CN108985234B - Bayes wavelet packet noise reduction method suitable for non-Gaussian signals - Google Patents

Bayes wavelet packet noise reduction method suitable for non-Gaussian signals Download PDF

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CN108985234B
CN108985234B CN201810796706.1A CN201810796706A CN108985234B CN 108985234 B CN108985234 B CN 108985234B CN 201810796706 A CN201810796706 A CN 201810796706A CN 108985234 B CN108985234 B CN 108985234B
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岳国栋
吴玉厚
崔修实
王丹
白晓天
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Shenyang Jianzhu University
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Abstract

The invention belongs to the technical field of signal processing, and particularly relates to a Bayesian wavelet packet noise reduction method suitable for non-Gaussian signals. The method is suitable for Bayesian wavelet packet noise reduction of non-Gaussian signals. The method comprises the following steps: the method comprises the following steps: and acquiring a noisy signal of the monitored object to construct a time sequence. Step two: and performing discrete wavelet packet decomposition on the time sequence, calculating grading indexes of the decomposition layer numbers according to the wavelet coefficients, and determining the optimal decomposition layer number according to the grading indexes. Step three: wavelet coefficients of the true signal at each decomposition level are estimated. Step four: and reconstructing the original signal by utilizing a wavelet packet inverse transformation formula according to the estimated wavelet coefficient so as to obtain the denoised signal.

Description

Bayes wavelet packet noise reduction method suitable for non-Gaussian signals
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to a Bayesian wavelet packet noise reduction method suitable for non-Gaussian signals.
Background
In order to ensure the safe and stable operation of mechanical equipment, the operation state of the equipment needs to be monitored, and the noise can seriously affect the system identification effect when the fault characteristics of the equipment are small. Therefore, it is important to effectively noise-reduce the test signal.
When the existing Bayesian wavelet packet noise reduction method is used, the prior information usually assumes Gaussian or mixed distribution with Gaussian, which cannot accurately describe non-Gaussian signals and seriously influences the use range of the existing method.
When the wavelet packet denoising method is used, the number of decomposition layers is an important parameter. If the number of the decomposition layers is too small, the separation degree of the real signal and the noise signal is insufficient; if the number of decomposition layers is too large, the distribution of the characteristic information in the wavelet coefficient is excessively dispersed, the real characteristic information is partially lost after noise reduction, and the subsequent characteristic extraction and analysis work is adversely affected. The existing method cannot get rid of manual intervention to select decomposition levels, and also influences the use of the wavelet packet noise reduction method in automatic monitoring.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a Bayesian wavelet packet noise reduction method suitable for non-Gaussian signals.
In order to achieve the purpose, the invention adopts the following technical scheme that the method comprises the following steps:
the method comprises the following steps: acquiring noisy signals of a monitored object to construct a time sequence y (t)i)。
Step two: for time series y (t)i) And carrying out discrete wavelet packet decomposition, calculating grading indexes of the decomposition layer numbers according to the wavelet coefficients, and determining the optimal decomposition layer number according to the grading indexes.
Step three: wavelet coefficients of the true signal at each decomposition level are estimated.
Step four: and reconstructing the original signal by utilizing a wavelet packet inverse transformation formula according to the estimated wavelet coefficient so as to obtain the denoised signal.
Preferably, the noisy signal in the step one is an acceleration signal.
Preferably, in the signal in the step one, the true signal and the noise signal have different probability distribution types.
Preferably, the base wavelet used in step two is an orthogonal wavelet.
Preferably, the maximum number of decomposition layers in step two is 2< J < 10.
Preferably, in step three, the probability distribution p of the wavelet coefficients describing the signal and noise as accurately as possibleGAnd pε，
If an orthogonal wavelet base, p, is adopted in the second stepGAnd pεAvailable x (t)i) And ε (t)i) Is represented by a probability distribution.
Preferably, in step four, the denoising effect is evaluated by using two methods, namely a signal-to-noise ratio (SNR) method and a time domain analysis method.
The specific mode of the first step is as follows:
based on the noisy signal measured by the sensor,intercepting a segment of a constructed time series y (t) containing signal featuresi). The sampling frequency is in accordance with Shannon sampling theorem; for convenience, let y (t)i)＝x(ti)+ε(ti),i＝1,2,…,2JWherein x (t)i) And ε (t)i) For time series of real and real noise signals, J ═ log2And N is added. The purpose of noise reduction is to obtain x (t)i) An estimate of (2).
The specific mode of the second step is as follows:
(1) selecting the number j of decomposition layers, for y (t)i) Carrying out wavelet packet decomposition to obtain a scaling coefficient s of the kth node of the original signaljkSum wavelet coefficient wjkThe following decomposition coefficient mjkDenotes, k ═ 1, 2, …, P. P represents the number of wavelet coefficients of the layer.
(2) Respectively calculating grading index K of each decomposition layer numberj、SjThe formula is as follows:
Figure GDA0001783469510000031
Figure GDA0001783469510000032
wherein the content of the first and second substances,
Figure GDA0001783469510000033
Figure GDA0001783469510000034
the reason for the grading index selection is as follows:
Kj、Sjcan reflect the difference of probability distribution of true value and noise value under the decomposition level, Kj、SjA larger value indicates a larger difference.
(3) Let the number of decomposition layers be JKTime, grade index KjObtaining a maximum value; number of decomposition layers JSTime, grade index SjThe maximum value is taken.
If J isK＝JSThen, J is selectedw＝JKFor optimal number of decomposition levels.
If J isK≠JSThen respectively make Jw＝JKAnd JS. The following steps are entered.
The third step is as follows:
(1) j (J is less than or equal to J)s) In the decomposition layer, the minimum value of the decomposition coefficient is min (m)j) Maximum value of max (m)j). For interval [2min (m) ]j)，2max(mj)]2P equally dividing is carried out, wherein P is equal to the number of wavelet coefficients of the layer.
(2) Let gjiIn the jth decomposition layer, the ith equally dividing point value is:
Figure GDA0001783469510000035
(3) computing
Figure GDA0001783469510000036
Obtaining an estimated value of the decomposition coefficient of the real signal, wherein the formula is as follows:
Figure GDA0001783469510000037
wherein p isG、pεWavelet coefficient probability distributions for the true signal and the noise signal, respectively.
The concrete mode of the step four is as follows:
(1) respectively calculating the pre-estimated scaling coefficient of the real signal by using the second step and the third step
Figure GDA0001783469510000041
And
Figure GDA0001783469510000042
(2) for time seriesThe final de-noising signal x (t) is obtained by line reconstructioni) The formula is as follows:
Figure GDA0001783469510000043
wherein
Figure GDA0001783469510000044
ψ(ti) Is the scale function of the wavelet and the wavelet basis function.
(3) In the second step, when J isK≠JSWhen the number of decomposition layers is J, it is calculated respectivelyK、JSTime series of reconstruction, note
Figure GDA0001783469510000045
Computing
Figure GDA0001783469510000049
And
Figure GDA00017834695100000410
is given by the formula:
Figure GDA0001783469510000046
selecting
Figure GDA0001783469510000047
And
Figure GDA0001783469510000048
and the medium SNR value is larger and is used as the finally obtained de-noising signal.
The invention has the beneficial effects.
Compared with the prior art mentioned in the background technology, the denoising method provided by the invention provides an optimal selection basis of the decomposition layer number of the wavelet packet, thereby better distinguishing a real signal from a noise signal without any manual intervention. The method has better filtering capability for non-Gaussian noise and can retain the characteristic frequency of the original signal.
Drawings
The invention is further described with reference to the following figures and detailed description. The scope of the invention is not limited to the following expressions.
Fig. 1 is a flowchart of a wavelet packet-based denoising method proposed by the present invention.
Fig. 2 is a waveform diagram of an attenuated oscillating signal according to an embodiment of the present invention and fig. 3 is a schematic structural diagram according to the present invention.
FIG. 3 is a waveform diagram and a spectrum diagram of an attenuated oscillator signal with white Gaussian noise according to an embodiment of the present invention.
FIG. 4 is a graph of hierarchical indices for each decomposition level computed in an embodiment of the present invention.
Fig. 5 is a waveform diagram and a frequency spectrum diagram after being processed by the denoising method provided by the present invention in the embodiment of the present invention.
Fig. 6 is a waveform diagram and a frequency spectrum diagram processed by using a bayesian wavelet packet noise reduction method of gaussian mixture prior.
Detailed Description
The invention is further illustrated in the following by reference to the figures and examples, as shown in figures 1-6. It should be apparent that the embodiments described in the detailed description are only a part of the embodiments of the present invention, and not all embodiments. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, belong to the scope of the present invention.
The embodiment of the invention is used for denoising an attenuated oscillation signal. The original signal is as follows:
Figure GDA0001783469510000051
wherein the carrier frequency f1＝3000Hz，f28000Hz, fig. 2 shows the time domain waveform and frequency spectrum. White gaussian noise with a standard deviation of 0.5 is added to the segment. FIG. 3 shows a noise-containingSignal time domain waveform and frequency spectrum. It can be seen that the characteristics of the signal are completely masked by the noise, and its carrier frequency cannot be found in the spectrogram.
The method comprises the following steps: constructing a time series y (t) from the noisy signali). The signal time length is 8ms and the sampling frequency fs＝25000Hz。y(ti) The SNR value of (a) is-0.75 dB.
Step two: let J equal 6, with db4 wavelet as the base wavelet, for y (t)i) Performing wavelet packet decomposition of 6 layers, and calculating grading index K of each decomposition layerj、SjSee fig. 4.
From FIG. 4, J is shownK＝JSThe optimal number of decomposition layers is selected to be 4 layers.
Step three: after the number of decomposition layers is selected, the maximum value max (m) of the decomposition coefficient of the layer is selectedj) To the minimum value min (m)j). In the interval [2min (m)j)，2max(mj)]The ith linear interpolation point is taken as gji。
Give pε、pGComputing each decomposition level
Figure GDA0001783469510000052
And
Figure GDA0001783469510000053
in particular, pGThe estimation formula is as follows:
Figure GDA0001783469510000061
where d is the standard deviation and α is the degree of slack in the signal, in this example d is 0.3 and α is 0.15.
Step four: and reconstructing the original signal according to the calculated decomposition coefficient to obtain a noise reduction signal.
Fig. 5 shows the noise reduction signal obtained after the above four steps, and it can be seen that, after the noise reduction processing is performed by using the method provided by the present invention, the SNR value is-1.35 dB, and the carrier frequency of the signal can be approximately identified to be 3000Hz and 8000 Hz. From time domain plot analysis, the amplitude approaches zero in 0-2ms and 5-8 ms.
FIG. 6 shows the signal processed by the Bayes wavelet packet denoising method using Gaussian mixture prior, and the waveform shows that a large amount of noise still exists in the signal, and the SNR value is-4.19 dB. From the time domain plot analysis, the amplitude is not zero in 0-2ms and 5-8 ms. This phenomenon may be due to the inability to accurately estimate the prior probability distribution of non-gaussian signals.
It should be understood that the detailed description of the present invention is only for illustrating the present invention and is not limited by the technical solutions described in the embodiments of the present invention, and those skilled in the art should understand that the present invention can be modified or substituted equally to achieve the same technical effects; as long as the use requirements are met, the method is within the protection scope of the invention.

Claims (8)

1. A Bayesian wavelet packet noise reduction method suitable for non-Gaussian signals is characterized by comprising the following steps:
the method comprises the following steps: acquiring a noisy signal construction time sequence of a monitored object;
step two: discrete wavelet packet decomposition is carried out on the time sequence, grading indexes of all decomposition layer numbers are calculated according to wavelet coefficients, and the optimal decomposition layer number is determined according to the grading indexes;
step three: estimating wavelet coefficients of real signals in each decomposition level;
step four: reconstructing the original signal by utilizing a wavelet packet inverse transformation formula according to the estimated wavelet coefficient so as to obtain a denoised signal;
the method comprises the following steps: based on the noise-containing signal actually measured by the sensor, intercepting a section of construction time sequence y (t) containing signal characteristicsi) (ii) a The sampling frequency is in accordance with Shannon sampling theorem; let y (t)i)＝x(ti)+ε(ti)，i＝1，2，...，2JWherein x (t)i) And ε (t)i) For time series of real and real noise signals, J ═ log2N, the purpose of noise reduction is to obtain x (t)i) A predicted value of (2);
the second step comprises the following steps:
(1) selecting the number j of decomposition layers for the time sequence y (t)i) Carrying out wavelet packet decomposition to obtain a scaling coefficient s of the kth node of the original signaljkSum wavelet coefficient wjkThe following decomposition coefficient mjkIs represented by k ═ 1, 2, …, P; p represents the number of wavelet coefficients of the layer;
(2) respectively calculating grading index K of each decomposition layer numberj、SjThe formula is as follows:
Figure FDA0003150365190000011
Figure FDA0003150365190000012
wherein the content of the first and second substances,
Figure FDA0003150365190000013
Figure FDA0003150365190000021
the grading index is as follows: kj、SjCan reflect the difference of probability distribution of true value and noise value under the decomposition level, Kj、SjA larger value indicates a larger difference;
(3) let the number of decomposition layers be JKTime, grade index KjObtaining a maximum value; number of decomposition layers JSTime, grade index SjObtaining a maximum value;
if J isK＝JsThen, J is selectedw＝JKThe optimal decomposition layer number is obtained;
if J isK≠JSThen respectively make Jw＝JKAnd JS。
2. The bayesian wavelet packet noise reduction method applicable to non-gaussian signals according to claim 1, wherein the method comprises the following steps: and the noisy signal in the step one is an acceleration signal.
3. The bayesian wavelet packet noise reduction method applicable to non-gaussian signals according to claim 1, wherein the method comprises the following steps: in the signal in the step one, the real signal and the noise signal have different probability distribution types.
4. The bayesian wavelet packet noise reduction method applicable to non-gaussian signals according to claim 1, wherein the method comprises the following steps: the base wavelet used in the step two is an orthogonal wavelet; and the maximum decomposition layer number 2 is more than d and less than 10 in the second step.
5. The bayesian wavelet packet noise reduction method applicable to non-gaussian signals according to claim 1, wherein the method comprises the following steps: in step three, the wavelet coefficient probability distribution p of signals and noise is described as accurately as possibleGAnd pεIf the orthogonal wavelet base, p, is adopted in the second stepGAnd pεThe available true signal x (t)i) And the true noise signal ε (t)i) Is represented by a probability distribution.
6. The bayesian wavelet packet noise reduction method applicable to non-gaussian signals according to claim 1, wherein the method comprises the following steps: and in the fourth step, evaluating the denoising effect by adopting two methods, namely a signal-to-noise ratio (SNR) and a time domain analysis.
7. The Bayesian wavelet packet noise reduction method for non-Gaussian signals as recited in claim 1, wherein the third step comprises:
(1) and J is not more than J in the jth decomposition layersThe minimum value of the decomposition coefficient is min (m)j) Maximum value of max (m)j) (ii) a For interval [2min (m) ]j)，2max(mj)]2P equally dividing, wherein P is equal to the number of wavelet coefficients of the layer;
(2) let gjiIs the ith in the jth decomposition layerDividing the point value equally:
Figure FDA0003150365190000031
(3) and calculating
Figure FDA0003150365190000032
Obtaining an estimated value of the decomposition coefficient of the real signal, wherein the formula is as follows:
Figure FDA0003150365190000033
wherein p isG、pεWavelet coefficient probability distributions for the true signal and the noise signal, respectively.
8. The Bayesian wavelet packet noise reduction method for non-Gaussian signals as recited in claim 1, wherein the fourth step comprises:
(1) respectively calculating the pre-estimated scaling coefficient of the real signal by utilizing the step two and the step three
Figure FDA0003150365190000034
And
Figure FDA0003150365190000035
(2) reconstructing the time sequence to obtain a final de-noising signal x*(ti) The formula is as follows:
Figure FDA0003150365190000036
wherein
Figure FDA0003150365190000037
ψ(ti) Is a scale function and a wavelet basis function of a wavelet;
(3) in the second step, when J isK≠JSWhen the number of decomposition layers is J, it is calculated respectivelyK、JSTime series of reconstruction, note
Figure FDA0003150365190000038
Computing
Figure FDA0003150365190000039
And
Figure FDA00031503651900000310
SNR, the formula is:
Figure FDA00031503651900000311
selecting
Figure FDA00031503651900000312
And
Figure FDA00031503651900000313
and the medium SNR value is larger and is used as the finally obtained de-noising signal.
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