Response of Periodically Varying Systems to Shot Noise - Application to Switched RC Circuits

Consider a circuit, shown in Fig. la, consisting of a resistance R shunted by a switch and condenser C. The circuit is driven by a Poisson shot noise current. The elementary charges q arrive at random at an average rate of v per second. The switch operates in a cycle with period T. It is closed during the intervals 11T t nT + aT and open during the intervals nT + aT t (n + I)T where n is an integer and 0 a 1. We are interested in the statistical properties of the voltage V(t) across the condenser. In particular, we want an expression for the two-sided power spectrum Wv(j) of V(t). This problem was encountered by D. D. Sell1 during the development of a new type of spectrophotometer. The determination of an exact expression for W v ( f ) turned out to be unexpectedly difficult, and led to the present investigation of the more general case in which 2221 2222 T H E BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1970 I(t) I N F I N I T E IMPEDANCE SHOT NOISE C U R R E N T GENERATOR v(t)