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In this paper, we conduct a thorough analysis of the nonlinear behavior of diode power probes and demonstrate how memory effects can alter power measurements of signals with wide modulation bandwidths and high values of peak-to-average power ratio. We show analytically, by simulations, and with measurements, that commonly used single-tone calibration procedures for diode power probes can provide erroneous values when measuring modulated signals used in many new wireless standards. We show that high values of peak-to-average-power ratio can degrade the calibration results due to the low-frequency response imposed by the power probe's baseband circuit impedance. This effect is first theoretically demonstrated by use of a Volterra series, and then validated by simulations and measurements with a diode power probe circuit. This study provides engineers with guidelines for techniques for correction of diode power probe measurements.