Complexity analysis of the Sphere Detector algorithm

The complexity analysis of the SD algorithm has been thoroughly investigated by researchers. For a few good examples refer to [71], [72], [73], [74], [75]. Finding the solution of the ILS problem or that of the CLPS problem is known to be NP-hard. A general conclusion is that the complexity of the SD algorithm is directly proportional to the number of lattice points explored. Furthermore, the number of lattice points examined throughout the SD algorithm is highly influenced by the sphere radius. The optimal radius, i.e., the coveringradius, requires a number of steps that grows exponentially [76]

with the dimension of the lattice. Thus, finding the optimal radius is not feasible for real systems.

The goal of this section is to give a brief overview on the work presented by Hassibi et al. in [71] where the expected complexity of the ILS problem averaged over the noise and over the lattice is studied. Moreover, in [72] it is demonstrated that the expected complexity is polynomial for a wide range of SNRs and number of antennas. By quanti-fying their results it is shown that the expected number of lattice points for a wide range of SNRs can be handled in real-time with modern computing architectures.

The first problem that has to be solved is the choice of the radius. As stated above, finding the covering radius based on the lattice generator matrix H is an NP hard problem. One solution is to set the sphere’s radius based on the noise statistics. A χ² random variable with M degrees of freedom is defined by scaling the noise vector as follows:

1

σ² · kvk² = 1

σ²ky−Hs_tk. (4.37)

In order to find a lattice point inside the sphere the integration of the probability density

DOI:10.15774/PPKE.ITK.2015.010

4.6. MAXIMUM LIKELIHOOD TREE-SEARCH BASED DETECTORS

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0

5 10 15 20 25 30 35 40

ǫ

d2

σ²= 0.3162 σ²= 0.5623 σ²= 0.7079 σ²= 1

Figure 4.5: The radius size of the bounding sphere for different εandσ² parameters.

function of χ² random variable with M degrees has to be close to one Z ^α·M

2

0

λ^M2−1

2^M2 Γ(^M₂ )e^−λ2 dλ= 1−ε (4.38) where Γ(x) is the gamma function, 0< ε1 andαis a scaling parameter used to define the sphere radius as follows:

d² =αM σ². (4.39)

If no lattice point is found 1 −ε can be further increased by lowering the value of ε that results in the increase of the α parameter, as a result the sphere radius d² is also increased. Figure 4.5 shows the change of the radius for various ε and σ² parameter configurations.

After determining the radius, the next question is the number of visited nodes on the tree for dimensions k = 1, . . . , N. Figure 4.4 shows a possible tree traversal, where on thek= 1,2 dimension two nodes and onk= 3 only a single node fullfills the conditions of Eq. 4.31. As a result, the complexity of the SD algorithm equals the number of nodes visited on every dimension times the floating point operations required to expand and evaluate a node on the specific dimension. The following formula gives a more precise

DOI:10.15774/PPKE.ITK.2015.010

4.6. MAXIMUM LIKELIHOOD TREE-SEARCH BASED DETECTORS

(expected # of nodes ink-dim sphere of radiusd)·(flops/node)

= The floating point operations performed for every visited node required by the SD algo-rithm based on the FP enumeration in [71] is defined asfp(k) = 2k+ 11. In the following the focus is on the derivation of E_p(k, d²).

The first step of determining the expected number of points inside the sphere is to determine how many lattice points lie within the sphere in every dimension. More formally, if s_t was transmitted andy=Hs_t+v was received what is the number of s_a arbitrary lattice points that satisfy

ky−Hs_ak ≤d² kv+H(s_t−s_a)k ≤d².

(4.41)

The resulting vector w=v+H(s_t − s_a) is an M-dimensional zero-mean Gaussian random vector where the (i, j) entry of the covariance matrix is E{w_iw_j}=δ_ij(σ²+ks_t−s_ak²). This implies that kwk²/2(σ²+ks_t−s_ak²) is a χ² ran-dom variable with M degrees of freeran-dom. As a result, the probability that s_a lies within a sphere of radius daroundy is given by its cumulative distribution function (CDF)

F M, d²

where γ(x, k) is the lower incomplete Gamma function defined as

γ d² In Eq. 4.42 the CDF is determined for N dimensional lattice points. However, the CDF has to be determined for smaller dimensions as well. The derivation of the probability for partial symbol vectors is shown in [71] and the resulting formula is:

F M−N+k, d²

4.6. MAXIMUM LIKELIHOOD TREE-SEARCH BASED DETECTORS and = 0.01,0.06,0.11,0.16,0.21 parameter values.

In order to find the estimated number of points in every dimension the CDFs presented in Eqs. 4.42 and 4.44 have to be evaluated for each pair of points

{(s_t,s_a)|s_t,s_a∈Ω^k,ks^k_t −s^k_ak² =l} (4.45) that is a computationally very intensive task. Instead of enumerating every pair of points, it is enough to count the number of points with the same argument of the gamma function. A modification of Euler’s generating function technique was proposed, so with the appropiate combination of well defined generating polynomials the number of point pairs belonging to the same signal set Ω and having the same Euclidean distance l can be counted easily. The generating polynomials for a four element signal set |Ω|= 4 are:

θ₀ = 1 +x+x⁴+x⁹ and θ₁ = 1 + 2x+x⁴. (4.46) The closed form expression for the expected complexity is given as follows:

C(N, σ², d²) = In Fig. 4.6 the expected number of nodes for a 4×4 MIMO system with a four element

DOI:10.15774/PPKE.ITK.2015.010

4.6. MAXIMUM LIKELIHOOD TREE-SEARCH BASED DETECTORS

signal set are shown. Note, that the solution might not be found, thus the search has to be restarted by increasing the size of the radius, however, the effects of possible further iterations are not accumulated. If= 0.01 is chosen, which means that the probability of finding a lattice point inside the sphere is 99%, and at 20 dB SNR the expected number of visited nodes is ∼ 6500 while at 30 dB SNR is less than 1000 nodes. This result is achieved with a SD using the FP enumeration. Algorithms with a lower complexity exist, i.e., algorithms based on the SE enumeration that achieve ML performance by visiting less nodes. Thus, the complexity formula shown in Eq. 4.47 can be regarded as an upper bound.

In many communication problems finding the ML solution reduces to solving an ILS problem. However, in the context of communication systems these problems are more operable because the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, the complexity of the algorithm has to be treated as a random variable as well. Based on these results it is possible to state that in case of high SNRs ML performance can be achieved for smaller sized MIMO configurations in real-time if the SD algorithm is suitable mapped on modern many-core architectures.