Comparison of Signal-to-Noise Ratios

The probability distribution (density) function for the experimental signal-to-noise ratio (SNR) defined as x̄/s, where x̄ is the sample mean and s is the customary sample standard deviation, has been derived and found to be in excellent agreement with accurate Monte Carlo simulation results. The SNR probability distribution function is a hypergeometric function which has no closed-form expression in elementary functions. The same applies to the probability distribution function for the relative standard deviation. In contrast, the probability distribution function for the approximate SNR defined by μ/s', where μ is the population mean parameter and s' ≡ s[(N − 1)/N]1/2, has a closed-form expression but is inaccurate for small numbers of measurements. The experimental SNR is a biased estimator of the true SNR, but the bias is easily correctable. Monte Carlo simulation methods were used to derive critical value tables for comparison of experimental SNRs and relative standard deviations. The critical value tables presented herein are accurate to about 1% for confidence levels of 75%, 90%, and 95%, and to about 5% for 99% confidence level.