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This was shown in [56] where the resulting decision metric
equations are rewritten here to establish a baseline in understanding and to be used in the performance
comparison study that follows. Lastly, it forms an alternative representation of DD/decoding. Let us
assume the transmitted phases are given as.
(4.97) fn  fnl  al1
i0
fni
max
f ^
k,f ^
k1
ReErk # r*k1 # ejf
^
k  rk1 # r*k2 # ejf
^
k1  rk # r*k2 # ej[f
^
kf
^
k1]F
f ^
2
f ^
0
 
p
4
, f ^
1 
3p
4
, f ^
2 
p
4
, f ^
3  
3p
4
MODULATION DETECTION TECHNIQUES 197
The output of an l-symbol DD is (  received phase at the nth symbol time) given as the sum of
l??“1 phase differences.
(4.98)
(4.99) cn(l )  al1
i0
fni
cn(l)  cn  cnl
cn
198 CHAPTER FOUR
Ts rk
( )*
.
.
.
Choose aj
and bj
for
largest
Re{xn}
Ts
( )*
( )*
e??“j(a0+b0)
e??“jb0
e??“jb1
e??“jb2
e??“ja0
e??“ja1
e??“ja2
e??“ja3
e??“jb3
e??“j(a0+b1)
e??“j(a0+b2)
e??“j(a3+b3)
.
.
.
x0
x15
??†fk, ??†fk??“1 ?† ?†
FIGURE 4.30 Multiple symbol DD (N  3 symbols) receiver block diagram.
We wish to maximize the a posteriori probability given as (where R  noise covariance matrix and
U  noise vector).


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