Precoding-Based NA-III Zi= milviX1tMi2 v2x2+Mi3v3x3 span(M12 V2)=span(M13 V3) rank (MulVI M12 V2)=2n+1 span(M23 v3)C span(M21 V1) rank(M21 V1 M22 V2)=2m+1 span(M3 V2)C span(M31 V1) rank(M31 V1 M33 V3)=2n+1 Alignment conditions Rank conditions 16
16 Precoding-Based NA - III Alignment conditions Rank conditions
Precoding-Based NA-Advantages Code design is simple Encoding decoding are predetermined regardless of topology Achievable rate x 2 min-cutliy 7+1 2n+1’2n+1’2n+1 17
17 Precoding-Based NA - Advantages • Achievable rate ≈ ½ min-cut[1] • Code design is simple Encoding & decoding are predetermined regardless of topology
Get a Better Understanding span(M12 V2)=span(M13 V3) M21 v2=M31 v3A span(M23 V3)C span(M21 V1) M32 V3=M12V,B> V3=M12M3V,B span(M32 V2)C span(M31 V1)E) M2V2-M13ViC> V2=,aMv,C TⅤ1C=V1BA T=M13M21M32M121M23M31 Vi can not be chosen freely! 18
18 Get a Better Understanding V1 can NOT be chosen freely!
Reformulated Feasibility Cond Condensed alignment cond TVICEVIBA rank(VI P1viC)=2n+1 Reformulated rank cond rank(VI P2ViC)=2n+1 rank(VI P3VICA)=2n+1 m13(x)721 P,= M3M2IM M P1 m1(x) p2(x) 3(x)m22 P,= M3M22M M m12(x)m23(x) m21(x)m3(x P.- Mo maaM mol 3(x)= 3(x)ma31( 7(x) m13(x)m21(x)m32(x) T=M13M21M32M12M21M31 19 m12(x)m23(x)m31(x)
19 Reformulated Feasibility Cond. Condensed alignment cond. Reformulated rank cond
Algebraic Formulation -I n(x)is not constant. V1 can not be arbitrary matrix Vi (w T' Tw 2=M13M23(wTw…Tn-lw) =M2M32(IwT2w…T"w) TVIC=VIBA V2=M13M53V,C V3=M12M=VB 20
20 Algebraic Formulation - I 𝜂(𝐱) is not constant. V1 can NOT be arbitrary matrix