In this article, the performance optimization of sub-Nyquist spectrum sensing via the Modulated Wideband Converter (MWC) is studied. The MWC allows to recover signals that are sparse in a wide bandwidth through an inverse sparse problem using mixtures of the signal sub-bands and a mixing matrix i.e. the dictionary obtained from pre-defined periodic sequences. The MWC architecture strongly influences the corresponding dictionary, which exhibits a very specific structure. Moreover, in presence of noise, the latter is folded in the mixtures so that a whitening step is beneficial. This step yields a challenging periodic signal selection in order to minimize the dictionary coherence which is a crucial criterion for greedy sparse recovery. We show explicitly the role played by the coherence of the dictionary on the recovery of the sub-band positions, denoted as support, and signal recovery in presence of noise when a greedy approach such as Simultaneous OLS is used. We then propose an original strategy to optimize the dictionary taking the specific structure of the MWC dictionary in presence of noise. Both numerical and theoretical expressions show the performance gain in optimizing the MWC periodic sequence coefficients.