Digital Signal Processing Reference
In-Depth Information
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Figure 12.13
More exotic constellations: irregular low-power 8-point constellation
(left panel) in which the outer point are at a distance of 1
+
3 from the origin ;
regular 19-point hexagonal-lattice constellation (right panel).
Transmission Reliability.
Let us assume that the receiver has eliminated
all the “fixable” distortions introduced by the channel so that an “almost
exact” copy of the symbol sequence is available for decoding; call this se-
quence
ˆ
[
]
. What no receiver can do, however, is eliminate all the additive
noise introduced by the channel so that:
ˆ
a
n
a
[
n
]=
a
[
n
]+
η
[
n
]
(12.16)
where
is a
complex
white Gaussian noise term. It will be clear later why
the internal mechanics of the receiver make it easier to consider a complex
representation for the noise; again, such complex representation is a conve-
nient abstraction which greatly simplifies the mathematical analysis of the
decoding process. The real-valued zero-mean Gaussian noise introduced by
the channel, whose variance is
η
[
n
]
2
0
, is transformed by the receiver into com-
plex Gaussian noise whose real and imaginary parts are independent zero-
mean Gaussian variables with variance
σ
2
0
σ
/
2. Each complex noise sample
η
[
]
n
is distributed according to
e
−
|z|
2
1
πσ
2
0
f
η
(
z
)=
σ
(12.17)
2
0
The magnitude of the noise samples introduces a shift in the complex
plane for the demodulated symbols
ˆ
with respect to the originally trans-
mitted symbols; if this displacement is too big, a decoding error takes place.
In order to quantify the effects of the noise we have to look more in detail
at the way the transmitted sequence is retrieved at the receiver. A bound
on the probability of error can be obtained analytically if we consider a sim-
ple QAM decoding technique called
hard slicing
. In hard slicing, a value
ˆ
a
[
n
]
a
by choosing the al-
phabet symbol at the minimum Euclidean distance (taking the gain
G
0
into
account):
[
n
]
is associated to the most probable symbol
α
∈
