A note on a sampling mixer under the coloured noise influence

In modern communication systems, one of the problems limiting the application of direct conversion receivers is the local oscillator port noise of sampling mixers. Ignoring the oscillator port noise contribution to the dynamical analysis of the sampling mixer will have influence on the trajectory estimation as well as its control. The intent of this note is to develop a general theory of sampling mixer under the coloured noise influence. The coloured noise process is exploited to model the oscillator port noise in lieu of the white noise process. Importantly, the coloured noise process confirms the real noise statistics. The coloured noise equation of the sampling mixer of this note is described by a scalar bilinear time-varying Stochastic Differential Equation (SDE). Subsequently, the extended phase space in combination with the notion of the Itô differential plays the role to develop the coupled dynamical equations of the estimated state trajectory of the sampling mixer. The numerical simulation in the MATLAB environment is demonstrated as well.