压缩多通道盲反卷积的可辨识性条件
Identifiability Conditions for Compressive Multichannel Blind Deconvolution
AI总结：
针对稀疏多通道盲反卷积的不适定可辨识问题，推导傅里叶域压缩测量下的确定性可辨识条件，提出核采样方案，以更少测量和通道实现辨识，助力低成本接收机构建。
摘要：
在多接收机雷达、超声阵列系统等应用中，观测信号通常可建模为代表发射脉冲的未知信号与描述稀疏目标场景的稀疏滤波器的线性卷积。辨识未知信号与稀疏滤波器的问题属于稀疏多通道盲反卷积（MBD）问题，通常是不适定的。本文研究稀疏MBD的可辨识性问题，证明与压缩感知类似，可通过输出序列的压缩测量值辨识稀疏滤波器。具体而言，我们考虑傅里叶域的可压缩测量，在确定性框架下推导了可辨识性条件。主要结果表明，仅需两个互质通道的2L²个傅里叶测量值，即可辨识L-稀疏滤波器；同时证明每个通道至少需要2L个测量值。当通道数渐近增加时，充分条件会进一步收紧，每个通道仅需采集约L个傅里叶样本即可满足要求。我们还提出了一种基于核的采样方案，可从数量相当的时间样本中采集傅里叶测量值。通过数值实验（包括对比实际重建算法），我们讨论了充分条件与必要条件之间的差距。与已有结果相比，本文提出的压缩MBD结果所需的测量值和通道数更少，有助于构建低成本接收机。
英文摘要：
In applications such as multi-receiver radars and ultrasound array systems, the observed signals can often be modeled as a linear convolution of an unknown signal which represents the transmit pulse and sparse filters which describe the sparse target scenario. The problem of identifying the unknown signal and the sparse filters is a sparse multichannel blind deconvolution (MBD) problem and is in general ill-posed. In this paper, we consider the identifiability problem of sparse-MBD and show that, similar to compressive sensing, it is possible to identify the sparse filters from compressive measurements of the output sequences. Specifically, we consider compressible measurements in the Fourier domain and derive identifiability conditions in a deterministic setup. Our main results demonstrate that $L$-sparse filters can be identified from $2L^2$ Fourier measurements from only two coprime channels. We also show that $2L$ measurements per channel are necessary. The sufficient condition sharpens as the number of channels increases asymptotically in the number of channels, it suffices to acquire on the order of $L$ Fourier samples per channel. We also propose a kernel-based sampling scheme that acquires Fourier measurements from a commensurate number of time samples. We discuss the gap between the sufficient and necessary conditions through numerical experiments including comparing practical reconstruction algorithms. The proposed compressive MBD results require fewer measurements and fewer channels for identifiability compared to previous results, which aids in building cost-effective receivers.
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