If the channel is nonstationary then the channel response to an impulse at time t
1
is not necessarily just
a shifted version of its response to an impulse at time t
2
, t
1
= t
2
, as shown in Figure 3.3.
Example 3.1: Consider a wireless LAN operating in a factory near a conveyor belt. The transmitter
and receiver have a LOS path between them with gain
0
, phase
0
and delay
0
. Every T
0
seconds a
metal item comes down the conveyor belt, creating an additional reflected signal path in addition to the
LOS path with gain
1
, phase
1
and delay
1
. Find the time-varying impulse response c(, t) of this
channel.
Solution: For t = nT
0
, n = 1, 2, . . . the channel impulse response corresponds to just the LOS path. For
t = nT
0
the channel impulse response has both the LOS and reflected paths. Thus, c(, t) is given by
c(, t) =
0
e
j
0
(
-
0
)
t = nT
0
0
e
j
0
(
-
0
) +
1
e
j
1
(
-
1
)
t = nT
0
(3.10)
Note that for typical carrier and Doppler frequencies, the nth multipath component will have (f
c
+
f
D
n
(t))
n
(t) >> 1. For example, with f
c
= 1 GHz, f
D
= 80 Hz, and
n
= 50 ns (a typical value for
an indoor system), (f
c
+ f
D
)
n
= 50 >> 1. Outdoor wireless systems have delay spreads much greater
than 50 ns, so this property also holds for these systems. If (f
c
+ f
D
n
(t))
n
(t) >> 1 then a small change
in the path delay
n
(t) can lead to a very large phase change in the nth multipath component with
phase
n
(t) = (f
c
+ f
D
(t))(t
-
n
). Rapid phase changes in each multipath component gives rise to
constructive and destructive addition of the multipath components comprising the received signal, which
in turn causes rapid variation in the received signal strength. This phenomenon, called fading, will be
discussed in more detail in subsequent sections. The multipath amplitude terms
n
(t) change much more
slowly, since the antenna gains and reflection coefficients are relatively constant, the path loss changes
slowly, and the shadowing changes on the order of the shadowing decorrelation distance. Similarly, the
multipath delay terms
n
(t) change slowly compared to the phase terms, since the scatterer locations are
fixed and the path delays associated with each scatterer change slowly relative to the distance traveled
by the transmitter or receiver.
The multipath delay spread of a time-invariant channel c( ) is a deterministic constant given by the
time-of-arrival difference between the first and last multipath component:
T
m
= max
n
n
- min
n
n
.
(3.11)
The minimum delay min
n
n
is often set to zero and the other multipath delays normalized with respect
to this minimum delay (the LOS multipath component delay when the channel has a LOS component).
If the channel is time-varying, then the delay spread also varies with time:
T
m
(t) = max
n
n
(t)
- min
n
n
(t).
(3.12)
For a stationary and ergodic time-varying channel we can obtain the probability distribution of T
m
from
samples of T
m
(t) over time. Let p
T
m
( ) denote this probability distribution. The average delay spread is
then µ
T
m
= E[T
m
] =
p
T
m
( )d and its standard deviation, also called the rms delay spread, is given
by
T
m
=
E[T
2
m
]
- (E[T
m
])
2
.
(3.13)
59