p-2a网28 SWR=PCAR》P.PE σ2(2P-1)V(R)2Nσ (R) (A VA y 4P(R) =3y.(A)Y(R) where (5.4) is defined as the nominal (or asymptotic)coding gain of A,and (R)=R 12P(R) (5.5) is defined as the shaping gain ofR The nominal coding gain (A)measures the increase in density of A over the baseline integer lattice Z".The shaping gain(R)measures the decrease in average energy ofR relative to an n-cube [-b.b]".It can be shown that, given any lattice constellation C.the nominal coding and shaping gains of any K-fold Cartesian product constellation C is the same as those of C Example 5.3.4:For the square constellation,n=2,shaping regionR is a 2rx2r square.Then (R)=,(R) 1.Since V(A)(A),we have A)=1.and so3 From the discussion above,the probability of block decoding error per two dimensions is (e)=K,(A)((A),(R)-3SNR.) (5.6 where K(A)=2K(A)/n is the normalized error coefficient per two dimensions. Note that the nominal coding gain is based solely on the argument of the (.)function in the UBE,which bec omes inf r dens onal latti ces s.On the other hand, as before,the effective coding gain is limited by the number of nearest neighbors. error coefficient Kmin(A).which becomes very large for high-dimensional dense lattices.In fact,the Shannon limit shows that no lattice can have a combined effective coding gain and shaping gain greater than 9dB at P(e)=10e-6.This limits the maximum possible effective 15 5-15 2 2 2 2 () log | ( , ) | log ( ) V C n nV ρ = Λ≈ Λ R R 2 / norm 2 2/ 2 ( ( , )) ( ) ( ) (2 1) ( ) n n PC V P SNR V ρ σ σ Λ Λ = ≈⋅ − R R R 2 / 2 min ( ) ( ) ( ) 3()( ) 4( ) n c s V d V P γ γγ ⎛ ⎞ Λ ⎜ ⎟ ⎝ ⎠ Λ ≈ =Λ R R R where 2 min 2 / ( ) ( ) ( ) c n d V γ Λ Λ = Λ (5.4) is defined as the nominal (or asymptotic) coding gain of Λ, and 2 / ( ) ( ) 12 ( ) n s V P γ = R R R (5.5) is defined as the shaping gain of R. The nominal coding gain γc(Λ) measures the increase in density of Λ over the baseline integer lattice Zn . The shaping gain γs(R) measures the decrease in average energy of R relative to an n-cube [-b, b] n . It can be shown that, given any lattice constellation C, the nominal coding and shaping gains of any K-fold Cartesian product constellation CK is the same as those of C. Example 5.3.4: For the square constellation, n=2, shaping region R is a 2r×2r square. Then 2 2 () 4 () 1 12 ( ) 12 / 3 s V r P r γ == = × R R R . Since 2 min V d () () Λ = Λ , we have γc(Λ)=1, and so γ=3. From the discussion above, the probability of block decoding error per two dimensions is P e K Q SNR s s cs () ( ) ( ) ( )3 ≈Λ Λ ⋅ ( γ γ R norm ) (5.6) where min ( ) 2 ( )/ K Kn s Λ= Λ is the normalized error coefficient per two dimensions. Note that the nominal coding gain is based solely on the argument of the Q(.) function in the UBE, which becomes infinite for dense n-dimensional lattices as n→∞. On the other hand, as before, the effective coding gain is limited by the number of nearest neighbors; i.e., the error coefficient Kmin(Λ), which becomes very large for high-dimensional dense lattices. In fact, the Shannon limit shows that no lattice can have a combined effective coding gain and shaping gain greater than 9dB at P(e)=10e-6. This limits the maximum possible effective