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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