Exact calculation of phase noise of RF digitally controlled oscillator with frequency resolution improved by dithering

We present the exact calculation of phase noise due to dithering in a digitally controlled oscillator (DCO) whose output frequency is determined by the switching of elementary capacitors. The instantaneous possible frequency values depend on possible values of capacitors. Frequency resolution is improved by a dithering technique consisting in switching two capacitors values at a high frequency rate. The output frequency depends on the resulting average capacitor value that is function of time percentages during which each capacitor is selected. We give the exact expression of the output power spectral density (psd) and show that it decreases as 1/f4.