Atomic norm denoising with applications to line spectral estimation
Atomic norm denoising with applications to line spectral estimation
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DOI：
10.1109/allerton.2011.6120177
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发表时间：
2011-09
期刊：
影响因子：
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通讯作者：
Badri Narayan Bhaskar;Benjamin Recht
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文献类型：
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作者：
Badri Narayan Bhaskar;Benjamin Recht
线谱的子奈奎斯特估计是信号处理中的一个经典问题，但目前流行的基于子空间的方法在存在噪声的情况下难以保证，而且依赖于系统模型阶数的先验知识。受最近反问题中原子规范研究的启发，我们提出了一种新的线谱估计方法，该方法为存在噪声且不需要预先知道模型顺序的均方误差性能提供了理论保证。我们提出了一种抽象的原子规范去噪理论，该理论专门用于估计具有均方误差保证界的复指数混合物的频率和相位的凸优化问题。一般来说，我们提出的优化问题没有已知的多项式时间解，但我们提供了一种有效的算法，称为DAST，基于快速傅里叶变换，实现几乎相同的错误率。我们将DAST与Cadzow的典型交替投影算法进行了比较，后者在高信噪比下表现略好，当模型顺序确切已知时，并通过实验证明DAST在广泛的信噪比范围内优于其他去噪技术，包括Cadzow的。
The sub-Nyquist estimation of line spectra is a classical problem in signal processing, but currently popular subspace-based techniques have few guarantees in the presence of noise and rely on a priori knowledge about system model order. Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectrum estimation that provides theoretical guarantees for the mean-square-error performance in the presence of noise and without advance knowledge of the model order. We propose an abstract theory of denoising with atomic norms which is specialized to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials with guaranteed bounds on the mean-squared-error. In general, our proposed optimization problem has no known polynomial time solution, but we provide an efficient algorithm, called DAST, based on the Fast Fourier Transform that achieves nearly the same error rate. We compare DAST with Cadzow's canonical alternating projection algorithm, which performs marginally better under high signal-to-noise ratios when the model order is known exactly, and demonstrate experimentally that DAST outperforms other denoising techniques, including Cadzow's, over a wide range of signal-to-noise ratios.