In this paper we consider the problem of recovering a signal $x \in \mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is $\sim\! N/4$ then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend earlier results of Wang and Xu \cite{arXiv:1310.0873}.

翻译：本文考虑从信号的功率谱中恢复信号 $x \in \mathbb{R}^N$ 的问题，假设该信号相对于 $\mathbb{R}^N$ 的某组通用基具有稀疏性。我们的主要结论是：若在该基下的稀疏度不超过 $\sim\! N/2$，则通用稀疏向量可由其功率谱在符号意义上唯一确定。我们还证明，当稀疏度为 $\sim\! N/4$ 时，每个稀疏向量均可由功率谱在符号意义上唯一确定。针对 $\mathbb{C}^N$ 中向量的功率谱，本文亦得到了类似的结论，这一结果拓展了Wang与Xu的早期工作 \cite{arXiv:1310.0873}。