Recovery of a lattice generator matrix from its gram matrix for feedback and precoding in MIMO

Either in communication or in control applications, multiple-input multiple-output systems often assume the knowledge of a matrix that relates the input and output vectors. For discrete inputs, this linear transformation generates a multidimensional lattice. The same lattice may be described by an infinite number of generator matrixes, even if the rotated versions of a lattice are not considered. While obtaining the Gram matrix from a given generator matrix is a trivial operation, the converse is not obvious for non-square matrixes and is a research topic in algorithmic number theory. This paper proposes a method to execute such a conversion and applies it in a novel MIMO system implementation where most of complexity is taken from the receiver to the transmitter. Additionally, given the symmetry of the Gram matrix, the number of elements required in the feedback channel is nearly halved.