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Ensemble-averaged randomly scaled radar or sonar pulses are considered and the statistical theory for the corresponding phase and amplitude modulations are developed. Random scaling expresses varying radar cross-sections for scattering objects or varying antenna gain of a sweeping emitter. The noise variance of the modulations depends on the distribution function of the scaling, and how to minimise the variance by rejecting pulses below a certain amplitude threshold is shown. The theory is asymptotic in the sense that it is more accurate for increasing signal-to-noise ratios (SNRs). In a test case with uniformly distributed scaling, sufficient accuracy is reached for an average SNR larger than ∼5 for the phase average and ∼15 for the amplitude average. [ABSTRACT FROM AUTHOR]