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A fully digital frequency synthesizer for RF wireless applications has recently been proposed. At its foundation lies a digitally controlled oscillator that deliberately avoids any analog tuning controls. When implemented in a digital deep-submicrometer CMOS process, the proposed architecture appears more advantageous over conventional charge-pump-based phase-locked loops (PLLs), since it exploits signal processing capabilities of digital circuits and avoids relying on the fine voltage resolution of analog circuits. An actual implementation of an all-digital PLL (ADPLL)-based local oscillator and transmitter used in a commercial 0.13-μm CMOS single-chip Bluetooth radio has recently been disclosed. The conventional phase/frequency detector, charge pump and RC loop filter are replaced by a time-to-digital converter and a simple digital loop filter. Due to the lack of the correlational phase detection mechanism, the loop does not contribute to the reference spurs. The measured close-in phase noise of -86 dBc/Hz is adequate even for Global System for Mobile communications (GSM) applications. In this paper, we present the mathematical description and operational details of the phase-domain ADPLL.