Square wave transform 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

$$sq(t) = { 1 \mbox{ if } \frac{\{t\}}{2\pi} &lt; 0.5, \mbox{ else} -1 }$$
$$cq(t) = sq(t + \pi/2)$$

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.
The problem with simply turning sine waves into square waves with "sign" is linear independence. Specifically, this set of square waves is not linearly independent. Pender and Covey (1992) came up with a square wave transform (SWT) composed of sets of phase-shifted waves that are linearly independent though not orthogonal. The transform is interesting because the forward transform is order N, like the Haar Transform, while the inverse transform is order N log N like the FFT and others. The Pender/Covey SWT is also very fast because it only uses sums and bit shifts.
The problem with simply turning sine waves into square waves with "sign" is linear independence. Specifically, this set of square waves is not linearly independent. Pender and Covey (1992) came up with a square wave transform (SWT) composed of sets of phase-shifted waves that are linearly independent though not orthogonal. The transform is interesting because the forward transform is order N, like the Haar Transform, while the inverse transform is order N log N like the FFT and others. The Pender/Covey SWT is also very fast because it only uses sums and bit shifts.