Digital Signal Processing Reference
In-Depth Information
the oscillator in the receiver has a phase offset of
θ
with respect to the trans-
mitter; when we retrieve the baseband signal
b
[
n
]
from
c
ˆ
[
n
]
we have
b
e
−j
(
ω
c
n−
θ
)
=
e
−j
(
ω
c
n−
θ
)
=
e
j
θ
[
n
]=ˆ
c
[
n
]
c
[
n
]
b
[
n
]
l
g
r
,
y
i
d
.
,
©
,
L
s
where we have neglected both distortion and noise and assumed
c
[
n
]=
c
. Such a phase offset translates to a rotation of the constellation points
in the complex plane since, after downsampling, we have
a
[
n
]=
a
[
n
]
e
j
θ
.
Visually, the received constellation looks like in Figure 12.22, where
[
n
]
θ
=
π/
9
◦
. If we look at the decision regions plotted in Figure 12.22, it is clear
that in the rotated constellation some points are shifted closer to the deci-
sion boundaries; for these, a smaller amount of noise is sufficient to cause
slicing errors. An even worse situation happens when the receiver's carrier
frequency is slightly different than the transmitter's carrier frequency; in this
case the phase offset changes over time and the points in the constellation
start to rotate with an angular speed equal to the difference between fre-
quencies. In both cases, data transmission becomes highly unreliable: car-
rier recovery is then a fundamental part of modem design.
20
=
Im
·
·
·
·
·
·
·
·
Re
·
·
·
·
·
·
·
·
Figure 12.22
Rational effect of a phase offset on the received symbols.
The most common technique for QAM carrier recovery over well-
behaved channels is a
decision directed loop
;justasinthecaseoftheadap-
tive equalizer, this works when the overall SNR is sufficiently high and the
distortion is mild so that the slicer's output is an almost error-free sequence
of symbols. Consider a system with a phase offset of
θ
; in Figure 12.23 the
