118)
An alternative way to view this is through the Cartesian coordinate system, where we have plotted the
received symbol (shown by a box) along with the quantized version (shown by a circle). Assume the
data symbol, 1 j 1, was transmitted and thus we have the following event described in Fig. 6.44.
Here a positive frequency offset is considered producing the error.
We notice in the absence of a frequency offset; the error signal is completely projected onto the
real axis. In the presence of a frequency offset the error signal has a projection onto the imaginary
axis, which can be used to indicate a frequency offset. This leads us to present the AFC-Two algorithm
shown in Fig. 6.45.
The corresponding error signal is written as follows which is accurate for small frequency offsets.
(6.119)
Note this error signal is basically the small angle approximation of the error signal used for the
AFC-One algorithm presented earlier.
We will also present two alternative forms of representing the phase discriminator for the AFCTwo
algorithm (see Fig. 6.46). These forms are intended to provide insight into the mechanisms used
to create the error signal.
e I1 # Q2 I2 # Q1
e ATANcI1 # Q2 I2 # Q1
I1 # I2 Q1 # Q2 d
RECEIVER DIGITAL SIGNAL PROCESSING 333
Differential
Detector
X
cos(x)
Ts
+
ATAN(Q2/I2) ATAN(Q1/I1)
RI(t)
??“
Quantizer
Loop Filter
VCO
RQ(t)
P/S
sin(??“x)
+
T
X
I2 + j Q2 I1 + j Q1
a
q2
q1
Phase
Discriminator
Output
Bits
Error Signal
FIGURE 6.
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