Lock-in frequency measurement with high precision and efficiency

This paper describes a new method of frequency measurement based on lock-in amplifiers (LIAs). In contrast to other frequency measurement methods, such as fast Fourier transformation (FFT), zero crossing, and scanning autocorrelation, this method is based on an adaptable LIA design for high-precision determination of not only the frequency but also the amplitude and phase of periodic signals, even when they are buried in heavy noise with low signal-to-noise ratios. Mathematical derivation of the local spectrum around the center frequency is performed, and the local frequency spectrum waveform of the sinusoidal signal, regardless of whether it is pure or noisy, is found to be exactly of a bell shape that can be described by a three-parameter sine function. Based on the principle of LIAs, the correct frequency can produce a peak amplitude in the local spectrum. As a result, the amplitudes of three frequency points around the target frequency can be used to precisely determine the peak frequency via sinusoidal fitting. The efficiency of the proposed method is log2(N) times that of FFT. Simulation results show that the new algorithm can reach the theoretical Cramer-Rao lower bound and remain below a lock-in upper bound. The new frequency measurement method has been implemented in an field-programmable gate array (FPGA)-based device and systematically tested for its dependence on the frequency, amplitude, and signal-to-noise ratio with typical noise types. Theoretical and experimental results show that the new method can be used in fine determination of the frequency if the user has prior knowledge of the approximate location of the frequency.