2.3 Complexity-Reduction Schemes for SD
2.3.2 Complexity-Reduction Schemes for K-Best SD
10−4 10−3 10−2 10−1 100 Eb/N0 (dB) BER 0 5 10 15 20 2510 2 103 104 105 106 BER and Complexity of Reduced−Complexity Depth−First SD in SDMA/OFDM Systems
Num. of Additions & Multiplications per Rx Signal Vector
SD SSD−ET SSD−UB Complexity 8x8 4QAM Complexity 4x4 16QAM BER 8x8 4QAM BER 4x4 16QAM Throughput=16bits/s
Figure 2.13: BER performance and computational complexity of reduced-complexity depth-first
SDs. The y-axis on the left quantifies the BER performance of the ML and SD algorithms using continuous lines, while the right y-axis quantifies the complexity versus the Eb/No, which is
plotted using broken lines. All system parameters were summarised in Table 2.1.
an evident complexity reduction, when the SNR is relatively low, while imposing only a slightly lower complexity than the original SD detector of [99] when the SNR is in excess of17.5dB. On
the other hand, when comparing two different SDMA/OFDM systems, we found that the(8×8)- antenna 4QAM system substantially outperforms the(4×4)-element 16QAM system in terms of the achievable BER, as a benefit of its higher diversity gain and its lower-density modulation con- stellation, while imposing an acceptable computational complexity. More specifically, for a given target BER of10−5, we have an SNR gain of about 9dB if the(8×8)-antenna 4QAM scheme is employed, rather than the(4×4)-element 16QAM arrangement. This is achieved at the cost of less than three times increased computational complexity, as quantified in terms of the number of real- valued additions and multiplications per received signal vector, when the updated-bound-assisted scheme is employed.
In addition to their reduced complexity, the search algorithm optimization schemes discussed in this section have a further benefit of rendering the complexity of the SD less sensitive to the specific choice of the ISR, which can be observed from Figure 2.14.
2.3.2 Complexity-Reduction Schemes forK-Best SD
2.3.2.1 Optimal Detection Ordering
Having discussed various complexity reduction schemes designed for the depth-first SD detector, let us now consider a range of complexity reduction schemes applicable to theK-best SD. The de-
2.3.2. Complexity-Reduction Schemes forK-Best SD 40 0 5 10 15 20 25 102 103 104 105
Initial Square Search Radius C
Num. of Additions & Multiplications per Rx Signal Vector
E b/No=4dB Eb/No=20dB SSD−UB SD SSD−TT 4x4 16QAM SDMA/OFDM
Figure 2.14: Complexity versus the square ISR of reduced-complexity depth-first SDs. All system
parameters were summarised in Table 2.1.
reducing the complexity of the depth-first SD, was found suitable also for theK-best SD, which
achieved a similar performance to that shown in Figure 2.11. For a rudimentary introduction to this scheme, please refer to Section 2.3.1.2.
2.3.2.2 Search-Radius-AidedK-Best SD
It becomes explicit based on the portrayal of the K-best SD in Sections 2.2.3 and 2.2.5, that its
computational complexity is controlled by the parameter K, for a certain modulation scheme and
a certain number of transmit antennas or users. This is in contrast to its depth-first counterpart, which achieves a low complexity, despite approaching the ML performance with the aid of the rapid shrinking the original search radius. Intuitively, if we can introduce a search radius for the employment in theK-best SD, its complexity can be further reduced by discarding the unlikely ML
candidate nodes which are located outside the sphere confined by the search radius, hence reducing the number of tentative nodes at each level. Consequently, since the partial Euclidean distances evaluated for some of the nodes exceed the radius, there may be less than K nodes that have to
be considered for each level, resulting in an additional complexity reduction. In contrast to the gradually reduced radius of the depth-first SD algorithm, the radius used for theK-best SD remains
unchanged during the entire search process, since it carries out the tree search in the downwards direction only and the search is ceased, whenever it reaches tree leaf level, namely the lowest level of the tree exemplified in Figure 2.5. Hence, exactly the same search radius selection problem is encountered by theK-best SD, as faced by the depth-first SD. In order to avoid having no lattice
points inside the sphere, which in turn results in a repeated search using an increased radius, the radius selection schemes used for the K-best SD should guarantee that at least one lattice point
is located in the search sphere. In this report, two radius selection schemes for K-best SD will
2.3.2. Complexity-Reduction Schemes forK-Best SD 41
0 5 10 15 20 25
102 103
Eb/N0 (dB)
Num. of Additions & Multiplications per Rx Signal Vector
No Radius−Aided LS Criterion Radius MMSE Criterion Radius K=16, 4x4 16QAM
Figure 2.15: Complexity versus SNR of the radius-basedK-best SD
schemes, while the latter was already discussed in the context of depth-first SD in Section 2.3.1.1. In Figure 2.15 we characterize these two radius-basedK-best SDs and the original K-best SD of
Secction 2.2.3 [61], where we find that a significantly lower complexity can be achieved by both of the radius-basedK-best SDs, compared to the original K-best SD of [61]. Hence, the radius-based K-best SD no longer exhibits an SNR-independent complexity as characterized in Figure 2.15,
because a higher complexity reduction can be attained when the SNR increases. On the other hand, the complexity of the MMSE-criterion-based radius scheme of Section 2.3.1.1 is evidently lower than that of the LS-criterion-based radius scheme, due to the fact that the former scheme is expected to operate using a smaller search radius, which is capable of reducing the number of nodes at each level that would be expanded.
2.3.2.3 Complexity-Reduction Parameterδ for Low SNRs
Although the complexity of K-best SD can be significantly reduced by introducing a search ra-
dius, it still exhibits a relatively high complexity when the SNR is low, as we can observe from Figure 2.15. Intuitively, when the noise level is high, i.e. at low SNRs, investing excessive de- tection efforts in terms of a large search space becomes futile. This will become more explicit by considering the ML detector, which has a high computational complexity and yet, hardly achieves any performance gain in comparison to the MMSE detector, for example, when the SNR is low. In order to mitigate the problem, we introduce a complexity-reduction parameterδ, which allows
us to reduce the complexity of the K-best SD, when the SNR is low. A similar parameter γ was
employed in the OHRSA detector of [59] in order to control its complexity, which will be discussed in Section 2.3.3.
The parameterδ is used as follows. When the SNR corresponding to the currently detected ith
signal component is lower thanδ, namely we have khik2
σ2
w
2.3.3. Optimized Hierarchy Reduced Search Algorithm 42