Raised cosine distribution

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The raised cosine PDF. [1]

The raised cosine distribution (RCD) is a bell-shaped distribution that is an approximation to the normal distribution. It is used in a variety of real-life situations, especially in signal processing.

Properties of the raised cosine distribution

The probability density function (PDF) for the standard raised cosine distribution is

standard raised cosine distribution pdf

which can also be written as

The cumulative distribution function (CDF) is

cdf raised cosine distribution

The raised cosine distribution type II (RCD II) has PDF [2]

rcd type II

When a = 0 and b = 1, this becomes the standard RCD.

There are many possible parameterizations of the RCD II: changing the values of the parameters a and b result in a variety of bell-shaped curves., which will have a mean of a. The figure below (plotted on Desmos) shows plots of the PDF for b = 1, 1.5, 2.5, 3, and 3.5 when a = 0:

parameterizations of the RCD II
Plots of the RCD Type II PDF for a = 0 and b = 1 (green), 1.5 (red), 2.5 (blue), 3 (purple), and 3.5 (black).

The moment generating function for the RCD Type II is

And the characteristic function is

Percentiles for the RCD cannot be expressed in explicit form [3].

The x in the raised cosine distribution is often an angle direction such as wind-speed direction, angular scattering of molecules in motion or angles of cornea curvature [4]. It can also be an angle in a transformed space as seen in automatic color recognition problems using color chroma in RGB coordinate systems [2].

Uses for the raised cosine distribution

The raised cosine distribution (RCD) has many applications in real world data. For example:

  • The raised cosine distribution can be used to model the shape of digital signals and model noise in communication channels.
  • It can approximate step functions and can also be used to avoid inter symbol interference in communications systems [4] — a form of distortion in digital communications systems that happens when pulses representing one symbol overlap with the pulses representing adjacent symbols.
  • In queuing theory, the RCD can model the waiting times in queues.
  • The RCD can be used to generate pseudorandom numbers [5].


[1] Graph of raised cosine PDF created with Desmos.com.

[2] Chattamvelli, R., Shanmugam, R. (2021). Cosine Distribution. In: Continuous Distributions in Engineering and the Applied Sciences – Part I. Synthesis Lectures on Mathematics & Statistics. Springer, Cham. https://doi.org/10.1007/978-3-031-02430-6_7

[3] Rinne, H., Location Scale Distribution: Linear Estimation and Probability Plotting Using MATLAB, Giessen, Germany.(2010).

[4] Shanmugam, R. (2020), What do angles of cornea curvature reveal? a new (Sinusoidal) probability density function with statistical properties assists

[4] Kyurkchiev, V., and Kyurkchiev, N., On the approximation of the step function by raised-cosine and laplace cumulative distribution functions. European International Journal of Science and Technology, 4(9), 75 – 84. (2016).

[5] stdlib-js/random-base-cosine

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