On the Basis of Overcomplete Dictionaries

An important case of multiuser channel estimation is considered here, the problem of identifying the crosstalk that disturbs a DSL signal. Crosstalk originates from signals transmitted on nearby pairs in a telephone cable, and couples over unknown pair-to-pair crosstalk coupling channels into the pair carrying the signal. While crosstalk is generally the dominant impairment for current DSL systems, only recently have papers appeared addressing the problem of multiuser crosstalk channel estimation [1], [2]. In [2], it was proposed to identify crosstalk sources by finding the maximum correlation with a “basis set” (dictionary) of representative measured coupling functions. It is shown here that this can be considered equivalent to finding an optimal sparse representation of a vector from an overcomplete set of vectors. A well-known algorithm that solves this problem is the Matching Pursuit (MP) algorithm [4], a greedy algorithm for choosing a subset of vectors from an overcomplete dictionary and finding a linear combination of that subset which approximates a given signal vector. A method based on singular value decomposition (SVD) for reducing the size of the dictionary is also discussed.