Method for comparing two time signals

Abstract

A method for comparing a first time-domain signal (T1) and a second time-domain signal (T2), involving determining the number of zero crossings for each signal, and optionally, the differential signals formed for each signal. From each number of zero crossings, frequency indices are determined, as well as spectral powers corresponding to each frequency index. A comparison index is then constructed, based on a comparison of each normalized number of zero crossings and each spectral power.

Description

Title of the invention: Method for comparing two time signals. Technical field

[0001] The technical field of the invention is signal processing, and more specifically the comparison of two time-domain signals. PRIOR ART

[0002] Time-domain signal comparison involves analyzing two or more time-domain signals to identify their similarities and differences. One of the objectives may be to compare a measured signal to a reference signal. Applications can be found in various fields, for example, medicine (comparison of biomedical signals for diagnostic assistance or patient monitoring), acoustics (comparison of audio signals), telecommunications (analysis of signal quality or reliability), or anomaly detection.

[0003] Different methods can be implemented, varying in complexity, ranging from simple visual comparisons to advanced analyses, for example in the spectral domain.

[0004] The publication Kediem B., "Spectral analysis and discrimination by zero crossings," describes a frugal method based on the number of zero crossings of a time-domain signal. Such a method allows the comparison of two time-domain signals with a low computational cost. This method was applied to the detection of anomalies in magnetic signals in the publication Sheinker, "Magnetic anomaly detection using high-order crossing method," IEEE Transactions on Geoscience and Remote Sensing, Vol. 50, No. 4, April 2012.

[0005] Zero crossing detection has been applied to electromyogram signals, for motion recognition purposes, in the Toledo Peres publication "A study of computing zero crossing methods and an improved proposition for EMG signs".

[0006] The invention described below is an improvement on the "zero crossing methods" implemented in the aforementioned publications. It enhances the discrimination performance between two signals, while being implementable with reduced computational cost. Description of the invention

[0007] A first object of the invention is a method for comparing a first time signal and a second time signal, the first time signal and the second time signal being centered and respectively formed of a first number and a second number of consecutive samples, each sample being associated with a moment;

[0008] the process comprising the following iterative steps, each step being associated with a rank & being an integer greater than or equal to 1: - a) determination of a first number of zero crossings of a first iteration signal and a second number of zero crossings of a second iteration signal, the first iteration signal and the second iteration signal being respectively: • during the first iteration, the first time signal and the second time signal; • at each iteration of rank greater than 1, a first differential signal and a second differential signal resulting from the previous iteration; - b) normalization of the first number of zero crossings and the second number of zero crossings resulting from step a) as a function of the first number of samples, and the second number of samples; - c) as long as a stopping criterion for the iterations has not been reached, calculation of a first differential signal and a second differential signal by calculating respectively a difference of the first iteration signal and the second iteration signal in different pairs of successive instants, then repetition of steps a) and b);

[0009] the process also comprising: - d) determination of a first frequency index and a second frequency index, for at least one iteration rank, or for each iteration rank, as a function of the first and second number of zero crossings normalized during step b);

[0010] the process being characterized in that it comprises: - e) calculation of at least a first and a second spectral power respectively of the first time signal and the second time signal in each first and second frequency index resulting from step d); - f) calculation of a comparison indicator, combining: • a frequency component, comprising a comparison of at least one first frequency index and a second frequency index determined for the same iteration rank; • a power component, comprising a comparison of at least one spectral power calculated respectively for the first time-domain signal and for the second time-domain signal, in at least one first frequency index and a second frequency index determined for the same iteration rank.

[0011] According to one possibility, step e) comprises: - ei) calculation of first and second Fourier coefficients respectively of the first time signal and the second time signal in at least one first and second frequency index, determined for the same iteration rank, or for each iteration rank; - e-ii) determination of the spectral power, respectively of the first time signal and of the second time signal at said first and second frequency index, from the first and second Fourier coefficients determined during ei);

[0012] steps a) to f) being implemented by a processing unit.

[0013] In step c), the successive moments are preferably consecutive. According to a Another possibility is that they can be offset by several time increments, but preferably by a small number of time increments.

[0014] The iteration stopping criterion can be a predetermined number of iterations. In one possibility, there is only one iteration. In this case, steps a) and b) are performed only once for each signal. The number of iterations can be between 1 and 5.

[0015] The comparison indicator may include a weighted sum of the frequency component and the power component. The frequency component and / or the power component may then be assigned a weighting factor.

[0016] According to one possibility: - the first signal is a reference signal; - the second signal is a signal resulting from a measurement.

[0017] The method may include a step g) of comparing the comparison indicator with a threshold, so that, depending on the comparison, the first time signal is considered to be similar to the second time signal.

[0018] A second object of the invention is a medium, configured to be read by a processing unit, and comprising instructions enabling implementation of steps a) to f) of a process according to the first object of the invention.

[0019] The invention will be better understood upon reading the description of the exemplary embodiments presented later in this description, in connection with the figures listed below. FIGURES

[0020] Fig. 1 describes the main steps of the method.

[0021] Figures 2A to 2C show a first example of implementation of the invention.

[0022] Fig. 2A shows a first signal, considered as a reference signal.

[0023] Fig. 2B shows a second signal, intended to be compared to the first signal shown in Fig. 2A.

[0024] Fig. 2C shows a spectral power density as a function of frequency indices of the reference signal, shown in Fig. 2A, and of the signal shown in Fig. 2B.

[0025] Figures 3A to 3C show a second example of implementation of the invention.

[0026] Fig. 3A shows a first signal, considered as a reference signal.

[0027] Figure 3B shows a third signal, intended to be compared to the first signal. represented in [Fig.3A].

[0028] Fig. 3C shows a spectral power density as a function of frequency indices of the reference signal, shown in Fig. 3A, and of the signal shown in Fig. 3B. PRESENTATION OF SPECIFIC IMPLEMENTATION METHODS

[0029] An objective of the method according to the invention is to compare a first time signal and a second time signal. By time signal, we mean a series of chronologically ordered samples, each sample corresponding to a specific instant. The first time signal may be a signal considered as a reference signal. The first time signal may result from a measurement, for example by means of a sensor, or from a theoretical model. The first time signal may be representative of a predetermined state, for example a normal state or an anomaly, of a situation that one wishes to analyze. The second time signal may be a signal intended to define a state of the situation. The comparison between the first time signal and the second time signal aims to determine whether the latter is representative of the situation corresponding to the first time signal.

[0030] Each time signal can be a measurement, or represent a measurement, from a sensor such as an optical, electrical, acoustic, magnetic, electrostatic, chemical, velocity, acceleration, or any other motion sensor, or an image sensor. Thus, the time signal can represent a physical or chemical quantity evolving over time.

[0031] Signal comparison is implemented by a processing unit, such as a computer, comprising a microprocessor, or other component, programmed to receive instructions enabling implementation of the method.

[0032] As described in relation to the prior art, frugal methods for comparing signals based on zero crossings have been developed. The principles are as follows:

[0033] We have a first time signal, whose successive samples are X1)-• -X»)-• •XaX and a second time signal j(])...fa). Each time signal is respectively defined on Nx and Ny instants 11. Each instant n is an integer such that 1 <n<Nx et \<n<Ny. Le premier signal temporelx et le deuxième signal temporel sont centrés. Le premier signal temporel et le deuxième signal temporel sont acquis respectivement selon une première et une deuxième fréquence d’échantillonnages fSJC et t fsr

[0034] Iterative steps are then implemented, each step being associated with an iteration rank k, where k is an integer between 1 and K corresponding to the total number of iterations. During each iterative step, the number of zero crossings of a first iteration signal xk and a second iteration signal yk is determined. The number of zero crossings can be calculated by multiplying two successive samples of each signal, each zero crossing corresponding to a negative product.

[0035] During the first iteration (^=1), the first iteration signal x1 is the first time signal x and the second iteration signal j' is the first time signal 4.

[0036] During each iteration: - we determine a first number of zero crossings D ( xk ) of the first iteration signal xk and a second number of zero crossings D(y^) of the second iteration signal yk. - We normalize the first number of zero crossings D(xk) and the second number of zero crossings D(yk). To do this, each number of zero crossings is normalized by a normalization term established respectively from the first number of samples Nx and the second number of samples Ny. We thus obtain a first normalized number of zero crossings R(x*) and a second normalized number of zero crossings R(yk) such that:

[0037] R(x£) - (l)etR(yt) - (1'), or, alternatively:

[0038]

[0039] We then calculate a first differential signal Vxk and a second differential signal Vy^ such that

[0040] Vx^n) = xk(n) -x*(n-1) (3) and Vj*(n) = ^(11) -yk(n-1) (3')

[0041] During the next iteration: xÆ+i= Vxfe (4) and y^in) = Vyk (4')

[0042] The steps described in relation to expressions (1) to (4) are then repeated, as well as (l')à(4').

[0043] The iterations continue up to a predetermined number of iterations with AM. It is common for K to be less than 5, for example equal to 2 or 3. It can be equal to 1 (only one iteration). We obtain a set of K ratios (R{xk) K (5) and {R(yk)}k=L^ (5 )

[0044] In the aforementioned publications, each value D(xk) or R(x^) can be translated into an angular frequency, by the relation _ ~ , , , (g) Thus the values ​​D(xk) ™k~ N 7iK\X ) or R(x*) are representative of the frequency components of the analyzed signals.

[0045] Next, differences are determined:

[0046] ô(xfc-') = when k = 1 and ô^(xfr) = R(xk} - when k > 1 (7);

[0047] as well as :

[0048] ôCy*-1) = RÇyk^) when k = 1 ; and ôA(yk) — R(yk) - XA) when k > 1 (7').

[0049] From the differences thus calculated, a comparison indicator Wxy of the first signalx and the second signal •? is calculated.

[0050] , _ yÆ (8) lPxy- W9

[0051] The comparison indicator V-Xy is confronted with a threshold, below which the signals x and y are considered comparable, in the sense that the second signal y can be considered representative of the first signal x.

[0052] The comparison indicator Vxy is representative of the prior art and more specifically of the previously cited Kediem publication. It is observed that the comparison indicator Vxy depends solely on the dominant frequencies of each signal to be compared. This may prove insufficient for comparing two signals, particularly when the dominant frequencies are close, and when the respective spectral powers of the first and second signals, at their respective dominant frequencies, are different. Furthermore, 7½ is not defined if 0^.( yk ) — 0.

[0053] An important aspect of the invention is to establish a comparison indicator that is more efficient than Wxy, taking into account not only the frequency aspect, but also the spectral power at one or more frequencies of the signals being compared.

[0054] The inventor proposes a method, the main steps of which are shown schematically on [Fig.1], and which allows for the determination of a more efficient comparison indicator, as shown by the experimental tests described below.

[0055] Step 100:

[0056] In this step, two centered time signals x and y are taken into account. As previously described, the first time signal is defined according to Nx consecutive instants and the second time signal is defined according to y consecutive instants.

[0057] Preferably, for each signal, the number of times is greater than 10, or even greater than 20 or 30. The number of times corresponds to the number of samples of each signal taken into account to perform the comparison.

[0058] According to one possibility, the first time signal x is a signal to be compared with the second time signal T. This can be defined over a number of instants much greater than x: Ny > Nx

[0059] Steps 110 to 140 aim to determine, through iterative steps, the number of zero crossings of the iteration signals xk and yk, as previously described. Each iteration is associated with a rank &, where & is an integer between 1 and K, and K denotes the number of iterations. The number of iterations K can be predetermined, based on trials or modeling. The number of iterations K is between 1 and 2V⁻¹. N is the lower of the values ​​Nx and Ny. As previously described, K is generally less than 5, for example, equal to 2 or 3.

[0060] Step 110: calculation of the number of zero crossings of the first iteration signal xk and the second iteration signal yk.

[0061] During the first iteration, xk=] = x and V^1 = y

[0062] The number of zero crossings of the first iteration signal and the second iteration signal are denoted D(xk) and D(yk).

[0063] Step 120: Normalization

[0064] During this step, each number of zero crossings is normalized by a normalization term established respectively from the first number of samples and the second number of samples Ny. This yields a first normalized number of zero crossings R(x^) and a second normalized number of zero crossings R(yk) as specified in (1), (1'), (2) and (2').

[0065] Step 130: Differentiation

[0066] As long as a stopping criterion for the iterations has not been reached, steps 110 and 120 are repeated. In this case, a differential signal is calculated, which will form the signal for the next iteration:

[0067] Vx^n) = xk(n) -xk(n-1) (3) and Vj*(n) = ^(11) -yk(n-1) (3')

[0068] Each differential signal corresponds to a difference of the first time signal or the second time signal at two successive instants, preferably consecutive, or offset by a predetermined time offset, and preferably small, for example less than 5 time increments.

[0069] The stopping criterion for the iterations can be a maximum rank of iterations corresponding to a predetermined limit value K.

[0070] Step 140: Reiterations

[0071] If a new iteration is carried out, the first iteration signal and the second iteration signal are, for the next iteration: xk+[ = Vx* (4) and yk+l{n) = Vyk (4').

[0072] Following steps 110 to 120, possibly repeated, we obtain a set of ratios {R ( xk )} and {R ( yk )} K relating to each time signal x and y.

[0073] Step 150: Definition of frequency indices

[0074] During this step, a frequency index is defined for each ratio, for the first time-domain signal and for the second time-domain signal, respectively according to: [°° 75 1 < 10 '>

[0076] The round operator denotes the integer part.

[0077] It is noted that there are as many different values ​​of frequency indices as there are different numbers of zero crossings determined during steps 110 to 140.

[0078] Step 160: calculation of coefficients of the Fourier transform of each signal at each frequency index.

[0079] During this step, the coefficient of the Fourier transform, or of the fast Fourier transform (FFT), is determined for each frequency index and m^r resulting from step 140.

[0080] Thus, with respect to the first temporal signal: j (H), and relative to the second time-domain signal-V, ^^J—ykn) (11')

[0081] In expressions (11) and (11'), J2 = -1 and each value or 2Æm^'” corresponds to a pulsation

[0082] Each frequency index can be related to a frequency by O?) n11 _ -^.f^ (12'). and are respectively the sampling frequencies of the first signalx and the second signal previously defined.

[0083] The frequencies and are characteristic frequencies of each time signal and y. The frequencies f ] V and resulting from the first iteration are dominant frequencies.

[0084] Step 170: Calculation of a spectral power at each frequency index.

[0085] During this step, a spectral power of the first time-domain signal and the second time-domain signal is determined for each frequency index etm^, according to:

[0086] / |__2_>y / x |2 (13) and / 1__2_.yy .(2(13)

[0087] Normalization by sampling frequency is optional if Lr = f«j.

[0088] Step 180: calculation of a comparison indicator of the first signalx and the second signal y, such that

[0089] -R k (y) I +4|p / *) -p / jO | ) (14)

[0090] and 22 are predetermined positive real weighting coefficients.

[0091] In general, the comparison indicator includes - a frequency component |R*(x) - R*(y)|, in the form of a comparison of zero-crossing numbers (D*(x), Dfc(j)) calculated during one or more iterations of the same rank, normalized by a normalization term containing the number of samples of each WNM signal - a spectral power component, | (X ) - p (y ) |, in the form of a comparison of spectral powers calculated relative to the first signal and the second signal, for frequency indices corresponding to the same iteration rank &.

[0092] The coefficients 2] and 22 are weighting coefficients, allowing the frequency component and the power component to be taken into account in the comparison indicator.

[0093] Step 190: confrontation with a threshold

[0094] During this step, the comparison indicator CXj is compared to a threshold Cth, according to which the first signal and the second signal are considered to be representative of each other, or different from each other.

[0095] For example, if Cx,y < Cth, the two signals x and are considered similar. And if Cx,y > Cth, the two signals x and are considered different.

[0096] When the first signal is considered a reference signal, representative of a normal situation, if the second signal is representative of the occurrence of an anomaly. The Ctk threshold is determined based on the acceptable rate of false positives (i.e., false alarms) or false negatives (i.e., undetected anomalies). Experimental trials

[0097] Figures 2A and 2B represent, respectively, a first time signal x and a second time signal y. The first time signal x is considered representative of normal operation. The second time signal y is also considered representative of normal operation.

[0098] The process described in connection with steps 100 to 180 was implemented, with = 2.

[0099] Figure 2C shows the power spectral densities (ordinate axis) depending on the frequency normalized by fw / 2 for the first signalJ and by f4j / 2 for the second signal -f. Thus, the value of each normalized frequency is between 0 and 1. Curve X corresponds to the first signal x and curve Y corresponds to the second signal

[0100] Figure 2C shows: - points XI and X2, whose coordinates are respectively (21¾^ Ah ' - points Y1 and Y2, whose coordinates are respectively (2m&=ij y(m fr Jet(2^ y( mfc2 J.

[0101] During the implementation of the process, Cx,y = 0.177 was obtained taking into account A = 1 and 22 = 1.1020. The respective values ​​of the frequency component and the spectral power component and of the comparison indicator Cx.j> are respectively 0.048 and 0.129.

[0102] In [Fig.2C], double arrows have been used to represent: - the difference Af 1 which respectively represents the difference between 2m*=u and . This corresponds to the frequency component of the comparison indicator for the frequencies fj^ and f, respectively normalized by 2 and fs,yl 2. The frequencies fi^ and fy. resulting from the first iteration for the signals x and Each frequency f]^ and f^ is a dominant frequency for the signal x and y respectively. - the difference Af2 which respectively represents the difference between 2mfc=aj and . Nx Ny This corresponds to the frequency component of the comparison indicator for frequencies f and f^, respectively normalized by Lu7 2 and Lj7 2. - the API difference which represents the gap between spectral powers ct F^m*-] y)- This difference is representative of the difference in spectral powers between the dominant powers of the signals x and - the AP2 difference which represents the gap between the spectral powers Ct. Taking into account the logarithmic scale of the axis of Ordered, AP2 is far inferior to API

[0103] .The prior art comparison method was implemented (cf. expression (8)), resulting in ^xy = 0.051.

[0104] Figure 3A represents the first signal x, as shown in Figure 2A. The Figure 3B represents a third signal corresponding to an anomaly.

[0105] The process described in connection with steps 100 to 180 was implemented, with K = 2.

[0106] Figure 3C shows the power spectral densities (ordinate axis) as a function of frequency normalized by the sampling frequency (abscissa axis), for the first time-domain signal (X curve) and for the third time-domain signal (Z curve). Figure 3C shows: - points XI and X2, whose coordinates are respectively (21¾^ 4¾. J) et x(m fc=2x )) - points ZI and Z2, whose coordinates are respectively f 2m*=^ ïVj » X m ^=u)) et 4¾¾)) -

[0107] During the implementation of the process, we obtained = 1.580, with = 1 and 22 = 1.1020. The respective values ​​of the frequency component and the spectral power component and of the comparison indicator Cx^ are respectively 0.072 and 1.503.

[0108] The values ​​of the frequency components of Cxy and Cx^ are 0.048 and 0.072, respectively. The values ​​of the spectral power components of Cx^ and Cx^ are 0.129 and 1.503, respectively. Thus, the difference between Cxy and Cx,z is essentially due to the variation in the spectral power component between the two indicators. Taking the spectral power component into account in the comparison indicator improves the ability of the comparison indicator to discriminate between two signals that are not representative of each other.

[0109] The prior art comparison method was implemented (see expression (7)), resulting in „ = 0.086. While wrv — 0.051. The relative difference between and Vxz is significantly smaller than the relative difference between Cx,y and Cxz. This shows that the comparison indicator, as explained in (14), is more relevant for discriminating between two different signals than the indicator explained in (8), resulting from the prior art.

[0110] Similarly to [Fig.2C], [Fig.3C] shows: - the difference Af 1 which respectively represents the difference between and 2m*=u. This corresponds to the spectral component of the comparison indicator for the frequencies f^ and f, respectively normalized by La- / 2 and f^ / 2. - the difference Af2 which represents respectively the difference between 2nWr and • This corresponds to the spectral component of the comparison indicator for the frequencies and , respectively normalized by / 2 and fs^ / 2. - the API difference which represents the gap between the spectral powers x(m£_ix) ct .)• This gap is representative of the gap in spectral powers between the dominant powers of the signals x and z. - the difference AP2 which represents the gap between the spectral powers ^k=2J and

[0111] The computational cost of each method was estimated, considering N = 30 and K = 1. The method resulting in the comparison indicator Vxy according to the prior art requires 60 additions and 30 multiplications. The method according to the invention, resulting in the comparison indicator Cx,.y, requires 90 additions and 60 multiplications. A discrimination method based on the complete calculation of the FFT of the 2 signals would require approximately 150 additions and 150 multiplications.

[0112] The implementation of the invention makes it possible to remain within a limited number of simple mathematical operations.

[0113] The invention can be applied in various fields, for example, the detection of a cyberattack, the time signal then being representative of a quantity of data passing through a node of a computer network. It can also be used for the detection of anomalies in machines following the implementation of non-destructive testing sensors (magnetic, optical, mechanical, or eddy current sensors). It can also be used for the detection of abnormal physiological behavior, the time signal being a physiological signal generated by a human body, for example, by an electrical, optical, or magnetic sensor.

Claims

Demands

1. A method for comparing a first time signal (x) and a second time signal (y), the first time signal and the second time signal being centered and respectively formed from a first number (Vx) and a second number (Ny) of consecutive samples, each sample being associated with a time (ny). the process comprising the following iterative steps, each step being associated with a rank & being an integer greater than or equal to 1: - a) determination of a first number of zero crossings of a first iteration signal and a second number of zero crossings of a second iteration signal, the first iteration signal and the second iteration signal being respectively: • during the first iteration, the first time signal and the second time signal; • at each iteration of rank greater than 1, a first differential signal and a second differential signal resulting from the previous iteration; - b) normalization of the first number of zero crossings and the second number of zero crossings resulting from step a) as a function of the first number of samples (Nxy) and the second number of samples (Ny); - c) as long as a stopping criterion for the iterations has not been reached, calculation of a first differential signal (Vx(w)fc) and a second differential signal (Vy(n)k) by calculating respectively a difference of the first iteration signal and the second iteration signal in different pairs of successive instants, then repetition of steps a) and b); the process also includes: - d) determination of a first frequency index ( and a second frequency index, for at least the same iteration rank, as a function of the first and second number of zero crossings normalized during step b); the process being characterized in that it comprises: - e) calculation of at least a first and a second spectral power respectively of the first time signal and the second time signal at each first and second frequency index resulting from step d); - f) calculation of a comparison indicator (Cv.y), combining: • a frequency component, comprising a comparison of at least a first frequency index and a second frequency index determined for the same iteration rank (&); • a power component, comprising a comparison of at least a spectral power calculated respectively for the first time signal and for the second time signal, at least a first frequency index and a second frequency index determined for the same iteration rank; steps a) to f) being implemented by a processing unit.

2. A method according to claim 1, wherein step e) comprises: - ei) calculation of first and second Fourier coefficients ), Y ( ) ) respectively of the first time signal and the second time signal at at least one first and second frequency index, determined for the same iteration rank; - e-ii) determination of the spectral power, respectively of the first time signal and the second time signal at each first and second frequency index, from the first and second Fourier coefficients determined during ei).

3. A method according to any one of the preceding claims, wherein the iteration stopping criterion is a predetermined number of iterations.

4. A method according to claim 3, wherein the number of iterations is between 1 and 5.

5. A method according to any one of the preceding claims, wherein the comparison indicator comprises a weighted sum of the frequency component and the power component.

6. A method according to claim 5, wherein the frequency component and / or the power component are affected by a weighting factor (¼ A)-

7. A method according to any one of the preceding claims, wherein - the first signal is a reference signal; - the second signal is a signal resulting from a measurement.

8. A method according to any one of the preceding claims, comprising a step g) of comparing the comparison indicator with a threshold (Cth\) such that, depending on the comparison, the first time signal is considered to be similar to the second time signal.

9. A medium, configured to be read by a processing unit, and comprising instructions enabling implementation of steps a) to f) of a method according to any one of the preceding claims.

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