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The facts concerning the relative merits of linear and square-law envelope detection of a sinusoidally modulated carrier accompanied by Gaussian noise are clarified in this paper. The comparison when the carrier is weakly modulated and has mean power substantially greater than that of the noise in the input circuit has been fully covered in prior technical literature. In this special case the signal-to-noise ratio (SNR) in the detected output is the same for the two methods, and the distortion of the signal is worse with square-law operation because of inherent second-harmonic production. Since noise in the detected output is typically measured in the absence of signal modulation, the fact that the relative performance changes when the index of modulation approaches 100 percent tends to be overlooked. It is shown in the present paper that the noise performance of the square-law detector suffers a penalty approaching 1.8 dB when the carrier is strongly modulated. Also, the distortion in the square-law case is characterized by a second harmonic only 12 dB down from the fundamental, while the second harmonic in the output of the linear detector is practically negligible. These results are important in the evaluation of techniques in diversity reception.