 ### JPL's Wireless Communication Reference Website

#### Chapter: Wireless Channels Section: Multipath fading Besides Rayleigh and Rician fading, refined models for the pdf of a signal amplitude exposed to mobile fading have been suggested.

## Nakagami Math

The distribution of the amplitude and signal power can be used to find probabilities on signal outages.
• If the envelope is Nakagami distributed, the corresponding instantaneous power is gamma distributed.
• The parameter m is called the 'shape factor' of the Nakagami or the gamma distribution.
• In the special case m = 1, Rayleigh fading is recovered, with an exponentially distributed instantaneous power
• For m > 1, the fluctuations of the signal strength reduce compared to Rayleigh fading.

The Nakagami fading model was initially proposed because it matched empirical results for short wave ionospheric propagation. In current wireless communication, the main role of the Nakagami model can be summarized as follows

## When does Nakagami Fading occur?

• It describes the amplitude of received signal after maximum ratio diversity combining.
After k-branch maximum ratio combining (MRC) with Rayleigh-fading signals, the resulting signal is Nakagami with m = k. MRC combining of m-Nakagami fading signals in k branches gives a Nakagami signal with shape factor mk.
• The sum of multiple independent and identically distributed (i.i.d.) Rayleigh-fading signals have a Nakagami distributed signal amplitude.
This is particularly relevant to model interference from multiple sources in a cellular system.
• The Nakagami distribution matches some empirical data better than other models
• Nakagami fading occurs for multipath scattering with relatively large delay-time spreads, with different clusters of reflected waves.
Within any one cluster, the phases of individual reflected waves are random, but the delay times are approximately equal for all waves. As a result the envelope of each cumulated cluster signal is Rayleigh distributed. The average time delay is assumed to differ significantly between clusters. If the delay times also significantly exceed the bit time of a digital link, the different clusters produce serious intersymbol interference, so the multipath self-interference then approximates the case of co-channel interference by multiple incoherent Rayleigh-fading signals.
• The Rician and the Nakagami model behave approximately equivalently near their mean value.
This observation has been used in many recent papers to advocate the Nakagami model as an approximation for situations where a Rician model would be more appropriate.
While this may be accurate for the main body of the probability density, it becomes highly inaccurate for the tails. As bit errors or outages mainly occur during deep fades, these performance measures are mainly determined by the tail of the probability density function (for probability to receive a low power).

## Further study, printable document Mathematical model presented in postscript format. Mathematical model presented in acrobat format.    