Best - Costas Loops: Theory, Design, and Simulation

Roland Best Costas Loops

Roland Best Costas Loops

Amplitude modulation (AM) Amplitude modulation with suppressed carrier Binary phase shift keying (BPSK) Carrier signal Differential encoding Highpass filter Loop filter Modulation index Phase detector Phase error Phase-locked loop (PLL) Voltage-controlled oscillator (VCO)

The Costas loop can be considered an extended version of the phase-locked loop (PLL) . The PLL has been invented in 1932 by French engineer Henri de Belleszice []. Assuming for the first moment that both signals U1 and U2 are sinusoidal, we can write where U10, U20 are the amplitudes of U1 and U2, respectively, 1 is the radian frequency of the input signal U1, and 1 and 2 are the zero phases of U1 and U2, respectively. Assume further that the system is already locked, i.e., both signals have the same frequency, but can have different phases. In this case, the signals u1 and U2 differ by 90 in the locked state; hence, it is reasonable to define U1 as a sine wave and U2 as a cosine wave. It can be shown that the output signal of the phase detector is proportional to sin(12) = sin(e), where e is called phase error. But the output signal of that type of phase detector also contains a high-frequency term, i.e., a sine term having radian frequency 2 1. This term is removed by the loop filter , which is mostly realized either as a lag-lead filter or as a PI filter (proportional and integral filter). More about loop filters later in this text. The output signal uf of the loop filter is applied to the input of the VCO . When there exists a phase error, the frequency of the VCO is adjusted such that finally the phase error becomes either 0 or is at least very small.