SD (sphere decoder)

The Sphere Decoder (SD) is an algorithm used in signal processing and communications to solve the problem of maximum likelihood (ML) detection in multi-dimensional signal constellations. It was first introduced by Alexei Viterbi and Jim K. Omura in 1989. The SD algorithm provides an efficient way to find the closest point in a high-dimensional lattice to a received signal, which is crucial in various communication systems, including wireless, satellite, and optical communications.

To understand the Sphere Decoder, let's consider a scenario where a transmitted signal is received in the presence of noise. The received signal is typically a combination of the transmitted signal and noise, and the task is to detect the transmitted signal accurately. The detection problem becomes more challenging as the dimensionality of the signal constellation increases.

A signal constellation is a set of points in a multi-dimensional space, where each point represents a possible symbol transmitted. In digital communications, these points are typically represented by complex numbers. For example, in a 16-QAM (Quadrature Amplitude Modulation) constellation, there are 16 possible points in the complex plane.

The Sphere Decoder algorithm operates on the assumption that the received signal can be expressed as the sum of the transmitted signal and Gaussian noise. The algorithm attempts to find the transmitted signal that best matches the received signal by searching within a bounded hypersphere in the signal space.

Here's a step-by-step explanation of the Sphere Decoder algorithm:

1. Initialization: Define an initial radius of the search sphere, which is typically set to a large value to encompass the entire signal space. Also, initialize a candidate symbol vector that will store the detected symbol.
2. Lattice Generation: Construct a lattice based on the signal constellation. A lattice is a discrete set of points with specific properties that facilitate efficient searching. The lattice structure is defined by the modulation scheme used, such as QAM or PSK (Phase Shift Keying).
3. Branch Enumeration: Start with the highest priority branch of the lattice, which corresponds to the signal point closest to the received signal in terms of Euclidean distance. Enumerate all possible branches at each level of the lattice, taking into account the previous candidate symbol vector.
4. Sphere Bound Calculation: For each branch, compute the Euclidean distance between the received signal and the corresponding signal point. If this distance is smaller than the current sphere radius, update the sphere radius to this value.
5. Pruning: If the current branch's Euclidean distance exceeds the sphere radius, prune the branch. Pruning eliminates branches that cannot potentially contain the ML solution, reducing the search complexity.
6. Recursive Search: Recursively perform steps 3 to 5 for the remaining branches, pruning the branches that do not satisfy the sphere bound condition.
7. Termination: Repeat the recursive search until all branches have been exhausted or the sphere radius becomes smaller than a predefined threshold. At termination, the candidate symbol vector with the smallest Euclidean distance to the received signal represents the ML estimate.
8. Output: Output the ML estimate as the detected symbol.

The Sphere Decoder algorithm exploits the structure of the lattice to efficiently search for the closest point in the signal constellation space. It reduces the complexity compared to exhaustive search methods while providing near-optimal performance. However, as the dimensionality of the signal constellation increases, the complexity of the Sphere Decoder also increases exponentially, making it computationally demanding for large constellations.

Researchers have proposed various techniques to optimize and approximate the Sphere Decoder algorithm, such as lattice reduction algorithms and suboptimal search strategies, to make it more practical for real-world applications.

In summary, the Sphere Decoder is an algorithm used to detect transmitted symbols accurately in multi-dimensional signal constellations. It operates by searching for the closest point in a bounded hypersphere within a lattice structure. By efficiently exploring the lattice branches and pruning infeasible paths, the Sphere Decoder provides an effective solution to the maximum likelihood detection problem in high-dimensional signal constellations.