The Properties of Time and Phase Variances in the Presence of Power Law Noise for Various Systems

This paper discusses the behavior of sample, standard, bandpass, and Allan variances of the time and phase error in the presence of negative power law or f^beta noise for a variety of systems. These systems include those in the digital, communications, signal processing, radar, ranging, and time transfer areas. A theory is presented which spectrally defines these variances by explicitly incorporating a system phase response function H_s(f) into the spectral integral. For many of the above systems, H_s(f) is shown to contain highpass as well as lowpass filtering properties, and these highpass properties, when present, are shown to enable both the sample and standard variances, as well as Allan variances, to be used in the presence of f^beta noise for beta>-4. It is also shown that the sample variance defined in this way can be used to justify the heuristic low and high frequency cut-offs that appear in the spectral definition of the bandpass variance (also known as the jitter). H_s(f) is further shown to fall into four general classes for the purposes of characterizing variance behavior. These classes we call: digital sampling, delay, delay with averaging, and PLL. The final part of the paper consists of a detailed discussion of the properties of these variances in the presence of negative power law noise for the above systems, organized by class of H_s(f)