Rayleigh_fading - Wikiparency

# Rayleigh fading **Rayleigh fading** is a statistical model for how a radio signal's amplitude fluctuates as it passes through a propagation environment containing many scattered paths and no single dominant path between transmitter and receiver. The model treats the received signal's in-phase and quadrature components as independent, zero-mean Gaussian random variables, an approximation justified whenever a great many reflected and scattered contributions of comparable strength add together, and the magnitude of their sum then follows the Rayleigh distribution that gives the model its name. The model is judged a reasonable description of tropospheric and ionospheric scatter propagation and of the heavily built-up urban environments in which a mobile receiver rarely has a clear line of sight to its base station, and it underlies the [[Fading|fading]] behaviour discussed generally in that article, arising in turn from the [[Multipath_propagation|multipath propagation]] responsible for the many paths the model assumes. One instance of that many-path picture, a large number of scattered rays combining into the characteristic Rayleigh-distributed envelope, is rendered by a three.js companion sketch elsewhere on this page. Where one propagation path, typically a direct line of sight, dominates the others, the same construction instead gives a Rician distribution, and Rayleigh fading is recovered as the special case in which that dominant path's strength falls to zero; more generally, Rayleigh fading is the fully diffuse limit of a two-wave-with-diffuse-power model that keeps two dominant paths rather than letting every dominant term vanish. The sections below develop the model's statistics, the properties that follow from them, and the practical methods used to generate a Rayleigh-faded waveform for simulation and test. ## The model Consider a signal that reaches a receiver by a large number of independent paths, each scaled by a random amplitude and shifted by a random phase, and add them as complex phasors. If no individual contribution dominates the sum, the central limit theorem makes the sum's real part, the in-phase component `I`, and its imaginary part, the quadrature component `Q`, approximately independent, zero-mean Gaussian random variables of equal variance `sigma^2`. The envelope of the sum, `R = sqrt(I^2 + Q^2)`, is then Rayleigh-distributed, with [[Probability_density_function|probability density]] `f(r) = (r/sigma^2) * exp(-r^2/(2*sigma^2))` for `r >= 0` and zero otherwise, a distribution named for Lord Rayleigh's nineteenth-century analysis of the resultant of many random vibrations of the same pitch and arbitrary phase.[^rayleigh1880] Its squared envelope, proportional to instantaneous received power, follows an exponential distribution instead. Because the derivation only assumes many comparable, independently phased contributions and never refers to the geometry that produced them, the same statistics turn up in any sufficiently rich scattering environment regardless of the physical mechanism, whether the scatterers are raindrops, building facades or irregularities in the ionosphere. The consequence is measured as well as theoretical: averaged over a Rayleigh-faded channel, a receiver's bit-error rate falls off only in proportion to the inverse of the signal-to-noise ratio, far more slowly than the exponential improvement the same modulation achieves on an unfaded channel, and combining several independently faded branches by maximal-ratio combining steepens that falloff toward the inverse signal-to-noise ratio raised to the number of branches combined.[^ell-mrc] ### Applicability Rayleigh fading is judged a reasonable model for tropospheric and ionospheric scatter propagation, and for the [[Multipath_propagation|multipath propagation]] a mobile receiver experiences in a heavily built-up urban environment where buildings block any direct line of sight to the transmitter. It is most applicable precisely when no single path dominates the sum described above; if a direct line-of-sight component survives alongside the scattered paths, that one strong term unbalances the sum and the envelope instead follows a Rician distribution, which has the Rayleigh case as its own limit once the line-of-sight component's strength is taken to zero. Rayleigh fading is, more generally, a special case of the two-wave-with-diffuse-power model, which keeps two dominant specular paths on top of a diffuse background rather than letting the dominant terms vanish entirely, so the three models form a family distinguished only by how much of the received power arrives along a small number of identifiable paths rather than a great many indistinguishable ones.[^twdp-cn2] ## Properties Three quantities describe how a Rayleigh-faded envelope behaves over time rather than only as a static distribution, and each depends on the same maximum Doppler shift `f_d = v/lambda`, set by the receiver's speed `v` and the carrier wavelength `lambda`, the same relationship worked through in the [[Doppler_effect|Doppler effect]] article. ### Level crossing rate The level crossing rate counts how often, per second, the envelope crosses a given threshold moving in one direction, typically downward through a level of interest. Normalising the threshold to `rho = R/R_rms`, the level relative to the root-mean-square envelope, the classical result is `N_R = sqrt(2*pi) * f_d * rho * exp(-rho^2)`, a rate that rises from zero at very low thresholds, peaks near `rho = 1/sqrt(2)`, and falls back toward zero at high thresholds the envelope rarely reaches.[^rice-cn] A receiver moving faster raises `f_d` and so raises the crossing rate at every threshold in direct proportion, without shifting the threshold at which the rate peaks. ### Average fade duration The average fade duration is the mean length of time the envelope spends below a given threshold once it has crossed down through it, `tau_bar = (exp(rho^2) - 1) / (rho * f_d * sqrt(2*pi))`, obtained by dividing the fraction of time the envelope spends below that threshold by the level crossing rate above.[^rice-cn] Average fade duration grows rapidly as the threshold is raised toward the typical envelope level and shrinks as the receiver speeds up, since a faster-moving receiver crosses the same threshold more often but spends correspondingly less time on each individual excursion below it: information a system designer uses directly to size the interleaving span needed to spread a fade's burst of errors thinly enough for a channel code to correct. ### Doppler power spectral density A single, unmodulated carrier reaching a moving receiver through a large number of scattered paths arriving from every direction in the plane of motion is spread, rather than shifted to one new frequency, into a continuous Doppler power spectral density `S(f) = 1 / (pi * f_d * sqrt(1 - (f/f_d)^2))` for `|f| <= f_d` and zero beyond: the classical U-shaped [[Spectral_density|spectrum]], finite in total power but rising without bound, in an integrable way, at its two edges `f = +f_d` and `f = -f_d`.[^rice-cn] Its width, `2*f_d`, is exactly the quantity that sets the level crossing rate and average fade duration above and the coherence time discussed in the [[Fading|fading]] article, so the same physical assumption, isotropic scattering seen by a moving receiver, ties every property in this section back to one number. ## Generating Rayleigh fading Simulating a Rayleigh-faded channel means producing a synthetic waveform whose envelope statistics, level crossing rate and Doppler spectrum all match the properties above, rather than merely drawing independent Rayleigh-distributed numbers, since a real fading process is correlated in time in exactly the way [[Multipath_propagation|multipath propagation]] makes it correlated in space. Three related approaches are in wide use. ### Jakes's model The construction generally known as Jakes's model builds the faded envelope from a small number of equal-amplitude sinusoids, spaced in frequency across the Doppler band and combined with carefully chosen phases and angles of arrival, so that the sum's statistics approximate the Rayleigh envelope and Doppler spectrum of an ideal, infinitely rich scattering environment using only a modest, fixed amount of computation. Because the construction is deterministic once its parameters are fixed, the same waveform can be reproduced exactly for repeated tests, a practical advantage over methods built on fresh random numbers each run. The method is usually attributed to a mobile-radio engineering handbook from the 1970s that built on an earlier statistical treatment of scattered reception, though the precise edition and page are not pinned down here.[^jakes-cn] ### Filtered white noise An alternative generates two independent white Gaussian noise sequences, passes each through an identical low-pass filter shaped to the desired Doppler spectrum, and treats the two filtered outputs as the in-phase and quadrature components of the faded signal. Because each output is a sum of many filtered noise samples, each remains approximately Gaussian, and the envelope of the resulting complex process is Rayleigh-distributed by exactly the central-limit argument used under The model above. The filter's shape, rather than any property of the underlying noise generator, is what fixes the simulated Doppler spectrum and every property derived from it, which is why choosing that filter carefully matters more than the choice of random-number generator feeding it. ### Butterworth filter as Doppler power spectral density The classical Doppler power spectral density above is awkward to implement directly because it diverges, if only mildly, at its two band edges, so a low-order Butterworth low-pass filter is often substituted in its place: a standard, easily designed filter whose smooth roll-off approximates the same overall bandwidth without reproducing the classical spectrum's edge peaks exactly. The substitution is explicitly illustrative rather than exact, trading a small, well-understood discrepancy in spectral shape for a filter that is simple to specify and implement, a compromise acceptable whenever the simulation's purpose is to reproduce realistic fade rates and durations rather than to match the idealised spectrum bin by bin. ## Microsims This article carries no p5.js sketch of its own. A three.js companion instead renders a large number of scattered rays arriving from every direction, summing them into the same Rayleigh-distributed envelope derived under The model above, so the statistical result can be watched building up from its individual, randomly phased contributions. *Try:* in the [[Doppler_effect]] sketch, raise the source speed and watch the two observer markers' readouts spread further apart: the same widening is what stretches the Doppler power spectral density described under Properties above across a wider band as `f_d` grows. *Try:* in the [[Radar]] sketch, freeze the sweep with [SPACE] and compare the glow left behind by several echoes at once: a crude, discrete stand-in for the many simultaneous scattered contributions this article's envelope sums continuously and at random. *Try:* in the [[Sonar]] sketch, set several targets at once and watch their echoes overlap on the A-scan: sonar keeps each scattered return separate and labelled, while the Rayleigh model above deliberately throws that same kind of sum together and keeps only its statistics. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Rayleigh_fading) : [Wikitube](https://en.wikitube.io/wiki/Rayleigh_fading) Skeleton mirrored at revision 1326474632. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Multipath_propagation]] - [[Fading]] - [[Doppler_effect]] - [[Radar]] - [[Sonar]] - [[Probability_density_function]] - [[Spectral_density]] ## References Standard probability results used above, including the central-limit derivation of Gaussian in-phase and quadrature components and the resulting Rayleigh and exponential distributions, are textbook material and are not separately footnoted here, per the Wikitube style guide's §6.1. Page numbers below are PDF pages of the open editions linked from the citations. [^rayleigh1880]: Strutt, J. W. (Lord Rayleigh). "On the Resultant of a Large Number of Vibrations of the Same Pitch and of Arbitrary Phase." Philosophical Magazine, 1880. [^ell-mrc]: Ellingson, S. *Radio Systems Engineering, Revised First Edition*. 2023, pp. 163-167 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^twdp-cn2]: Citation needed: a primary source establishing the two-wave-with-diffuse-power (TWDP) model's original formulation, and the precise conditions under which it reduces to the Rayleigh and Rician cases, would fix exact attribution. [^rice-cn]: Citation needed: the classical level-crossing-rate, average-fade-duration and Doppler-spectrum results credited to S. O. Rice's statistical theory of random noise, as applied to fading radio channels, would fix the precise paper, year and page for each formula. [^jakes-cn]: Citation needed: the original statement of the sum-of-sinusoids method for generating Rayleigh fading, commonly attributed to William C. Jakes's 1970s mobile-radio handbook building on R. H. Clarke's earlier statistical model, would fix the precise edition and page. ## External links This article carries no p5.js or three.js sketch of its own with a published external URL; the interactive sketches referenced above are external-linked from their own articles.