4 MIMO systems with 16-QAM modulation. (a) SER. (b) Complexity. K = 8 for K -Best SDAs.
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10−3 10−2 10−1 100
SNR(dB)
SER
Conventional K−Best SDA (K=8) (4x4) Proposed K−Best SDA (K=8) (4x4) SE SDA (4x4)
(a)
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103 104
SNR(dB)
Complexity
SE SDA (4x4)
Conventional K−Best SDA (K=8) (4x4) Proposed K−Best SDA (K=8) (4x4)
(b)
Figure 2-13: Performance and complexity of SDA for 4x4 MIMO systems with 64-QAM modulation. (a) SER. (b) Complexity. K = 8 for K -Best SDAs.
16-QAM case. This is because the probability of the ML solution being dropped increases as the modulation symbol alphabet becomes larger [17]. In contrast, the proposed CML search strategy keeps all possible candidates in the preceding layers, significantly reducing the probability of the ML solution being dropped. Comparing Figure 2-12(b) with Figure 2-13(b), the proposed K-Best SDA has higher complexity than that of the SE SDA in the high SNR regime. This is due to the fact that the proposed K-Best SDA visits more candidate symbols than the SE SDA when the number of layers N is smaller. As shown in the simulation cases, the complexity of the SE SDA varies with SNR. This is not desirable in practice, because a steady SNR is not achievable in realistic wireless environments, such that the decoding throughput of the SE SDA cannot be stable. In contrast, the proposed K-Best SDA provides a nearly fixed throughput, with excellent performance and low complexity. The proposed efficient architecture reduces the number of path weight evaluations and sorting operations in each layer. As a result, the proposed K-Best SDA exhibits near-ML performance and reduces 46.62% and 58.14% complexity respectively over the conventional K-Best approach using the same K.
Figure 2-14(a) and Figure 2-14(b) show the 8 8´ 16-QAM simulations of SER and complexity with K =14. The normalized deviation of SER is set as d =15% and the threshold Tr is set as 0.833 and the corresponding probability of performing the proposed ML search is 43.9%. The probability of performing the ML search is higher than that of the 4 4´ case because the probability of the ML solution being dropped in the K-Best SDA is higher in the 8 8´ case. Again, the performance of the proposed K-Best SDA is better than the conventional K-Best SDA, and the complexity of the proposed K-Best SDA is lower than that of the SE SDA and the conventional K-Best SDA. Compared with the conventional K-Best SDA, the proposed method decreases complexity by more than 41.75%. Because the number of search layers is larger in this
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SNR (dB)
SER
Conventional K−Best SDA (K=14) (8x8) Proposed K−Best SDA (K=14) (8x8) SE SDA (8x8)
(a)
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104 105 106
SNR (dB)
Complexity
SE SDA (8x8)
Conventional K−Best SDA (K=14) (8x8) Proposed K−Best SDA (k=14) (8x8)
(b)
Figure 2-14: Performance and complexity of SDA for 8x8 MIMO systems with16-QAM modulation. (a) SER. (b) Complexity. K = 14 for K -Best SDAs.
case, the proposed method reduces more complexity in path evaluations and sorting operations. In the 8 8´ case, the value of K must be set larger to reduce the probability of the ML solution being dropped in the preceding layers. Hence, the gap in complexitybetween the proposed K-Best SDA and the conventional K-Best SDA is smaller than that in the 4 4´ case. We can further improve the performance of the proposed K-Best SDA by choosing a higher threshold.
Figure 2-15(a) and Figure 2-15(b) show the 8 8´ 64-QAM simulations of SER and complexity with K = 32. The normalized deviation of SER is set as d =15% and the threshold T is set as 1.143 and the corresponding probability of performing the ML r search is 65.8%. In this case, the proposed K-Best SDA still works better than the conventional K-Best SDA. The gap in complexity between the proposed K-Best SDA and the conventional K-Best SDA is smaller than that in the 8 8´ 16-QAM case. This is because a higher threshold value causes the ML search operations to occur more frequently, though the proposed efficient sorting method reduces much more complexity for a larger modulation alphabet. The proposed CML search procedure significantly reduces the amount of path evaluations, but induces extra memory read/write and table access operations. Nevertheless, the proposed K-Best SDA still has lower average complexity than the conventional one. Finally, under the same channel conditions, the conventional K-Best SDA requires K =52 to achieve near-ML performance.
The configuration increases the computational complexity and decoding latency. The proposed decoder with K =32 can provide nearly the same performance, reducing 53.45% computational complexity over the conventional K-Best SDA with K =52.
This section simulates the SER performance and complexity of the proposed SDA and compares it with the SE SDA and the conventional K-Best SDA. Although the value of r does not directly reflect the channel condition in all cases, the proposed criterion o i i, ,
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Figure 2-15: Performance and complexity of SDA for 8x8 MIMO systems with 16-QAM modulation. (a) SER. (b) Complexity. K = 32 for proposed K -Best SDA; K =
32 and 52 for conventional K -Best SDAs.
does help the decoder successfully produce a near ML solution over poor channels, without always performing the ML search in the preceding layers. This systematic approach thus requires fewer ML search layers than previous methods [23], [27]. The simulation results confirm that the proposed decoder exhibits excellent performance and requires lower complexity than the conventional K-Best SDA. It is also worth noting that the performance of the proposed decoder is close to that of the SE SDA (i.e., ML performance).