I sent a perfect 1000 Hz signal from a soundblaster output into a POTS
phone that was directly connected to another POTS phone and a
microphone soundblaster input.

The waveform I got in is at:

http://server3.uploadit.org/files/mike7411-1kclear2.JPG

Any idea what the small bumps are?

I realize there are harmonics at the 1000s, but I don't know about the
small bumps.

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ups.com...
I sent a perfect 1000 Hz signal from a soundblaster output into a POTS
phone that was directly connected to another POTS phone and a
microphone soundblaster input.

The waveform I got in is at:

http://server3.uploadit.org/files/mike7411-1kclear2.JPG

Any idea what the small bumps are?

I realize there are harmonics at the 1000s, but I don't know about the
small bumps.

Try a much larger FFT. 1024 is really small.