Digital Signal Processing Reference
In-Depth Information
receiver. The extent in which a single impulse applied to the input of the chan-
nel is spread over time at the output is described by the
delay spread
parameter
of the channel. Rather than describing the delay spread as the time interval
between the first and the very latest arriving signal component, it is more con-
venienttousetherms delay spread. Formally, the rms delay spread (
σ
τ
,
τ
rms
)
is defined as 'the square root of the second central moment of the power de-
lay profile' [Hol99]. The power delay profile (pdp), in his turn, is a statistical
distribution which describes the average expected power that arrives at the an-
tenna as a function of the excess time delay, starting from the moment the
impulse is injected into the channel by the transmitter.
Let us explain this more carefully using Figure 4.1. The time-varying impulse
response h
(τ, t
1
)
of the wireless link is defined as the response of the channel at
time
τ
(which is the propagation delay from transmitter to receiver) to a single
impulse applied at time stamp
t
1
. The mathematical expression for the power
delay profile is then given by the average power at the receiving antenna as a
function of the propagation delay
τ
(4.1):
pdp
(τ )
e
|
h
(τ, t)
|
,
2
=
(4.1)
where e[
·
] is the expectation operator.
Channel impulse responses
Calculation of RMS delay spread
|h
avg
(τ
i
)|
2
h(
τ
i
,t
1
)
tx
power delay profile:
average |h|
2
over t
1
...t
n
tx
los
0
τ
[
μ
s]
experiment at t
1
spatial position 1
τ
[
μ
s]
nlos
h(
τ
i
,t
2
)
P(
τ
i
)
probability density
of τ using the PDP
as weight function.
tx
1.0
∑
P(
τ
i
)=1
i
0
τ [μs]
experiment at t
2
spatial position 2
0
τ
[
μ
s]
mean:
〈τ〉=∑ τ
i
.P(
τ
i
)
P(
τ
i
)
h(
τ
i
,t
n
)
stddev: σ=
∑ (τ
i
-〈τ〉)
2
.P(τ
i
)
tx
〈τ〉
〈τ〉−σ
〈τ〉+σ
0
σ
τ
[
μ
s]
this is called RMS
delay spread (
experiment at t
n
spatial position n
τ
RMS
).
0
τ
i
−〈τ〉
Figure 4.1.
Calculation of the rms power delay spread using a discrete-
time tapped delay line model of the wireless channel.
