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:
where x1 and x2 are directions between 0 and 360 degrees, we write
μ = arg ( exp[jπ x1 /180] + exp[jπ x2 /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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