A Feedback Correction De-Noising Algorithm Based on the Wavelet Coefficients

The localization of satellite interference sources under the condition of low SNR has been a hot research topic at current, where the signal de-noising is important in the following positioning work. With the help of the wavelet analysis theory, this paper proposes a de-noising algorithm based on the wavelet coefficients in low SNR. After correcting and reconstructing the signal by the correction coefficients, we could obtain the SNR and Root mean square error (RMSE), and then we could correct the signal for the second time. The simulation results show that this de-noising algorithm can effectively remove the noise of the signal in low SNR, which would have a good effect on the estimation of time delay parameter in the localization of satellite interference sources.

SECTION I. Introduction

In the satellite interference sources localization system, the difference of signal intensity transmitted to the surface of the earth between the interfered main star and the neighboring star is generally about 40dB. In order to locate the interfer-ence sources, the weak signal should be detected at first. So the signal de-noising is indispensable during the signal processing part. So far various de-noising methods have been widely used, such as Wiener filtering, Kalman filtering, spectrum subtraction method, etc. The wavelet analysis is an excellent mathematical tool which has been developed these years, and the WT (Wavelet transform) can get the multi-resolution description of the signal, which overcomes the shortcoming of unable to describe the detailed characteristics of the signal of Fourier transform. As any detail of the signal can be observed by using WT, so it is an essential part of the wavelet transform to use wavelet to de-noise. There are three commonly used de-noising methods which process the wavelet coefficients in nonlinear way at present: modulus maximum method, spatial correlation filtering, and the threshold de-noising algorithm, where the threshold de-noising algorithm is the most commonly used one, and the selection of the threshold function and the con-firmation of the threshold are two fundamental issues of the algorithm.

Soft and hard threshold functions are the two most commonly used threshold de-noising methods. However, the hard threshold function is not continuous, which may directly cause some oscillation in the signal after de-noising. This phenomenon is particularly obvious when the noise level is high ^[1]. Although the soft threshold function is continuity on the whole, there is always a constant deviation between the processed coefficients and the original ones when the wavelet coefficients are higher than the threshold, which directly affects the approximation degree of the reconstructed signal and real signal, and causes inevitable errors to the reconstruction of the signal ^{[2], [3]}. The adaptive threshold function method ^[4], which de-noises effectively in high SNR, has been developed in recent years, but the result is not good enough when the SNR is under -10dB. There are also some other de-noising algorithms based on correction threshold and threshold function ^[5] which could achieve good effect when the SNR is high. But when the SNR is under -10dB, the result turns bad.

When the SNR is low, the de-noising methods above wouldn't perform well. The lower the SNR is, the higher the noise level becomes. When the signal has a high frequency, the wavelet coefficients of noisy signal are more close to the noise coefficients, leading to some part of the useful high-frequency signal being removed easily by the method of threshold function. Meanwhile, the existing methods do not process the wavelet approximation coefficients, so parts of the noise still remain in the reconstructed signal ^[6]. Considering for low SNR, the method presented in this paper could correct both approx-imation coefficients and detail coefficients of levels, which has a better de-noising effect than the methods above.

SECTION II. Theory of Wavelet De-Noising

The wavelet transform can effectively de-noise because of these following features^[7]:

Low Entropy

The sparse distribution of wavelet coefficients lower the entropy after wavelet transform.

Multiresolution

As the wavelet transform uses the multiresolution analysis, it can describe non-stationary of the signal very well so as to remove the noise according to the distribution of the signal and noise at different resolution.

Decorrelation

The wavelet transform can decorrelate the signal and whiten the noise, so de-noising can perform better in wavelet domain than in time domain.

Flexible Choice of Base Function

The wavelet trans- form has the flexibility of choosing the base function according to different situation. It can choose either M-band wavelet or wavelet packet as well according to the signal feature and the de-noising demands.

The wavelet de-noising process is divided into three steps:

① Wavelet decomposition. Decompose the signal after choosing a wavelet function and confirming the decomposition levels N.

② The threshold of wavelet coefficients. Choose the threshold function and threshold to process the detail coefficients of levels. Generally the approximation coefficients of level $N$ are not processed.

③ Signal reconstruction. Reconstruct the wavelet coefficients after thresholding to de-noising and extracting the weak signal.

The common wavelet functions are as follows:

Haar Wavelet

The property is bad as a basic wavelet function because of the discontinuity in time domain. But it has some advantages such as simple calculation and better or-thogonality.

Daubechies (DBN) Wavelet

It has the properties of short support in time domain and $N$-degree zero at $r_{n}$ in frequency domain. The wavelet function is orthonormalized with its integer shift.

Mexican Hat Wavelet

It keeps the fine time-frequency localization property but does not possess the character of or-thogonality.

Symlet (SYMN) Wavelet

It is the improvement of db wavelet that makes the wavelet function nearly symmetric.

In signal de-noising, the selection of wavelet function should take the factors of orthogonality, symmetry, short sup-port into consideration. The simulation of this paper picks db3 wavelet as the wavelet function that is commonly used in signal de-noising.

SECTION III. The New Algorithm of De-Noising

The usual soft threshold and hard threshold methods have obtained ideal results in high SNR. But in low SNR, the effect of common threshold de-noising method is not good enough. In this paper, our new de-nosing algorithm for low SNR is proposed. And the following simulation is operated under low SNR as well.

1. Process of Feedback Correction De-Noising Algo-Rithm

After analyzing kinds of wavelet coefficients correction de-noising[8], [9], the method of coefficient feedback correction is proposed. This method mainly includes coefficient ratios pre-operation, wavelet coefficients correction, feedback correction, etc.

① Decompose the reference signal and extract the approx-imation coefficients and the detail coefficients of levels.

② Pre-operate the coefficients of every level and obtain the correction coefficient ratios $(r_{n})$. ③ Decompose the signal to be processed and extract the approximation coefficients and the detail coefficients of levels, then correct them by $r_{n}$ and reconstruct the corrected wavelet.

④ Analyze the results of SNR and RMSE of the feedback reconstruction signal. If it is possible for improvement, select proper correction coefficients to correct the signal again according to this SNR.

RMSE (Root mean square error) is used to measure the effect of de-noising. The lower RMSE is, the better de-noising result gets. The definition of RMSE is as follows: $$\begin{equation*}\{\frac{1}{N}\sum_{n=1}^{N}[x(n)-\hat{x}(n)]^{2}\}^{1/2}\end{equation*}$$ where $x(n)$ is the original signal, and $\hat{x}(n)$ is the de-noised signal.

2. Correction Coefficient Ratios Pre-Operation

In the interference sources localization system, the accurate value of the correction coefficient ratios, corresponding to the SNR of the weak signal obtained, are unknown. So we need to use reference signals whose frequency spectrum or the signal feature is similar to the unknown signals to pre-operate the correction coefficients. Assume that $x$ is the wavelet coefficient of reference signal, $n$ is the coefficient of noise, and $x_{n}$ is the coefficient of noisy signal. For different levels, we divide $x$ by $x$ to obtain the coefficient ratio $x/n$ of sampling points, then average the value $x/n$ and get $r$. Because the noisy signal is simply the superposition of signal and noise, so the coefficient $x_{n}$ equals to the summation of $x$ and $n$. We can obtain the correction coefficient ratio $x/x_{n}$ from $x/n$. $$\begin{align*} &1+x / n=(x+n) / n=x_{n} / n\tag{1}\\ &\qquad r=\text { mean value }(x / n)\tag{2}\end{align*}$$

We get $1-n/x_{n}=x/x_{n}$ from Eqs.(l) and (2)

Then the correction ratio coefficient $r_{n}$ can be derived from Eq.(3), $$\begin{equation*} r_{n}=x/x_{n}=1-[1/(1+r)]\tag{3}\end{equation*}$$ then multiply the correction coefficient ratio $r_{n}$ by the wavelet coefficients of noisy signal $x_{n}$, that is $$\begin{equation*} x_{n}\cdot r_{n}=x_{n}\cdot\frac{x}{x_{n}}=x\end{equation*}$$

In this way the wavelet coefficient $x$ of the signal can be recovered.

From Eq. (3), the approximation coefficient $r_{na}$ and the detail coefficient $r_{nd}$ of every level can be acquired according to the reference signal in different SNR (for example, -5dB, -10dB, -20dB, -30dB, -40dB). In different SNR, the approximation coefficients and the detail coefficients of levels are different. After acquiring the SNR of the noisy signal, we take $r_{na}$ and $r_{nd}$ under that SNR as the correction ratio coefficients to correct the noisy signal in proportion.

3. Signal Correction, Reconstruction and Feedback Reconstruction

The BPSK signal is used for transmitting signal of inter-ference sources in this simulation, where the baseband signal frequency is 900Hz, the carrier frequency is $f_{c}=800\text{Mhz}$, and the sampling frequency is $f_{s}=4f_{c}$. The db3 wavelet function is used for de-noising and the decomposition levels are 5. The noise is white Gaussian noise. The db3 wavelet function is used for de-noising and the decomposition levels are 5. The noise is white Gaussian noise.

Decompose the noisy reference signal and obtain the ap-proximation coefficient ratio $r_{na}$ and the detail coefficient ratio $r_{nd}$ of every level (these coefficients result from the mathemat-ical average of a great many samplings) in different low SNR, from OdB to -40dB, and then correct the noisy signal in pro-portion. After pre-operation in Section III.2, the correction coefficient ratios are shown in Table 1.

Table 1. Correction coefficient ratios

The coefficient of noisy signal $x_{n}$ should be correspondingly multiplied by the correction coefficient ratio $r_{n}$ to get the corrected coefficient. Then we reconstruct the signal by these coefficients and get the corrected signal.

The feedback of SNR and RMSE of the corrected signal can be got and the signal can be corrected for the second time. During the second correction, since the SNR of the first de-noising signal is known, we select the appropriate correction coefficient ratios according to the acquired SNR and correct the signal again in pre-correction way.

Take the following simulation as an example. After cor-recting the noisy signal of -30dB SNR, the SNR becomes -12.41dB, with RMSE 2.69, and then we calculate the co-efficient ratios again according to that SNR and multiplied it by the corrected coefficients. After the feedback correction and the reconstruction, the de-noising effect can be improved again. The SNR of this example turns out to be -8.69dB after the second correction while the RMSE is 1.83.

SECTION IV. Simulation Results

1. Comparison of De-Noising Effects

Compare the method suggested in this paper with common methods of soft threshold, hard threshold de-noising and the improved threshold de-noising^[4]. In low SNR, the simulation is operated from -10dB to -40dB. Table 2 shows the results of de-noising.

Table 2. Comparison of the results after de-noising in low SNR

Take -20dB as an example, the BPSK signal above and the noisy signal are shown in Figs.1 and 2.

Fig. 1.

BPSK signal

Fig. 2.

Noisy signal

The de-noising effects of de-noising algorithm in this paper and soft threshold, hard threshold de-noising, improved threshold de-noising in Ref. [4] are shown inFigs.3–​6.

The RMSE comparison of the above four methods is shown in Fig. 7.

Fig. 3.

Feedback correction de-noising algorithm

Fig. 4.

Soft threshold de-noising method

Fig. 5.

Hard threshold de-noising method

Fig. 6.

Improved threshold de-noising method

Fig. 7.

RMSE comparison

The RMSE of soft threshold and hard threshold de-noising are similar so that the two lines are almost coincident.

As can be seen from the simulation results of Table 2 and Fig. 7, in low SNR, the de-noising algorithm suggested in this paper has better effects than the other 3 methods in SNR and RMSE.

TheFigs.3–​6 show that the soft threshold and hard threshold de-noising methods have removed high frequency part of the useful signal, and the effect of improved threshold de-noising is quite limited. While for the method in this paper, it has achieved more effective de-noising result. The waveform is more similar to the original BPSK signal in terms of magnitude.

2. Possibility of Interference Sources Localization

Take the BPSK signal above as the simulation signal. SNR1 is the ratio of the interference signal from the earth station transmitted by the main star, while SNR2 is that of the signal by the neighboring star. Assume that the time delay is 1μs (accurate to ns). Then we estimate the time delay by Generalized cross correlation (GCC).

SNR1 = OdB, SNR2 = -l0dB. The estimation results of feedback correction de-noisingsoft threshold and hard threshold and improved threshold de-noising are shown in Fig. 8.

SNR1 = OdB, SNR2 = -20dB. Fig. 9 shows the estimation results of the four de-noising methods.

SNR1 = OdB, SNR2 = -30dB. Fig.10 shows the estimation results of the four de-noising methods.

Fig. 8.

Time delay estimation with SNR difference -10dB

Fig. 9.

Time delay estimation with SNR difference -20dB

Fig. 10.

Time delay estimation with SNR difference -30dB

The experiment results show that when the SNR of the interference signal transmitted by neighboring star is -l0dB, the soft threshold and hard threshold de-noising couldn't accurately estimate the time delay because of removing the useful signal of high frequency part. When the SNR reduces to -20dB, the improved method and the feedback correction de-noising method could both estimate the time delay effectively. When the SNR drops to -30dB, the feedback correction de-noising method is the only method that can accurately estimate the time delay. Therefore, the method in this paper could estimate the time delay more effectively than other methods.

SECTION V. Conclusion

It can be concluded from the above analysis and the simu- lation that the feedback correction de-noising method presents higher ability to remove the noise in low SNR and it can accurately estimate the time delay of interference sources.