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The so called “augmented” statistics of complex random variables has established that both the covariance and pseudocovariance are necessary to fully describe second order properties of noncircular complex signals. To jointly decorrelate the covariance and pseudocovariance matrix, the recently proposed strong uncorrelating transform (SUT) requires two singular value decompositions (SVDs). In this correspondence, we further illuminate the structure of these matrices and demonstrate that for univariate noncircular data it is sufficient to diagonalize the pseudocovariance matrix-this ensures that the covariance matrix is also approximately diagonal. The proposed approach is shown to result in lower computational complexity and enhanced numerical stability, and to enable elegant new formulations of performance bounds in widely linear signal processing. The analysis is supported by illustrative case studies and simulation examples.