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Rayleigh
fading
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Chapter: Wireless Propagation
Channels
Section: Multipath
Fading | 
Fade DurationThe mobile Rayleigh
or Rician radio channel is characterized by
rapidly changing
channel characteristics. As the amplitude of a signal received over
such a channel also fluctuates, the receiver will experience periods
during which the signal can not be recovered reliably. If a certain
minimum (threshold) signal level is needed for
acceptable communication performance, the received signal will experience
periods of
- sufficient signal strength or "non-fade intervals", during which the
receiver can work reliably and at low bit error rate
- insufficient signal strength or "fades", during which the bit error
rate inevitably is close to one half (randomly guessing ones and zeros)
and the receiver may even fall out of lock.
It is of critical
importance to the performance of digital mobile networks that the block
length or packet duration is chosen taking into
account the expected duration of fades and non-fade intervals. One of two
approaches can taken:
- make the block length at least an order of magnitude longer than the
average fade / non-fade period, and rely on error correction to cope
with burst errors. This approach can be used for mobile reception of
digital broadcast signals (e.g. DAB),
particularly if the effect of fading is mitigated through using a wide
transmission bandwidth and appropriate signal processing. This approach
would be impractical in indoor office
communication (wireless LANs) with high bit
rates and extremely small Doppler
spreads, i.e., with very long fade / non-fade periods.
- make the block length shorter than the average fade / non-fade
period and retransmit lost data. This approach works best in full duplex
mobile data systems and random access data systems.
If the data block length is larger than the average non-fade period,
almost all blocks will experience a signal fade and a corresponding burst
of bit errors. This may result in an excessive packet dropping rate,
unless powerful error correction codes are used. If the system supports a
feedback signal with acknowledgments of received blocks, it is mostly
advantageous to use only limited error correction coding, but to rely on
retransmission of lost blocks. To minimize the number of retransmissions,
one should choose the block length shorter than the average fade and
non-fade period.
Figure: fade and non-fade duration for a sample of a fading signal.
Average Fade DurationWe use:
Outage Probability = Average number
of fades per second * Average fade duration
where the average number of fades per second is called the threshold
crossing rate.
Expressions for Average (Non-) Fade DurationIn a Rayleigh fading
channel with fade margin M, the average nonfade duration (ANFD) is
SQRT(M)
ANFD = --------------
SQRT(2 pi) fD
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| where fD is the Doppler
spread. M is the ratio of the local-mean
signal power and the minimum (threshold) power
needed for reliable communication.
Average non-fade duration in Rayleigh-fading channel versus fade
margin for n = 1, 2, 3, 4, 5 and 6 Rayleigh-fading interfering
signals. Normalized by dividing by the Doppler Spread.
The curve for n = 6 closely resembles the curve the ANFD in an
interference-free but noise-limited channel.
Thus
- The ANFD is proportional to the speed of the mobile user. Channel
fading occurs mainly because the user moves. If the user is stationary
almost no time variations of the channel occur (except if reflecting
elements in the environment move)
- The ANFD increases proportional with the square root of the fade
margin.
- The non-fade duration is not so sensitive to whether the signal
experiences fades below a constant noise-floor or a fading interfering
signal.
Calculation of the distribution of non-fade periods is
tedious, but has been elaborated by Rice. Because of the shape of the
Doppler spectrum, fade durations that coincide with a motion of about half
a wavelength are relatively frequent.
 Exercise
A subscriber of an analog NMT cellular system (900
MHz) connects a 1200 bit/s modem to his cellular phone. He drives his car
at 36 km/h. The signal experiences wide-sense stationary noise. Experiment
with the Javascript
spreadsheet to find the fade margin required to ensure an average
nonfade duration enough to pass 600 bits uninterrupted.
Answer
The average fade duration (AFD) is
SQRT(M)
AFD = -------------- [ exp{ 1/M} -1]
SQRT(2 pi) fD
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| Thus
- The AFD is proportional to the speed of the mobile user.
- The fade durations rapidly reduce with increasing fade margin, but
the time between fades increases much slower.
Average fade duration in Rayleigh-fading channel versus fade margin
for n = 1, 2, 3, 4, 5 and 6 Rayleigh-fading interfering signals.
Normalized by dividing by the Doppler Spread.
Experiments revealed that at large fade margins, the fade durations are
approximately exponentially distributed around their mean value.
How do systems avoid long fades when the user is stationary?
Analog
- Handover if possible
- Disconnect user
GSM
- Slow frequency hopping
- Handover, if appropriate
- Power control
DECT
- Diversity at base station
- Best channel selection by handset
IS95 Cellular CDMA
- Wide band transmission avoids most deep fades (at least in
macro-cells with large delay spread)
- Power control
Wireless LANs
- Frequency Hopping
- Antenna Diversity
 
  
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