Digital Signal Processing Reference
In-Depth Information
based on Monte Carlo averaging over one thousand runs with
T
B
= 400,
T
s
= 25 µs and
varying Doppler spreads. (The results were obtained following the procedure in [61].) For
a fixed value of
Q
, DPS-BEM provides the best fit, whereas CE-BEM (no oversampling)
yields minor improvements with increasing
Q
. On the other hand, the basis functions in
oversampled CE-BEM are not mutually orthogonal, leading to “analytical” difficulties.
It is interesting to note that there exists a vast literature based on CE-BEM (no oversam-
pling) where the large modeling errors are completely ignored and it is assumed that
physical channel is accurately described by CE-BEM for both analysis and simulations;
2.2.2 Time-Invariant Channels
After some processing (matched filtering, for instance), the continuous-time received
signals are sampled at the baud (symbol) or higher (fractional) rate before being pro-
cessed for channel estimation or equalization. It is therefore convenient to work with an
equivalent baseband discrete-time white noise channel model [42, section 10.1]. For a
baud-rate sampled system, the equivalent baseband channel model is given by
L
∑
0
yn
()
=
hlsn l
()(
− +
)
vn
()
,
(2 . 21)
l
=
where {
v
(
n
)} is a white Gaussian noise sequence with variance σ
2
; {
s
(
n
)} is the zero-mean,
independent and identically distributed (i.i.d.), information (symbol) sequence, possi-
bly complex, taking values from a finite set; {
h
(
l
)} is an FIR linear filter (with possibly
complex coefficients) that represents the equivalent channel, including the effects of the
noise whitening filter; and {
y
(
n
)} is the (possibly complex) equivalent baseband received
signal. A tapped delay line structure for this model is shown in Figure 2.3.
The model (2.21) results in a single-input single-output (SISO) complex discrete-time
baseband-equivalent channel model. The output sequence {
y
(
n
)} in (2.21) is discrete-
time stationary. When there is excess channel bandwidth [bandwidth >
2
× (baud rate)],
baud-rate sampling is below the Nyquist rate, leading to aliasing and, depending upon
the symbol timing phase, in certain cases, causing deep spectral notches in the sampled,
aliased channel transfer function [15]. Linear equalizers designed on the basis of the
s
(
n
)
s
(
n
−2)
s
(
n
−
L
)
z
−1
z
−1
z
−1
×
×
×
h
(0)
h
(1)
×
h
(2)
h
(
L
)
v
(
n
)
y
(
n
)
+
+
+
+
FIgure 2.3
Tapped delay line model of the frequency-selective but time-non
-
selective baud-
rate channel.
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