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Suppose I have a sequence of functions $f_N(t)= a(\frac t N ) e^{i \omega t}$ where $\omega$ real and $a$ is a compactly supported smooth function.

I would like to determine $a$ and $\omega$ by using the Fourier transform. (N.B., there may be other, better ways of doing it. I am not interested in those.)

My idea is to pick some (smooth, compactly supported) window function $w$, with $w(0)=1$, and compute $X_N(v,\tau) = \int f_N(t) w_N(t-\tau) e^{i v t} dt$, a short-time Fourier transform of $f_N$. Here $w_N(t) = w(\frac {t} {\sqrt{N}})$.

The functions $f_N(t)$ will, for large $N$, look like a complex exponential with an amplitude that is slowly changing. My intution tells me that a short-time Fourier transform at the time $N \tau$ ought to therefore resemble $a(\tau) \delta_w$. By increasing window size this resemblance ought to become stronger.

I therefore conjecture that for fixed $\tau$ the measures on $\mathbb{R}$ induced by $X_N(v,N*\tau)$ converges (weakly) towards the measure $a(\tau) \delta_w$. Am I right? Or is my intuition completely wrong?

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