Detection and correction of level echo based on generalized S-transform and singular value decomposition

ZHU Tianliang， WANG Xiaopeng， WANG Qi

(School of Electronics and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070， China)

Abstract： The echo of the material level is non-stationary and contains many singularities. The echo contains false echoes and noise, which affects the detection of the material level signals, resulting in low accuracy of material level measurement. A new method for detecting and correcting the material level signal is proposed, which is based on the generalized S-transform and singular value decomposition (GST-SVD). In this project, the change of material level is regarded as the low speed moving target. First, the generalized S-transform is performed on the echo signals. During the transformation process, the variation trend of window of the generalized S-transform is adjusted according to the frequency distribution characteristics of the material level echo signal, achieving the purpose of detecting the signal. Secondly, the SVD is used to reconstruct the time-frequency coefficient matrix. At last, the reconstructed time-frequency matrix performs an inverse transform. The experimental results show that the method can accurately detect the material level echo signal, and it can reserve the detailed characteristics of the signal while suppressing the noise, and reduce the false echo interference. Compared with other methods, the material level measurement error does not exceed 4.01%, and the material level measurement accuracy can reach 0.40% F.S.

Key words： echo signal； false echo； generalized S-transform； singular value decomposition (SVD)； level measurement

0 Introduction

The frequency modulation (FM) radar level gauge is based on the principle of FM radar ranging, and it has the advantage of less influenced by temperature. It has been widely used in industrial fields. In practical application, the false material level echo is generated because the material will splash or dust when materials fall. Also, false echoes, random spikes and noise generated by the structure of the sensor can cause false level echoes and interference of noise.Therefore the detection and denoising of echo signals is the key to improve the accuracy of material level measurement. In Ref.[1], the target echo signal is segmented and equivalent to a continuous pulse signal, and the target echo signal is detected and located by pulse compression and Doppler principle. In Ref.[2], the echo signal is subjected to short-time fractional Fourier transform, and the enveloped time and amplitude plane and the frequency and amplitude plane are enveloped to obtain the joint estimation of the delay and Doppler shift of the dynamic target echo signal, thereby achieving the recognition of the target radar echo. An approximation function of the mean square error function is constructed, and the optimal value in the sense of mean square error is obtained. Although the method can effectively filter the noise in the signal when the signal is mixed with white noise and discontinuous points, the computational complexity is high[3-5]. Denoising by modulus maxima has a good effect, but it produces pseudo-extreme points and loses some essential local singularities[6]. The peak point of the original signal can be better preserved, and the calculation is faster after denoising by wavelet threshold. However, its denoising effect depends on the signal-to-noise ratio and threshold selection in Ref.[7]. The above method can effectively denoise, but it is easy to cause the signal detail information to be smooth so that the signal singularity feature cannot be retained.

In order to accurately detect the level echo signal and suppress the noise interference, a method of detecting and correcting the level echo signal based on the combination of generalized S-transform and singular value decomposition (SVD) is proposed. Firstly, the time-frequency coefficient matrix of the echo signal is obtained by the generalized S-transform, and then the matrix is reconstructed by using the SVD. This method can filter the false echo in the echo signal, retain the singularity of the signal and avoid the pseudo-Gibbs oscillation.

1 Analysis of level echo signal

The echo signal includes the radar’s transmit signal, the reflected material level echo signal and the noise. When the material is in canned, the surface of the material in the tank will have a weak displacement (splash, dust, etc.). The accuracy of the level measurement is affected because the vibration caused by the falling of the material produces a small Doppler shift. The radar’s transmit signal is a continuous FM signalST(t)

(1)

whereAis the amplitude of the transmitted signal;fcis the frequency of the transmitted signal;θiis the initial phase of the transmitted signal;μis the linear modulation frequency. The reflected echo signal produces a Doppler effect due to the height variation of the material level. So the reflected echo signal is

SR(t)=kAcos

(2)

wherekAis the echo signal amplitude;φis the additional phase shift produced by the Doppler effect;φiis the phase shift caused by surface vibration when the material falls;tdis the propagation delay,td=2R/c.

(3)

Thematerial level echo signal can be expressed as[4]

(4)

wherex(t) is the noise signal that power spectral density function obeys the Gaussian distribution. Fig.1 shows the echo spectrum of the material level with noise. It can be seen that there is a difference in the frequency between the material level echo, the noise and the false echo. This difference is used to detect level echo signals and remove noise.

Fig.1 Level echo signal spectrum

2 Correction of echo signal

2.1 Generalized S-transform

S-transform is a time-frequency analysis tool, which is based on the improvement of short-time Fourier transform and wavelet transform. Stockwell proposed a Gaussian window that varied with frequency (f)[8-9]. The window function could be expressed as

(5)

The S-transform of the non-stationary signal is

(6)

S-transform is a linear, multi-resolution, lossless and reversible time-frequency analysis method. It can detect the high-frequency component of the signal effectively and avoid the cross-term interference of Cohen distribution. However, under specific frequencies, the time-frequency window of S-transform is slender, which leads to the improvement of time resolution and the decrease of frequency resolution. The phenomenon of inaccurate resolution in frequency domain occurs in the higher frequency band. In order to improve the resolution of S-transform in the frequency domain at specific frequencies, adjusting parametersλandpare introduced to improve the Gauss window, which is used to adjust the time-width and convergence trend of the window function. At this point, the window function is

(7)

Then the generalized S-transform of the non-stationary signalh(t) can be defined as

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exp(-i2πfτ)dτ.

(8)

Then the one-dimensional discrete generalized S-transform of the non-stationary signalh(t) can be defined as

(9)

whereTis the sampling interval of the signalh(t), andNis the number of sampling points,j,n=0,1,2,…,N-1. The echo signal is transformed into a two-dimensional matrix by generalized S-transform. The column of thetmatrix is the time-domain coefficient of the echo signal; the row of the matrix is the frequency domain coefficient of the echo signal. The matrix reflects the relationship between signal amplitude, time and frequency. S-transformation time-frequency analysis of echo signals is shown in Fig.2.

Fig.2 S-transform time-frequency distribution of level echo signal

It can be seen from the Fig.2 that the high-frequency components are not clearly distinguished in the time domain. Fig.3 is a generalized S-transform that adjusts the window coefficientsλandpaccording to the frequency distribution characteristics of the level echo signal. As seen from the Fig.3, the frequency domain resolution of the high-frequency portion of the echo signal is improved without changing the time resolution.

Fig.3 GST time-frequency distribution of level echo signal

2.2 Denoising based on generalized S-transform and SVD

The singular value obtained by SVD decomposition can reflect the intrinsic properties of the signal, so it can eliminate most of the noise in the signal and improve the signal-to-noise ratio. The time-frequency coefficient matrix of the echo signal is obtained by generalized S-transform of the echo signal, which is used as the target matrix to denoise and reconstruct the signal matrix. LetHbe a real matrix of typem×n, and then there are orthogonal matrices of order 1 and 2. There is an orthogonal matrixUof ordermand an orthogonal matrixVof ordern.There are

H=UΛVT,

(10)

whereΛis a diagonal matrix,Λ=diag(σ1,σ2,…,σn). The diagonal element isσi>0(i=1,2,…,n), which is singular value of descending order of the matrixH. SVD can represent a matrix (m×n) of rankkas the sum ofksub-matrices (m×n) of rank 1. The sub-matrices are multiplied by two eigenvectors and weights, which is

(11)

wherekis the rank of the matrix;uiandviare the singular value eigenvectors of the columniofUandV;σiis theisingular value of matrix;Hiis a submatrix containinguiandvi. In practical applications, the time-frequency matrix of the echo signal is decomposed to obtain a series of singular values and singular value vectors corresponding to the time-frequency subspace. The transmitted signal is a modulated signal, so the frequency period of the echo signal does not change. After the generalized S-transform, the main energy of the echo signal is concentrated in a particular time-frequency domain, and the frequency-domain distribution of the material level echo signal is relatively concentrated. In contrast, the energy distribution of the noise and the false echo signal are relatively scattered. SVD is used to extract time-frequency features of relatively high energy, and eliminate the time-frequency characteristics of relatively dispersed energy. The time-frequency matrix of the echo signal is analyzed to obtain that the singular value corresponding to the material level echo is mainly distributed between 1 andkw(kw

1) The echo signal is subjected to a generalized S-transform to obtain a transformed time-frequency coefficient matrix.

2) Singular value decomposition of the time-frequency matrix obtained by generalized S-transformis is used to obtain the singular value of matrixAand arrange the singular values in decreasing order, that isaandb.

3) Calculating the one-sided maximum of the singular difference spectrum to determine the active rank order of the reconstructed signal, and then reconstructing the S-transform time-frequency coefficient matrix.

4) Performing the inverse transform on the reconstructed time-frequency coefficient matrix to obtain a corrected echo signal.

3 Results and discussion

In order to test the echo detection effect and material level measurement accuracy of the method in this paper, the simulation is performed based on the MATLAB2016 platform. The method in this paper is compared with the wavelet packet decomposition, ensemble empirical mode decomposition (EEMD) and S-transform and singular value decomposition (ST-SVD) for noise effect and measurement accuracy.

3.1 Detection and analysis of level echo signal

A plane wave is used to simulate the radar transmission signal. The carrier frequency is set to 20 GHz, the bandwidth is 500 MHz, the incident azimuth elevation angle is 90°, and the azimuth offset is approximately 0°. From Fig.4, it can be seen that energy of the reflected echo signal is concentrated in a particular area, and there are many interferences of Gauss noise and spike noise around it, which makes it impossible to detect the reflected echo of material level accurately. After the processing of GST-SVD (Fig.5), the Gauss noise and peak noise around the echo signal reflected by the material are expertly filtered, and the echo signal of the material level is accurately detected.

Fig.4 Unprocessed level echo time-frequency diagram

Fig.5 Reconstructed level echo time-frequency diagram

3.2 Test and analysis of denoising simulation

In order to verify the denoising effect of the method in this paper, Gauss noise (SNR=8 dB) is added to the given echo signalSS(t). Peak noise is added to the range of sampling points [2 400, 2 500]. Then wavelet packet decomposition, EEMD, ST-SVD and GST-SVD are used to denoise the level echo. The comparison result is shown in Fig.6. It can be seen that the wavelet packet decomposition (Fig.6(c)) effectively suppresses Gaussian noise. However, the filtering effect of the spike noise is poor, and pseudo-Gibbs oscillation appears in the peak portion of the signal. EEMD (Fig.6(d)) effectively filters out the spike noise, but the suppression effect on Gaussian noise is poor, resulting in low signal-to-noise of the echo signal. ST-SVD (Fig.6 (e)) has a good suppression effect on spike noise. However, because the S-transform window is fixed, it has a poor filtering effect on Gaussian noise and generates more pseudo-Gibbs oscillations at the peak portion of the signal. GST-SVD (Fig.6(f)) can effectively suppress the noise amplitude. Pseudo-Gibbs oscillation is increased, and signal details are better-preserved. The running time of the above four methods is shown in Table 1. The processing time of the echo signal in this paper is shorter than that of the EMMD algorithm, more than that of ST-SVD and wavelet packet decomposition, but still meets the real-time requirements.

(a) Original echo signal

Table 1 Signal-to-noise ratio and operation time

3.3 Measurement and analysis of level accuracy

In order to test the measurement accuracy of this method and simulate the filling situation of material in the tank, the measured values of level gauges with different correction methods are compared with the actual set values, and the relative errors are calculated.The measurements of FM radar level gauge using wavelet transform, traditional filtering and the method in this paper are given in Table 2. The detection error of the wavelet transform correction method is 6.73%, the traditional filter correction method is 19.04%, and the detection error of this method is 4.01%, which shows that proposed method has less detection error. Due to the reduction of echo noise, the convergence speed of the measured value tending to the real value is accelerated, the output of the measured value is relatively stable, the relative error of material level is small, the maximum relative error is not more than 4.01%, and the measurement accuracy reaches 0.40% F.S.

Table 2 Measurement accuracy of FM radar level meter

4 Conclusions

Considering the problem that false echo affects the accuracy of material level measurement in echo signal, the time-frequency matrix is obtained by generalized S-transform of the captured radar echo signal, and the material level echo signal is detected. Time-frequency matrix of the echo signal is reconstructed based on SVD, and the inverse transform is carried out to get the echo signal. In addition to the denoising in the echo signal, the false echo interference can be reduced, and the accuracy of material level measurement can be improved. The simulation results show that the relative error of the proposed method is smaller and the accuracy can reach 0.40% F.S compared with the existing measurement and correction methods. Under the condition of accurate measurement of material level, the singularity of the echo signal can be well preserved, and the pseudo-Gibbs oscillation can be avoided.