Digital Signal Processing Reference
In-Depth Information
spect to the symbol times, then this signal could be used for synchronism;
indeed, from
sin
2
n
sin
2
K
n
π
f
b
F
s
l
g
r
,
y
i
d
.
,
©
,
L
s
p
[
n
]=
=
we would have
p
KD
0. If this component was present in the signal,
then the block
{·}
would be a simple resonator
with peak frequencies
at
[
n
]=
ω
=
±
K
, as described in Section 7.3.1.
Now, consider more in detail the baseband signal
b
2
π/
[
n
]
in (12.4); if we
always transmitted the same symbol
a
,then
b
[
n
]=
a
i
g
[
n−iK
]
would
be a periodic signal with period
K
and, therefore, it would contain a strong
spectral line at 2
K
which we could use for synchronism. Unfortunately,
since the symbol sequence
a
π/
[
n
]
is a balanced stochastic sequence we have
that:
E
b
]
=
[
[
[
]]
[
]=
n
E
a
n
g
n
−
iK
0
(12.46)
i
and so, even on average, no periodic pattern emerges.
(8)
The way around
this impasse is to use a fantastic “trick” which dates back to the old days
of analog radio receivers, i.e. we process the signal through a
nonlinearity
which acts like a diode. We can use, for instance, the square magnitude
operator; if we process
b
[
n
]
with this nonlinearity, it will be
E
b
2
]
E
a
]
g
a
∗
[
[
n
=
[
h
]
i
[
n
−
hK
]
g
[
n
−
iK
]
(12.47)
h
i
Sincewehaveassumedthat
a
[
n
]
is an uncorrelated i.i.d. sequence,
E
a
]
=
σ
2
a
[
h
]
a
∗
[
i
δ
[
h
−
i
]
and, therefore,
E
b
[
n
]
=
σ
a
i
g
[
n−iK
]
2
2
(12.48)
Thelasttermintheaboveequationisperiodicwithperiod
K
and thismeans
that, on average, the squared signal contains a periodic component at the
frequency we need. By filtering the squared signal through the resonator
above (i.e. by setting
]
=
x
[
n
]
x
), we obtain a sinusoidal com-
2
[
n
ponent suitable for use by the PLL.
(8)
Again, a rigorous treatment of the topic would require the introduction of cyclostation-
ary analysis; here we simply point to the intuition and refer to the bibliography for a
more thorough derivation.
