formulas assume that and thus that the bandwidth of the observation time is much lower than the instruments bandwidth. When this condition is not met, all noise forms depend on the instrument's bandwidth. α–μ mapping The detailed mapping of a phase modulation of the form where or frequency modulation of the form into the Allan variance of the form can be significantly simplified by providing a mapping between α and μ. A mapping between α and Kα is also presented for convenience: {| class="wikitable" |+ Allan variance α–μ mapping |- !α !β !μ !Kα |- | −2 | −4 | 1 | |- | −1 | −3 | 0 | |- | 0 | −2 | −1 | |- | 1 | −1 | −2 | |- | 2 | 0 | −2 | |} General conversion from phase noise A signal with spectral phase noise with units rad2/Hz can be converted to Allan Variance by Linear response While Allan variance is intended to be used to distinguish noise forms, it will depend on some but not all linear responses to time. They are given in the table: {| class="wikitable" |+ Allan variance linear response |- ! Linear effect ! time response ! frequency response ! Allan variance ! Allan deviation |- | phase offset | | | | |- | frequency offset | | | | |- | linear drift | | | | |} Thus, linear drift will contribute to output result. When measuring a real system, the linear drift or other drift mechanism may need to be estimated and removed from the time-series prior to calculating the Allan variance. Time and frequency filter properties In analysing the properties of Allan variance and friends, it has proven useful to consider the filter properties on the normalize frequency. Starting with the definition for Allan variance for where Replacing the time series of with the Fourier-transformed variant the Allan variance can be expressed in the frequency domain as Thus the transfer function for Allan variance is Bias functions The M-sample variance, and the defined special case Allan variance, will experience systematic bias depending on different number of samples M and different relationship between T and τ. In order to address these biases, the bias-functions B1 and B2 has been defined and allows conversion between different M and T values. These bias functions are not sufficient for handling the bias resulting