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In compressed sensing it is generally assumed that the dictionary matrix constitutes a (possibly over-complete) basis of the signal space. In this paper, the authors consider dictionaries that do not span the signal space, i.e., rank deficient dictionaries. They show that in this case the Signal-to-Noise Ratio (SNR) in the compressed samples can be increased by selecting the rows of the measurement matrix from the column space of the dictionary. As an example application of compressed sensing with a rank deficient dictionary, they present a case study of compressed sensing applied to the Coarse Acquisition (C/A) step in a GPS receiver.
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