convolution(a(t),b(t)) in time domain is equivalent to multiply(A(s),B(s)) in complex frequency domain, and vice versa.
Sure. Thing is, if you have a chirp, a frequency sweep, and you window it (with overlapping windows) with a suitable size, and do an FFT of that, you need to account for the effect of the windowing function on the computed FFT, to get a representation of the original signal.
Even then, that FFT is a frequency-domain representation of the time-domain signal during that time window. Because the frequency changes continuously in the time domain, the frequency domain magnitude spike is always spread out a bit. (It is also spread out a bit due to the windowing et cetera, but the changing frequency exacerbates that.) If the rate of frequency change is sufficiently low compared to the FFT size, then the effect is small.
One can take the windowed FFT centered at any point in time, even from consecutive samples. But even then, the FFT represents the frequency domain of the signal
within the FFT time window, not at a particular sample.
The question is, what does one want to find out? FFT is a poor tool if you want to measure e.g. attenuation or phase change of a known signal, but a good tool if you are interested in the frequency spectrum of the measured signal.
Constructing a shaped pulse of a specific frequency yields an analytically known easily calculated frequency spectrum. The pulse is self-windowing, so that taking the FFT of the measured signal in one chunk, from silence before to silence after, yields directly the frequency spectrum,
exactly because we are not interested in any time-dependent variance in it. The two can be compared one-to-one, without any convolution needed (to undo the effect of a windowing function). And taking the FFT of the real-world generated signal, and comparing that to the computed Fourier transform, tells how well the signal generation and capture matches the expected; basically gives a simple test/calibration approach, albeit at a single frequency per test pulse.
Mathematically this is rather lightweight, because the Fourier transform can be calculated as needed (it does not need to be precalculated or stored).
Convolution is just one (complex) multiplication
sum per FFT bin for each FFT, so it isn't that costly; that's not the issue. The issue is its practical effect on the FFT. Consider input time-domain data consisting of 8-bit samples. The quantization noise creates a noise floor at about 44-48 dB (it is more or less flat, similar to white noise). If you then apply any kind of convolution to compensate for the effect of the windowing function on the spectrum, it will apply to the quantization noise as well. Then, if your compensated samples are mapped back to integers, you have a second set of quantization errors, except this time in the frequency domain (as the convolution then does not exactly match the intended one). It gets quite fuzzy fast, so omitting the entire step can be a big win in the frequency domain accuracy.
(Edited to add the missing crucial
sum in the above paragraph.)
Now, it is important to realize that any signal can be reconstructed from its Fourier transform. This means that if you do an FT over a single, complete chirp, the FT is a complete description of it: it does describe how it progresses through the frequency, and you can reproduce the original time-domain signal using the FT. (I do believe this is also algebraically known, if the frequency changes during the chirp in an easily described manner, say linearly or exponentially.)
It is only that it is difficult to see or determine how the time-domain signal evolved during the FFT window from the FFT data itself.
(I don't know if it is even possible in any other way except actually reconstructing the time-domain signal, and observing its characteristics.)
Because of this, a single FFT over an entire chirp, compared to the computed (or measured at signal generator), does describe the full frequency-domain response, but it is very difficult to say which extraneous frequencies were caused by which input frequencies, because that information (albeit embedded within the FFT, since it can reconstruct the original signal) is difficult if not impossible to extract directly from the FFT itself.
To recover that, the measured signal is windowed, into many overlapping windows, with window length corresponding to the lowest frequency one is interested in (taking into account the windowing function effect on the resulting FFT), and FFT taken of each window separately.
What I found interesting, and why I posted these few messages, is that generating a different signal –– a modulated constant-frequency pulse ––, there would be no need for either windowing (because the signal itself is naturally "one window"), nor compensating for windowing function effects. It would reveal details related to only a very narrow slice of the entire possible input frequency domain, but what it would reveal, would be clear and straightforward, standalone description.
Now, I just hope I didn't waste time for anyone who read this, because while I find it interesting, I haven't checked it out in practice; only done "back of the envelope" calculations and estimates to verify it indeed is interesting, and not just a random idea that popped into my mind. Hopefully, the above explains what and why I found interesting in it.