Coding on Flag Manifolds for Limited Feedback MIMO Systems

Renaud-Alexandre Pitaval

Research output: ThesisDoctoral ThesisCollection of Articles


The efficiency of the physical layer in modern communication systems using multi-input multi-output (MIMO) techniques is largely based on the availability of channel state information (CSI) at the transmitter. In many practical systems, CSI needs to be quantized at the receiver side before transmission through a limited rate feedback channel. This is typically done using a codebook-based precoding transmission, where the receiver transmits the index of a codeword from a pre-designed codebook shared with the transmitter. To construct such codes one has to discretize complex flag manifolds. For single-user MIMO with a maximum likelihood receiver, the spaces of interest are Grassmann manifolds. With a linear receiver and network MIMO, the codebook design is related to discretization of Stiefel manifolds and more general flag manifolds. In this thesis, coding in flag manifolds is studied. In a first part, flag manifolds are defined as metric spaces corresponding to subsurfaces of hyperspheres. The choice of distance defines the geometry of the space and impacts clustering and averaging (centroid computation) in vector quantization, as well as coding theoretical packing bounds and optimum constructions. For two transmitter antenna systems, the problem reduces to designing spherical codes. A simple isomorphism enables to analytically derive closed-form codebooks with inherent low-implementation complexity. For more antennas, the concept of orbits of symmetry groups is investigated. Optimum codebooks, having desirable implementation properties as described in industry standardization, can be obtained using orbits of specific groups. For large antenna systems and base station cooperation, a product codebook strategy is also considered. Such a design requires to jointly discretize the Grassmann and Stiefel manifolds. A vector quantization algorithm for joint Grassmann-Stiefel quantization is proposed. Finally, the pertinence of flag codebook design is illustrated for a MIMO system with linear receiver.
Translated title of the contributionCoding on Flag Manifolds for Limited Feedback MIMO Systems
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Tirkkonen, Olav, Supervising Professor
  • Tirkkonen, Olav, Thesis Advisor
Print ISBNs978-952-60-5491-9
Electronic ISBNs978-952-60-5492-6
Publication statusPublished - 2013
MoE publication typeG5 Doctoral dissertation (article)


  • MIMO systems
  • quantization
  • coding
  • flag
  • Stiefel
  • Grassmann
  • manifolds

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