Combining pulses to create a longer pulse

Question

This might be a silly question but i am very new to DSP. From my understanding using FFT on a small amount of very narrow pulses results in very poor frequency estimates because when the pulse is on there usually isnt enough samples to obtain an accurate estimate (i could very well be wrong here). My question is about combining pulses together to create one longer pulse. If you did some thresholding and acquired just the IQ samples of the actual pulse(when it is on), could you add these together to make one longer pulse? If not why not? Does it have something to do with the phase? Apologies if this doesnt make sense its still a bit unclear to me

Answer 1

It is correct that the frequency estimates are related to the duration of the pulse, assuming the signal is stationary over the pulse duration. The resolution bandwidth of each bin in the FFT is equal to the inverse of the FFT duration, or the pulse duration, whichever is shorter. To ideally combine multiple pulses, we first must assume the stationarity has been maintained from pulse to pulse and that we can align the phase such that the phase is coherent when the pulse are stitched together. Consider the simple case of a single sine wave-- if the sine wave is continuous throughout its duration then we can achieve the stated resolution. However if there are abrupt transitions when the pulses are stitched together, the discontinuities will result in many other frequencies in the Fourier Transform and significant inaccuracies in an estimate of the actual tone. This is similar to a disruptive phase modulation on this original tone; a worst case condition would be if the phase were to effectively flip back and forth 180 degrees which result in carrier suppression causing the tone at the original frequency to completely disappear, and be replaced by sidebands at the rate of the sub-pulse durations.

Below shows an example of stitching together four arbitrary pulses in the time domain as given in the first plot: the upper plot is what we would get if we had a pulse aligned in phase throughout its duration, and the lower plot is stitching together with arbitrary phase (note the 3 transitions). Below that is the FFT result for each in the frequency domain, with the orange trace showing the disruption that can occur.