I always wondered why log amps were not used. Now, as you surmise, with DSPs and good ADCs it ought to be possible to do true video integration, which is widely used in radar signal processing. Anyway, the following approach suggests itself to me:
1. Use a random pattern of interpulse intervals, to decorrelate all manner of background noise. This would eliminate the need to vary the pulse frequency, and is easy if one uses computer control, which is likely if one also uses DSPs.
2. Sum the received signal of at least 100 identical successive pulses. The analog-digital converter should be set up to start converting at a fixed offset from the pulse end (coil-current turnoff), and to convert at a fixed sample rate, so the resulting time series can be summed sample by sample. If wide pulses and narrow pulses are used, sum wide pulses only with wide pulses, and narrow with narrow. Timing from turn-off allows the resulting summed time series for wide pulses and for narrow pulses to be more easily compared, as the turn-off is the same for all pulse widths.
This process will cause most noise to average out. The signal voltage will grow linearly with the number of pulses included in the sum, while uncorrelated noise will grow with the square root of the number of included pulses. So, the signal to noise ratio will grow as the square root of the number of pulses, increasing by 10:1 for 100 pulses.
3. Take the logarithm of the sums, yielding a vector of logarithms. The noise-reduction from the prior step of summing multiple successive pulses will greatly reduce the noise.
4. Form the reverse cumulative sum of the vector of logarithms. This will preserve the shape of exponential-decay slopes, while further reducing the noise. (I posted a long-winded description of the reverse cumulative sum approach some months ago.)
Joe Gwinn