We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{{\bf i}\alpha \cdot }\varphi), \alpha \ne 0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{{\bf i}\alpha \cdot }\varphi$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability 1, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability 1, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.

中文翻译：

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平移不变空间中因果信号的随机无相采样：零分布视角

我们证明了线性相位调制移不变空间 (SIS) 中的无相采样 (PLS) $V(e^{{\bf i}\alpha \cdot }\varphi), \alpha \ne 0,$ 即使实值函数是不可能的 $\varphi$ 享有充分的火花属性（也是 $e^{{\bf i}\alpha \cdot }\varphi$）。换句话说，复杂生成的 SIS 中的 PLS 与实际生成的 SIS 中的 PLS 本质上不同。受此启发，我们首先在复值生成器上建立条件$\phi$ 使得不可分离因果 (NC) 信号的 PLS $V(\phi)$可以通过随机抽样来实现。该条件是从广义 Haar 条件 (GHC) 的角度建立的。基于所提出的重建方法，证明如果 GHC 成立，那么概率为 1，随机采样密度（SD）$=3$对复杂生成的 SIS 中 NC 信号的 PLS 来说是足够的。对于实值情况，我们还证明，如果 GHC 成立，那么概率为 1，随机 SD$=2$对于实际生成的 SIS 中实值 NC 信号的 PLS 来说是足够的。对于诸如啁啾之类的高振荡信号的局部重建，需要大量的确定性样本。与确定性抽样相比，所提出的随机方法不仅具有更大的抽样灵活性，而且样本数量要少得多。为了验证我们的结果，进行了数值模拟以重建啁啾调制 SIS 中的高振荡 NC 信号。