Time-frequency analysis method for non-stationary signals under coexistence of harmonic waves and pulses

Abstract

The invention relates to the technical field of signal processing, in particular to a time-frequency analysis method for a non-stationary signal under coexistence of harmonic waves and pulses, which comprises the following steps: acquiring a linear frequency modulation signal in which a harmonic component and a pulse component exist at the same time, and solving a threshold value compared with a chirp rate; calculating the chirp rate of each time frequency point of the linear frequency modulation signal based on a chirp rate estimator; when the chirp rate of a certain time-frequency point is less than or equal to a threshold value, considering that the time-frequency point tends to the harmonic characteristic, and redistributing the time-frequency point in the frequency direction; when the chirp rate of a certain time-frequency point is greater than a threshold value, considering that the time-frequency point tends to pulse characteristics, and redistributing the time-frequency point in the time direction; and adding the two redistribution results to obtain the time-frequency representation of the linear frequency modulation signal. According to the method, the time-frequency representation with concentrated energy can be generated for the harmonic and pulse components in the signal at the same time.

Description

Technical Field

[0001] The present invention relates to the technical field of signal processing, and more particularly to a time-frequency analysis method for non-stationary signals in the coexistence of harmonics and pulses. Background Art

[0002] For harmonic signals, synchrosqueezing transform can accurately estimate their two-dimensional instantaneous frequency distribution, and use the synchrosqueezing operator along the frequency direction to squeeze the time-frequency energy to the true instantaneous frequency position. The synchrosqueezing transform is only effective when the signal can be locally approximated as a harmonic. When there are pulse components in the signal, the sharp change in the instantaneous frequency of the pulse signal causes the synchrosqueezing transform to be unable to generate a time-frequency representation with concentrated energy.

[0003] For pulse signals, time-reassignment synchrosqueezing transform can effectively sharpen their time-frequency representation. Although the time-reassignment synchrosqueezing transform can generate a time-frequency representation with concentrated energy for pulse signals, it is only effective when the signal can be locally approximated as a pulse. When there are harmonic components in the signal, the sharp change in the group delay of the harmonic signal causes the time-reassignment synchrosqueezing transform to be unable to generate a time-frequency representation with concentrated energy.

[0004] Although the synchrosqueezing transform and the time-reassignment synchrosqueezing transform can respectively generate time-frequency representations with concentrated energy for harmonic signals and pulse signals, when there are both harmonic and pulse components in the signal, a single method cannot generate time-frequency representations with concentrated energy for these two components. Summary of the Invention

[0005] In view of this, the present invention provides a time-frequency analysis method for non-stationary signals in the coexistence of harmonics and pulses, which can simultaneously generate time-frequency representations with concentrated energy for the harmonic and pulse components in the signal.

[0006] In order to achieve the above object, the present invention adopts the following technical scheme:

[0007] A time-frequency analysis method for non-stationary signals in the coexistence of harmonics and pulses, comprising the following steps:

[0008] Obtain a chirp signal with both harmonic components and pulse components, and obtain a threshold for comparison with the chirp rate;

[0009] Calculate the chirp rate of each time-frequency point of the chirp signal based on a chirp rate estimator;

[0010] When the chirp rate of a certain time-frequency point is less than or equal to the threshold, it is considered that the time-frequency point tends to have harmonic characteristics, and it is redistributed in the frequency direction;

[0011] When the chirp rate of a certain time-frequency point is greater than the threshold, it is considered that the time-frequency point tends to have pulse characteristics, and it is redistributed in the time direction;

[0012] Add the two reassignment results to obtain the time-frequency representation of the chirp signal.

[0013] Furthermore, the expression of the chirp rate estimator is:

[0014]

[0015] where represents the chirp rate at a certain time-frequency point (t, ω), t is time, and ω is frequency; represents the representation of the instantaneous frequency estimate in the time-frequency space; represents the representation of the group delay estimate in the time-frequency space; Inf means that when the chirp rate estimator is infinite, and in this case, the chirp rate cannot be represented by a finite value.

[0016] Furthermore, obtaining the threshold for comparison with the chirp rate includes:

[0017] For any time-frequency point in the chirp signal, calculate the error between the instantaneous frequency estimate and the true instantaneous frequency at this time-frequency point

[0018] Calculate the error between the group delay estimate and the true group delay at this time-frequency point

[0019] Take the chirp rate at as the boundary, and at this time the chirp rate threshold is expressed as: |c| = β -2 / 3 , where the short-time Fourier transform is performed in the form of a window function, and β represents the parameter of the window function.

[0020] Furthermore, the error between the instantaneous frequency estimate and the true instantaneous frequency

[0021]

[0022] reflects the concentration degree of the synchrosqueezing transform result, and its expression is: represents the representation of the instantaneous frequency estimate in the time-frequency space, and its expression is:

[0023]

[0024] Furthermore, the error between the group delay estimate and the true group delay

[0025]

[0026] Among them, (ω - b) / c represents the group delay of the signal; represents the representation of the group delay estimation in the time - frequency space, and its expression is:

[0027]

[0028] Furthermore, if less than or equal to then the result obtained by using the synchrosqueezing transform is closer to the time - frequency trajectory of the signal than the result obtained by using the time - rearranged synchrosqueezing transform. At this time, the signal is more suitable to be processed by the synchrosqueezing transform; otherwise, the signal is suitable to be processed by the time - rearranged synchrosqueezing transform.

[0029] Furthermore, according to the comparison results of the chirp rates and the threshold of each time - frequency point in the chirp signal, the short - time Fourier transform results are divided into two parts, namely:

[0030]

[0031] Among them, G F (t, ω) represents the part to be subjected to the squeezing transform in the frequency direction, G T (t, ω) represents the part to be subjected to the squeezing transform in the time direction, and G(t, ω) represents the short - time Fourier transform.

[0032] Furthermore, the final time - frequency representation of the chirp signal is:

[0033]

[0034] Among them, two variables in (u, η), (u, ω), (t, η) both represent time and frequency; represents the two - dimensional instantaneous frequency estimation operator, is the synchrosqueezing operator for synchrosqueezing in the frequency direction, where δ() is the Dirac function, that is, δ(0) = 1, δ(non - 0) = 0; F S (u, η) represents the result after the synchrosqueezing transform in the frequency direction; represents the two - dimensional group delay estimation operator; represents the synchrosqueezing operator for synchrosqueezing in the time direction; T S (u, η) represents the result after the synchrosqueezing transform in the time direction.

[0035] It can be seen from the above - mentioned technical solutions that, compared with the prior art, the present invention has the following beneficial effects:

[0036] The present invention adopts two-way rearrangement squeezing transformation and adaptively selects an appropriate direction to reallocate time-frequency coefficients by using the magnitude of the local chirp rate of the signal. When the local chirp rate of the signal is less than a specific threshold, the local signal is considered to tend to harmonic characteristics, so the time-frequency coefficients are reallocated in the frequency direction. When the local chirp rate of the signal is greater than the specific threshold, the local signal is considered to tend to pulse characteristics, so the time-frequency coefficients are reallocated in the time direction. Finally, the results of the two reallocations are added together to obtain the final result. The present invention adopts a "divide and conquer" strategy for the signal, and different parts of the signal are reallocated in a reasonable direction according to their own characteristics, so that energy-concentrated time-frequency representations can be generated for both harmonic and pulse components in the signal. Description of the Drawings

[0037] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required to be used in the description of the embodiments or the prior art. Obviously, the drawings in the following description are only embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained according to the provided drawings.

[0038] Figure 1 is a flowchart of the method of the present invention;

[0039] Figure 2 is the acoustic signal and its envelope at a constant engine speed, where (a) is the signal and (b) is the envelope;

[0040] Figure 3 is the result of applying different time-frequency analysis methods to the envelope; (a) is the short-time Fourier transform, (b) is the synchrosqueezing transform, (c) is the time rearrangement synchrosqueezing transform, and (d) is the two-way rearrangement squeezing transform;

[0041] Figure 4 is the ridge extraction result of the two-way rearrangement squeezing transform result;

[0042] Figure 5 is the acoustic signal and its envelope at a variable engine speed; where (a) is the signal and (b) is the envelope;

[0043] Figure 6 is the result of applying different time-frequency analysis methods to the envelope, (a) is the short-time Fourier transform, (b) is the synchrosqueezing transform, (c) is the time rearrangement synchrosqueezing transform, and (d) is the two-way rearrangement squeezing transform;

[0044] Figure 7 is the ridge extraction result of the two-way rearrangement squeezing transform result. Detailed Embodiments

[0045] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0046] As Figure 1 shown, an embodiment of the present invention discloses a time-frequency analysis method for non-stationary signals under the coexistence of harmonics and pulses, including the following steps:

[0047] S1. Obtain a chirp signal with both harmonic components and pulse components, and obtain a threshold for comparison with the chirp rate;

[0048] S2. Calculate the chirp rate of each time-frequency point of the chirp signal based on a chirp rate estimator;

[0049] S3. When the chirp rate of a certain time-frequency point is less than or equal to the threshold, it is considered that this time-frequency point tends to have harmonic characteristics, and reallocate it in the frequency direction;

[0050] When the chirp rate of a certain time-frequency point is greater than the threshold, it is considered that this time-frequency point tends to have pulse characteristics, and reallocate it in the time direction;

[0051] Add the two reallocation results to obtain the time-frequency representation of this chirp signal.

[0052] Next, the above steps will be further described.

[0053] S1. Obtain a chirp signal with both harmonic components and pulse components, and obtain a threshold for comparison with the chirp rate.

[0054] Before obtaining the threshold, first explain the working principles of the synchrosqueezing transform and the time-rearranged synchrosqueezing transform.

[0055] For harmonic signals, the synchrosqueezing transform can accurately estimate their two-dimensional instantaneous frequency distribution, and use the synchrosqueezing operator along the frequency direction to squeeze the time-frequency energy to the true instantaneous frequency position. Here, taking the harmonic signal as an example to illustrate the working principle of the synchrosqueezing transform, first introduce the short-time Fourier transform, which is expressed as:

[0056]

[0057] where \(s(u)\) and \(g(u)\) represent the analysis signal and the window function respectively. According to Parseval's theorem, Equation (1) can also be expressed as:

[0058]

[0059] Among them, and respectively represent the frequency-domain forms of the analysis signal and the window function. The time-domain expression of the harmonic signal used in the present invention is:

[0060]

[0061] The amplitude of this signal is A, and the instantaneous frequency is ω 0 ; j represents the imaginary unit. Performing a short-time Fourier transform on this signal, its expression is:

[0062]

[0063] The synchrosqueezing transform constructs an instantaneous frequency estimation operator by using the relationship between the short-time Fourier transform and the partial derivative of the short-time Fourier transform with respect to time. Taking the partial derivative with respect to time of the time-frequency result of the short-time Fourier transform in Equation (4), the following relationship can be obtained:

[0064]

[0065] When the instantaneous frequency estimation can be expressed in the time-frequency space as:

[0066]

[0067] Substituting Equation (4) and Equation (5) into Equation (6) gives:

[0068]

[0069] It can be seen that the instantaneous frequency of the signal s h (t) can be accurately located. On this basis, the synchrosqueezing transform constructs a synchrosqueezing operator along the frequency direction to squeeze the extended time-frequency energy to the position of the true instantaneous frequency, and its expression is:

[0070]

[0071] The synchrosqueezing transform performs a secondary rearrangement of the time-frequency energy distribution of the short-time Fourier transform along the frequency direction to achieve the purpose of enhancing the time-frequency energy concentration. The time-frequency distribution result after the synchrosqueezing transform has a reconstruction property, as shown in the following formula:

[0072]

[0073] The synchrosqueezing transform is only effective when the signal can be locally approximated as a harmonic. When there are pulse components in the signal, the sharp change in the instantaneous frequency of the pulse signal causes the synchrosqueezing transform to be unable to generate a time-frequency representation with concentrated energy.

[0074] For pulse signals, the time-rearranged synchrosqueezing transform can effectively sharpen their time-frequency representations. The group delay of a Dirac signal is a constant value, and this signal is a typical pulse signal. Here, the Dirac signal is taken as an example to illustrate the working principle of the time-rearranged synchrosqueezing transform. The time-domain expression of the Dirac signal is:

[0075] sδ(t) = A·δ(t - t 0 ) (10)

[0076] The group delay of this signal is t = t 0 . Performing a short-time Fourier transform on this signal, its expression is:

[0077]

[0078] where u represents the time variable. The time-rearranged synchrosqueezing transform constructs a group delay estimation operator using the relationship between the short-time Fourier transform and the partial derivative of the short-time Fourier transform with respect to frequency. Taking the partial derivative of the time-frequency result of the short-time Fourier transform in Equation (11) with respect to frequency, the following relationship can be obtained:

[0079]

[0080] When , the group delay estimation can be expressed in the time-frequency space as:

[0081]

[0082] where Im() represents taking the imaginary part. Substituting (11) and (12) into (13) gives:

[0083]

[0084] It can be seen that the group delay of the signal s δ (t) can be accurately located. On this basis, the time-rearranged synchrosqueezing transform constructs a synchrosqueezing operator along the time direction to squeeze the extended time-frequency energy to the true group delay position, and its expression is:

[0085]

[0086] The time-rearranged synchrosqueezing transform performs a secondary rearrangement of the time-frequency energy distribution of the short-time Fourier transform along the time direction to achieve the purpose of enhancing the time-frequency energy concentration. The time-frequency distribution result after the time-rearranged synchrosqueezing transform also has a reconstruction property, as shown in the following equation:

[0087]

[0088] Although the time-rearranged synchrosqueezing transform can generate a time-frequency representation with concentrated energy for pulse signals, it is only effective when the signal can be locally approximated as a pulse. When there are harmonic components in the signal, the sharp change in the group delay of the harmonic signal causes the time-rearranged synchrosqueezing transform to be unable to generate a time-frequency representation with concentrated energy.

[0089] In response to this, the present invention proposes a new time-frequency post-processing method named bidirectional rearrangement and squeezing transform. The chirp rate of the harmonic signal, that is, the first derivative of the instantaneous frequency with respect to time, is small, and the chirp rate of the pulse signal is very large. Therefore, the proposed method compares the chirp rate of each time-frequency point of the signal with a specific threshold. If the chirp rate is less than the threshold, it is considered that this time-frequency point is more inclined to the harmonic characteristic, so it is redistributed in the frequency direction. If the chirp rate is greater than the specific threshold, it is considered that this time-frequency point is more inclined to the pulse characteristic, so it is redistributed in the time direction. Then the results of the two redistributions are added together to obtain the final result. Using the proposed method, both the harmonic components and the pulse components in the signal can obtain a time-frequency representation with concentrated energy.

[0090] The key of the present invention lies in obtaining the threshold for comparison with the chirp rate of the signal. The specific obtaining process includes:

[0091] Assume that the chirp signal is expressed as:

[0092]

[0093] where, b + ct and c are the instantaneous frequency and the chirp rate respectively. Assume that the window function has the following form:

[0094]

[0095] Substitute Equation (17) into Equation (6) to obtain the representation of the instantaneous frequency estimate in the time-frequency space Its expression is:

[0096]

[0097] The instantaneous frequency is the first derivative (derived with respect to time) of the phase of the expression of the signal in the time domain. From Equation (17), it can be seen that the instantaneous frequency is b + ct, while Equation (19) shows that the result obtained by estimating the instantaneous frequency is not b + ct. Therefore, it can be seen that the instantaneous frequency of the chirp signal cannot be accurately located. The objective of the present invention is to construct a new estimation operator such that after substituting the signal shown in Equation (17), the obtained instantaneous frequency estimate is exactly b + ct. At this time, it shows that the estimation operator proposed by the present invention can accurately locate the instantaneous frequency of the chirp signal.

[0098] According to Equation (19), the error between the estimated instantaneous frequency and the true instantaneous frequency at any time-frequency point in the chirp signal can be obtained. It is expressed as:

[0099]

[0100] Where reflects the concentration degree of the synchrosqueezing transform result, b + ct represents the instantaneous frequency, and c represents the chirp rate.

[0101] When the signal is a harmonic wave, That is the instantaneous frequency can be accurately located. Therefore, the synchrosqueezing transform can generate a time-frequency representation with concentrated energy. When the chirp rate increases, increases, resulting in the distance from the instantaneous frequency increases, and the time-frequency representation of the synchrosqueezing transform becomes blurred. Therefore, the concentration degree of the synchrosqueezing transform result decreases with the increase of the chirp rate.

[0102] Next, the relationship between the concentration degree of the time-rearranged synchrosqueezing transform result and the chirp rate is described. s f (t) can be represented in the frequency domain as:

[0103]

[0104] Where, (ω - b) / c represents the group delay of the signal.

[0105] Based on Equation (13), the representation of the estimated group delay at any time-frequency point in this signal in the time-frequency space can be obtained:

[0106]

[0107] The group delay is the first derivative of the phase of the expression of the signal in the frequency domain (derived with respect to frequency). It can be seen from Equation (21) that the group delay is (w - b) / c, while Equation (22) shows that the result obtained from the group delay estimation is not (w - b) / c. Therefore, it can be seen that the group delay of the frequency modulation signal cannot be accurately located. The objective of the present invention is to construct a new estimation operator such that after substituting the signal shown in Equation (21), the obtained group delay estimation is exactly (w - b) / c. At this time, it shows that the estimation operator proposed by the present invention can accurately locate the group delay of the chirp signal.

[0108] According to Equation (22), the error between the estimated group delay and the true group delay can be obtained as:

[0109]

[0110] Reflects the concentration of the results of the time-rearranged synchrosqueezing transform. When the chirp rate is small, is large, indicating that the distance from the true group delay is large, so the time-frequency representation of the time-rearranged synchrosqueezing transform is ambiguous. As the chirp rate increases, decreases, indicating that the distance from the true group delay decreases, and the concentration of the time-frequency representation of the time-rearranged synchrosqueezing transform becomes higher. Therefore, the concentration of the results of the time-rearranged synchrosqueezing transform increases with the increase of the chirp rate.

[0111] It can be seen that and respectively reflect the concentration of the synchrosqueezing transform and the time-rearranged synchrosqueezing transform. Therefore, these two can be used as the criteria for choosing which method. For example, if is less than then the difference between the instantaneous frequency estimate and the true instantaneous frequency is less than the difference between the group delay estimate and the true group delay, which results in the result obtained by using the synchrosqueezing transform being closer to the time-frequency trajectory of the signal than the result obtained by using the time-rearranged synchrosqueezing transform. Therefore, when is less than , the signal is more suitable to be processed by the synchrosqueezing transform, and vice versa, it is suitable to be processed by the time-rearranged synchrosqueezing transform. This means that the chirp rate at this time can be used as a boundary, that is to say, taking the chirp rate obtained when as the chirp rate threshold, that is, when equations (20) and (23) are equal, simplifying it can obtain the following formula:

[0112] |c| = β -2 / 3 (24)

[0113] where the form of the window function used in the short-time Fourier transform is β represents the parameter of the window function.

[0114] S2. Calculate the chirp rate of each time-frequency point of the chirp signal based on the chirp rate estimator; in practical applications, for the chirp signal described by equation (17), its chirp rate is not known in advance. For this reason, a chirp rate estimator is proposed. Based on equations (19) and (22), find the partial derivatives of the frequency ω with respect to time t and Taking the partial derivatives of (19) and (22) can obtain:

[0115]

[0116] Calculating according to equation (25) can obtain c, that is, the chirp rate, and the chirp rate estimator can be expressed as:

[0117]

[0118] Among them, represents the chirp rate at a certain time-frequency point (t, ω), where t is time and ω is frequency; represents the representation of the instantaneous frequency estimate in the time-frequency space; represents the representation of the group delay estimate in the time-frequency space; Inf means that when the chirp rate estimator is infinite, and in this case, the chirp rate cannot be represented by a finite value.

[0119] S3. Compare the chirp rates of each time-frequency point of the chirp signal with the threshold respectively. The short-time Fourier transform result can be divided into two parts, namely:

[0120]

[0121] Among them, G F in (t, ω), else where represents the case of, G T in (t, ω), elsewhere represents the case of. It can be understood in this way that G F (t, ω) is the part that retains the chirp rate less than or equal to β -2 / 3 in the short-time Fourier transform of the signal, and the rest is set to zero. G T (t, ω) is the part that retains the chirp rate greater than β -2 / 3 in the short-time Fourier transform of the signal, and the rest is set to zero.

[0122] The above formula means that the short-time Fourier transform is divided into two parts by the chirp rate operator. Among them, G F (t, ω) represents the part to be subjected to the squeezing transformation in the frequency direction, and G T (t, ω) represents the part to be subjected to the squeezing transformation in the time direction, and G(t, ω) represents the short-time Fourier transform.

[0123] Take the sum of the two squeezing operations as the final time-frequency representation of the chirp signal:

[0124]

[0125] When the chirp rate is greater than the threshold, that is, the time-frequency coefficients of this part of the short-time Fourier transform are regarded as suitable for redistribution in the frequency direction. When the chirp rate is less than or equal to the threshold, that is, The time-frequency coefficients of this part of the short-time Fourier transform are regarded as suitable for redistribution in the time direction. After dividing the two parts of the short-time Fourier transform, frequency-direction redistribution and time-direction redistribution are performed on them respectively to obtain F s and T s . Adding the two gives R(u,η), see Figure 1 .

[0126] Among them, the variables in (u,η) represent time and frequency in the time-frequency representation from left to right. (u,ω) is the same as (u,η), except that different variables are used, but they have the same meaning; represents a two-dimensional instantaneous frequency estimation operator, is a synchrosqueezing operator for synchrosqueezing in the frequency direction, where δ() is the Dirac function, i.e., δ(0)=1, δ(non-0)=0, and F S (u,η) represents the result after synchrosqueezing transformation in the frequency direction. (t,η) has the same meaning as (u,ω) and (u,η), except that different variables are used, but they represent the same meaning. represents a two-dimensional group delay estimation operator, is a synchrosqueezing operator for synchrosqueezing in the time direction, and T S (u,η) represents the result after synchrosqueezing transformation in the time direction.

[0127] The method of the present invention divides the time-frequency coefficients of the short-time Fourier transform into two parts, and performs synchrosqueezing transformation (time or frequency) on each part only in a single direction. Therefore, each time-frequency point is synchrosqueezed only in one direction. Since the original signal can be reconstructed from the results of synchrosqueezing transformation in a single direction, the original signal can be obtained by reconstructing the two synchrosqueezing results respectively, that is:

[0128]

[0129] Among them, s(u) represents the original signal; g(t) is the window function, and g(0) represents the value of the window function at time 0; represents the value of the Fourier transform of the window function at time 0; T s (υ,η) represents the result after synchrosqueezing transformation in the time direction in the method of the present invention; (υ,η) represents (time, frequency); for (u,η), (υ,η), (t,η), (u,ω), as long as they are time-frequency transformation results, they all represent (time, frequency).

[0130] Next, a set of acoustic signals of the engine at constant speed and variable speed are used to verify the performance of the method proposed in the present invention. By arranging microphones around the automotive engine, detailed information about the engine's acoustic characteristics can be obtained. As acoustic sensors, microphones achieve precise capture of the engine's acoustic phenomena by converting the vibrations of surrounding sound waves into corresponding electrical signals. These acoustic signals contain key information such as the engine speed, which is crucial for the engine's condition monitoring. The sampling frequency of the signals is 48000Hz, and the sampling time is 3s.

[0131] Experiment 1

[0132] First, analyze the acoustic signals of the engine operating at a constant speed. The signal waveform and its envelope are as Figure 2 shown. Figure 3 shows the results produced by different time-frequency analysis methods when analyzing the signal envelope. Due to the limitation of the Heisenberg uncertainty principle, there is a serious time-frequency energy diffusion phenomenon in the time-frequency distribution results processed by the short-time Fourier transform. The time-frequency results processed by the synchrosqueezing transform and the bidirectional reassignment synchrosqueezing transform can significantly reduce the range of small-time-frequency energy diffusion, which enables the precise capture of the frequency variation law of the signal. In contrast, there is a serious divergence phenomenon in the time-frequency results processed by the time-reassigned synchrosqueezing transform because the constant-speed signal presents a time-frequency characteristic similar to that of a harmonic wave, while the time-reassigned synchrosqueezing transform is more suitable for processing pulse-like signals. Then, use the ridge extraction algorithm to extract the ridges in the results of the bidirectional reassignment synchrosqueezing transform. The results are as Figure 4 shown. The results show that the engine rotates at a constant speed of 25Hz, which indicates that the bidirectional reassignment synchrosqueezing transform method has extremely high accuracy in extracting ridges and can accurately locate the rotational speed frequency of the engine.

[0133] Experiment 2

[0134] Next, analyze the acoustic signals of the engine operating at a variable speed. The signal waveform and its envelope are as Figure 5 shown. Figure 6 shows the results produced by different time-frequency analysis methods when analyzing the signal envelope. It can be seen from the figure that there are obvious differences in the time-frequency representation effects of different time-frequency analysis methods on the engine variable-speed signals. There is still a serious energy divergence phenomenon in the results of the short-time Fourier transform. Due to the drastic change in the instantaneous frequency and the high chirp rate in the variable-speed section (1 - 2 seconds) of the engine, the concentration degree of the time-frequency representation of the synchrosqueezing transform in the variable-speed section is poor, while the time-reassigned synchrosqueezing transform can obtain a time-frequency representation result with concentrated energy. The bidirectional reassignment synchrosqueezing transform can obtain a time-frequency representation with concentrated energy in both the constant-speed section and the variable-speed section because this method can select different squeezing directions according to the magnitude of the chirp rate, thus more effectively aggregating the time-frequency energy. The ridge extraction results of the bidirectional reassignment synchrosqueezing transform are shown in Figure 7It can be clearly seen from the results that the frequency starts to rise rapidly after 1 second, reaches a peak of approximately 43 Hz at 2 seconds, and then begins to decline and stabilizes near 40 Hz.

[0135] It can be known that under the condition of constant engine speed, the envelope of the acoustic signal presents time-frequency characteristics similar to harmonics, which enables the synchrosqueezing transform to obtain a time-frequency representation with concentrated energy, while the result of the time-rearranged synchrosqueezing transform diverges severely. Under the condition of variable engine speed, the frequency of the signal envelope changes sharply, the chirp rate is large, and the signal presents characteristics similar to pulses, which enables the time-rearranged synchrosqueezing transform to obtain a time-frequency representation with concentrated energy, while the concentration degree of the result of the synchrosqueezing transform is poor. The bidirectional rearranged synchrosqueezing transform can generate a sharpened time-frequency representation of the signal envelope under both constant and variable engine speed conditions, which enables it to better capture the dynamic changes of the engine speed and is crucial for engine condition monitoring and fault diagnosis.

[0136] In this specification, each embodiment is described in a progressive manner. The key point of each embodiment is to illustrate the differences from other embodiments. For the same or similar parts among the embodiments, reference can be made to each other. For the devices disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple. For the relevant parts, reference can be made to the description in the method part.

[0137] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be obvious to those skilled in the art. The general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but will be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses, characterized in that: The following steps are involved: Acquire a linear frequency modulation signal having both harmonic components and pulse components, and obtain a threshold value for comparison with the chirp rate; Calculating the chirp rate of each time-frequency point of the linear frequency modulation signal based on a chirp rate estimator; When the chirp rate of a certain time-frequency point is less than or equal to the threshold, it is considered that the time-frequency point tends to have harmonic characteristics and is redistributed in the frequency direction; When the chirp rate of a certain time-frequency point is greater than the threshold, it is considered that the time-frequency point tends to have pulse characteristics and is redistributed in the time direction; The two redistribution results are added together to obtain the time-frequency representation of the linear frequency modulation signal.

2. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 1, characterized in that: The expression of the chirp rate estimator is: in, It represents the chirp rate at a certain time-frequency point (t, ω), where t is time and ω is frequency; represents the representation of instantaneous frequency estimate in time-frequency space; represents the representation of the group delay estimate in the time-frequency space; Inf means when When , the chirp rate estimator is infinite, in which case the chirp rate cannot be represented by a finite value.

3. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 2, characterized in that: The obtaining of a threshold value for comparison with the chirp rate comprises: For any time-frequency point in the linear frequency modulation signal, calculate the error between the instantaneous frequency estimate and the true instantaneous frequency at that time-frequency point Calculate the error between the estimated group delay and the actual group delay at this time-frequency point Will The chirp rate at this time is used as the boundary. The chirp rate threshold is expressed as: |c| = β -2 / 3 , where short-time Fourier transform is performed in the form of a window function, and β represents the parameter of the window function.

4. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 3, characterized in that: The error between the instantaneous frequency estimate and the true instantaneous frequency The concentration of the synchronous extrusion transformation result is expressed as: Where b+ct represents the instantaneous frequency, and c represents the chirp rate; represents the representation of instantaneous frequency estimation in time-frequency space, and its expression is:

5. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 3, characterized in that: Error between estimated group delay and true group delay The expression that reflects the concentration of the time-rearranged synchronous squeezing transformation result is: Where (ω-b) / c represents the group delay of the signal; represents the representation of group delay estimation in time-frequency space, and its expression is:

6. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 3, characterized in that: if Less than or equal to When , the result obtained by using synchronous squeezing transform is closer to the time-frequency trajectory of the signal than the result obtained by using time-rearranged synchronous squeezing transform. In this case, the signal is more suitable for synchronous squeezing transform processing. Otherwise, the signal is suitable for processing using the time-rearranged synchrosqueezing transform.

7. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 3, characterized in that: According to the comparison result of the chirp rate of each time-frequency point in the linear frequency modulation signal and the threshold, the short-time Fourier transform result is divided into two parts, namely: Among them, G F (t,ω) represents the part to be squeezed in the frequency direction, G T (t,ω) represents the part to be squeezed in the time direction, and G(t,ω) represents the short-time Fourier transform.

8. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 1, characterized in that: The final time-frequency representation of the linear frequency modulation signal is: Among them, the two variables in (u,η), (u,ω), and (t,η) represent time and frequency; represents the two-dimensional instantaneous frequency estimation operator, is the synchronous squeezing operator for synchronous squeezing in the frequency direction, where δ() is the Dirac function, i.e., δ(0) = 1, δ(non-0) = 0; F S (u,η) represents the result of synchronous squeezing transformation in the frequency direction; represents a two-dimensional group delay estimation operator; represents the synchronous extrusion operator for synchronous extrusion in the time direction; T S (u,η) represents the result of synchronous extrusion transformation in the time direction.

9. The time-frequency analysis method for non-stationary signals with coexistence of harmonics and pulses according to claim 1, characterized in that: Also includes: The two redistribution results are reconstructed and added to obtain the original signal of the linear frequency modulation signal.

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