Estimation of the complex frequency of a harmonic signal based on a linear least squares method

In this study, we propose a simple linear least squares estimation method(LLS) based on a Fourier transform to estimate the complex frequency of a harmonic signal. We first use a synthetically-generated noisy time series to validate the accuracy and effectiveness of LLS by comparing it with the commonly used linear autoregressive method(AR). For an input frequency of 0.5 m Hz, the calculated deviations from the theoretical value were 0.004‰and 0.008‰ for the LLS and AR methods respectively; and for an input 5 10 6attenuation,the calculated deviations for the LLS and AR methods were 2.4% and 1.6%. Though the theory of the AR method is more complex than that of LLS, the results show LLS is a useful alternative method. Finally, we use LLS to estimate the complex frequencies of the five singlets of the0S2 mode of the Earth’s free oscillation. Not only are the results consistent with previous studies, the method has high estimation precisions, which may prove helpful in determining constraints on the Earth’s interior structures.

Behaviors of harmonic signals in wavelength-modulated spectroscopy under high absorption strength

Behaviors of harmonic signals in wavelength modulation spectroscopy （WMS） for gas detection with Lorentzian line under high absorption strength are investigated. Approximate analytic expressions of the second, fourth, and sixth harmonics on the strength ave presented in concise forms. Simulations show that the expressions ave in agreement with the Fourier expansion by numerical integration. It is expected theoretically and experimentally in a WMS system for methane detection that there ave not only a maximum, but also a null point in the harmonics versus strength relations, which should be of practical importance in methane sensing applications.

The frequency estimation of harmonic signals embedded in multiplicative and additive noise

A method to separate a harmonic signal from multiplicative and additive noises is proposed. The method is to square the signal x(t), which consists of a harmonic signal embedded in multiplicative and additive noises, to form another signal y(t) = x2(t)-E[x2(t)]. After y(t) having been gotten, the Fourier transform is imposed on it. Because the information of x(t) (especially about frequency) is included in y(t), the frequency of x(t) can be estimated from the power spectrum of y(t). According to the simulation, under the condition where frequencies divided by resolution dω are integer, the maximum relative error of estimated frequencies is less than 0.4% when the signal-to-noise ratio (SNR) is greater than -23 dB. If frequencies divided by resolution dω are not integer, the maximum relative error will be less than 2.9%. But it is still small in terms of engineering.

Harmonicity Spectrum

Perceiving harmonic information (especially weak harmonic information) in time series has important scientific and engineering significance. Fourier spectrum and time-frequency spectrum are commonly used tools for perceiving harmonic information, but they are often ineffective in perceiving weak harmonic signals because they are based on energy or amplitude analysis. Based on the theory of Normal time-frequency transform (NTFT) and complex correlation coefficient, a new type of spectrum, the Harmonicity Spectrum (HS), is developed to perceive harmonic information in time series. HS is based on the degree of signal harmony rather than energy or amplitude analysis, and can therefore perceive very weak harmonic information in signals sensitively. Simulation examples show that HS can detect harmonic information that cannot be detected by Fourier spectrum or time-frequency spectrum. Acoustic data analysis shows that HS has better resolution than traditional LOFAR spectrum.

Stochastic resonance in an under-damped bistable system driven by harmonic mixing signal

Stochastic resonance（SR） is studied in an under-damped bistable system driven by the harmonic mixing signal and Gaussian white noise. Using the linear response theory（LRT）, the expressions of the spectral amplification at fundamental and higher-order harmonic are obtained. The effects of damping coefficient, noise intensity, signal amplitude, and frequency on spectral amplifications are explored. Meanwhile, the power spectral density（PSD） and signal-to-noise ratio（SNR） are calculated to quantify SR and verify the theoretical results. The SNRs at the first and second harmonics exhibit a minimum first and a maximum later with increasing noise intensity. That is, both of the noise-induced suppression and resonance can be observed by choosing proper system parameters. Especially, when the ratio of the second harmonic amplitude to the fundamental one takes a large value, the SNR at the fundamental harmonic is a monotonic function of noise intensity and the SR phenomenon disappears.

Euclidean distance harmonic method for establishing theoretical MAPK/Erk signaling pathway in treated breast cancer line MCF-7

Hierarchical clustering algorithms, such as Pearson＇s correlation, Euclidean distance, Euclidean distance harmonic, Spearman rank correlation, Kendall＇s tau, and City-block distance, were used to find the best way to establish theoretical MAPK/Erk signaling pathway on the basis of breast cancer line MCF-7 gene expressions. The algorithm constructs a hierarchy from top to bottom on the basis of a self-organizing tree. It dynamically finds the number of clusters at each level. It was found that only Euclidean distance harmonic is fit for the analysis of the cascade composed from a RAF1 （c-Raf）, a MKNK1, a MAPKK （MEK1/2） to MAPK （Erk） in breast cancer line MCF-7. The result is consistent with the biological experimental MAP/Erk signaling pathway, and the theoretical MAPK/Erk signaling pathway on breast cancer line MCF-7 is set up.