Yes you can do it with spectrum analysis software. In the early 90s back in Georgia we had a guy in our group with the software running on a lap top. He used a directional microphone and would record a fly-by. Using frequency and doppler shift he could calculate engine rpm and aircraft speed. He recorded the data in ram. I don't know how many data samples he took nor how much band width he used. It took his old lap top a few minutes to do the calculations but it seemed to be very accurate.