One wouldn’t typically record bird sounds within a city environment, but go to a wooded are or a meadow. You could make an assumption that a bird call occupies primarily the mid-high frequencies, while wind and distant highway or idling train engines emit a low rumble. Then use an equalizer with a “highpass filter” to remove a long-term average voltage leaving the faster fluctuations.
A more elaborate approach would be using a “noise gate” that is triggered by sound above a certain threshold, to smoothly fade the remaining background noise in and out at a rate that matches the rise and decay the wanted sound. This would result in the noise breathing.
The bandwidth can be split into several ranges using stacked filters and each of those bands adjusted by a separate noise gate. Now the bands further away from the tones of the bird call can be reduced by a bigger amount. When overdone, this approach yields a result similar to mp3 compression with more noticeable tonal components breathing or suddenly popping out of unnatural silence.
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