Digital Signal Processing Reference
In-Depth Information
Q
∑
1
j
ω
n
hnl
()
;=
h le
()
,=, +,,+−,
n
1
T
1
(2.15)
q
nn
n
q
kk
k
B
q
=
where one chooses (
l
= 0, 1,
…
,
L
, and
K
is an integer)
T TK
B
:= , ≥1
(2.16)
Qf T
ds
≥
2
+ ,
1
(2.17)
2
π
−
( )
, =,, ,,
ω
:=
qQ
12
q
12
Q
(2.18)
q
T
L
:=
τ /
ds
.
(2.19)
The BEM coefficients
h
q
(
l
) remain invariant during this block, but are allowed to
change at the next block, and the Fourier basis functions {
e
j
ω
q
n
} (
q
= 1, 2,
…
,
Q
) are com-
mon for each block. If the delay spread τ
d
and the Doppler spread
f
d
of the channel (or at
least their upper bounds) are known, one can infer the basis functions of the CE-BEM
[36]. Treating the basis functions as known, estimation of a time-varying process is
reduced to estimating the invariant coefficients over a block of length
T
B
symbols. Note
that the BEM period is
T
=
KT
B
, whereas the block size is
T
B
symbols. If
K
> 1 (e.g.,
K
= 2
or
K
= 3), then the Doppler spectrum is said to be oversampled [32] compared to the case
K
= 1, where the Doppler spectrum is said to be critically sampled. In [13, 36] only
K
= 1
(henceforth called CE-BEM) is considered, whereas [32] considers
K
≥ 2 (henceforth
called oversampled CE-BEM).
CE-BEM has a finite impulse response (FIR) structure in both time and frequency
domains [46]. This unique time-frequency duality makes it a widely used model depict-
ing the temporal variations of wireless channels. For
K
= 1, the rectangular window
of this truncated discrete Fourier transform (DFT)-based model introduces spectral
leakage [43]. The energy at each individual frequency leaks to the full-frequency range,
resulting in significant amplitude and phase distortion at the beginning and the end of
the observation window [61]. To mitigate this leakage, the oversampled CE-BEM with
K
= 2 or 3 has been considered in [32].
Equation (2.15) applies to single-input single-output systems—one user and one
receiver with symbol-rate sampling. It is easily modified to handle multiuser, multiple-
transmit and -receive antennas, and higher-than-symbol-rate sampling—the basic rep-
resentation remains essentially unchanged.
The representation
h
(
n
;
l
) in (2.15) is a special case of a more general representation:
Q
∑
1
hnl
()
;=
=
h l
() ()
φ ,
n
(2.20)
q
q
q
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