One can define an analogous transform to the Fourier transform that uses square waves as the basis instead of sinusoids. Everything seems to work out in parallel and I imagine one can even come up with a FFT analogue.
We can define the square wave functions in a number of ways. The most direct seems to be
$sq(t) = { 1 \mbox{ if } \frac{\{t\}}{2\pi} < 0.5, \mbox{ else} -1 }$ $cq(t) = sq(t + \pi/2)$
here $sq(t)$ and $cq(t)$ are analogous to $sin(t)$ and $cos(t)$ resp.
The spectrograms are almost identical(well, there is some artifacting in the square wave case) and I was thinking they would be drastically different(the square wave case being more condensed).
Anyone think they can come up with an FFT version or have some fast hardware they could do some computations of various test cases and send me the spectrums if they are different? Or anyone can prove that they should be identical(or close)?
I am using
$\int f(\tau) w(t - \tau) (cq(\tau) + isq(\tau)) d\tau$
as the transform.