Advances in Visual Data Compression and Communication

i

i

i

i

i

i

i

i

62 3 Communication

16-QAM. It can carry M = 16 different symbols and a symbol consists of 4 bits.

Thus the modulation rate is ρ = 4.

In digital communications, we are usually interested in the minimum probability

of error (MPE) decision rule: given a received vector

ˆ

y, choose the signal point y

∗

∈

A to minimize the probability of decision error Pr{E}. Since the probability that a

decision y

∗

is correct is simply the a posteriori probability p(y

∗

|

ˆ

y), the MPE rule

is equivalent to the maximum a posteriori (MAP) rule: choose the y

∗

∈ A such that

p(y

∗

|

ˆ

y) is maximum among all p(y

j

|

ˆ

y),y

j

∈ A . By Bayes law,

p(y

j

|

ˆ

y) =

p(

ˆ

y | y

j

)p(y

j

)

p(

ˆ

y)

. (3.41)

If