# The Nature of Circles

# What do you mean?

When calculating the mean of a list of numbers, the obvious approach is to sum them and divide by how many there are.

Suppose I give you a list of two numbers:

- 0
- 359

What is their mean? The obvious answer is 179.5.

If I told you that the numbers were compass bearings in degrees, what would your answer be then? Does 179.5 seem correct?

In the case of compass bearings, 0 is the same direction as 360. When talking about angles in the DSP world, we often talk about angles between -π and +π (in radians).

This conundrum is related to Steve Smith's **Nuisance 7**.

# Circular Reasoning

This problem is well-studied [1] and there is a clear solution to the problem [2]: use of vectorial (or phasor) addition for finding the mean. Instead of writing:

_{1}+ x

_{2}

where x_{1} and x_{2} are directions between 0 and 360 degrees, we write

μ = arg ( exp[*j*π x_{1 }/180] + exp[*j*π x_{2} /180 ] ).

where **exp**( ) is the exponential function, *j *is the square root of -1 and **arg**( ) is the argument (complex angle or phase) of the result.

In effect, this is a phasor average rather than a linear average.

# References

[1] Kanti V. Mardia and Peter E. Jupp, "Directional Statistics," Wiley, 1999, ISBN-10: 0471953334.

[2] Lovell, Brian C. and Kootsookos, Peter J. and Williamson, R. C. (1991) The Circular Nature Of Discrete-Time Frequency Estimates. In * IEEE International Conference on ASSP*, May, 1991, pages 3369-3372, Toronto.

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