(4.81)
Since this is a noncoherent technique, the channel phase offset is unknown to us, so we wish to
remove it from the above equation in the following manner:
(4.82)
(4.83)
Here we have assumed the channel phase offset is a uniformly distributed random variable with a
probability density function (PDF) equal to p(u). The details of the integration are left to the reader
and so we simply state the result. (Io(x) modified, zeroth order Bessel function)
(4.84)
In order to obtain the maximum likelihood estimate (MLE) we make use of the following
approximation:
(4.85)
Hence maximizing the a posteriori probability of r given s is equivalent to maximizing the following
expression:
(4.86)
(4.87)
We now have an MLE metric that is dependent on the possibilities of the transmitted phases. We wish
to have a term resembling the differentially encoded data. This is accomplished as follows. First we
notice that adding a phase ambiguity, say , to all the estimated phases has an identical decision rule.
(4.88)
where is assumed to be uniform. If we let then the metric becomes
(4.89)
And assuming differential encoding of phases, , the decision rule becomes
(4.90)
max .
??†
??†
f
f
r e k i
j
i
N k i m
m
N i
??’
??’
=
??’
??’ ??’
=
??’ ??’
??‘
??‘
0
2
0
1
2
fki fki fki1
max
f PaN1
i0
rki # ej[fkifkN1] P2
ua fkN1 ua
max
f PaN1
i0
rki # ej[fkiua] P2
ua
max
f PaN1
i0
rki # ejfki P2
max
f PaN1
i0
rki # s*ki P2
ln [Io(x)] > x2
4
P r s e
n
N
r s
n
k i k i
( )
( )
??’
??›
???
???
???
??
???
??’ ??’
1
2 2
1
2 2
2 2
ps
s .
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