The digital noise power due to transmission errors is determined for 8-bit μ-law PCM, μ = 255, signals transmitted viaM-level QAM,M= 16, 64 and 256, over Gaussian and Rayleigh fading channels. The analytical results apply for various laws that map the 8-bit μ-law PCM signal into the QAM format, and for different binary Gray codes used to represent the QAM signal points. In addition to the theoretical results expressed as overall speech signal/noise ratio (SNR) as a function of channel SNR for different input speech signal levels, simulations for Gaussian and mobile radio channels using speech signals are also presented. From our deliberations we can conclude that the 256-level QAM system employing the best mapping law and Gray code has a gain of about 6dB in overall speech SNR compared to transmission without co-ordination of the speech bits over a Gray-coded link. Corresponding gains for 16-level and 64-level QAM are marginally smaller. By increasing the number of QAM levels from 4 to 256, the required increase in channel SNR per bit to maintain an overall speech SNR of 30 dB is approximately 12 dB. The bandwidth efficiency increases by a factor of four. The theoretical analysis in the paper is approximate. Only the effect of the most likely channel errors (single errors) has been taken into account. The theoretical results coincides well with the simulations, where such approximations were made.