Digital Signal Processing Reference
In-Depth Information
y
(
t
)
c
y
(
n
)
y
(
n
)
x
(
n
)
x
(
n
)
H
(
j
ω
)
H
(
z
)
d
D/C
C/D
T
c
(a)
(b)
T
Figure 4.3
. The sampling identity. (a) A continuous-time filter
H
c
(
jω
) sandwiched
between the D/C and C/D building blocks, and (b) equivalent digital filter system.
The impulse response of
H
d
(
z
)is
h
d
(
n
)=
h
c
(
nT
).
it follows that
y
(
n
)=
y
c
(
nT
)=
k
kT
)
=
k
x
(
k
)
h
c
(
t
−
x
(
k
)
h
c
(
nT
−
kT
)
.
t
=
nT
This can be written as a discrete-time convolution
y
(
n
)=
k
x
(
k
)
h
d
(
n − k
)
,
(4
.
7)
where
h
d
(
n
) =
h
c
(
nT
)
.
(4
.
8)
Thus a continuous-time LTI system sandwiched between the C/D and D/C
operators as in Fig. 4.3(a) is equivalent to a discrete-time LTI system (Fig.
4.3(b)) with transfer function
∞
∞
h
d
(
n
)
z
−n
=
h
c
(
nT
)
z
−n
.
H
d
(
z
)=
(4
.
9)
n
=
−∞
n
=
−∞
This result will be referred to as the
sampling identity
, and is similar to the
polyphase identity of Sec. 3.2.3.
4.3 Discrete-time representations of channels
In this section we explain how digital communication systems can be represented
entirely in terms of discrete-time transfer functions. Such representations also
give rise to optimal transceiver problem formulations, with the prefilter and
equalizer defined entirely in the discrete-time domain.
4.3.1 Digital communication over a SISO channel
The digital communication system is reproduced in Fig. 4.4. Ignoring noise
for a moment, the system from
s
(
n
)to
s
(
n
) can be regarded as an LTI system
(sampling identity, Fig. 4.3). Its transfer function has the form
H
d
(
z
)=
n
h
d
(
n
)
z
−n
,
(4
.
10)
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