From Chapter 2,the probability that an ML detector selects s;instead of s given that s;is the transmitted vector is equal to d ps→s,)=92N。 (4.42) where d=s,-s,P Ifs,and s differ in d coordinates,i.e.d()=d,then d=2(sa-s尸=dlE=4dE, (4.43) Substituting (4.43)into (4.42)yields s→-R-o (4.44) This is the so-called pairwise error probabiliry (PEP)between the two codewords with Hamming distance d.A useful property is that Applying the union bound,the probability of decoding error given that e is the transmitted codeword is given by Pele)sB(c-→c) where An is the number of codewords in C with Hamming distance d from c.din is the minimum Hamming distance of code C.For a convolutional code,d=d.Hence,the average probability of error over all codewords is upper-bounded by P(e)=∑Pc)P(elc) s(d) (4.45) where=P(e)A is the average number of codewords c'e at distance d from e.Due to the linearity,A is also the number of codewords with Hamming weightd. 4384-38 From Chapter 2, the probability that an ML detector selects sj instead of si given that si is the transmitted vector is equal to           0 2 2 2 ( ) N d P Q ij i j s s (4.42) where 2 2 | | ij i j d  s  s If si and sj differ in d coordinates, i.e., d H (ci , c j)  d ，then   s s n k dij (sik sjk ) d 2 E 4dE 2 1 2 2       (4.43) Substituting (4.43) into (4.42) yields 2 2 0 2 ( ) () s i j dE P Pd Q N         s s (4.44) This is the so-called pairwise error probability (PEP) between the two codewords with Hamming distance d. A useful property is that 0 / 2 2 1 ( ) dEs N P d e  Applying the union bound, the probability of decoding error given that c is the transmitted codeword is given by    c c P e P ' 2 ( | c) (c c')   N d d Ad cP d min ( ) | 2 where Ad|c is the number of codewords in  with Hamming distance d from c, dmin is the minimum Hamming distance of code . For a convolutional code, dmin  d f . Hence, the average probability of error over all codewords is upper-bounded by ( ) ( ) ( | ) c Pe P Pe   c c   N d d AdP d min ( ) 2 (4.45) where | ( ) A PA d d   c c c is the average number of codewords c’c at distance d from c. Due to the linearity, Ad is also the number of codewords with Hamming weight d