number of points in the constellation.) Example 5.3.2:An M-PAM constellation 3.(M-1))is a one-dimensional lattice constellation C(2Z,R)with A+t=2Z+1 and R=[-M.Ml. Some 2D lattice constellations are shown in Fig.12. Fig.12 Two-dimensional constellations based on the integer lattice The key geometric properties of the region R are (a)its volume V(R)=[dx; (b)the average energy P(R)per dimension of a uniform probability density function overR A-1念 (5.3) For performance analysis of large lattice constellations,we use the following approximations Fomey call this the contimous approximation). The continuous approximation: (a)The size of the constellation C(A,R)(i.e.,the number of signal points in C(A,R))is well approximated by V(R)/V(A). (b)When the number of points in C(A,R)is large,a uniform discrete distribution of the points over C(A,R)is well approximated by a uniform continuous distribution over 5-13 5-13 number of points in the constellation.) Example 5.3.2: An M-PAM constellation {±1, ±3,., ±(M-1)} is a one-dimensional lattice constellation C(2Z, R) with Λ+t =2Z+1 and R=[-M, M]. Some 2D lattice constellations are shown in Fig. 12. 16-QAM 64-QAM 32-QAM 256-QAM 128-QAM Fig.12 Two-dimensional constellations based on the integer lattice Z2 . The key geometric properties of the region R are (a) its volume ( ) V d = ∫ x R R ; (b) the average energy P(R) per dimension of a uniform probability density function over R: 2 || || ( ) ( ) d P n V = ∫ x x R R R (5.3) For performance analysis of large lattice constellations, we use the following approximations (Forney call this the continuous approximation). „ The continuous approximation: (a) The size of the constellation C(Λ, R) (i.e., the number of signal points in C(Λ, R)) is well approximated by ( ) / ( ) V V R Λ . (b) When the number of points in C(Λ, R) is large, a uniform discrete distribution of the points over C(Λ, R) is well approximated by a uniform continuous distribution over