X. Giraud, ``Constellations designed for fading channels,'' PhD., ENST (France), June 1994
Advisor : J.C. Belfiore
This dissertation is devoted to the design of constellations matched
to
fading channels. In the first part of this work we define a general
theoretical
frame for the lattice design problem applicable to fading channels.
For the
Rician channel, the lattice design problem reduces to the packing of
cross-polytopes whereas for the Rayleigh fading channel, we are led
to
determine what number geometry recognizes as dense admissible
lattices of
This approach quantitatively establishes the properties that a
lattice should have on the fading channels and why the packing
formulation
does not apply to the Rayleigh channel. We provide a list of the
totally real number fields, which produce the
densest possible lattices that this method can give up to dimension
8, together with a generating matrix for each of these lattices.
Computer simulation confirmed that the n-dimensional scheme offers
an
nth order diversity. Finally, the optimality of such lattices is
evaluated
indicating that further gain can be obtained when these
constellations are
combined with appropriate coding.
In the second part of the work, we attempt to improve the performance of uncoded modulation schemes by combining them with coset codes in order to increase the gain. On the Rician channel, two families of lattices are proposed. The first one results from the use of negacyclic codes since we prove that good codes with respect to the Lee metric are desirable; the second family is obtained in a similar way as the Barnes-Wall lattices by using a multi-level construction with Reed-Muller codes. On the Rayleigh fading channel we can not use any norm to evaluate symbol isolation. Algebraic construction of the constellation matched to the Rayleigh fading channel simplifies the design of coset codes to be combined with them. Using tools from number field theory we increase the minimal coordinate product while keeping the constellation expansion under control. The overall coding gain of TCM is evaluated. Simple convolutional codes are considered to illustrate the method. Theoretical results, confirmed by computer simulations, show that substantial gains are achievable.
In the last part, two detection algorithms are described. Both are available for any constellation; the second one though is most suited to circulant lattices as defined in the first part, because it takes advantage of the possible symmetries of the constellations.
Author's adress : Department of Communications. ENST. 46, rue Barrault 75013 PARIS - France. e-mail : giraud@com.enst.fr