BENEFITS OF ISI IN THE INDOOR ENVIRONMENT - Ultra-Wideband Pulse-based Radio - page 81

modulation depth that would be necessary to achieve this goal can simply not
be met. In practice, the inability of hardware to keep up with the wide dynamic
range of possible channel conditions is eventually translated into either a higher
transmission power or a reduced information throughput.
Continuing the example of the nlos channel, suppose that the delay spread
of the channel allows the receiver to resolve n multipath components, instead
of the single path from above. Remember that each of the resolved symbol
streams still suffers from (independently) Rayleigh fading as a result of inter-
ference within a data symbol. The benefit of a rake receiver becomes clear
when the signal power of several of these multipath components is combined
to form a new symbol stream. The combination of n independent Rayleigh-
distributed random variables - in this particular case the vector magnitude of
the different resolved symbol streams - results in a new randomly distributed
variable. The probability density of this variable can be calculated using the
n -fold convolution of the pdf's of the summed variables [Lun74]. The mathe-
matical expression for the sum of n random distributed variables is not straight-
forward to compute and is out of the scope of this text. However, a few inter-
esting characteristics that are worth to be underlined.
First of all, the sum of two Rayleigh-distributed random variables does not
have a Rayleigh pdf any more. According to the central limit theorem [Fel45],
the sum of independent arbitrary distributed random variables will approach a
normal Gaussian distribution if a large number of such variables is combined.
The combination of only a few Rayleigh distributed variables holds the middle
ground between a Rayleigh shaped and a Gaussian shaped distribution. Also,
it turns out that the point of maximum population density shifts from a rather
low value as a result of the Rayleigh-like distribution, to the mean value ( 1)

when more resolved multipath signal components are combined and the pdf
shape becomes more Gaussian (Figure 4.7).
In order to make a comparison between rake and non-rake receivers, the
total combined energy from all multipath components has been normalized
to the average received power. The rationale behind this is that the received
energy per symbol must remain constant (E b / N 0 =

μ =

constant) for a fair com-
parison. In a nlos environment were all paths approximately share the same
average power, resolving n multipath components implies not only that the
average total power is divided over those n resolved paths, but the standard
deviation of the power in a path becomes also n times smaller. Many engineers
know by heart that the variance of the sum of uncorrelated random variables is
given by the sum of their variances [Bei01], so the following expression for the