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IIRC, Hamming's book "Digital Filters" (1989) has a section on FFTs with only the sign of the coefficient being used. It performed surprisingly well.

What is the sign of a complex number? Do you mean the phase?

AFAICT, both the real and imaginary components are from (-1, 0, +1) only. No single sign, but only 8 directions and the center.

You mean Fast Hadamard Transform?