An even more interesting example of partial periodicity

Continuing onto what I've said in message #102004, this example may be even more interesting than the previous ones although not as easy to understand. This time, there's a rolloff meaning that the intensity is inversely proportional to frequency. It's a stereo sound where one channel contains frequencies of 1:5:9:13:17:... while the other contains frequencies of 3:7:11:15:19:...

http://dl.dropbox.com/u/8497979/1vs3_over_4_rolloff.wav

Although only one of them actually contains a sine wave of 1Hz, the general periodicity of 1Hz is obvious in both of them. And although none of them contains a sine wave of 4Hz, the partial periodicity of 4Hz is also pretty apparent in both of them.

What's more, since the impulses which are 250 ms apart are just 90-degree or -90-degree phase-shifted copies of each other, then summing the two channels results in a square wave while subtracting them results in an approximated phase-shifted square wave delayed by 250 ms. Of course, it's mathematically impossible to exactly phase-shift a "step" or a "switch" because the resulting impulse would run into infinitely high amplitudes and infinite lengths. But the periodic approximations given here are very good anyway.

Petr