This paper addresses recovery of a kernel $\boldsymbol{h}\in \mathbb{C}^{n}$ and a signal $\boldsymbol{x}\in \mathbb{C}^{n}$ from the low-resolution phaseless measurements of their noisy circular convolution $\boldsymbol{y} = \left \rvert \boldsymbol{F}_{lo}( \boldsymbol{x}\circledast \boldsymbol{h}) \right \rvert^{2} + \boldsymbol{\eta}$, where $\boldsymbol{F}_{lo}\in \mathbb{C}^{m\times n}$ stands for a partial discrete Fourier transform ($m<n$), $\boldsymbol{\eta}$ models the noise, and $\lvert \cdot \rvert$ is the element-wise absolute value function. This problem is severely ill-posed because both the kernel and signal are unknown and, in addition, the measurements are phaseless, leading to many $\boldsymbol{x}$-$\boldsymbol{h}$ pairs that correspond to the measurements. Therefore, to guarantee a stable recovery of $\boldsymbol{x}$ and $\boldsymbol{h}$ from $\boldsymbol{y}$, we assume that the kernel $\boldsymbol{h}$ and the signal $\boldsymbol{x}$ lie in known subspaces of dimensions $k$ and $s$, respectively, such that $m\gg k+s$. We solve this problem by proposing a blind deconvolution algorithm for phaseless super-resolution (BliPhaSu) to minimize a non-convex least-squares objective function. The method first estimates a low-resolution version of both signals through a spectral algorithm, which are then refined based upon a sequence of stochastic gradient iterations. We show that our BliPhaSu algorithm converges linearly to a pair of true signals on expectation under a proper initialization that is based on spectral method. Numerical results from experimental data demonstrate perfect recovery of both $\boldsymbol{h}$ and $\boldsymbol{x}$ using our method.

翻译：这张纸将解析 $\ boldsymbol{ h{ h} in\ mathb{ C} $ 和信号$\ boldsymology{x}\ mathb{ C} 美元,因为低分辨率的测量 他们的噪音环形变换$\ boldsymbol{y} = left\ rver\ boldsymbol{ F ⁇ } (\ bysymbol{x} listymsl{x} comsymbl{cl{clb} 美元 和信号$xxxxxxxlmathb} 美元。 美元是部分离散的Fouriery 变换(美元) 美元, $\ boldsymallsy_ blorxxx dismlations a froomy_ matter_ maxy_ blor_ max max max max max mail_mail_max max mail_max max max max max max max max mox mox mox max mox mox max max mox max max mox mox mox max max mox mox mox mox mox mox mox mox mox mox mox max mox mox mox mox mox mox mox mox mox mox mox mox mox mox mox mox mox max mox mox mox mas mox mox mox mox mox mox mas mos mos mos mos mos mos mos mos mos mos mos mos mos mos mos mox mos mos mos mos mos