Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 188

M− 1

p ( s k r ) f ( r ) d r

=1 −

R k

k =0

p ( s k r ) f ( r ) d r ,

=1 −

R

where R is the union of the disjoint regions R k . The third equality is obtained

by using Bayes' rule, that is, f ( r s k ) p ( s k )= p ( s k r ) f ( r ). So we have proved
that the average error probability is

p ( s k r ) f ( r ) d r . P e =1 − (5 . 55)

R

It is now clear that this is minimized by maximizing p ( s k r )foreach k .Sothe
MAP estimate also minimizes average error probabilities . Since the ML estimate
agrees with the MAP estimate when all p ( s k )areequal,the ML estimate also
minimizes error probabilty when all s k have identical probabilities.

5.5.4 The ML estimate in the Gaussian case

The maximum likelihood (ML) method finds an estimate s est from the mea-
surement r such that the conditional probability f ( r s est ) is maximized. Now

assume that we have an additive white Gaussian noise or AWGN channel, that
is,

|

r ( n )= s ( n )+ q ( n ) , (5 . 56)
where q ( n ) is zero-mean white Gaussian noise with variance σ q . For a fixed
transmitted symbol s ( n )= s k , we see that r ( n ) is a Gaussian random variable
with mean s k . Its density function is therefore
f ( r s k )= exp −

,

s k ) 2
2 σ q

1
2 πσ q

( r −

(5 . 57)

where the time argument ( n ) has been omitted for simplicity. Maximizing this
quantity is therefore equivalent to minimizing

D 2 ( r, s k )=( r s k ) 2 . − (5 . 58)

That is, given the received sample r ( n )attime n , the best estimate of the symbol
s ( n )attime n would be that value s k in the constellation which minimizes the
distance D ( r, s k ) . Similarly, suppose we have received a sequence of K samples

r =[ r (0) r (1) ... r ( K − 1) ] (5 . 59)

and want to estimate the first K symbols

s =[ s (0) s (1) ... s ( K − 1) ] (5 . 60)

such that the conditional pdf

f ( r s k )

(5 . 61)