A Costas loop is a phase-locked loop (PLL) based circuit which is used for carrier frequency recovery from suppressed-carrier modulation signals (e.g. double-sideband suppressed carrier signals) and phase modulation signals (e.g. BPSK, QPSK). It was invented by John P. Costas at General Electric in the 1950s. Its invention was described as having had "a profound effect on modern digital communications". The primary application of Costas loops is in wireless receivers. Its advantage over other PLL-based detectors is that at small deviations the Costas loop error voltage is as compared to . This translates to double the sensitivity and also makes the Costas loop uniquely suited for tracking Doppler-shifted carriers, especially in OFDM and GPS receivers. Classical implementation In the classical implementation of a Costas loop, a local voltage-controlled oscillator (VCO) provides quadrature outputs, one to each of two phase detectors, e.g., product detectors. The same phase of the input signal is also applied to both phase detectors, and the output of each phase detector is passed through a low-pass filter. The outputs of these low-pass filters are inputs to another phase detector, the output of which passes through a noise-reduction filter before being used to control the voltage-controlled oscillator. The overall loop response is controlled by the two individual low-pass filters that precede the third phase detector, while the third low-pass filter serves a trivial role in terms of gain and phase margin. The above figure of a Costas loop is drawn under the "locked" state, where the VCO frequency and the incoming carrier frequency have become the same due to the Costas loop process. The figure does not represent the "unlocked" state. Mathematical models In the time domain In the simplest case . Therefore, does not affect the input of the noise-reduction filter. The carrier and voltage-controlled oscillator (VCO) signals are periodic oscillations with high-frequencies . The block is an analog multiplier. A linear filter can be described mathematically by a system of linear differential equations: where is a constant matrix, is a state vector of the filter, and are constant vectors. The model of a VCO is usually assumed to be linear: where is the free-running frequency of the VCO and is the VCO gain factor. Similarly, it is possible to consider various nonlinear models of VCO. Suppose that the frequency of the master generator is constant Equation of VCO and equation of filter yield The system is