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Just looking for general guidance, as an absolute newbie in the field. How would I go about generating and detecting a narrow pulse with SDR (say, HackRF One) in one of the ISM bands?

Let's say I am aiming for a 1 microsecond wide gaussian pulse. Am I correct in assuming that HackRF (with its 20 MHz sampling rate) would be able to adequately sample the pulse and allow me to determine its time of arrival?

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Let's say I am aiming for a 1 microsecond wide gaussian pulse

you're sampling at 20 MS/s, so that pulse would be 20 samples long. So, that would be very easy to detect, e.g. simply with a treshold, or by correlating with the same shape (or with an appropriate high- or bandpass filter, or a difference of Gaussians approach – many many options, depending on what you know about your pulse shape, specifically).

Note that you don't need to think of this as "narrow pulse" (it really isn't that narrow in relationship to pulse rates of most communication systems on these bands), but simply as a ca 1 MHz wide signal.

allow me to determine its time of arrival?

Well, down to the accuracy limits that math sets, yes.

(if you get to choose what you transmit, a Gaussian pulse might – usually – not be the pulse shape of choice for a ToA estimator)

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