the region R.Thus.the average energy per dimension of a signal constellation is P(C(A,R)》≈P(R) (c)The average number of nearest neighbors to any point in C(A,R)isK(A). Example 5.3.3:For R=-M.M],the parameters are V(R)=2M,P(R)=M/3 When n=2,and the shaping gain is an 2rx2r square,we have VR)=(2r),and V(R)=2.If X=(X1,X2)is uniformly distributed over the circle,then P(R)=E[llX]P1/2 5.3.I Shaping and Coding Gain Under the continuous approximation,the coding gain for a lattice code is separable into hen),which depends on the reaive spacing of points in. but is independent of 2)The shaping gain(R).which is determined by the choice of the signal constellation bounding regionR Consider the probability of decoding error for a lattice constellation A used over an AWGN channel.The union bound estimate(UBE)is P(e)K(A) d2() V 4 where is the noise power per dimension. Sirce)c where 4G2 4P(C(A,R)2(2”-) 4P(C(,R) is a parameter of the constellation,the UBE can be written as P(e)≈K=()(VP.SNR】 Using the continuous approximation,we have 5-145-14 the region R. Thus, the average energy per dimension of a signal constellation is PC P ( ( , )) ( ) Λ R R ≈ (c) The average number of nearest neighbors to any point in C(Λ, R) is ≈ min K ( ) Λ . Example 5.3.3: For R=[-M, M], the parameters are V(R) = 2M, P(R) = M2 /3. When n=2, and the shaping gain is an 2r×2r square, we have V(R)=(2r) 2 , and 2 2 2 2 1 2 12 1 1 ( ) 24 3 r r r r r x x dx dx r − − =× + = ∫ ∫ . When the shaping region R is a circle with radius r, V(R) =πr 2 . If X=(X1, X2) is uniformly distributed over the circle, then P(R) = E[||X||2 ]/2 = 2 1 E X[ ], where 2 21 2 21 2 2 2 1 1 21 2 1 [ ] 4 r rx r rx r E X x dx dx πr − − −− = = ∫ ∫ . 5.3.1 Shaping and Coding Gain Under the continuous approximation, the coding gain for a lattice code is separable into two parts: 1) The fundamental coding gain γc(Λ), which depends on the relative spacing of points in Λ, but is independent of R. 2) The shaping gain γs(R), which is determined by the choice of the signal constellation bounding region R. Consider the probability of decoding error for a lattice constellation Λ used over an AWGN channel. The union bound estimate (UBE) is 2 min min 2 ( ) () ( ) 4 d Pe K Q σ ⎛ ⎞ Λ ≈ Λ ⎜ ⎟ ⎝ ⎠ where σ 2 is the noise power per dimension. Since 2 2 min min 2 2 norm ( ) ( )(2 1) ( ( , )) 4 4 ( ( , )) (2 1) d d PC SNR P C ρ ρ γ σ σ Λ Λ− Λ = ⋅ =⋅ Λ − R R , where 2 min ( )(2 1) 4 ( ( , )) d P C ρ γ Λ − = Λ R is a parameter of the constellation, the UBE can be written as P e K Q SNR () ( ) ≈Λ ⋅ min norm ( γ ) Using the continuous approximation, we have