Carrying out the above
mathematical operations leads us to the following relationship:
(2.23)
(2.24) Qk ak # Qk1 bk # Ik1
Ik ak # Ik1 bk # Qk1
which can be represented in matrix notation as
(2.25)
Stated another way, the symbols to be presently transmitted are represented in vector notation as
and are generated by projecting the present information signal vector onto the phase matrix.
(2.26)
Observe that the phase matrix determinant is expressed as .
Similarly, we can show the demodulator as in Fig. 2.42, using a symbol time (Ts) delay block and a
conjugate operation.
det (I ) I 2
k1 Q2k1 0
i I # m
m(k)
i(k)
c Ik
Qk d c Ik1 Qk1
Qk1 Ik1 d # cak
bk d
76 CHAPTER TWO
X
Ts
ak + j bk Ik + j Qk
Ik??“1 + j Qk??“1
FIGURE 2.41 Conventional baseband DQPSK modulator.
X
Ts
ak + j bk Ik + j Qk
( )*
Ik??“1 ??“ j Qk??“1
FIGURE 2.42 Conventional DQPSK demodulator.
After carrying out the above mathematical operations, we arrive at the following relationship:
(2.27)
(2.28)
which can also be represented in matrix notation as
(2.29)
which can also be written using the previously defined phase matrix:
(2.30)
as
(2.31)
Note that we have assumed, without loss of generality, that the transmitted signal constellation has a
constant envelope, when sampled at the symbol rate.
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