Model Inversion

Given our model, we may now consider its inversion. That is, given FFT magnitude and phase characteristics that could be generated by equations 92 and 93 (or 105 and 106), we claim that we have a linear frequency chirp signal. By the uniqueness of the non-aliased FFT, we claim that only a chirp signal characterized by a specific $\alpha$ could generate the same FFT characteristics.

Because both the phase and magnitude approximations for our model depend on $\alpha$, we may attempt to invert either the phase equation or the magnitude equation to estimate $\alpha$. To determine which will yield a more accurate estimate, we examine the way errors will affect either approximation. Up to this point, we have only considered errors as they effect the real and imaginary parts of the signal, not as they affect the phase and magnitude. Using standard error propagation analysis [7], it becomes evident that obtaining the magnitude (via squaring, addition, and square root operations) propagates error in a less data-degrading way than obtaining the second order phase difference (via division, arctangent, and two differencing operations). Empirical observation supports this claim: inverting the magnitude model as described below yields generally more accurate estimates of $\alpha$ than inverting the phase model. We provide details and results for both algorithms below.