Digital Signal Processing Reference
In-Depth Information
That is, the sampled output
r
(
n
) of the receiver filter contains information
only about the
h
0
(
t
) part. The
h
1
(
t
) part, which is in the orthogonal com-
plement
V
g
,
is lost. An example is shown in Fig. 5.13. In this case we can
reconstruct from the sampled signal
r
(
n
) only the part
n
s
(
n
)
h
0
(
t
−
nT
)
,
which is the orthogonal projection of
n
s
(
n
)
h
(
t
−
nT
) onto the subspace
g
∗
(
nT
V
g
.
Summarizing, if
spans a certain space
V
g
,
then the sampled
output
r
(
n
) of the receiver filter
g
(
t
) contains all the information about
y
(
t
) (the output of
h
(
t
)) if and only if
h
(
t
)
{
−
t
)
}
∈
V
g
,
or equivalently
V
h
⊂
V
g
,
where
V
h
is the span of
{
h
(
t
−
nT
)
}
.
2.
Receiver filter space should minimize noise
. Given any
h
(
t
)
∈
V
h
it is clear
that the space spanned by
should include
V
h
, but should it
be exactly identical to
V
h
? The answer is, this is the optimal choice when
there is noise and interference. In general any interfering signal that enters
g
(
t
) has components in
V
h
and
V
h
. Choosing
g
(
t
) such that
{
g
∗
(
nT
−
t
)
}
g
∗
(
nT
}
is a basis for
V
h
implies that the portion of interference present in
V
h
is
suppressed.
{
−
t
)
s
(
n
)
y
(
t
)
g
(
t
)
r
(
n
)
h
(
t
)
D/C
C/D
T
T
1
t
t
T/
2
−
T
0.5
h
(
t
)
0
t
T
0.5
h
(
t
)
1
t
T
T/
2
Figure 5.13
. Example of a situation where the space spanned by
{g
∗
(
nT −t
)
}
includes
part of
h
(
t
) but not all of it. With
h
(
t
) written as
h
0
(
t
)+
h
1
(
t
), the part
h
1
(
t
)is
suppressed by the receiver filter and sampler.
5.5 Optimal estimates of symbols and sequences
The process of estimating a transmitted symbol stream from received noisy sam-
ples is central to the successful operation of digital communication receivers. In
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