Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model

We consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input. From these observations, we wish to estimate the impulse responses of each of the channels. We show that this problem is well-posed if the chann...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 64; no. 7; pp. 4792 - 4818
Main Authors: Lee, Kiryung, Tian, Ning, Romberg, Justin
Format: Journal Article
Language:English
Published: New York IEEE 01.07.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Asymptotic methods
Blind deconvolution
Channel estimation
Channels
Computational geometry
Convexity
Convolution
Deconvolution
eigenvalue decomposition
Eigenvalues
Eigenvalues and eigenfunctions
Eigenvectors
Empirical analysis
Estimates
Iterative methods
Mathematical analysis
Mathematical model
Mathematical models
Matrix methods
Model accuracy
non-convex optimization
Optimization
Quadratic equations
sensitivity analysis
Signal processing algorithms
Well posed problems
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:We consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input. From these observations, we wish to estimate the impulse responses of each of the channels. We show that this problem is well-posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low-dimensional subspace but with each channel modulated by an independent gain. Under this model, we show how the channel estimates can be found by minimizing a quadratic function over a non-convex set. We analyze two methods for solving this non-convex program, and provide performance guarantees for each. The first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems. The second is a truncated power iteration, which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model. As with most non-convex optimization algorithms, the performance of both of these algorithms is highly dependent on having a good starting point. We show how such a starting point can be constructed from the channel measurements. Our performance guarantees are non-asymptotic, and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy. Our analysis uses a model with a "generic" subspace that is drawn at random, and we show the performance bounds hold with high probability. Mathematically, the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean. We also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2018.2840711