Multiple Rayleigh-fading signals
In wireless system, typically interference from multiple transmitters is
experienced. Each signal may experience multipath fading and shadowing.
Cumulation of multiple Rayleigh-fading
signals requires investigation of the nature of the signals contributing to the interference.
We consider the signal behaviour during an
observation interval of duration T which is short compared to the rate of channel fading This implies that the fading does not affect the amplitudes and phases of the signals.
However, modulation can affect amplitudes and phases during T.
Two extreme cases are distinguished:
- Coherent (or phasor) cumulation:
This occurs if, during the observation interval, the
phase fluctuations caused by the modulating signals are sufficiently small, and the carrier frequencies
of the signals are exactly equal.
The joint signal behaves as a Rayleigh phasor, with Gaussian inphase and quadrature phase components.
The instantaneous power is exponentially
distributed. The local-mean
power is equal to the sum of the local-mean powers of the individual
signals.
Coherent cumulation can occur only if
- phase modulation with a very small deviation is
applied, or
- if the observation interval is taken short with respect to the rate of modulation.This
occurs for instance in digital systems if the joint
interference signal is studied during one bit interval or during the lock-in of a carrier-recovery loop
in a synchronous detector.
- Incoherent (or power) addition:
if the phases of each of the individual signals substantially
fluctuate due to mutually independent modulation, the signals add incoherently.
The interference power experienced during the observation interval is the power sum of the individual
signals.
With coherent
addition, the joint interference signal may exhibit deep fades, caused by mutual cancelling of phasors
from the signals. This cannot continue for a sustained period due to the phase variations caused by
angle modulation of each signal or by slightly different carrier frequencies due to Doppler shifts and
free-running oscillators.
With incoherent cumulation, the joint interference signal behaves as a band-limited Gaussian noise
source if the number of components is sufficiently large. Moreover, any fade of one of the signals is
likely to be hidden by the other interfering signals. Hence, the joint interference signal tends to
exhibit less multipath fluctuations per unit of time than the signal from one individual interferer.
Multiple incoherent Rayleigh-fading signals with equal mean-power
If the interference is caused by the power sum of n Rayleigh-fading signals, with identical local-mean
power , the pdf of the joint interference power is the n-th convolution of the exponential distribution
of the power of an individual interfering signal.
The pdf of the joint interference power is found to have a gamma distribution.
It may not be fully appropriate to speak of the envelope of such a joint interference signal,
but if one defines the amplitude to be proportional to the square root of the power, then one finds that the amplitude has a Nakagami distribution.
The pdf of the joint interference power caused by interfering signals with different
local-mean powers can be approximated by a gamma distribution.