5.3 Lattice Constellations It is from Shannon's capacity theorem that an optimal block code for a bandwidth-limited AWGN channel consists of a dense packing of code points within a sphere in a high-dimensional Euclidean space.Most of the densest known packings are lattices. An n-dimensional (n-D)lattice A is an infinite discrete set of points(vectors,n-tuples)in the real Euclidean n-space R"that has the group property. Example 5.3.1:The set of all integers,=,is a one-dimensional lattice,since is a discrete subgroup of R.The set 22 of all integer-valued two-tuples(m,n)with n Z is a 2-dimensional lattice.More generally,the set Z"of all integer-valued n-tuples is an n-D lattice. The lattice RZ.whereR1 1-1A is obtained by rotating Z2 by /4 and scaling by √5.Clearly,R2z2=2z Definition I:Let g1sjsm be a set of linearly independent vectors in R"(so that msn).The set of points ==2,e2 (5.1) is called an m-dimensional lattice,and gsmis called a basis of the lattice. That is,A是基向量的整数线性组合。The matrix withg,as rows g G Lg. is called a generator matrix for the lattice.在后续讨论中，we will deal with full-rank lattices. i.e.,m=n.So a general n-D lattice that spans R"may be expressed as A=x=aGaeZ" (5.1) 例如，the lattice2 has the generator-0] 10 A coset of a lattice A,denoted by A+x,is a set of all points obtained by adding a fixed point x to all lattice points aA.Geometrically,the coset A+x is a translate of A by x.If 文105-10 5.3 Lattice Constellations It is known from Shannon’s capacity theorem that an optimal block code for a bandwidth-limited AWGN channel consists of a dense packing of code points within a sphere in a high-dimensional Euclidean space. Most of the densest known packings are lattices. An n-dimensional (n-D) lattice Λ is an infinite discrete set of points (vectors, n-tuples) in the real Euclidean n-space Rn that has the group property. Example 5.3.1: The set of all integers, {0, 1, 2,.} Z = ± ± , is a one-dimensional lattice, since Z is a discrete subgroup of R. The set Z2 of all integer-valued two-tuples (n1, n2) with i n ∈Z is a 2-dimensional lattice. More generally, the set Zn of all integer-valued n-tuples is an n-D lattice. The lattice RZ2 , where 1 1 1 1 R ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ − , is obtained by rotating Z2 by π/4 and scaling by 2 . Clearly, R2 Z2 = 2Z2 . Definition 1: Let { ,1 } j g ≤ ≤j m be a set of linearly independent vectors in Rn (so that m n ≤ ). The set of points 1 m jj j j a a = ⎧ ⎫ Λ= = ∈ ⎨ ⎬ ⎩ ⎭ x g ∑ Z (5.1) is called an m-dimensional lattice, and { ,1 } j g ≤ j m≤ is called a basis of the lattice. That is, Λ是基向量的整数线性组合。The matrix with gj as rows 1 2 m G ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ g g g # is called a generator matrix for the lattice. 在后续讨论中，we will deal with full-rank lattices, i.e., m=n. So a general n-D lattice that spans Rn may be expressed as { }n Λ= = ∈ xaa G Z (5.1) 例如，the lattice Z2 has the generator 1 0 0 1 G ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . „ A coset of a lattice Λ, denoted by Λ+x, is a set of all points obtained by adding a fixed point x to all lattice points a∈Λ. Geometrically, the coset Λ+x is a translate of Λ by x. If