This paper presents complex-valued free running oscillators derived from recursive digital filter structures. Four alternative approaches of complex direct-form and coupled-form oscillator structures are discussed leading to a multiple-output direct-form oscillator as the best solution from the viewpoint of computational efficiency. Two inherent disadvantages of direct-form structures can be avoided by appropriate means. The problem of poor frequency resolution at low frequencies is solved by a new method called two's-complement improvement (TCI). An oscillator frequency displacement is discovered which is caused by correlation between one state variable and the quantization error using finite arithmetic. This frequency error can be substantially reduced by appropriate error noise shaping. Finally, some telecommunication applications like fixed-frequency complex oscillators for baseband signal processing, complex baseband FSK modulators, and multiple-phase oscillators for multipath filters are discussed.
