Grassmannian Constellations Toolbox

A Matlab toolbox with implementations of different Grassmannian constellation designs for noncoherent communications.

Noncoherent communication schemes alleviate the drawbacks of training-based methods in scenarios dominated by fast fading or massive MIMO systems, where getting an accurate channel estimate would require pilots to occupy a disproportionate fraction of the communication resources.

Coherent versus noncoherent transmission schemes over a block-fading channel.

In particular, Grassmannian signaling appears to be a promising approach to this problem when considering block-fading channels, which are likely to arise in many future networks. The basic idea is to encode the messages to be transmitted into different subspaces. When the coherence time of the channel, understood as the time that the channel remains approximately constant, is greater than the number of transmit antennas, the transmitted subspaces (represented by a semi-unitary matrix) are invariant to the MIMO channel, allowing the receiver to decode the received signal without the need of knowing the channel.

Existing Grassmannian constellation designs can be generically categorized into two groups: unstructured and structured. The first group uses numerical optimization tools to solve packing problems on the Grassmannian.

Numerical optimization methods on the Grassmann manifold.

On the other hand, structured Grassmannian constellations impose particular structure on the constellation to facilitate low complexity mapping and/or demapping.

Representation of structured Grass-Lattice constellation.

This toolbox includes algorithms to obtain both structured and unstructured Grassmannian constellation designs and testcripts to evaluate their performance.

Maintainer: Diego Cuevas, Universidad de Cantabria, Spain

Contributors:

• Ignacio Santamaria, Universidad de Cantabria, Spain
• Carlos Beltran, Universidad de Cantabria, Spain
• Javier Alvarez-Vizoso

Official web: https://github.com/diegocuevasfdez/grassbox

Sample run

Figures generated by StructuredConstellations/Testscript_Structured_M1.m.

SER curves of Grass-Lattice, Exp-Map and Cube-Split constellations.

BER curves of Grass-Lattice, Exp-Map and Cube-Split constellations.

Directories included in the toolbox

StructuredConstellations/ - testscripts for SER/BER evaluation of structured Grassmannian constellations

StructuredConstellations/figures - figures obtained with testscripts for structured constellations

StructuredConstellations/functions/ - structured Grassmannian constellation designs implementation

UnstructuredConstellations/ - testscripts for SER evaluation of unstructured Grassmannian constellations

UnstructuredConstellations/BestPackings - Grassmannian packings with the minimum chordal distance achieved by method in [1]

UnstructuredConstellations/figures - figures obtained with testscripts for unstructured constellations

UnstructuredConstellations/functions - algorithms implementation for designing unstructured Grassmannian constellations

Citing GRASSBOX

If you use this toolbox in your research please cite "Advanced Grassmannian Constellation Designs for Noncoherent MIMO Communications":

@phdthesis{CuevasThesis2024,
      author = {Cuevas, Diego},
      title = {Advanced {Grassmannian} Constellation Designs for Noncoherent {MIMO} Communications},
      school = {Universidad de Cantabria, Spain},
      year = {2024},
      month = nov,
      note = {Software available at \url{https://github.com/diegocuevasfdez/grassbox/}}
    }

Included algorithms and designs

• GrassmannOpt-Chordal (GMO-Chordal) algorithm, as proposed in:

  [1] D. Cuevas, C. Beltran, I. Santamaria, V. Tucek and G. Peters, "A Fast Algorithm for Designing Grassmannian Constellations," 25th International ITG Workshop on Smart Antennas (WSA), French Riviera, France, Nov. 2021.

• GrassmannOpt-Coherence (GMO-Coherence) algorithm, as proposed in:

  [2] J. Alvarez-Vizoso, D. Cuevas, C. Beltran, I. Santamaria, V. Tucek and G. Peters, "Coherence-based Subspace Packings for MIMO Noncoherent Communications," 30th European Signal Processing Conference (EUSIPCO), Belgrade, Serbia, Aug. 2022.

• Grass-Lattice constellation design, as proposed in:

  [3] D. Cuevas, J. Alvarez-Vizoso, C. Beltran, I. Santamaria, V. Tucek and G. Peters, "A Measure Preserving Mapping for Structured Grassmannian Constellations in SIMO Channels,", IEEE Global Communications Conference (GLOBECOM), Rio de Janeiro, Brazil, Dec. 2022.

  [4] D. Cuevas, J. Alvarez-Vizoso, C. Beltran, I. Santamaria, V. Tucek and G. Peters, "Constellations on the Sphere with Efficient Encoding-Decoding for Noncoherent Communications," in IEEE Transactions on Wireless Communications, vol. 23, no. 3, pp. 1886-1898, Mar. 2024.

  [5] D. Cuevas, C. Beltran, M. Gutierrez, I. Santamaria and V. Tucek, "Structured Multi-Antenna Grassmannian Constellations for Noncoherent Communications," IEEE 13rd Sensor Array and Multichannel Signal Processing Workshop (SAM), Corvallis, OR, USA, Jul. 2024.

• Exp-Map constellation design, as proposed in:

  [6] I. Kammoun, A. M. Cipriano and J. -C. Belfiore, "Non-Coherent Codes over the Grassmannian," in IEEE Transactions on Wireless Communications, vol. 6, no. 10, pp. 3657-3667, Oct. 2007.

• Cube-Split constellation design, as proposed in:

  [7] K. -H. Ngo, A. Decurninge, M. Guillaud and S. Yang, "Cube-Split: A Structured Grassmannian Constellation for Non-Coherent SIMO Communications," in IEEE Transactions on Wireless Communications, vol. 19, no. 3, pp. 1948-1964, Mar. 2020.

Funding

• Grant FPU20/03563, funded by Ministerio de Universidades (MIU), Spain
• Grant PID2022-137099NB-C43 (MADDIE), funded by MICIU/AEI /10.13039/501100011033 and FEDER, UE
• Project GRASSCOM, funded by Huawei Technologies, Sweden

License

This source code is released under the BSD 3-Clause License.