< 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 cumulative distribution function (CDF) is

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] 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).