Fixed-Complexity Sphere Decoder (FSD)

In order to solve the random complexity and sequential nature of SD, a new complexity

fixed detection algorithm termed FSD was proposed for SEFDM in [39]. Unlike the

depth-first search of SD, FSD is a breadth-first detection algorithm. The FSD fixes the

complexity of SD by restricting the search space to a fixed size subspace, which means,

at each level, a fixed number of nodes, termed tree-width, are examined. Therefore, it

is more practical in terms of hardware implementation. The FSD has been studied in

detail in [116] and its rapid prototyping was reported in [117]. Later, this technique was

extended to a Turbo-MIMO system [118] to obtain likelihood information. However,

FSD doesn’t guarantee an optimal solution like SD since it enumerates a fraction of

points within the sphere search space. The FSD estimate is expressed as

SF SD= arg min

S∈ON_{, S∈M}kR − CSk

2_{≤ g}

F SD (2.46)

where M is the search subspace which is determined by the tree-width, gF SD = kR − CSZFk 2

determines the constrained space and if finally no node is found within the sphere, the

initial constrained estimate SZF is taken as the solution as expressed in the equality

below:

SF SD= SZF (2.47)

Fig. 2.12 illustrates the FSD tree search for a 4 sub-carrier system with BPSK

symbols. In this diagram, the tree-width TW is set to 2. The figure illustrates that

at each level, only two nodes are reserved, while the rest of the nodes are discarded

with their children nodes. Based on the aforementioned analysis, FSD has a fixed

complexity (i.e. not dependent on noise) and is practical for hardware implementation.

architecture. This means resource utilization of FSD is relatively higher than that of

SD. However, the throughput of FSD is fixed and relatively higher than that of SD. In

terms of BER performance, since FSD search is restricted in a fixed search space, only

a fraction of SD search space is examined, therefore, the performance is sub-optimal

and is worse than SD. It should be noted that the choice of the subset M is crucial

to the performance of FSD. This subset can be configured based on targeted BER

performance or design complexity.

Figure 2.12: FSD tree search diagram for a 4 sub-carrier SEFDM system with BPSK symbols.

In order to cope with challenges arising from the use of FSD, several optimization

methods were proposed. Work in [119] introduced an early termination FSD which

makes full use of the features in FSD while preserves advantages of SD such as branch

termination and radius update. Therefore, unnecessary branches are pruned leading

to reduced complexity while the throughput is still fixed. The other technique named

staggered SD was reported in [120] which can simultaneously search along the depth

and breadth of the tree. In other words, this technique combines advantages of both

SD and FSD. Results present that the throughput per unit power and per unit area is

greatly improved over the parallel and sequential methods.

to FSD. Its main idea is to keep K best nodes at each level. The K-best decoding

algorithm was described in detail in [121][122]. Based on the theoretical work, a radius

adaptive K-best SD with early termination was proposed in [123]. The main idea of

this algorithm is to decompose the whole tree into several sub-trees and define a new

tree-width to each sub-tree. The search sphere in one sub-tree is updated and used

for other sub-trees. Therefore, redundant search can be avoided and the throughput is

improved. Moreover, a complexity reduced technique termed sort-free K-best SD was

reported in [124]. Instead of extending all the children nodes in parallel, only the child

node with the minimum metric inherited from its parent node is extended. The process

continues following the same criterion until all K paths have been found.

A comparison between FSD and SD are listed below:

• BER Performance. SD outperforms FSD substantially. FSD restricts a fixed

number of searching branches in each detection and there is no update for the

search space. Such detection simplification results in performance degradation.

For SD, the radius is shrinking gradually in order to speed up the detection.

• Throughput. FSD has a fixed throughput while SD has a variable throughput.

It should be noted that FSD has a higher throughput than SD as well.

• Complexity. The main advantage of FSD is its parallel processing structure.

However, a sort unit has to be used at each level in order to find the best TW

nodes. This unit consumes a large amount of resources as mentioned in our

previous work [49]. Furthermore, with the increase of the number of sub-carriers

and the order of modulation scheme, the resource consumption of FSD is increased

exponentially. However, SD has a lower complexity level since no sort unit is used

at each level. On the other hand, it is noted that FSD has fixed complexity while

SD has random complexity.

• Power Efficiency. SD is more power efficient than FSD. FSD searches redundant