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The MUSIC algorithm is a method based on the autocorrelation matrix feature decomposition and uses the orthogonality of the signal subspace and the noise subspace to estimate the signal with high resolution. In this paper, the MUSIC algorithm is applied to the processing of the vortex signal to obtain high precision. Vortex signal frequency points.

1 Signal-to-noise characteristics of vortex flowmeter

1.2 Vortex signal model

In an actual industrial site, vortex signals may be mixed with various types of noise such as Gaussian white noise, periodic pulse noise, and harmonic noise. Therefore, the following discrete random signal model is established

(2)

Among them, n = 0, 1, ···, N-1, k is the total number of fundamental waves and harmonics, and k≤3;, respectively, the amplitude, frequency, and phase of the fundamental waves and harmonics; fs is the sampling Frequency; w (n) and φ (n) are Gaussian white noise and impulse noise, respectively.

2 MUSIC algorithm

Let the sequence x (n) be composed of M complex sines plus noise, and the autocorrelation function Rx (k) is

(3)

If there are (p + 1) Rx (k) forming correlation matrix

(4)

Define the signal variable ei

(5)

Then

(6)

By formula (6) as a feature decomposition, we can get

(7)

In the formula, V1 is the main eigenvector and is orthogonal to each other. It can be seen that V1, ..., VM constitute a signal subspace with a characteristic value of λ2 + σ2; VM + 1, ..., Vp + 1 constitute a noise subspace with a characteristic value of λ2. The function that defines the spectral estimation of the MUSIC method is

(8)

The estimation of the signal angular frequency ω can be determined by the M peak positions of the function Pmusic (ω). The peak position of the spectral function Pmusic (ω) reflects the frequency value of the signal, but it is not the power spectrum of the signal, and it is generally called the MUSIC spectrum.

In order to improve the accuracy of spectral estimation at low signal-to-noise ratio, a weighting coefficient λk is introduced

(9)

3 Simulation research

3.1 Simulation under Gaussian white noise background

Let the Gaussian white noise sequence be W (n), which meets the Gaussian distribution of (0, σ2), and the original sinusoidal signal sequence is S (n), then the vortex signal x1 (n) can be expressed as

(10)

The Matlab software is used to generate the vortex signal x1 (n), where the original sinusoidal signal frequency fo = 200Hz, signal-to-noise ratio SNR = 10dB, and sampling frequency fs = 3072Hz. The simulation results are shown in Figure 1. The comparison shows that the spectral curve obtained by the MUSIC method is smooth, the frequency resolution is significantly improved, and the signal-to-noise ratio is significantly improved.

3.2 Simulation in the context of periodic impulsive noise

ɑStable distribution model is a mathematical model that better describes impulse noise. Its characteristic function is

(11)

(12)

(13)

Among them, ɑ is a characteristic index, which is used to measure the thickness of the distribution tail; β is a symmetric parameter, β = O is a symmetrical ɑ stable distribution; γ is a dispersion coefficient, which measures the width of the distribution; μ is a position parameter. The smaller the ɑ, the heavier the tail.

A Matlab software is used to generate a vortex signal x2 (n) composed of a sinusoidal signal S (n) and a pulse signal φ (n), where the frequency of the sinusoidal signal s (n) f0 = 200Hz and the pulse signal φ (n) obey Stable distribution, ɑ = 2, β = γ = 0, the frequency of the pulse signal fφ = 50Hz, and the sampling frequency fs = 3072Hz. The simulation results are shown in Figure 2. By comparison, it can be seen that the spectrum obtained by the MUSIC method still has a good resolution of the spectral peaks of the sinusoidal signal.

3.3 Simulation in the context of harmonic noise

Certain harmonic noise often exists in practical applications. In order to verify the effectiveness of the MUSIC algorithm, simulation research is performed in the above noise environment. The disturbed vortex signal x3 (n) can be expressed as

(14)

Among them, the sinusoidal signal frequency fo = 200Hz, the harmonic frequencies f1 and f2 are 400, 600Hz, and the sampling frequency fs = 3072Hz. The simulation results are shown in FIG. 3. It can be seen that both of them are accurate in resolving frequency points, but the periodogram method is poor in resolving the amplitude of harmonic components.

The above results show that, compared with the traditional periodic graph method, the MUSIC algorithm shows better performance in a complex noise environment, and it has important significance for improving the anti-interference ability of the vortex flowmeter.

From Zhang Qiongdan, Meng Jianbo.Vortex signal processing method based on MUSIC algorithm [J] .Sensors and Microsystems, 2015,34 (5), 38-40.

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