kT/C noise: Covariance matrix calculation of instationary noise in time-varying systems

The voltage variance at a capacitance C with a noisy resistance in parallel is kT/C, even if the resistance is infinite. This so-called kT/C noise may dominate in switched-capacitor circuits. In this paper kT/C noise is treated analytically, using explicit and implicit notations of differential equations. The resulting algorithms can be implemented in any circuit simulator with transient analysis, including cases where there are capacitor islands without capacitive paths to ground. The general approach allows to take into account high integrator bandwidth, slow switching and other methods to mediate kT/C noise. A numerical example shows that an extremely high integrator bandwidth is needed to reduce kT/C noise.