# Raised cosine distribution

< Probability distributions list > Raised cosine distribution

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 [2]

The PDF can also be written as

where hvc is the havercosine.

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

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.

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].

## References

[1] Graphed with Desmos.

[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] 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).