Calculate amplitude of noise added Sine Gaussian signal from frequency domain
Random noise doesn't have an amplitude, it has statistical properties such as standard deviation and variance.  Variance (standard dev. squared) is interpreted as the AC power of the noise.  In the freq domain you can estimate the total power of a signal and the total 'power' (variance) of the noise.  The ratio of the signal power over the noise power is your signal-to-noise ratio.  Multiplying the log-base-ten of that ratio by ten gives you the signal-to-noise ratio measured in dB.  [-Rick-]