Digital Signal Processing Reference
In-Depth Information
y
(
t
)
r
(
n
)
s
(
n
)
s
(
n
)
y
(
t
)
2
j
ω
*
H
(
j
ω)
H
(
j
ω)
A
(
e
)
H
(
j
ω)
D/C
T
C/D
T
D/C
T
digital filter
channel
matched filter
Figure 5.3
. Reconstruction of
y
(
t
) from the sampled output
r
(
n
) of the matched
filter
H
∗
(
jω
) when the channel
H
(
jω
) is such that (
|H
(
jω
)
|
2
)
↓T
>
0 everywhere.
Defining the reconstruction filter
B
(
jω
)=
A
2
(
e
jωT
)
H
(
jω
)
,
the reconstructed
signal (Fig. 5.4(a)) is:
Y
1
(
jω
)=
R
(
e
jωT
)
A
2
(
e
jωT
)
H
(
jω
)
⎧
⎨
S
(
e
jωT
)
A
2
(
e
jωT
)
A
2
(
e
jωT
)
H
(
jω
)for
H
(
jω
)
=0
0
=
⎩
otherwise
S
(
e
jωT
)
H
(
jω
)=
Y
(
jω
)
.
=
That is,
y
1
(
t
)=
y
(
t
) indeed.
By using the appropriate noble identity from Sec. G.2 in Appendix G, we can
redraw the reconstruction system of Fig. 5.3 in various equivalent forms, as
in Fig. 5.4. Thus the periodic filter (or transversal filter)
A
2
(
e
jωT
)canbe
lumped either with the matched filter
H
∗
(
jω
) (Fig. 5.4(b)), or with the final
reconstruction filter
H
(
jω
) on the right (Fig. 5.4(a)). It can also be split equally
into two factors
A
(
e
jωT
)
≥
0 as shown in Fig. 5.4(c).
Pulse matching is not sufficient
Now consider Fig. 5.5, where we have shown a transmitting pulse
p
(
t
)and
a channel
h
(
t
). The effective channel seen by the receiver is the convolution
(
p
h
)(
t
)then
the information about the channel output
y
(
t
) is preserved in the samples
r
(
n
).
But in the system shown, the receiver filter is matched only to the transmitted
pulse
p
(
t
). Is this also su
cient to preserve information about
y
(
t
)?
The answer in general is
no
. There might be loss of information. To see this
observe that the impulse response of the cascade is the convolution of
p
(
t
)
,h
(
t
)
,
and
p
∗
(
∗
h
)(
t
). As shown in Lemma 5.1, if the receiver filter is matched to (
p
∗
t
), which is equal to the convolution of
h
(
t
)with
R
pp
(
t
) (autocorrelation
of the pulse). The impulse response
h
d
(
n
) of the discrete-time LTI system from
s
(
n
)to
r
(
n
) is the sampled version of this convolution:
h
d
(
n
)=
∞
−∞
−
h
(
t
)
R
pp
(
nT
−
t
)
dt.
(5
.
6)
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