Rayleigh Distribution
Figure 1: The Rayleight distribution function
Rayleigh Distribution
The Rayleigh distribution, named for William Strutt, Lord Rayleigh, is a continuous probability distribution whose components are independent, positive random variables with a normal distribution. Up to the rescaling, it coincides with the chi distribution with two degrees of freedom. In statistics, two degrees of freedom means that two values in the calculation can vary freely.
Mathematically, the Rayleigh distribution results from two Gaussian distributions px(x) and py(y), which are independent of each other. Consider two independent Gaussian distributed random variables x and y, each with a symmetric bell-shaped curve about the mean value:
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(1)
The joint probability density function (PDF) of two independent variables is the product of the individual density functions because of the statistical independence of px(x) and py(y):
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(2)
If x and y represent noise on the real (In-Phase) and imaginary (Quadrature) parts of a complex signal, we are interested in the probability density function of the magnitude, ρ² = x² + y². Transform to polar coordinates (ρ, ϕ):
Figure 2: The real magnitude of ρ is independent of ϕ
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(3)
In polar form the probability density function is independent of ϕ:
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(4)
Therefore, the ϕ integration simply gives a factor of 2π.
The Rayleigh probability density function is:
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(5a)
(or in another notation)
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(5b)
- σ is the “mode”, or most probable value, and the maximum PDF ( pmax) is:
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- the expected value of p is
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(noise power) and the variance is
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Applications in radar signal processing
Rayleigh distribution is used:
- for the probability density function of the radar cross-section for the Swerling type 1 and 2 models;
- for the amplitude density of bandpass noise.
- The Rayleigh distribution is suitable for describing meteorological clutter, ground clutter of low resolution radar (Antenna beam width greater than 2°, pulse width larger than 1 µs). Generally, it is the most classical model for describing the amplitude distribution of range-heterogeneous clutter. When the radar clutter in a clutter cell contains a number of independent but no strong scattering objects (like chaff interference), the clutter envelope obeys the Rayleigh distribution. These interferences are then also called Rayleigh Clutter.
- Models with Rayleigh distribution are often used to describe diffuse fading situations.