3.4 Codes with Many Codewords-the Union Bhattacharyya Bound and Gallager 3.4.1 Union Bhattacharyya Bound We now consider generalizing the bound in (3.12)to a code C=with many codewords.Since the complement ofD.is given byand noting that D,i=l2,Mare disjoint decision regions,we have(3.5)也可直接写为(3.l5) Pa(elm)=PryeUD sent =2eDkm叫 -∑RwIx) (3.15) 点 Note that,for ML decoding. y∈D.→Pw(ylxm)2Px(ylxm) Since every y Dr can also be put into the decoding region D:of an ML decoder for the code (x2)=(x)with only two codewords,we see that 是R,AR)-盆P.=BD (3.16 veD Say x Assuming m'=1 Say x2 D PutyeD into D D D D Say x Say x Decision regions for Decision regions for C=X C'={,x}={xm,xm} Figure 3.2 Illustration of observation space and decision regions Invoking the Bhattacharyya bound(cf.(3.10))in (3.16),we have 3-63-6 3.4 Codes with Many Codewords – the Union Bhattacharyya Bound and Gallager Bound 3.4.1 Union Bhattacharyya Bound We now consider generalizing the bound in (3.12) to a code 1 2 { , ,., } = M C xx x with many codewords. Since the complement of Dm is given by m m m ′ ′≠ ∪ D , and noting that , 1,2,., i D i M = are disjoint decision regions, we have ((3.5)也可直接写为(3.15)) ' ' ( | ) Pr B mm m m P e m sent ≠ ⎛ ⎞ = ∈ ⎜ ⎟ ⎝ ⎠ y x ∪ D ( ' ) ' Pr | m m m m sent ≠ = ∈ ∑ y x D ' ' 1 ' (| ) m M N m m m m P = ∈ ≠ = ∑ ∑ y y x D (3.15) Note that, for ML decoding, ' ' (| ) (| ) y y ∈⇒ ≥ Dm NmN m P P x y x Since every y ∈ Dm’ can also be put into the decoding region D2 ′ of an ML decoder for the code 12 ' (, ) ( , ) m m xx x x ′ ′ = with only two codewords, we see that '22 1 ( | ) ( | ) ( | ) ( |1) m Nm Nm N B D D P P P Pe ∈ ∈∈ ′ ′ ∑∑∑ ≤ =≡ ′ ′ y yy yx yx yx D (3.16) D1 D2 Dm 1 D′ 2 D′ 1 Assuming ' 1 m = Say x Say m x Say 1 x′ Say 2 x′ 1 2 { , ,., } = M C xx x 12 ' '{, }{ , } m m C = xx x x ′ ′ = Put into m' 2 y ∈D D′ Figure 3.2 Illustration of observation space and decision regions Invoking the Bhattacharyya bound (cf. (3.10)) in (3.16), we have