Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.