The extension to this is to use N symbols to make a decision on
N-1 symbols. Let us now formulate the problem and rederive the decision metric to be used in the
receiver (given earlier in [52]).
Let us assume the transmitted signal is given as (notice the constant envelop notation)
(4.75)
where f(k) is the modulation phase at the time instant k, for our p/4-DQPSK modulation scheme the
phases are
(4.76)
The received signal is given as follows (assuming an AWGN channel)
(4.77)
where the channel has inserted a phase offset of u(k) and we have further assumed the phase is constant
over the observation window and n(k) is the AWGN component of the received signal. This obviously
depends on the actual system design, but in general is a descent assumption.
(4.78)
We can write the a posteriori probability of r given s and as follows (assuming a sequence of length
N samples)
(4.79)
where the noise variance. Please note we have dropped the dependency on the time variable, k,
in the above equations as well as the equations that follow, for the simple reason of not cluttering the
equations.
The above expression can be expanded and rewritten to equal.
(4.80) P r s e
n
N
r s r
n
k i k i k
,u
ps
s
( ) =
( )
?‹…
??’
( ) 1
2 2
1
2
2 2
2 2
i k i
i
N
i
N
s?‹… ?‹… [ ]
??§
???
???
??©
???
??«
??¬
???
??
???
??‘ ??‘
=
??’
* cos
0
1
0
1
u a
s2n
i rs # e j u i 2
2s
2
n P(rZs, u)
1
(2ps2
n )N
# e
u
r(k) ej[f(k)u(k)] n(k)
r(k) s(k) # eju(k) n(k)
(m 0, 1, 2, c, 7)
f(k)
p
4
# m
s(k) ejf(k)
194 CHAPTER FOUR
where the following phase variable has been defined.
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