Nakagami fading

Unfortunately, mobile radio links are subject to severe multipath fading due to the combination of randomly delayed, reflected, scattered, and diffracted signal components. Fading leads to serious degradation in the link carrier to noise ratio (CNR), leading to higher bit error rate (BER). Rayleigh and Rician fading models have been widely used to simulate small scale fading environments. M. Nakagami observed this fact and then formulated a parametric gamma function to describe his large scale experiments on rapid fading in high frequency long distance propagation.

The model

The Nakagami fading model was initially proposed because it matched empirical results for short ionospheric propagation. The Nakagami distribution or the Nakagami −"m" distribution is a probability distribution related to the gamma distribution. This more general fading distribution was developed whose parameters can be adjusted to fit a variety of empirical measurements. The Nakagami distribution described the magnitude of the received envelope by the probability density function:

: "P"("z") = 2"m"^"m""z"^{(2"m" − 1)}/Γ("m") Ω^"m" exp [−"mz"²/Ω] , "z" ≥ 0, Ω ≥ 0, "m" ≥ 0.5

: Ω = E(z²)is the average received power or average CNR

Γ(.) is the Gamma function.

: m = E(z²)/var(z²) is the fading figure or the shape factor.

The probability density function (PDF) are primarily known as first order characteristics and mainly used to obtain static metrics associated with the channel, i.e. bit error rate (BER).

When does Nakagami fading occur?

Nakagami fading occurs for multipath scattering with relatively larger time-delay spreads, with different clusters of reflected waves. Within any one cluster, the phases of individual reflected waves are random, but the time delays are approximately equal for all the waves. As a result the envelope of each cluster signal is Rayleigh Distributed. The average time delay is assumed to differ between the clusters. If the delay times significantly exceed the bit period of the digital link, the different clusters produce serious intersymbol interference. [3]

Properties of Nakagami fading

In current wireless communication, the main role of the Nakagami model can be summarized as follows: [3]

• If the envelope is Nakagami Distributed, the corresponding power is Gamma distributed.• The parameter m is the fading figure or shape factor that denotes the severity of the fading.• In the special case m=1, the distribution reduces to Rayleigh fading.• For m>1, the fluctuations of the signal strength are reduced as compared to Rayleigh Fading• For m = 0.5, it becomes one-sided Gaussian distribution• For m = ∞, the distribution becomes an impulse, i.e. AWGN with no fading.• The sum of multiple independent and identically distributed Rayleigh-fading signals has Nakagami Distributed signal amplitude.• The Rician and Nakagami model behave approximately equivalently near their mean value. While this may be true for main body of the probability density, it becomes highly inaccurate for tails. The outage mainly occurs during the deep fades, these quality measures are mainly determined by the tail of the probability density function (For the probability to receive less power).• The Rician distribution can be closely approximated by using the relationship between the Rice Factor ‘K’ and Nakagami Shape Factor ‘m’

$K\; =\; frac\{sqrt\{m^2-m\{m-sqrt\{m^2-m$

$(m\; >\; 1)$

$m\; =\; frac\{(K+1)^2\}\{2K+1\}$

• Note that some empirical measurements support values of the m-parameter less than unity, in which case the Nakagami fading causes more sever performance degradation than Rayleigh fading.• The power distribution for Nakagami fading, obtained by a change of variables is given by

p(x) = mmxm-1/Γ(m) Ωm exp [-mx/Ω]

Γ(.) is the Gamma function.

Ω = E(x2)is the average received power or average CNR

m = E(x2)/var(x2) is the fading figure or the shape factor.

Level Crossing Rate (LCR) and Average Fade Duration (AFD)

The envelope level crossing rate L is defined as the expected rate (in crossings per second) at which the signal envelope crosses the level Z in the downward direction. Another way to define it is the number of times per unit duration that the envelope of a fading channel crosses a given value in the negative direction.Denoting the time derivative ‘r’ and ‘R’ and Level Crossing Rate as R, the LCR occurring at a certain level ‘R’ defined as [3]

N = ∫0∞ r’ p(r’ r = R) dr’

Average Fade Duration corresponds to the average length of time the envelope remains under the threshold value once it crosses a given value in the negative direction. The Average Fade Duration is denoted as [3]

T = prob(r≤ R)/NF(r) = r∫0 P(a) da is the characteristic function of a channelT = F(r) / N(r)

The analysis of LCR and AFD enables one to get the statistics of the burst errors occurring on the fading channel. The LCR and AFD combined give a useful means of characterizing the severity of the fading over time. This statistics provides useful information for the design of the error–correcting codes which are complicated by the presence of error bursts. [3]

Channel capacity

Maximal-ratio diversity combining (MRC) is the optimal diversity scheme, and therefore provides the maximum capacity improvement relative to all combining techniques. It requires that the individual signals from each branch be co phased, weighted by their own CNR, then summed. Let Ωk denote the CNR on the kth branch. For independent branch signals and equal average branch CNR Ω. [4]

Ωk = Ω for all k ε {1,2,3,…,L},

The PDF of the received CNR at the output of a perfect L-branch MRC combiner is given by [4] pmrc(x) = mLmxLm-1/Γ(Lm) ΩLm exp [-mx/Ω] x≥0The assumptions are that the combiner output CNR Ω is tracked perfectly by the receiver as well as the variation in Ω is sent back to the transmitter via error-free feedback path. The time delay in this feedback path is also assumed to be negligible compared to the rate of the channel variation.

Given an average transmit power constraint, the channel capacity of the fading channel with received CNR distribution p(Ω) and optimal power and rate adaptation [4]

= B ∫xo+∞ log2 (Ω / Ω o) p(Ω) d Ω Bit/s

where B [Hz] is the channel bandwidth and Ω o is the optimal cutoff CNR level below which data transmission is suspended. This optimal cutoff must satisfy the equation: [4] ∫ Ω o+∞ (1/ Ω o - 1/ Ω) p(Ω) d Ω = 1To achieve the capacity, the channel fade level must be tracked at both the receiver and the transmitter. The transmitter has to adapt its power and rate accordingly, allocating high power levels and rates for good channel conditions (Ω large), and lower power levels and rates for unfavorable channel conditions (Ω small). [4]

ee also

Fading