Computational Complexity

Under the assumption that the conventionalK-best SD iterative receiver dispensing with the ALT

technique generates the candidate list only once at the first iteration, which is stored in the memory for the extrinsic LLR calculation of the forthcoming iterations, the performance gains attained by both the center-shifting scheme introduced in Section 3.2.3 and the ALT scheme are achieved at the cost of an acceptable computational complexity investment, since the candidate list has to be regenerated at each iteration. However, the memory requirement imposed is expected to be reduced, since there is no need to store the resultant candidate list. The complexity imposed by invoking the ALT scheme can be viewed in Figure 3.25(a), where the overall computational complexity quantified in terms of the number of PED-evaluations per channel use imposed by the system operating both with and without the aid of the ALT scheme (ALT=7) are plotted for (8×4)- element rank-deficient 4QAM SDMA/OFDM system scenario. Specifically, since SD has to be carried out only once per channel use, regardless of how many iterations have been carried out, the receiver not benefitting from the ALT technique exhibits the same computational complexity of

3.3.2. Features of the ALT-AssistedK-Best SD Receiver 85 2 3 4 5 6 7 8 9 10 11 12 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 E b/N0 (dB)

Total Complexity/Channel Use (# of PED−Evaluations)

Non−ALT (K=1024) Non−ALT (K=128) ALT−Aided (K=128) (a) 1 2 3 4 5 6 7 8 0 500 1000 1500 2000 2500 Iterations

Computational Complexity per Channel Use (# of PED−Evaluations)

E b/N0=7 dB E b/N0=8 dB E b/N0=9 dB E b/N0=10 dB (b)

Figure 3.25: Histogram of the candidate list generation related computational complexity imposed

by the ALT-aidedK-best SD in the(8×4)-element rank-deficient 4QAM SDMA/OFDM system

scenario: (a) the overall computational complexity per channel use for differentEb/N0values

(ALT=7); (b) computational complexity per channel use of each iteration (K=128, ALT=7). Note:

the maximum number of iterations for all differentEb/N0value was fixed to 8 and the iterative

detection was terminated as soon as there is no more iteration gain can be achieved, i.e. the resultant EXIT trajectory line reached the convergence point of[IA, IE] = [1, 1].

2,388 PED-evaluations per channel use forK =128, regardless of the channel SNR. By contrast,

the number of PED-evaluations required by the ALT-assisted receiver differs for different SNRs. To be specific, observe in Figure 3.25(a) that the complexity increases steadily as the SNR increases from 2 dB to 7 dB, peaking at about 17,000 PED-evaluations per channel use. Beyond 7 dB, the complexity decays steadily as the SNR increases further, but levels out around 5,000 PED- evaluations at an SNR of 12 dB. Upon inspecting both Figure 3.24 and Figure 3.25(a), we observe that no performance gain is achieved by the ALT-aided receiver when the SNR is lower than 6 dB, despite the additional computational efforts of regenerating the candidate list at each iteration. This is not unexpected, since it is unlikely that the a priori LLRs gleaned by the outer channel decoder become higher than the threshold of the ALT scheme, because the intersection of the inner and outer EXIT curve occures at a low IAvalue in Figure3.42(a). Hence, it is unwise to activate the ALT scheme, when the SNR is low, since it may impose an increased complexity without any performance improvements.

On the other hand, as seen in Figure 3.24, with the advent of the ALT scheme the K-best SD

becomes capable of achieving a near-MAP performance by settingK = Ncand = 128 instead of

1024. Recall our arguments on the complexity of the extrinsic LLR calculation for the list SD outlined in Section 3.2.1.2 that the corresponding complexity is linearly proportional toNcand, as explicitly expressed in Eq.(3.16). From this perspective, given a fixed target BER performance, the

3.3.2. Features of the ALT-AssistedK-Best SD Receiver 86 computational complexity imposed by the extrinsic LLR calculation of theK-best SD can be con-

siderably reduced by employing the ALT scheme. Furthermore, with reference to Figure 3.25(a), the candidate list generation complexity of the ALT-aided receiver is well below that of its ‘non- ALT-aided’ counterpart for the SNR range spanning from 2 dB to 12 dB except for SNRs in the immediate vicinity of 7 dB, if our aim is to achieve the near-MAP BER performance quantified in Figure 3.24, which can be attained by havingK= Ncand=1024 for the system operating without

the ALT technique or by setting K = Ncand = 128 in the presence of the ALT scheme. More

specifically, the number of PED-evaluations per channel use carried out by the non-ALT-aided sys- tem using Ncand = 1024 remains as high as 13,652, regardless of the SNR and the number of

iterations. On the other hand, in the presence of the ALT scheme, the candidate list has to be re- generated at each iteration, but nontheless, the total complexity imposed is substantially reduced, except for SNRs in the immediate vicinity of 7 dB. There are two reasons for this phenomenon. 1) When the SNR is low, the number of iterations providing a useful gain is low, because there is no open tunnel between the EXIT curves of the inner and the outer decoder, unless the SNR is sufficiently high. 2) By contrast, when the SNR is high, the resultant stair-case shaped decoding trajectory can readily pass through the widely open EXIT tunnel and reaches the point of perfect convergence at [IA, IE] = [1, 1] after a low number of iterations. Furthermore, when the SNR is

high, the number of PED-evaluations carried out at each iteration is expected to decrease, as the iterations continues, as oberved in Figure 3.25(b). This is due to the fact that the a priori LLRs fed back from the outer decoder of Figure 3.1 to the SD are likely to become higher than the LLR threshold after the first few iterations, and this allows the ALT-assisted SD to directly truncate the low-probability branches, hence leading to a reduced constellation size, which in turn results in a reduced complexity. More specifically, the complexity histogram of Figure 3.25(b) indicates that the higher the SNR, the more sharply the complexity drops as the iterations continue. Actually, when the SNR is relatively high, the complexity imposed becomes more modest after a few it- erations, since the majority of the a priori LLRs fed back from the outer decoder to the SD of Figure 3.1 becomes higher than the LLR threshold. From a different perspective, this observation also explains the reason why we experience a complexity peak at the moderate SNR of 7 dB, where the ALT-related complexity does not decrease sufficiently substantially as the iterations continue and hence a high number of iterations are required to attain the maximum achievable iteration gain, since only a rather narrow EXIT tunnel was created between the EXIT curves of the inner and the outer decoder.