EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

Fig. 21 – Circular synchronization index. r = magnitude of coherence
22 – Phase straightening
As mentioned previously phase angle has an intrinsic discontinuity,
where 0 and 360 are at opposite ends while in the circular distribution 0 0 =
360 0 (Jammalamadaka and SenGupta, 2001). A method to remove
discontinuities due to the mathematical limit of the arctangent is a procedure
called “phase straightening” by Otnes and Enochnson (1972, p. 238). The
procedure involves checking for a large jump which happens when the phase
goes from + 180 0 to – 180 0 and then adding or subtracting 360 0 depending
on the direction of sign change. For example, ∆θ = (180 – ε) 0 + (180 – ε) 0 =
360 0 - 2ε which is the same as 2ε since -(180 – ε) 0 = 180 + ε. This
procedure results in phase being a smooth function of time or frequency and
removes the discontinuities. The programmer needs only to keep track of
the number of winds around the circle also called the “winding number” if
absolute phase differences are needed.