CN105547700B - Retainer outer arc based on relevant denoising spectrum diagnostic method - Google Patents

Abstract

The invention discloses a kind of retainer outer arc based on relevant denoising spectrum diagnostic method, relate to the method for testing technical field of bearing.Described method comprises the steps: the signal post of two identical acceleration transducer collections is designated as vibration signal one and vibration signal two；Vibration signal one is carried out analysis based on peak factor, vibration signal two is carried out analysis based on degree of bias index；The resonant belt that latter two signal of selection analysis is identical, processes the resonant belt signal corresponding high Q band filter of feeding；The output signal of two-way filter is done cross correlation process, does Hilbert transform afterwards and by doing FFT spectrum analysis after anti-mixing filter, extract failure-frequency.Described method can not only adaptively selected optimum resonant belt, and the signal to noise ratio of signal can be improved, reduce the operand of Hilbert transform, meet the real-time of bearing failure diagnosis, improve the accuracy to the spectrum diagnosis of retainer outer arc significantly.

Description

Retainer outer arc based on relevant denoising spectrum diagnostic method

Technical field

The present invention relates to the method for testing technical field of bearing, particularly relate to a kind of retainer based on relevant denoising Outer arc spectrum diagnostic method.

Background technology

The importance of retainer is: rolling element can be made to keep suitable distance each other, prevent adjacent rolling Between kinetoplast directly contact, with by friction and therefore and the heat produced is maintained at floor level；Rolling element is made to be evenly distributed In whole bearing, load is made more uniformly to be distributed and reducing noise；Rolling element is guided, to improve in bearing in no-load district Rolling condition and prevent the slip of damageability；For the bearing of divergence type, installing or removing one of them bearing During lasso, rolling element can be maintained at one.

Diagnostic method such as resonance and demodulation method that is the most perfect to the method for retainer outer arc spectrum at present, that often use Deng, not only self adaptation can not choose optimum resonant belt, and not have can to remove targetedly in resonant belt substantial amounts of high-strength Degree noise, the outer arc of retainer is easily composed the situation falling into oblivion or similar outer ring failure-frequency frequency multiplication occur by this, it is difficult to Cause the attention of technical staff.

Summary of the invention

The technical problem to be solved is to provide the spectrum diagnosis of a kind of retainer outer arc based on relevant denoising Method, described method can not only adaptively selected optimum resonant belt, and the signal to noise ratio of signal can be improved, significantly Improve the accuracy to the spectrum diagnosis of retainer outer arc.

For solving above-mentioned technical problem, the technical solution used in the present invention is: a kind of bearing based on relevant denoising is protected Hold frame outer arc spectrum diagnostic method, it is characterised in that comprise the steps:

1) use acceleration transducer to gather the vibration information of retainer, the acceleration transducer that two identical is adopted The signal post of collection is designated as vibration signal one and vibration signal two；

2) vibration signal one is carried out analysis based on peak factor, vibration signal two is carried out based on degree of bias index point Analysis；

3) resonant belt that latter two signal of selection analysis is identical, sends resonant belt signal into corresponding high Q band filter Process；

4) output signal of two-way filter is done cross correlation process, the signal after cross correlation process is done Hilbert and becomes Change and by doing FFT spectrum analysis after anti-mixing filter, extract failure-frequency, carrying out the spectrum diagnosis of retainer outer arc.

Further technical scheme is: described carries out the method for conversion based on peak factor such as by vibration signal one Under:

The peak value x of signal_peakComputing formula as follows:

x_peak=max (x_i)

Wherein x is the Discrete signal collected, and i is the footnote i.e. sequence of Discrete signal of Discrete signal Number, wherein that amplitude maximum is x_peak；

Virtual value x of signal_rmsComputing formula as follows:

$x_{rms}=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\Σ}}{(x_i-\stackrel{\‾}{x})}^2}$

For time-domain signal x_iThe meansigma methods of amplitude；

The computing formula of peak factor C is as follows:

$C=\frac{x_{peak}}{x_{rms}}.$

Further technical scheme is: the method for described conversion based on degree of bias index is as follows:

The computing formula of degree of bias index P is as follows:

α is the degree of bias；

$P=\frac{\mathrm{\α}}{x_{rms}^3}$

During P=0, signal presents symmetrical；During P ≠ 0, the probability distribution of signal is asymmetric；Wherein, as P > 0 time, signal Positively biased；When P < when 0, signal negative bias；The absolute value of degree of bias index is the biggest, illustrates that the deviation of signal distributions form is symmetrical the tightest Weight.

Further technical scheme is: the method that two described signals do relevant treatment is as follows:

Signal x (t) is defined as with the cross-correlation function of y (t)

It is A (t)=a (t)+s (t), wherein a (t) by the signal of self adaptation height Q band filter based on peak factor For selected resonant belt signal, s (t) is noise；It is B (t) by the signal of self adaptation height Q band filter based on degree of bias index =b (t)+v (t), wherein b (t) is selected resonant belt signal, and v (t) is noise；Can be obtained by signal cross-correlation function definition:

$\begin{array}{c}{R}_{AB}\left(\mathrm{\&tau;}\right)=\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}A\left(t\right)B(t+\mathrm{\&tau;})dt\\ =\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^T\&lsqb;a\left(t\right)+s\left(t\right)\&rsqb;\&lsqb;b(t+\mathrm{\&tau;})+v(t+\mathrm{\&tau;})\&rsqb;dt\\ =\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}a\left(t\right)b(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}a\left(t\right)v(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}s\left(t\right)b(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}s\left(t\right)v(t+\mathrm{\&tau;})dt\\ =R_{ab}\left(\mathrm{\&tau;}\right)+R_{av}\left(\mathrm{\&tau;}\right)+R_{sb}\left(\mathrm{\&tau;}\right)+R_{sv}\left(\mathrm{\&tau;}\right)\end{array}$

Noise and the usual non-correlation of signal, i.e. R_av(τ)=0, R_sb(τ)=0；Signal is through the wave filter of different parameters The noise comprised is the most different, i.e. R_sv(τ)=0, therefore R_AB(τ)=R_ab(τ)。

Further technical scheme is: it is as follows that described signal does Hilbert transform method:

The Hilbert transform of continuous signal x (t)Definition is as follows:

$\hat{x}\left(t\right)=x\left(t\right)*\frac{1}{\mathrm{\&pi;}t}=\frac{1}{\mathrm{\&pi;}}{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}\frac{x\left(\mathrm{\&tau;}\right)}{t-\mathrm{\&tau;}}d\mathrm{\&tau;}$

Thus can get the analytic signal of x (t)

$z\left(t\right)=x\left(t\right)+j\hat{x}\left(t\right)=A\left(t\right)e^{\mathrm{\&theta;}\left(t\right)}$

Wherein θ (t) is the phase place of z (t), and A (t) is the amplitude of z (t), namely the envelope of signal x (t).

Use produced by technique scheme and have the beneficial effects that: 1) described method use two paths of signals respectively through time Peak factor in domain analysis selects identical resonant belt with degree of bias index, and be so generally selected is optimum resonant belt.2) institute State the resonant belt using time-domain analysis to determine in vibration signal in method, can reach certainly for different bearing band filters The effect adapted to.3) two paths of signals that described method uses is owing to using different Time Domain Analysis, causes in resonant belt Noise is the most different, is completely eliminated effect of noise when using cross-correlation analysis, prominent vibration signal, improves significantly Accuracy to the spectrum diagnosis of retainer outer arc.

Accompanying drawing explanation

Fig. 1-2 is the flow chart of the method for the invention；

Fig. 3 a is the time-domain diagram of the original vibration signal that first acceleration transducer collects

Fig. 3 b is the frequency domain figure of the original vibration signal that first acceleration transducer collects；

Fig. 4 a is the time-domain diagram of the original vibration signal that second acceleration transducer collects

Fig. 4 b is the frequency domain figure of the original vibration signal that second acceleration transducer collects；

Fig. 5 a is the filtered time-domain diagram of the adaptive bandpass filter through analyzing based on peak factor；

Fig. 5 b is the filtered frequency domain figure of the adaptive bandpass filter through analyzing based on peak factor；

Fig. 6 a is through the filtered time-domain diagram of adaptive bandpass filter based on degree of bias index analysis

Fig. 6 b is through the filtered frequency domain figure of adaptive bandpass filter based on degree of bias index analysis；

Fig. 7 a is the cross-correlation analysis time-domain diagram of two-way original vibration signal；

Fig. 7 b is the cross-correlation analysis time-domain diagram of the two-way vibration signal respectively through adaptive-filtering；

Fig. 8 is that the cross-correlation analysis result of the two-way vibration signal respectively through adaptive-filtering is after Hilbert transform Time-domain diagram；

The extraction result figure of Fig. 9 outer arc spectrum.

Detailed description of the invention

Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete Ground describes, it is clear that described embodiment is only a part of embodiment of the present invention rather than whole embodiments.Based on Embodiment in the present invention, it is every other that those of ordinary skill in the art are obtained under not making creative work premise Embodiment, broadly falls into the scope of protection of the invention.

Elaborate a lot of detail in the following description so that fully understanding the present invention, but the present invention is all right Using other to be different from alternate manner described here to implement, those skilled in the art can be without prejudice to intension of the present invention In the case of do similar popularization, therefore the present invention is not limited by following public specific embodiment.

Overall, as shown in Figure 1-2 the flow chart of described method (A in A with Fig. 2 in Fig. 1 collectively form after being connected), The invention discloses a kind of retainer outer arc based on relevant denoising spectrum diagnostic method, comprise the steps:

1) use acceleration transducer to gather the vibration information of retainer, the acceleration transducer that two identical is adopted The signal post of collection is designated as vibration signal one and vibration signal two, as shown in Fig. 3 a-3b and 4a-4b；

2) vibration signal one is carried out analysis based on peak factor, vibration signal two is carried out based on degree of bias index point Analysis；

3) resonant belt that latter two signal of selection analysis is identical, sends resonant belt signal into corresponding high Q band filter Process；

4) output signal of two-way filter is done cross correlation process, the signal after cross correlation process is done Hilbert and becomes Change and by doing FFT spectrum analysis after anti-mixing filter, extract failure-frequency, carrying out the spectrum diagnosis of retainer outer arc.

Concrete:

1) based on the peak factor analysis to resonant belt

Peak value: be applicable to the fault diagnosis with temporary impact, particularly early stage bearing surface damaged, such as impression, stripping Fall, bearing surface cut, crackle equivalent damage, be very easy to by peak value change-detection out.But it is to having rolling element to holding The transient vibration that the reasons such as the impact of frame and sudden external interference cause is more sensitive.Peak value x_peakThe following institute of computing formula Show:

$x_{peak}=\underset{i}{\max }\left(x_i\right)$

Virtual value (RMS): along with the development of fault and monotone increasing, generic failure bearing signal is than fault-free bearing signal RMS value high, the bearing signal of many places fault is higher than the bearing signal RMS value of single fault.RMS applies effect in trend analysis Fruit is preferable, and to there being the abnormal the most sensitive of face crack random vibration waveform.Virtual value x_rmsThe following institute of computing formula Show:

$x_{rms}=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\&Sigma;}}{(x_i-\stackrel{\&OverBar;}{x})}^2}$

Wherein that amplitude maximum is x_peak；

For time-domain signal x_iThe meansigma methods of amplitude；

Peak factor: dimensionless group peak factor considers the pass of dimensional parameters peak value and virtual value in the calculation System.Generally, in normal bear vibration, its peak factor is about 5, and when there is the superficial failures such as crackle when bearing, peak factor can To reach more than 10, peak factor carries out detection to bearing can not be affected by bearing rotating speed and size, is i.e. used in detection In transducer sensitivity change also will not on certainty of measurement produce impact.The computing formula of peak factor C is as follows:

$C=\frac{x_{peak}}{x_{rms}}$

Figure after conversion is as shown in Fig. 5 a-5b.

2) based on the analysis to resonant belt of the degree of bias index

Degree of bias index can reflect discrete signal probability mass function or the symmetry of continuous signal probability density function.The degree of bias The computing formula of index P is as follows:

$\mathrm{\&alpha;}=\frac{1}{N}\underset{i=1}{\overset{N}{\&Sigma;}}{(x_i-\stackrel{\&OverBar;}{x})}^3$

$P=\frac{\mathrm{\&alpha;}}{x_{rms}^3}$

During P=0, signal presents symmetrical；During P ≠ 0, the probability distribution of signal is asymmetric.Wherein, as P > 0 time, signal Positively biased；When P < when 0, signal negative bias.The absolute value of degree of bias index is the biggest, illustrates that the deviation of signal distributions form is symmetrical the tightest Weight.

The technical specification of band filter can be obtained by the analysis result of peak factor and degree of bias index.Although additionally, The resonant belt that band filter selects is identical, but the band filter technical parameter that different analyses obtains has trickle difference Not.Therefore by effective vibration signal contained in the signal resonant belt after band filter is identical, noise differs, and becomes Figure after changing is as shown in Fig. 6 a-6b.

3) cross-correlation analysis

Signal x (t) is defined as with the cross-correlation function of y (t)

It is A (t)=a (t)+s (t), wherein a (t) by the signal of self adaptation height Q band filter based on peak factor For selected resonant belt signal, s (t) is noise；It is B (t) by the signal of self adaptation height Q band filter based on degree of bias index =b (t)+v (t), wherein b (t) is selected resonant belt signal, and v (t) is noise.Can be obtained by signal cross-correlation function definition:

$\begin{array}{c}{R}_{AB}\left(\mathrm{\&tau;}\right)=\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}A\left(t\right)B(t+\mathrm{\&tau;})dt\\ =\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^T\&lsqb;a\left(t\right)+s\left(t\right)\&rsqb;\&lsqb;b(t+\mathrm{\&tau;})+v(t+\mathrm{\&tau;})\&rsqb;dt\\ =\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}a\left(t\right)b(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}a\left(t\right)v(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}s\left(t\right)b(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}s\left(t\right)v(t+\mathrm{\&tau;})dt\\ =R_{ab}\left(\mathrm{\&tau;}\right)+R_{av}\left(\mathrm{\&tau;}\right)+R_{sb}\left(\mathrm{\&tau;}\right)+R_{sv}\left(\mathrm{\&tau;}\right)\end{array}$

General, noise and the usual non-correlation of signal, i.e. R_av(τ)=0, R_sb(τ)=0；Signal is through different parameters The noise that wave filter comprises is the most different, i.e. R_sv(τ)=0, therefore R_AB(τ)=R_ab(τ).Learnt by the characteristic of cross-correlation function: with The periodic signal of frequency or include the signal of periodic component of same frequency, cross-correlation function is still periodic signal, its cycle Constant and phase information is not lost.So doing cross correlation process to have reached the effective vibration signal of reinforcement and the effect of attenuating noise, As shown in Fig. 7 a-7b.

4) fault characteristic frequency is extracted

(1) Hilbert transform of continuous signal x (t)Definition is as follows:

$\hat{x}\left(t\right)=x\left(t\right)*\frac{1}{\mathrm{\&pi;}t}=\frac{1}{\mathrm{\&pi;}}{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}\frac{x\left(\mathrm{\&tau;}\right)}{t-\mathrm{\&tau;}}d\mathrm{\&tau;}$

Thus can get the analytic signal of x (t)

$z\left(t\right)=x\left(t\right)+j\hat{x}\left(t\right)=A\left(t\right)e^{\mathrm{\&theta;}\left(t\right)}$

Wherein θ (t) is the phase place of z (t), and A (t) is the amplitude of z (t), namely the envelope of signal x (t).

Signal retains low frequency fault-signal through anti-mixing filter after Hilbert transform, carries out fault-signal Fft analysis, obtains fault characteristic frequency, as shown in Figure 8.

5) outer arc spectral technology

During some retainer Initial operation, the thing of losing of some copper ashes lost and damage of the bearing may be had to enter raceway, Unstable adheres to outer shroud in short-term, is rolled by roller, produces the parasitic fault of outer shroud seemingly but spectral line is isolated (the most multistage Property) impact.Outer arc spectral technology loses thing, with indirect identification retainer fault by identification retainer incipient failure.Application Outer arc spectral technology should be noted that problems with:

1) outer arc spectrum is the feature that there is hardware impurity in bearing.

2) outer arc spectrum is also to maintain the feature of frame damage (losing material), occurs that the bearing majority that outer arc is composed has accordingly The modulation impact in retainer characteristic information-retainer cycle.

3) the outer arc spectrum that the retainer local broken fault initial stage occurs is the best period finding fault, is illustrated in figure 9 The extraction result figure of outer arc spectrum.

Described method uses two paths of signals to select identical respectively through the peak factor in time-domain analysis with degree of bias index Resonant belt, be so generally selected is optimum resonant belt.Described method use time-domain analysis determine the resonance in vibration signal Band, can reach adaptive effect for different bearing band filters.The two paths of signals that described method uses is owing to adopting With different Time Domain Analysis, cause the noise in resonant belt the most different, be completely eliminated when using cross-correlation analysis and make an uproar The impact of sound, prominent vibration signal, improve the accuracy to the spectrum diagnosis of retainer outer arc significantly.

Claims (1)

1. retainer outer arc based on a relevant denoising spectrum diagnostic method, it is characterised in that comprise the steps:

1) use acceleration transducer to gather the vibration information of retainer, two identical acceleration transducers are gathered Signal post is designated as vibration signal one and vibration signal two；

2) vibration signal one is carried out analysis based on peak factor, vibration signal two is carried out analysis based on degree of bias index；

The described method that vibration signal one carries out analysis based on peak factor is as follows:

The peak value x of signal_peakComputing formula as follows:

x_peak=max (x_i)

Wherein x is the Discrete signal collected, and i is the footnote i.e. sequence number of Discrete signal of Discrete signal, its Middle amplitude maximum is x_peak；

Virtual value x of signal_rmsComputing formula as follows:

$x_{rms}=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\&Sigma;}}{(x_i-\stackrel{\&OverBar;}{x})}^2}$

For time-domain signal x_iThe meansigma methods of amplitude, N is the sample points of discretization signal；

The computing formula of peak factor C is as follows:

$C=\frac{x_{peak}}{x_{rms}};$

The method of described analysis based on degree of bias index is as follows:

The computing formula of degree of bias index P is as follows:

$\mathrm{\&alpha;}=\frac{1}{N}\underset{i=1}{\overset{N}{\&Sigma;}}{(x_i-\stackrel{\&OverBar;}{x})}^3$

α is the degree of bias, and N is the sample points of discretization signal；

$P=\frac{\mathrm{\&alpha;}}{x_{rms}^3}$

During P=0, signal presents symmetrical；During P ≠ 0, the probability distribution of signal is asymmetric；Wherein, as P > 0 time, signal is just Partially；When P < when 0, signal negative bias；The absolute value of degree of bias index is the biggest, illustrates that the deviation of signal distributions form is symmetrical the most serious；

3) resonant belt that latter two signal of selection analysis is identical, is carried out the resonant belt signal corresponding high Q band filter of feeding Process；

4) output signal of two-way filter is done cross correlation process, the signal after cross correlation process is done Hilbert transform also By doing FFT spectrum analysis after anti-mixing filter, extract failure-frequency, carry out the spectrum diagnosis of retainer outer arc；

The described method that the output signal of two-way filter is done cross correlation process is as follows:

Signal x (t) is defined as with the cross-correlation function of y (t)

Being A (t)=a (t)+s (t) by the signal of self adaptation height Q band filter based on peak factor, wherein a (t) is institute Selecting resonant belt signal, s (t) is noise；It is B (t)=b by the signal of self adaptation height Q band filter based on degree of bias index T ()+v (t), wherein b (t) is selected resonant belt signal, and v (t) is noise；Can be obtained by signal cross-correlation function definition:

$\begin{array}{c}{R}_{AB}\left(\mathrm{\&tau;}\right)=\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}A\left(t\right)B(t+\mathrm{\&tau;})dt\\ =\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^T\&lsqb;a\left(t\right)+s\left(t\right)\&rsqb;\&lsqb;b(t+\mathrm{\&tau;})+v(t+\mathrm{\&tau;})\&rsqb;dt\\ =\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}a\left(t\right)b(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}a\left(t\right)v(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}s\left(t\right)b(t+\mathrm{\&tau;})dt+\underset{T\&RightArrow;\mathrm{\&infin;}}{\lim }\frac{1}{T}{\&Integral;}_0^{T}s\left(t\right)v(t+\mathrm{\&tau;})dt\\ =R_{ab}\left(\mathrm{\&tau;}\right)+R_{av}\left(\mathrm{\&tau;}\right)+R_{sb}\left(\mathrm{\&tau;}\right)+R_{sv}\left(\mathrm{\&tau;}\right)\end{array}$

Noise and the usual non-correlation of signal, i.e. R_av(τ)=0, R_sb(τ)=0；Signal comprises through the wave filter of different parameters Noise is the most different, i.e. R_sv(τ)=0, therefore R_AB(τ)=R_ab(τ), wherein R_ab(τ) cross-correlation function of a (t) and b (t) is represented； R_av(τ) cross-correlation function of a (t) and v (t) is represented；R_sb(τ) cross-correlation function of s (t) and b (t) is represented；R_sv(τ) s is represented The cross-correlation function of (t) and v (t)；

It is described that signal after cross correlation process is done Hilbert transform method is as follows:

The Hilbert transform of continuous signal x (t)Definition is as follows:

$\hat{x}\left(t\right)=x\left(t\right)*\frac{1}{\mathrm{\&pi;}t}=\frac{1}{\mathrm{\&pi;}}{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}\frac{x\left(\mathrm{\&tau;}\right)}{t-\mathrm{\&tau;}}d\mathrm{\&tau;}$

Thus can get the analytic signal of x (t)

$z\left(t\right)=x\left(t\right)+j\hat{x}\left(t\right)=A\left(t\right)e^{\mathrm{\&theta;}\left(t\right)}$

Wherein θ (t) is the phase place of z (t), and A (t) is the amplitude of z (t), namely the envelope of signal x (t).