Digital Signal Processing Reference
In-Depth Information
QAM.
The simplest mapping strategies are one-to-one correspondences
between binary values and signal values: note that in these cases the sym-
bol sequence is uniformly distributed with
p
a
2
−M
for all
.For
example, we can assign to each group of
M
bits
(
b
0
,...,
b
M−
1
)
the signed bi-
nary number
b
0
b
1
b
2
(
α
)=
α
∈
l
g
r
,
y
i
d
.
,
©
,
L
s
2
M−
1
and 2
M−
1
(
b
0
is the sign bit). This signaling scheme is called
pulse amplitude modula-
tion
(PAM) since the amplitude of each transmitted symbol is directly deter-
mined by the binary input value. The PAM alphabet is clearly balanced and
the inherent power of the mapper's output is readily computed as
(4)
···
b
M−
1
which is a value between
−
2
M−
1
2
M
2
M
(
+
3
)+
2
2
2
−M
2
σ
α
=
α
=
24
α
=
1
Now, a pulse-amplitude modulated signal prior tomodulation is a base-
band signal with positive bandwidth of, say,
ω
0
(see Figure 12.9, middle
panel); therefore, the
total
spectral support of the baseband PAM signal is
2
ω
0
. After modulation, the total spectral support of the signal actually dou-
bles (Fig. 12.9, bottom panel); there is, therefore, some sort of redundancy
in the modulated signal which causes an underutilization of the available
bandwidth. The original spectral efficiency can be regained with a signaling
scheme called
quadrature amplitude modulation
(QAM); in QAM the sym-
bols in the alphabet are complex quantities, so that
two
real values are trans-
mitted simultaneously at each symbol interval. Consider a complex symbol
sequence
G
0
α
]
=
[
]=
[
]+
α
[
[
]+
[
]
a
n
n
j
n
a
I
n
ja
Q
n
I
Q
Since the shaper is a real-valued filter, we have that:
]=
a
I
,
KU
]
+
j
a
Q
,
KU
]
=
b
[
n
∗
g
[
n
∗
g
[
n
b
I
[
n
]+
jb
Q
[
n
]
so that, finally, (12.7) becomes:
Re
b
e
j
ω
c
n
s
[
n
]=
[
n
]
=
b
I
[
n
]
cos
(
ω
c
n
)
−
b
Q
[
n
]
sin
(
ω
c
n
)
In other words, a QAM signal is simply the linear combination of two pulse-
amplitude modulated signals: a cosine carrier modulated by the real part of
the symbol sequence and a sine carrier modulated by the imaginary part of
the symbol sequence. The sine and cosine carriers are
orthogonal
signals,
so that
b
I
can be exactly separated at the receiver via a sub-
space projection operation, as we will see in detail later. The subscripts
I
(4)
A useful formula, here and in the following, is
n
=
1
n
2
[
n
]
and
b
Q
[
n
]
=
N
(
N
+
1
)(
2
N
+
1
)
/
6.
