Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 186

Since this is a sum over all k, it does not depend on the transmitted symbol s k . If
we make the assumption that all transmitted symbols have identical probabilities ,
that is, p ( s k )=1 /M for all k ,then
p ( s k r )= cf ( r s k )

(5 . 46)

for constant c . This implies that symbol estimates based on the MAP and ML
criteria are identical, whenever p ( s k ) is the same for all symbols.

5.5.2 Symbol estimate based on a sequence of received samples

In general there is intersymbol interference, which means that the received signal
r ( n ) depends not only on the present value of the transmitted symbol s ( n ) , but
also on some of the past values s ( n

m ) . Assuming that the discrete-time channel
H d ( z ) (i.e., the sampled version of the cascade F ( jω ) H ( jω ) G ( jω )) is FIR with
order L , the received signal r ( n ) depends on s ( n ) and the past values

−

s ( n − 1) ,s ( n − 2) ,...,s ( n − L ) . (5 . 47)

Thus a received sample r ( n ) has information about the present and L past
symbols. Putting it another way, the estimate of the present symbol s ( n ) will
benefit from the present and future observations

[ r ( n ) r ( n +1) ... r ( n + L )] .

Note also that the past received samples r ( n k ) ,k > 0 , can be correlated to

the current symbol s ( n ) because of memory in the sequence s ( . ) itself. Thus,
assuming an arbitrary starting time n =0 , an estimate of s ( n ) based on the
vector of present, past, and L future observations

−

r =[ r (0) r (1) ... r ( n + L )] (5 . 48)

will be more accurate than estimating it from the sample r ( n ) alone. Similar to
what we did in Sec. 5.5.1 we can now define the quantity
p ( s k r ) . (5 . 49)

This is the a posterior probability that s k wastransmittedattime n ,giventhe
received samples r in Eq. (5.48). Given the received signal samples r ,ifwe
choose the estimate s est of the transmitted symbol such that
p ( s est r ) p ( s k r ) ≥ (5 . 50)

for all possible choices of s k , then the symbol estimate s est is the MAP estimate
based on the sequence of observations (5.48). Next, the conditional pdf that
the received signal samples are r , given that s k was transmitted at time n ,is
denoted by

f ( r s k ) .

(5 . 51)