Digital Signal Processing Reference
In-Depth Information
∫
∞
−
j
2
π
ft
Hf
(
;= ;
−∞
τ
)
ht
()
τ
exp
dt
.
(2.2)
If |
H
(
f
;τ)| ≈ 0 for |τ| > τ
d
, then τ
d
is called the (multipath) delay spread of the channel.
If |
H
(
f
;τ)| ≈ 0 for |
f
| >
f
d
, then
f
d
is called the Doppler spread of the channel. Equation
(2.1) is the most general form of a mobile channel discussed in this chapter.
In order to capture the complexity of the physical interactions characterizing the
transmission through a real channel,
h
()
;τ is typically modeled as a two-dimensional
zero-mean random process. If
h
()
;τ is wide-sense stationary in variable
t
, and
h
(
;τ
1
)
is uncorrelated with
h
(
;τ
2
f o r τ
1
≠ τ
2
and any
t
, one obtains the well-known wide-sense
stationary uncorrelated scattering (WSSUS) channel [2, 42, section 14].
In this chapter we will confine our attention to deterministic modeling of
h
()
)
;τ ,
which may be thought of as capturing realizations of the underlying random process.
2.2.1.1 Tapped Delay Line Model
We now consider a discrete-time channel model. If a linear modulation scheme is used,
the baseband transmitted signal can be represented as
∞
∑
st
()
=
skgtkT
() (
−
)
,
(2.3)
T
s
k
=−∞
where {
s
(
k
)} is the information sequence and
g
T
(
t
) is the transmit (low-pass) filter (typi-
cally a root raised cosine filter). Therefore, the baseband signal incident at the receiver
is given by
∞
∑ ∫
α
∞
xt
()
=
sk
()
ht
()(
;
gtkT
−− +
α α ).
)
dwt
(
(2.4)
T
s
−∞
k
=−∞
After filtering with a receive filter with impulse response
g
R
(
t
), the received baseband
signal is given by
∞
∑
∫∫
ββαβ
∞
∞
yt
()
=
sk
()
gt h
(
− ;
)
(
)
g
(
−
k
TTd dvt
s
−
ααβ
)
+
( ),
(2.5)
R
T
−∞
−∞
k
=−∞
where
∫
τ
vt
()
=
g
() (
wt
−
τ τ
)
d
.
R
If the continuous-time signal
y
(
t
) is sampled once every
T
s
seconds, we obtain the
discrete-time sequence
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