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A rather simple approach to approximating an instantaneous (unwrapped) phase $φ$ from a time series is as follows:

  • Define some a appropriate marker events (e.g., upwards zero crossings) $t_0 < … < t_m$.
  • Define $φ(t_j):=2πj$.
  • For every other point, define $φ$ such that it linearly interpolates between the existing points.

This approach is often attributed to S.O. Rice and called the Rice phase. More precisely his paper Mathematical analysis of random noise¹ is cited for it, but all I could find in this publication are some analyses of the rate of zero crossings for some stochastic processes. Given that this paper has 163 pages, I might have missed something, however.

My question is thus: Who did first propose this concept? Note that I am only interested in “direct“ evidence, i.e., you have to be able to point me to the very point in some publication where it happens. This in particular holds, if your answer is Rice.


¹ S.O. Rice – Mathematical analysis of random noise, Bell System Technical Journal 23/24, 1944/1945 (first part, second part)

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