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Chapter: Wireless ChannelsSection: Multipath Fading, Nakagami fading

Rice and Nakagami fading are two generalisations of the model for Rayleigh fading. In literature, often a Nakagami model is used for analytical simplicity incases where Rician fading would be a more appropriate model. In contrast to common belief, the Nakagami model is not an appropriate approximation forRician fading. It has an essentially different behaviour for deep fades, such that results on outage probabilities or error rates can differ by orders of magnitude.

Rician Models

To describe microcellular propagation, the Rayleigh model lacked the effect of adominant line-of-sight component, and Rician model appeared to be moreappropriate. For analytical and numerical evaluation of system performance, theexpressions for Rician fading are less convenient, mainly due to the occurrence of aBessel function in the Rician probability density function of received signalamplitude. Approximations by a Nakagami distribution, with simpler mathematicalexpressions have become popular.

Nakagami Models

The Nakagami fading model was initially proposedbecause it matched empirical results for short waveionospheric propagation. In current wirelesscommunication, the main role of the Nakagami modelcan be summarized as follows

It describes the amplitude of received signal aftermaximum ratio diversity combining.The sum of multiple independent and identicallydistributed (i.i.d.) Rayleigh-fading signals have aNakagami distributed signal amplitude. This isparticularly relevant to model interference frommultiple sources.The Nakagami distribution matches someempirical data better than other modelsThe Rician and the Nakagami model behaveapproximately equivalently near their mean value.This observation has been used in many recentpapers to advocate the Nakagami model as anapproximation for situations where a Ricianmodel would be more appropriate.

Comparison

In the analysis of outage probabilities or error rates, it is the behavior of the model for signals in deep fades that has the determining effect. As the behavior ofthe probability density functions for amplitudes near zero differ significantly, approximations based on behavior near the mean are inappropriate.

Rician and Nakagami models have a fundamentally different density for deep fades. Modeling a Rician fading signal by a Nakagami distribution of theamplitude leads to overly optimistic results, and discrepancies can be many orders of magnitude. That is, we challenge the accuracy of the last application ofthe model in the above list.

Distribution of received power

A typical radio channel exhibits multipath reception, which causes fading. The mathematical evaluation of this paper addresses narrowband systems, in whichthe channel transfer function is sufficiently constant over the signal bandwidth. This corresponds to the assumption that Intersymbol Interference does not playa major role in the performance of the radio links. However, the models also play a role in wideband systems, in which each resolvable bin of reflected wavescan be modeled to exhibit flat fading.

Rice

Rician fading assumes a dominant line-of-sight component and a large set of i.i.d.reflected waves. Reflected waves arrive with a random phase offset and theaccumulation can be modeled as a phasor addition of signals with random amplitudeand phase. This assumption leads to a Gaussian Inphase and Quadrature component,and a corresponding Rician amplitude. It has been shown that the instantaneous power

pi received from the i-th user, with pi = ri2/2 and ri the signal amplitude, has the

probability density function (pdf)

where the Rician K-factor is defined as the ratio of the power in the dominant

component and the scattered (multipath) power, is the total local-mean power in thedominant and scattered waves, and I0(.) denotes the modified Bessel function of thefirst kind and order zero.

Nakagami

For Nakagami fading, the instantaneous power has thegamma pdf

where G(m) is the gamma function, with G(m +1) = m! for integer shape factors m. The mean value is

. In the special case that m = 1, Rayleighfading is recovered, while for larger m the spread ofthe signal strength is less, and the pdf converges to adelta function for increasing m.

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In the special case that the dominant component is zero (K = 0) or m = 1, Rayleigh fading occurs, with an exponentially distributed power, viz.,

The Nakagami model is sometimes used to approximate the pdf of the power of a Rician fading signal. Matching the first and second moments of the Rician andNakagami pdfs gives

which tends to m = K/2 for large K.

For Rician fading, the probability distribution at small powers is

For Nakagami fading, we have

Here denotes the probability that pi < pth .

The results are strikingly different for m larger than one. As the relation between K and m was based merely on the first and second moments, it is likely to bemost accurate for values close to the mean. Outage probabilities however highly depend on the tail of the pdf for small power of the wanted signal. Theprobability of deep fades (small pi) differs for these two models, so an approximation the pdf of a Rician-fading wanted signal by a Nakagami pdf can be highlyinaccurate: Results differ even in first-order behavior. For Rician fading, the slope of the outage probability versus C/I is the same as for Rayleigh fading. ForNakagami fading, the slope is steeper, similar to that of m-branch diversity reception of a Rayleigh fading signal.

Method for Link Evaluation

The nakagami approximation appeared less suitable to approximate a Rician channel. Nevertheless other methods are available to make reasonableapproximations in numerical evaluations of the outage probability in interference-limited situations, while restricting the complexity of calculations.

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