For this example the error signal driving the VCO is proportional to the phase error doubled.
A modified version of the Costas loop is given in Fig. 4.5, in fact some performance results for
Gaussian Filter MSK (GMSK) modulation are given in [13] and referred to as a closed loop coherent
detector.
y(t) K # A2(t) sin [2f(t) 2f ^
(t)]
x(t)
1
4
A2(t) sin [2u(t) 2f(t) 2f ^
(t)]
x(t) I(t) # Q(t)
Q(t)
1
2
A(t) sin [u(t) f(t) f ^
(t)]
I(t)
1
2
A(t) cos [u(t) f(t) f ^
(t)]
f ^
(t)
176 CHAPTER FOUR
FIGURE 4.4 Costas loop receiver block diagram.
BPF
LPF
LPF
VCO
90?°
I(t)
Q(t)
LNA
LPF
r(t)
z(t)
y??™(t)
( )2
( )2
PLL
v(t) y(t)
+
??“
Quadrature Demodulator Error Signal Generation
BPF
LPF
LPF
VCO
90?°
I(t)
Q(t)
LNA
LPF
r(t)
x(t) y(t)
Quadrature Demodulator Error Signal Generation
S
L
I
C
E
R
FIGURE 4.5 Modified Costas loop (closed loop) receiver block diagram.
The baseband signals are given as
(4.7)
(4.8)
The first mixer/multiplier output is
(4.9)
The subtractor output error signal component is
(4.10)
The second mixer/multiplier output is
(4.11)
The LPF removes the modulation and we now have the following
(4.12)
Here we have the error signal that is proportional to the phase error, but this time quadrupled.
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