Binary Signal Perfect Recovery From Partial DFT Coefficients

Soo-Chang Pei, Kuo-Wei Chang

Abstract: How to perfectly recover a binary signal from its discrete Fourier transform (DFT) coefficients is studied. The theoretic lower bound and a practical recovery strategy are derived and developed. The concept of ambiguity pair is introduced. This pair of signals has almost the same DFT coefficients except for some positions. It can prove that when the signal length is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> , then at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\tau (N)$</tex-math></inline-formula> DFT coefficients must be sampled, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\tau (N)$</tex-math></inline-formula> is the number of factors of the signal length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> . A recovery algorithm is proposed and implemented. It can achieve the lowed bound for length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N=2^{m}, m \leq 6$</tex-math></inline-formula> . To overcome the length limitation problem, a more practical recovery method is also proposed and implemented for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N=2^{m}, m&gt;6$</tex-math></inline-formula> . We can sample 11% of the total DFT coefficients to perfectly recover the binary signal. We also extend the concept of ambiguity pair to other discrete transforms (DCT and WHT) and two-dimensional DFT cases.