The noise conversion method for oscillatory systems

We present a method for calculating the amplitude and phase power spectral density (PSD) functions of an oscillatory system (resonator, oscillator, bandpass filter, selective circuit, etc) via the PSD of its intrinsic noise sources and relevant transformation coefficients. A systematic description of the method is given for the scalar and vector noise. As an illustration, the noise transformation coefficients are derived for a piezoelectric series branch with fluctuating motional inductance, capacity, and losses. where static capacity is disregarded. We clarify the rules regarding the shaping of either PSD function. The importance of this method resides in the fact that it enables us to study particular finite ranges of the PSD function without using the differential equations.