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Recently, a number of papers have been published that show significant performance gains can be obtained by accounting for the fact that communication signals can be improper. In this paper, we derive a general result comparing the performance of conventional processing, which ignores the improper nature of signals, with processing that takes it into account. In particular, for an estimation and a detection problem, we find that the performance gain, as measured by mean-squared error and deflection, respectively, can be as large as a factor of 2, but no larger. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3 dB gain over non-coherent processing.