Characterization of Frequency Stability: Effect of RF Filtering

This paper shows that the RF filtering of a signal generated by an oscillator affects the characterization of the frequency stability in a manner that differs from the filtering of the phase of this signal. The effect on both the frequency domain and the time domain measurement is considered. It is shown that the spectral density of the phase fluctuations of the filtered signal is a function of the spectral densities of the amplitude fluctuations and of the phase fluctuations of the original signal. The corresponding expression for the two-sample variance (Allan variance) is then also a function of these two contributions. Detailed calculation is made for the case of a signal with frequency stability limited by white phase noise and filtered by a first-order, low-pass filter. It is found that the frequency stability is improved by more filtering if the cutoff frequency, fc, is higher than the signal frequency, f0, as is the case for phase filtering. However, the stability will degrade with more filtering if fc < f0. An optimum frequency stability is reached when fc = f0/√2. Experimental measurements confirm these theoretical predictions.