Rectangular Fourier Transform and its Properties

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Is there a name for a transformation using the orthonormal base

$$s_k(x)=\lceil \sin kx \rceil,\: c_k(x) = \lceil \cos kx \rceil \quad ?$$

So basically a Fourier transform or Fourier series using periodic rectangles. What are the properties? Is there some kind of convolution theorem?

 

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I found some answers. The Walsh-transform looks very similar. I noticed that the functions are not orthogonal so sign(sin(kx)) and sign(cos(kx)) is probably a better choice.