I have encountered a formulation simplification problem in engineering.
Consider integral: $$h(x) = \left| \int_{-\infty}^{\infty}\! e^{\mathrm{j} x \omega} F(\omega) e^{\mathrm{j} g(x - t\omega)} \, \mathrm{d} \omega \right|^2$$ where $x$ and $\omega$ are Fourier duals in $\mathbb{C}$, $F(\omega)$ is the Fourier transform of $f(x)$, $g(x)$ is an unknown function in $\mathbb{R}$, and $0 \leq t \leq 1$.
The engineering problem is, given $h(x)$, $|f(x)|^2$ and $t$, estimate $g(x)$.
I know:
If $g(x-t\omega) = \text{const}$, $h(x) = |f(x)|^2$.
If linearize $g(x-t\omega)$ as:
$$g(x - t\omega) \approx g(x) - t g'(x) \omega$$ then $$ h(x) = \left| e^{\mathrm{j} g(x)} \int_{-\infty}^{\infty}\! e^{\mathrm{j} [x - tg'(x)] \omega} F(\omega) \, \mathrm{d} \omega \right|^2 = \left| f(x - tg'(x)) \right|^2 $$ i.e. consequence of the phase-shift theorem in Fourier transforms. By tracking the movements between case 1 and case 2, $g'(x)$ can be obtained, follow by an integration to get $g(x)$. However this model is not accurate for larger $\omega$ because $t\omega$ will be significantly large beyond locality, thus this linearization simplification is only valid for low frequencies (i.e. small $\omega$'s).
My questions:
Except doing linearization, what else can I do to simplify the original integral? A neat formulation for probable estimation for $g(x)$ is preferable.
If linearization is the only way, how should I deal with the higher frequencies (large $\omega$'s)?