MULTIPATH FADING
Various Features in a Wireless Channel
Path Loss
Multipath fading
Interference
Shadowing
Excess Delay
The propagation delay relative to that of the
shortest path
t=8ms
t=0
t=47ms
Strength Variation
As the vehicle moves, the strength of each path varies
because the surfaces are complex
The Channel is a Filter
The multipath channel can be represented as linear, time-
varying bandpass filter
Transmitted
Signal
Received
Signal
h(t , )
x(t)
y(t)
y (t ) =
x(t )h(t , )d
Measured Data from Darmstadt,
[Molisch, 01]
Germany
Baseband Impulse Response
More convenient to work with baseband signals
{
}
x(t ) = Re{ c(t )e }
y (t ) = Re{ r (t )e }
h(t , ) = Re hb (t , )e
j c t
j c t
j c t
1
r (t ) = c(t )hb (t , )d
2
The factor of
ensures that baseband
average power
equals passband
average power
Path Model
The channel is assumed to comprise N
discrete paths of propagation (rays)
Each path has an amplitude (t), a phase (t)
and a propagation delay
hb (t , )
N =5
0 (t ) e j ( t )
1 (t )e j1 (t ) (t )e j 2 (t )
2
0
t)
j 4 ( t )
3 (t )e j (
(
t
)
e
4
3
delay
Probing the Channel
The channel may be probed or sounded by
transmitting a pulse p(t) and recording the
response at the receiver
The response is the convolution of p(t) with the
channel impulse response
N 1
1
j i ( t )
r (t ) = i (t )e
p(t i )
2 i =0
Pulse Width >> max
Suppose a pulse much wider than the length
p(t)
of the impulse response is transmitted at time
t=0
g
hb (t , )
0 1 2 3 4
Tp
delay
Instantaneous Power
The magnitude squared of any sample in the interval t4 and
t0 + Tp will equal
2
2
r (t ) =
r(t)
2
4
N 1
j ( t )
(
t
)
e
i
i
i =0
hb (t , )
0 1 2 3 4
We say, the multipath is not resolved
delay
Time Variation of the Probe Response
If one or both of the terminals moves, the path phases
change because the path lengths change
The path amplitudes do not change much
These changes yield large changes in the magnitude
of the received waveform
|r(t)| in dBm
[not real data]
Narrowband Fading
This same type of fading happens to a digital waveform if
the symbol period is much larger than (>10 times) the
channel length
Such long symbol periods correspond to narrowband
signals
Average Power for Narrowband Signals
Assuming the channel is ergodic, the ensemble
average may be approximated by a time average:
= E r (t )
t+
T
2
T
t
2
j ( s )
(
s
)
e
ds
i
i
i =0
where the interval [t-T/2,t+T/2] corresponds to a local
area
Uncorrelated Scattering
Assume that the phases of different paths are
uncorrelated and that the energy of the pulse
is one
Then the time average simplifies to
= E r (t )
where
1
T
2
i
t+
}
T
2
2
i
i =0
i (s)ds
t
T
2
Pulse Width << max
Now consider a small pulse width
p(t)
g
t
Tp
hb (t , )
delay
Multipath Resolved
Pulses do not overlap
r (t , )
0 (t )e j ( t )
t)
j 2 ( t )
1 (t )e j1 (
2 (t )e
0
(t )
j 4 ( t )
3 (t )e j
(
t
)
e
4
3
Wideband Signal
The Fourier Transform of such a narrow pulse has a wide
spectrum
P(f)
p(t)
g
Tp
F.T.
Power Delay Profile (PDP)
The PDP is a time-average of |r(t,)|2 over a
small interval (assuming the terminal is moving)
P( )
1
P( )
T
2 02
212
2 22
t+
T
2
r ( s, )
t
ds
T
2
2 32
2 42
t
W. Mohr, Modeling of wideband mobile radio channels based on propagation measurements,
in Proc. 16th Int. Symp. Personal, Indoor, Mobile Radio Communications, vol. 2, pp. 397-401, 1995
Average Power for Wideband Signals
The average power is the integral of the PDP
PAVG = P( )d
0
N 1
= i2 =
i =0
Local Average Powers Are The Same
Narrowband and Wideband averaged powers are equal
Moments of the PDP
Channels are often described by their rms delay spread
To compute rms delay spread, normalize the PDP to
make it like a PDF for a random variable (unit area) and
then find its standard deviation
Must you use excess delay to compute rms delay spread?
Mean Delay
Must first compute the mean delay
+
P( )d
0
+
P( )d
0
For this to be mean excess delay, the origin of
the axis needs to be the time of the first
arriving path
Second Moment
Next need the second moment of this PDF
P( )d
0
P( )d
0
RMS Delay Spread
Recall that standard deviation is the square root of
variance and variance is the second moment minus the
first moment squared
Variance
rms delay spread
()
( )
Example Data
[Rappaport, 02]
How RMS Delay Spread Can Be Used
If <<symbol period, assume narrowband fading
effects
If >>symbol period, assume wideband fading effects
(will need an equalizer, CDMA or OFDM)
The Frequency Domain View
<<symbol period implies that the frequency response of
the channel,
H (t , f ) = h(t , ) exp( j 2f )d ,
doesnt vary much with frequency over the bandwidth of
the transmitted signal
Narrowband Case
The channel appears flat to the signal
H (t , f )
Transmitted
Signal Spectrum
900MHz
Flat Fading
When <<symbol period, we say the signal undergoes
flat fading
The channel frequency response is approximately flat
over the signal bandwidth
Wideband Case
>> symbol period implies that the frequency
response of the channel varies significantly with
frequency over the bandwidth of the transmitted
signal
H (t , f )
Transmitted
Signal Spectrum
900MHz
Frequency Selective Fading
When >> symbol period, we say the signal undergoes
frequency selective fading
The channel frequency response is strong for some
frequencies and not for others within the signal bandwidth
Complete Narrowband Statistical
Fading Model
= r (t ) = xy is the signal envelope. Recall E{ }= =
2
N 1
i =0
x represents the small-scale or multipath fading, and has either
the Rayleigh or Rician distribution with unit mean square value, i.e.
Therefore,
{ }
{ } { }
y represents the large-scale or shadow fading, and has a
lognormal distribution. Recall E{ y }= is local average
power
E x2 = 1
E 2 = E y2
2
Then z is N ( p
(d ), ) (the zs are the
(
)
points on the scatter plot)
Let
z = 10 log10 y.2
p (d )is the path loss, with the model
d
p (d ) = p(d o ) n10 log10
d o
2
i
Complete Wideband Statistical
Fading Model
N 1
Recall
Each path of the wideband model has
i =0 i .
independent small-scale fading: = x y
xi is the small-scale or multipath fading on each path, and has
either the Rayleigh or Rician distribution with mean square value
{ } { }
E xi2 = E i2 /
represents the large-scale or shadow fading, as before, and
has a lognormal distribution.
Let z
( )
= 10 log10 y 2
. Then
points on the scatter plot)
is
N p (d ), 2
p (d ) is the path loss, with the model
)(the zs are the
d
p (d ) = p(d o ) n10 log10
d o
Fade Margin
RX
Power
Decoding
Threshold
Extra power (i.e. a fade
margin) is required, to keep
above the decoding
threshold
Fade Margin
One channel with
multipath fading
Summary
The multipath channel model has a discrete number of
propagation paths
Each path has amplitude, phase and delay
The PDP is the local average of the magnitude squared of
the impulse response of the channel
Average power of the channel is the integral of the PDP
Average power is same for narrowband and wideband
channels
The fading is flat or frequency selective depending on
the comparison between rms delay spread and the
symbol period
References
[Rapp, 02] T.S. Rappaport, Wireless
Communications, Prentice Hall, 2002
[Molisch, 01] Andreas F. Molisch (ed), Wideband
Wireless Digital Communications, Prentice Hall
PTR, 2001.
W. Mohr, Modeling of wideband mobile radio
channels based on propagation measurements,
in Proc. 16th Int. Symp. Personal, Indoor, Mobile
Radio Communications, vol. 2, pp. 397-401, 1995
