A model of the probability distribution of the signal-to-noise ratio estimated from BER measurements

For additive white Gaussian noise (AWGN) channels, first we show that the conditional probability distribution of the symbol error ratio (BER), in n transmitted symbols, can be modeled as lognormal in BPSK, QPSK and QAM transmission schemes, for a given average probability of symbol error p, and then we show that the channel signal-to-noise ratio (SNR), conditioned to p, can be modeled as a Gaussian random variable, as long as p >> 1, n >> 1, np >= 1000. The theoretical results could be useful in assessing bounds to the estimated SNR in terrestrial, satellite or deep-space AWGN channels. Moreover, from the estimated SNR, it should be possible to estimate the fading due to slow tropospheric phenomena, such as rain, water vapor, oxygen attenuation.