Digital Signal Processing Reference
In-Depth Information
complex low-pass equivalent vector representation of the impulse response experienced
by subcarrier
k
of the
u
th
user, for a receiver equipped with
M
antennas, is
P
k
u
()
=
()
u
ρ
t
∑
1
()
−
(
( )
k
u
u
u
Ht
G
,
t
δτ
t
t
.
(10. 5)
k
kp
kp
,
()
()
e
u
r
t
p
=
where ρ
u
(
t
) and (
r
u
)
e
(
t
) model the effects of shadowing and path loss, respectively,
r
u
(
t
)
is the distance from the
u
th
user to the base station, and
e
is the path loss exponent. We
assume their variations in time to be very slow, and hence nearly constant over several
symbol durations. The
M
-dimensional complex vector G
k,p
(
t
)* denotes the fading and
the array response from the user to the antenna elements of the receiver, and τ
k,p
(
t
) rep-
resents the propagation time delay along the
p
th
path. We note here that the large-scale
path loss that includes free-space path loss and shadowing is the same for all subcarriers
of the same user. Moreover, the number of resolvable paths and their propagation time
delays depend on the reflecting objects and scatterers and can be assumed equal for all
subcarriers [28]. Therefore, we omit the index
k
from ρ,
P
, and τ (ρ
k
= ρ
u
,
P
k
=
P
u
, and
τ
k,p
= τ
u
p
) and reformulate equation (10.5) as
()
=
()
P
u
∑
u
ρ
t
()
−
(
( )
u
u
u
Ht
G
,
t
δτ
t
t
.
(10.6)
k
kp
p
()
()
e
u
r
t
p
=
1
A frequency-domain channel model for a multicarrier system can be characterized by
the coherence bandwidth† [28]:
B
c
1
2π∆
.
(10.7)
When the frequency separation λ/
T
MC
is less than
B
C
, the MC-CDMA system is sub-
ject to correlated fading over different subcarriers. Fades across taps (multipaths) are
mutually independent for the same carrier. However, fading for the same tap across
different carriers is correlated. The envelope correlation coefficient between subcarrier
k
and subcarrier
k
′ for user
u
is [28]
E
2
1
λ
λ
π
( )
+
−
kk
,
′
1
+
λ
k
,
k
2
kk
,
′
=
() ( )
u
u
u
ρ
E
G
t
G
t
τ
=
,
(10.8)
kk
,
′
kpm
,
,
kpm
′
,
,
π
2
−
2
* We may characterize G
k,p
(
t
) in a space manifold parameterized by angles of arrivals [27].
However, a space characterization requires perfect antenna calibration and adequate sensor
positioning.
† The coherence bandwidth is defined as the bandwidth over which the envelope correlation is
above 0.5 [28].
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