Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. It assumes that the power of a signal that has passed through such a transmission medium (also called a communications channel) will vary randomly, or fade, according to a Rayleigh distribution — the radial component of the sum of two Gaussian random variables. It is a reasonable model for tropospheric and ionospheric signal propagation as well as the effect of heavily built-up urban environments on radio signals.^[1][2] Rayleigh fading is most applicable when there is no line of sight between the transmitter and receiver. If there is a line of sight, Rician fading is more applicable. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Look up model in Wiktionary, the free dictionary. ... A wave is a disturbance that propagates through space, often transferring energy. ... Wireless is an old-fashioned term for a radio receiver, referring to its use as a wireless telegraph; now the term is used to describe modern wireless connections such as in cellular networks and wireless broadband Internet. ... A transmission medium is any material substance, such as fiber-optic cable, twisted-wire pair, coaxial cable, dielectric-slab waveguide, water, or air, that can be used for the propagation of signals, usually in the form of modulated radio, light, or acoustic waves, from one point to another. ... Channel, in communications (sometimes called communications channel), refers to the medium through which information is transmitted from a sender (or transmitter) to a receiver. ... Fading (or fading channels) are mathematical models for the distortion that a carrier-modulated telecommunication signal experiences over certain propagation media. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... Probability density function of Gaussian distribution (bell curve). ... A random variable is a term used in mathematics and statistics. ... Tropospheric scatter (or troposcatter) is the scattering of distant TV and FM radio stations by the troposphere so that they travel farther than the line of sight. ... Ionospheric reflection: Of electromagnetic waves propagating in the ionosphere, a redirection, bending--by a complex process involving reflection and refraction--of the waves back toward the Earth. ... An urban area is a term used to define an area where there is an increased density of human-created structures in comparison to the areas surrounding it. ... When viewing a scene, as in optics, photography, or even hunting, the line of sight is the straight line between the observer and the target. ... In telecommunication, multipath is the propagation phenomenon that results in radio signals reaching the receiving antenna by two or more paths. ...

The model

Rayleigh fading is a reasonable model when there are many objects in the environment that scatter the radio signal before it arrives at the receiver. The central limit theorem holds that, if there is sufficiently much scatter, the channel impulse response will be well-modelled as a Gaussian process irrespective of the distribution of the individual components. If there is no dominant component to the scatter, then such a process will have zero mean and phase evenly distributed between 0 and 2π radians. The power, or envelope, of the channel response will therefore be Rayleigh distributed. Calling this random variable R, it will have a probability density function In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ... Central limit theorems are a set of weak-convergence results in probability theory. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... Generally, the word gaussian pertains to Carl Friedrich Gauss and his ideas. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, the uniform distributions are simple probability distributions. ... The radian (symbol: rad, or a superscript c ( half circle)) is the SI unit of plane angle. ... An envelope detector is a device which is used to demodulate AM signals. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

where σ = E(R²). Often, the gain and phase elements of a channel's distortion are conveniently represented as a complex number. In this case, Rayleigh fading is exhibited by the assumption that the real and imaginary parts of the response are modelled by independent and identically distributed zero-mean Gaussian processes so that the amplitude of the response is the sum of two such processes. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

Applicability

Densely-built Manhattan has been shown to approach a Rayleigh fading environment.

One second of Rayleigh fading with a maximum Doppler shift of 10Hz.

One second of Rayleigh fading with a maximum Doppler shift of 100Hz.

The requirement that there be many scatterers present means that Rayleigh fading can be a useful model in heavily built-up city centres where there is no line of sight between the transmitter and receiver and many buildings and other objects attenuate, reflect, refract and diffract the signal. Experimental work in Manhattan has found near-Rayleigh fading there.^[3] In tropospheric and ionospheric signal propagation the many particles in the atmospheric layers act as scatterers and this kind of environment may also approximate Rayleigh fading. If the environment is such that, in addition to the scattering, there is a strongly dominant signal seen at the receiver, usually caused by a line of sight, then the mean of the random process will no longer be zero, varying instead around the power-level of the dominant path. Such a situation may be better modelled as Rician fading. Lower Manhattan, from the Brooklyn Bridge, 2005 spotteddogs. ... Lower Manhattan, from the Brooklyn Bridge, 2005 spotteddogs. ... Image File history File links Rayleigh_fading_doppler_10Hz. ... Image File history File links Rayleigh_fading_doppler_10Hz. ... Image File history File links Rayleigh_fading_doppler_100Hz. ... Image File history File links Rayleigh_fading_doppler_100Hz. ... Near line of sight. ... Attenuation is the decrease of the amount, force, magnitude, or value of something. ... Spheres reflecting the floor and each other. ... The straw seems to be broken, due to refraction of light as it emerges into the air. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The Borough of Manhattan, highlighted in yellow, lies between the East River and the Hudson River. ... Tropospheric scatter (or troposcatter) is the scattering of distant TV and FM radio stations by the troposphere so that they travel farther than the line of sight. ... Ionospheric reflection: Of electromagnetic waves propagating in the ionosphere, a redirection, bending--by a complex process involving reflection and refraction--of the waves back toward the Earth. ... Radio signals, like all electromagnetic radiation, usually travel in straight lines. ... In telecommunication, multipath is the propagation phenomenon that results in radio signals reaching the receiving antenna by two or more paths. ...

Note that Rayleigh fading is a small-scale effect. There will be bulk properties of the environment such as path loss and shadowing upon which the fading is superimposed. Path loss: In a communication system, the attenuation undergone by an electromagnetic wave in transit between a transmitter and a receiver. ... Shadows on a pavement A shadow is a region of darkness where light is blocked. ...

How rapidly the channel fades will be affected by how fast the receiver and/or transmitter are moving. Motion causes Doppler shift in the received signal components. The figures show the power variation over 1 second of a constant signal after passing through a single-path Rayleigh fading channel with a maximum Doppler shift of 10Hz and 100Hz. These Doppler shifts correspond to velocities of about 6km/h (4mph) and 60km/h (40mph) respectively at 1800MHz, one of the operating frequencies for GSM mobile phones. This is the classic shape of Rayleigh fading. Note in particular the 'deep fades' where signal strength can drop by a factor of several thousand, or 30-40 dB. The Doppler effect, named after Christian Andreas Doppler, is the apparent change in frequency or wavelength of a wave that is perceived by an observer moving relative to the source of the waves. ... This article or section may be confusing for some readers, and should be edited to be clearer or more simplified. ... The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. ...

Properties

Since it is based on a well-studied distribution with special properties, the Rayleigh distribution lends itself to analysis, and the key features that affect the performance of a wireless network have analytic expressions. In mathematics, an analytic expression is a mathematical expression, constructed using well-known operations that lend themselves readily to calculation. ...

Note that the parameters discussed here are for a non-static channel. If a channel is not changing with time, clearly it does not fade and instead remains at some particular level. Separate instances of the channel in this case will be uncorrelated with one another owing to the assumption that each of the scattered components fades independently. Once relative motion is introduced between any of the transmitter, receiver and scatterers, the fading becomes correlated and varying in time.

Correlation

The autocorrelation function of the 10Hz Doppler Rayleigh fading channel.

The normalised autocorrelation function of a Rayleigh faded channel with motion at a constant velocity is a zeroth-order Bessel function of the first kind:^[4] Image File history File links Jakes_rayleigh_autocorr_10Hz_doppler. ... Image File history File links Jakes_rayleigh_autocorr_10Hz_doppler. ... Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ... In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ...

at delay when the maximum doppler shift is f_d. The autocorrelation function of the Rayleigh fading channel shown above with 10Hz maximum Doppler shift is shown in the figure. It is periodic in delay and its envelope decays slowly after the inital zero-crossing.

Level crossing rate

The level crossing rate is a measure of the rapidity of the fading. It quantifies how often the fading crosses some threshold, usually in the positive-going direction. For Rayleigh fading, the level crossing rate is:^[5]

where f_d is the maximum Doppler shift and is the threshold level normalised to the root mean square (RMS) signal level: In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...

.

Average fade duration

The average fade duration quantifies how long the signal spends below the threshold . For Rayleigh fading, the average fade duration is:^[5]

.

The level crossing rate and average fade duration taken together give a useful means of characterising the severity of the fading over time.

For a particular normalised threshold value ρ, the product of the average fade duration and the level crossing rate is a constant and is given by

.

Doppler power spectral density

The normalized Doppler power spectrum of Rayleigh fading with a maximum Doppler shift of 10Hz.

The Doppler power spectral density of a fading channel describes how much spectral broadening it causes. This shows how a pure frequency e.g. a pure sinusoid, which is an impulse in the frequency domain is spread out across frequency when it passes through the channel. It is the Fourier transform of the time-autocorrelation function. For Rayleigh fading with a vertical receive antenna with equal sensitivity in all directions, this has been shown to be:^[4] Image File history File links Rayleigh_Doppler_PSD_10Hz. ... Image File history File links Rayleigh_Doppler_PSD_10Hz. ... In applied mathematics and physics, the spectral density is a general concept applied to a signal which may have any physical dimensions or none at all. ... t is the time duratio specific impulse [ ...

,

where is the frequency shift relative to the carrier frequency. Clearly, this equation is only valid for values of between ; the spectrum is zero outside this range. This spectrum is shown in the figure for a maximum Doppler shift of 10Hz. The 'bowl shape' or 'bathtub shape' is the classic form of this doppler spectrum.

Generating Rayleigh fading

As described above, a Rayleigh fading channel itself can be modelled by generating the real and imaginary parts of a complex number according to independent normal Gaussian variables. However, it is sometimes the case that it is simply the amplitude fluctuations that are of interest (such as in the figure shown above). There are two main approaches to this. In both cases, the aim is to produce a signal which has the Doppler power spectrum given above and the equivalent autocorrelation properties.

Jakes' model

In his book,^[6] Jakes popularised a model for Rayleigh fading based on summing sinusoids. Let the scatterers be uniformly distributed around a circle at angles α_n with k rays emerging from each scatterer. The Doppler shift on ray n is A sine wave or sinusoid is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C where A is the amplitude, ω is the angular frequency (2Ï€/P where P is the wavelength), α is the phase shift, and C is the vertical offset. ...

and, with M such scatterers, the Rayleigh fading of the k^th waveform over time t can be modelled as:

.

Here, and the and are model parameters with usually set to zero, chosen so that there is no cross-correlation between the real and imaginary parts of R(t):

and used to generate multiple waveforms. If a single-path channel is being modelled, so that there is only one waveform then can be zero. If a multipath, frequency-selective channel is being modelled so that multiple waveforms are needed, Jakes suggests that uncorrelated waveforms are given by:

.

In fact, it has been shown that the waveforms are correlated among themselves — they have non-zero cross-correlation — except in special circumstances.^[7] The model is also deterministic (it has no random element to it once the parameters are chosen). A modified Jakes' model^[8] chooses slightly different spacings for the scatterers and scales their waveforms using Walsh-Hadamard sequences to ensure zero cross-correlation. Setting The term deterministic may refer to: the more general notion of determinism from philosophy, see determinism a type of algorithm as discussed in computer science, see deterministic algorithm scientific determinism as used by Karl Popper and Stephen Hawking deterministic system in mathematics deterministic system in philosophy deterministic finite state machine... In mathematics, a Walsh matrix is a square matrix, with dimensions a power of 2, the entries of which are +1 or -1. ...

and ,

results in the following model, usually termed the Dent model or the modified Jakes model:

.

The weighting functions A_k(n) are the k^th Walsh-Hadamard sequence in n. Since these have zero cross-correlation by design, this model results in uncorrelated wavforms. The phases can be initialised randomly and have no effect on the correlation properties.

The Jakes' model also popularised the Doppler spectrum associated with Rayleigh fading, and, as a result, this Doppler spectrum is often termed Jakes' spectrum.

Filtered white noise

Another way to generate a signal with the required Doppler power spectrum is to pass a white Gaussian noise signal through a filter with a frequency response equal to the square-root of the Doppler spectrum required. Although simpler than the models above, and non-deterministic, it presents some implementation questions related to needing high-order filters to approximate the irrational square-root function in the response and sampling the Guassian waveform at an appropriate rate. For other uses of the term white noise, see white noise (disambiguation). ... In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ...

See also

• Fading
• Rician fading
• Non line-of-sight
• Line-of-sight
• Wireless

Fading (or fading channels) are mathematical models for the distortion that a carrier-modulated telecommunication signal experiences over certain propagation media. ... In telecommunication, multipath is the propagation phenomenon that results in radio signals reaching the receiving antenna by two or more paths. ... Near line of sight. ... When viewing a scene, as in optics, photography, or even hunting, the line of sight is the straight line between the observer and the target. ... Wireless is an old-fashioned term for a radio receiver, referring to its use as a wireless telegraph; now the term is used to describe modern wireless connections such as in cellular networks and wireless broadband Internet. ...

References

1. ^ John G. Proakis (1995). Digital Communications, 3rd edition, 767–768, Singapore: McGraw-Hill Book Co. ISBN 0-07-113814-5.
2. ^ Bernard Sklar (July 1997). "Rayleigh Fading Channels in Mobile Digital Communication Systems Part I: Characterization". IEEE Communications Magazine 35 (7): 90–100. DOI:10.1109/35.601747 ISSN 0163-6804.
3. ^ Dmitry Chizhik, Jonathan Ling, Peter W. Wolniansky, Reinaldo A. Valenzuela, Nelson Costa, and Kris Huber (April 2003). "Multiple-Input–Multiple-Output Measurements and Modeling in Manhattan". IEEE Journal on Selected Areas in Communications 21 (3): 321–331. DOI:10.1109/JSAC.2003.809457.
4. ^ ^a b R. H. Clarke (July–August 1968). "A Statistical Theory of Mobile Radio Reception". Bell Systems Technical Journal 47 (6): 957–1000.
5. ^ ^a b T. S. Rappaport (December 31, 2001). Wireless Communications: Principles and Practice, 2nd edition, Prentice Hall PTR. ISBN 0130422320.
6. ^ William C. Jakes, Editor (February 1, 1975). Microwave Mobile Communications. New York: John Wiley & Sons Inc. ISBN 0471437204.
7. ^ Von Eckardstein, S. and Isaksson, K. (December 1991). Kanalmodeller för radiotransmission (Channel models for radio transmission) (Master's thesis) (in Swedish), Stockholm, Sweden: Royal Institute of Technology.
8. ^ P. Dent, G. E. Bottomley and T. Croft (24 June 1993). "Jakes Fading Model Revisited". Electronics Letters 29 (13): 1162–1163.