Complexity Analysis

Analyzing the complexity of Sphere Decoding can only be carried out in terms of statistics. This is due to the fact, that the complexity of Sphere Decoding is a random variable. Henceforth, complexity is often analyzed in terms of average number of nodes the algorithm visits. This analysis is rather of high level as the number of multiplications or flops is not considered.

However, visiting a node requires calculating the PEDs of the corresponding A0 leafs of that node. Thus, the number of visited nodes scales in the same manner as the number of multiplications required. To demonstrate the complexity of Sphere Decoding, we consider solving (3.7) for a single activity LLR. Calculating this activity LLR requires solving two instances of the under-determined and penalized integer least-squares problem. The penalty term reflecting the prior knowledge is subsumed via α. In the following analysis we restrict the calculation to a single instance of this

−40 −30 −20 −10 0 10 20 30 40 0.01

0.04 0.07 0.10

Ln

pLn(Ln)

K_best = 10 K_best = 50 K_best = 700 SD

Figure 3.15: PDF of the activity LLRs for K-Best detection in a system with M = 5, N = 20, pa = 0.2 at a SNR of ¹/σ²_w = 30dB

optimization problem. Thus, the following results have to be multiplied by two in order to obtain the complexity for calculating one activity LLR.

Fig. 3.16 plots the average number of visited nodes over the SNR for different numbers of observations. In the low SNR range the number of visited nodes is 20 for all cases considered. This is due to the fact that in low SNR α dominates the optimization problem due to the dependency ofσ_w². Thus, the all-zero vector is always the optimal solution to the optimization problem and the PEDs are dominated by the impact ofα. In this case the Sphere Decoding traverses the tree only once. In higher SNR the number of visited nodes heavily depends on the overloading of the system. For the fully determined system, the number of visited nodes exhibits a peak and decreased again down to 20. This does not hold not true for systems with lower number of observations. Here, the number of visited nodes increases in the same manner in the mid range SNR, but the decrease cannot be observed. The reason for this lies in the prior knowledge that depends on the noise power.

In the low and mid SNR range the prior knowledge is exploited to implicitly regularize the under-determined system matrix. With decreasing α, the lower M −N, ..., N rows in the matrix R converge to zero. If this happens, the PEDs for the layers N, ..., M −N converge to zero D_N−M → 0. Thant means the Sphere Decoder algorithm cannot reasonably estimate the sub-vector x^(N−M). For the first Sphere Decoding iterations the estimate for the sub-vector x^(N−M) is merely a random guess. This guess is most likely incorrect and causes interference at the lower layers and the Sphere Decoding

−10 0 10 20 30 40 50 10⁰

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

1/σ_w² in dB

Averagenoofnodesvisited

M = 5 M = 10 M = 15 M = 20

Figure 3.16: Average number of visited nodes for Sphere Decoding in a system with N = 20, pa = 0.2 with varying number of observations M, with implicit regularization.

algorithm has to run longer until the interference is canceled. In the extreme case withα = 0 the Sphere Decoding still finds the optimal solution but brute forces the layers N, ..., M + 1, leading to increased complexity, especially if the system is highly overloaded such as M = 5.

Sphere Decoding vs. K-Best Detection

The K-Best algorithm instead allows for a fixed complexity, which solely depends on the dimensions of the underlying tree. This tree is only deter-mined by the number of layers, corresponding to N and the cardinality of the augmented modulation alphabet |A⁰|. At layer n, the K-Best algorithm visits at most K_best nodes. If the number of nodes at layer n is lower, the K-Best algorithm only visits |A⁰|^N−n nodes, corresponding to the maximum number of nodes at layer n. The number of nodes for K-Best detection thus reads

N n=1

min

K_best,|A0|^N−n*

, (3.45)

which, is in contrast to Sphere Decoding, a fixed number. With (3.45) and the complexity analysis for Sphere Decoding, a certain break-even point where K-Best detection outperforms Sphere Decoding in terms of complexity exists.

We therefore consider the Pareto frontiers given in Fig. 3.14 and consider the value K_best where K-Best detection achieves the same performance as Sphere Decoding. Further, we define the K_best-SD break even point,

10⁰ 10¹ 10² 10³ 10⁴ 10⁰

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

6800 1500

18 14

K_best

Numberofnodesvisited

SD: M = 5 SD: M = 10 SD: M = 15 SD: M = 20

Figure 3.17: Number of visited nodes versus K_best for a system with N = 20 nodes and a varying number of observations.

denoting the value Kbest where a K-Best detector has the same complexity as a Sphere Decoder. If the number of nodes a K-Best detector visits is below this point, we assume that K-Best detection is feasible. Fig. 3.17 plots the number of nodes visited versus K_best for a system with N = 20 nodes according to (3.45), which is a straight line. In contrast to that, the complexity of Sphere Decoding depends on the number of observations M and the particular SNR. Thus, the points on the line in Fig. 3.17 denote the maximum average number of nodes a Sphere Decoder requires for a given M. Hence, the corresponding value on the x-axis denotes theKbest-SD break-even point.

These values are summarized in Table 3.1. In all cases considered, the Kbest that is required to nearly achieve the performance of Sphere Decoding lies below the break-even point, showing that K-Best detection is indeed a viable approach to nearly achieve the performance of Sphere Decoding with lower complexity.

M K_best required according to Fig. 3.14 K_best-SD break-even

5 > 700 6800

10 100 1500

15 10 18

20 10 14

Table 3.1: Comparison of K_best required and the K-best Sphere Decoding break-even point for different numbers of observations M.