Other Monotonic Nonstationary Signals

Now, we apply the algorithm to nonlinear instantaneous frequency functions using only the logic that such functions might be well modeled as quasi-linear. In figures 18 through 21, we show exponential, quadratic, cubic, and logarithmic frequency functions $q(n)$, as well as our system's guess of a parameter $\alpha$, estimated using inversion techniques identical to those used in the linear frequency chirp case above. To determine how close the algorithm is to optimal for these cases, a least squares fit of an affine function passing through the same detected center frequency is also shown in the third subplot (dot-dash). The corresponding $a$ value, $a_{ls}$, is shown above the plot with the other estimates. All other plots are presented in the same manner as for the linear frequency case. We make an interesting observation that the phase inversion approximation generally tracks the frequency trajectory more closely than than the magnitude inversion approximation. In most cases, we see that this approximation is within 3% of $a_{ls}$, showing that our blind algorithm is, in fact, nearly optimal. In future research, we will explore the underlying cause of this success for the phase inversion approximation.

Figure: $q(n) = .08\cdot\exp((n-20)/40)$

Figure 19: $q(n) = .003\cdot (n+(K-1)/2)^2$

Figure 20: $q(n) = .0025\cdot (n+(K-1)/2)^3$

Figure: $q(n) = \ln(0.03\cdot(n+(K-1)/2))$