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Rayleigh
fading
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PDF of received signal amplitudeIn order to obtain the
probability density function (PDF) of the signal amplitude r of a Rayleigh
fading signal, we observe the random processes of the inphase and
Quadrature components, I(t) and Q(t) , respectively, at one
particular instant t0. If N becomes very large
and are i.i.d., the central limit theorem says that
I(t0) and Q(t0) are (zero-mean)
Gaussian each with variance s2. The
received signal
has a Rayleigh amplitude r(t), with
 and a uniform phase q(t) between 0 and
2 p. The pdf of the amplitude is r r2
fr(r) = --- exp(- --)
s2 2s2
for r > 0.
 Exercise
Let z = x + jy with x
and y i.i.d. Gaussian with zero mean and variance s2.
- Show that
 is Rayleigh distributed. answer
- Show that
 , thus about 1.2533 s.
- Show that Er2 = 2 s2.
- Show that the variance is s2
(4-p)/2.
Simulations have shown that the Rayleigh pdf appropriately describes the
fading of the amplitude if N is larger than 6. Measurements over
non-line-of-sight paths at UHF frequencies in urban environments confirmed
the accuracy of the Rayleigh pdf.
PDF of Received PowerThe instantaneous power p, with
 , thus, averaged over one RF-cycle, has the exponential pdf
where s2 is the local-mean power.
ExerciseFind the cumulative distribution of the received
power. In a cellular voice channel, a signal outage occurs if the
instantaneous signal-to-noise ratio is less than z = 10 (10 dB).
Find the required local-mean power to ensure an outage probability of less
than 1%. Find the corresponding fade margin, defined as the excess power
above the threshold z N with N the noise power.
ExerciseWrite a computer program that plots the probability
that a Rayleigh fading signal drops below a certain threshold. Show that
the curve can be plotted as a function of a fade margin, i.e., the ratio
between the local mean power and the threshold.
If the receiver can
choose between L (L= 1,2, ..) different antennas and each antenna
sees an independently fading signal, how does this improve the
probability? (Hint: see diversity)
Solution.
If the set of reflected waves are dominated by one strong component, Rician fading is a more appropriate model.
   
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