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首页> 外文期刊>IEEE Transactions on Information Theory >Blind Deconvolution Using Convex Programming
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Blind Deconvolution Using Convex Programming

机译：使用凸规划进行盲反卷积

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摘要

We consider the problem of recovering two unknown vectors, ${mbi w}$ and ${mbi x}$, of length $L$ from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension $N$ and the other with dimension $K$. Although the observed convolution is nonlinear in both ${mbi w}$ and ${mbi x}$, it is linear in the rank-1 matrix formed by their outer product ${mbi w}{mbi x}^{ast}$. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that, for “generic” signals, the program can deconvolve ${mbi w}$ and ${mbi x}$ exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is, we show that if ${mbi x}$ is drawn from a random subspace of dimension $N$, and ${mbi w}$ is a vector in a subspace of dimension $K$ whose basis vectors are spread out in the frequency domain, then nuclear norm minimization recovers ${mbi w}{mbi x}^{ast}$ without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length $N$, which we code using a random $Ltimes N$ coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length $K$, then the receiver can recover both the channel response and the message when $Lgtrsim N+K$, to within constant and log factors.
机译：我们考虑从圆形卷积中恢复长度为LL的两个未知向量$ {mbi w} $和$ {mbi x} $的问题。我们进行结构假设，即两个向量都是已知子空间的成员，一个向量的维数为$ N $，另一个向量的维数为$ K $。尽管在$ {mbi w} $和$ {mbi x} $中观察到的卷积都是非线性的，但在由其外积$ {mbi w} {mbi x} ^ {ast} $形成的1级矩阵中它是线性的。该观察结果使我们能够从线性测量中将反卷积问题重铸为低秩矩阵恢复问题，其自然凸松弛是核标准最小化程序。我们通过证明对于“一般”信号，该程序可以使$ {mbi w} $和$ {mbi x} $去卷积，恰好在$ N $和$ K $的最大值几乎等于$ L $的订单。也就是说，我们表明如果$ {mbi x} $是从维度为$ N $的随机子空间中得出的，而$ {mbi w} $是维度为$ K $的子空间中的向量，其基础向量分布在在频域上，然后核规范最小化就可以正确地恢复$ {mbi w} {mbi x} ^ {ast} $。我们在通信中的盲信道估计的背景下讨论此结果。如果我们有一条长度为$ N $的消息，并使用一个随机的$ Ltimes N $编码矩阵进行编码，并且编码后的消息通过了一个未知的线性的时不变的，最大长度为$ K $的信道，则接收方可以恢复$ Lgtrsim N + K $时，通道响应和消息必须在常数和对数因子之内。

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