Nonmonotonic Nonstationary Signals

Finally, we apply our algorithm to the case where the instantaneous frequency function is not monotonically increasing or decreasing. Examples of this case are seen in figures 22 through 24. We make the very important observation that the phase inversion model does not in any of these examples show a clear positive or negative characteristic. As a result, the averaging procedure often causes the mean phase concavity to be artificially close zero, so when we apply equation 117, we get a very large $\alpha$ estimate. This is of course not in agreement with the magnitude inversion approximation, which generates a much smaller $\alpha$ estimate since the peak width is still narrow. This can be exploited in a chirp detection setting: if the phase concavity is ambiguous (averages to a very small value, causing a very large $\alpha$ estimate) while the peak width is wider than for a quasistationary sinusoid (but still generates a much smaller $\alpha$ than the phase inversion technique), we claim that we do not have a monotonic instantaneous frequency function.

Figure 22: $q(n) = 2\cdot 2\pi /F\vert n-0.5\vert$

Figure 23: $q(n) = (0.003\cdot n)^2$

Figure: $q(n) = \ln(0.00008\cdot n^2+1.01)$