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 $N$, then at least $\tau (N)$ DFT coefficients must be sampled, where $\tau (N)$ is the number of factors of the signal length $N$. A recovery algorithm is proposed and implemented. It can achieve the lowed bound for length $N=2^{m}, m \leq 6$. To overcome the length limitation problem, a more practical recovery method is also proposed and implemented for $N=2^{m}, m>6$. 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.

中文翻译：

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从部分 DFT 系数中完美恢复二进制信号

研究了如何从离散傅里叶变换 (DFT) 系数中完美地恢复二进制信号。推导出和开发了理论下限和实际恢复策略。引入歧义对的概念。除了某些位置外，这对信号具有几乎相同的 DFT 系数。可以证明当信号长度为$N$，那么至少$\tau (N)$必须对 DFT 系数进行采样，其中$\tau (N)$是信号长度的因子个数$N$. 提出并实现了一种恢复算法。它可以达到长度的下限$N=2^{m}, m \leq 6$. 为了克服长度限制问题，还提出并实施了一种更实用的恢复方法$N=2^{m}, m>6$. 我们可以采样 11% 的总 DFT 系数来完美地恢复二进制信号。我们还将模糊对的概念扩展到其他离散变换（DCT 和 WHT）和二维 DFT 情况。