Parameters of Exterior Ballistic Feature Points Extraction in Radar Measurement Data by EMD

Wanjun Zhang, Kailin Wang, Xiaoying Wu, Guohui Li and Hongtian Liu

(Department of Arms Engineering, Academy of Army Armored Forces, Beijing 100072, China)

Abstract: The problem of measuring exterior ballistic feature points is always difficult to solve and it is essentiale on exterior ballistic measurement. By analysis of radar reflection characteristics and non-stationary echo signals of exterior ballistic feature points, the echo data of exterior ballistic feature points is measured by using the continuous wave radar. The parameters of feature points are extracted by the empirical mode decomposition method (EMD) of Hilbert-Huang transform (HHT) spectrum analysis technique. The radar echo signal model and EMD extraction model are established to analyze the exterior ballistic mutation point detection and EMD extraction method of aliasing echo signal. Typical feature point parameters of exterior ballistic in rocket flight tests are carried out and the effectiveness of the method is verified. A new method of measuring the parameters of exterior ballistic feature point is therefore presented.

Key words: exterior ballistic characteristic points; empirical mode decomposition method (EMD); continuous wave radar; information extraction

At present, radar measurement parameters of exterior ballistic can be categorized into two kinds, one is the general exterior ballistic parameters, including projectile position, velocity, and other conventional high range ballistic performance parameters; The other is feature point parameters of exterior ballistic, including burst point time, coordinate and mutation moment, such as shapnel cabin opening, rocket separation. By processing radar measurement data of these mutation points, it is possible to calculate the precise moment of exterior ballistic feature (mutation or singularity) points.

In fact, the projectile (especially some special bombs) for scientific research tests, will incur multi-target projectile parameters disintegration, such as shapnel cabin opening. Requirements of measuring equipment can complete the multi-target flight parameter measurement. This multi-target measurement technology with great difficulty, wide range, urgent demand is listed as one of the key technologies of measurement in 21th century by American shooting researchers.

Most continuous wave radar tracking can only track a signal target, but by using spectrum analysis technology, signal analysis and processing can not only deal with the target information of strongest echo signal, but also with weak echo signal of other targets. The radar has a multi-target processing capability. Therefore, the continuous wave radar is proposed to measure the data of exterior ballistic feature points, extracting the characteristic parameters by empirical mode decomposition method(EMD), accordingly, when the ballistic of measured target appears separation, disintegration, fracture and other mutations, such as the rocket wing dropping out. It can be found in the spectrum analysis.

1 Analysis of Feature Point Echo Signal and EMD Extraction Model

When radar radiation source is reflected by exterior ballistic feature points, echo signal will be inevitably affected by the event, then the mutation occurs. At the same time, echo signal is aliased or remodulated due to feature point environment complexity of electromagnetic wave and target intersection.

1.1 Analysis of characteristic point echo signal

In radar test of exterior ballistic, velocity and signal to noise ratio are directly measured elements. When projectile explosion, separation, cabin opening and other events happen, detonation will give the fragment and the separator a certain initial velocity instantly. It makes trajectory feature point’s velocity suddenly rise or fall. Thus, velocity curve shows multiple or discontinuous phenomena, in addition spectrum shows strong noise with a certain band width.

The origin of exterior ballistic feature points is the mutation of target force, which is easy to identify target acceleration mutation in the time domain. However, the acceleration usually is a calculation element of radar measurement. The changes of the position and velocity need to be integrated for a short time. At this time, the feature points become gentle and not conducive to accurate determination of characteristic points. In the frequency domain, the feature points’ frequency is multi-component and transient. Sometimes the transition is steeper or flat, even non-stationary. An obvious feature is shown as the instantaneous frequency (IF).

Owing to mutation of the ballistic, the echo signal contains a variety of frequency components. To effectively estimate the IF of non-stationary signal, it is necessary to decompose this kind of multi-component signals. That is to say, when the multi-component echo signal of exterior ballistic feature points isr(t), the mathematical model can be established. According to the Boashash,

(1)

In Eq.(1),ri(t) is the echo single component signal,nis the number of single component signals, andN(t) is the nondeterministic noise signal. Although it is known that the echo signal is a hybrid signal consisting of a single component signal and noise, how to determine a single component signal?

In fact, in the radar measurement, the ballistic velocity parameter is a closed-time continuous single-valued function. In the case of low data rate, when the original echo data interval is 10 ms or more, the characteristic moment is obvious and easy to distinguish; However, in the case of high data rate, if the data interval is 1 ms, the characteristic moment will become blurred. Even if in the tracer bomb event, it can be seen on some of the mutation information in the frequency domain. However, the frequency domain curve has no time concept and it is impossible to give mutation information of multi-component echo signal. How can the transient feature of the exterior ballistic feature points be extracted?

1.2 EMD extraction model of echo signal characteristic points

Therefore, we put forward to making use of HHT analyzing echo signal, which innovation is that it does not directly analyze echo signal for Hilbert transform (HT) or wavelet transform, but rather define the single component signal with an intrinsic function of the intrinsic mode function (IMF). EMD will be any composite signal decomposition into the sum of the IMF. Finally, in order to obtain IF. We need to derive analytic phase function derivatives for each IMF.

That is, echo signalr(t) of any feature points is decomposed into the IMF sum by EMD:

(2)

where IMFi(t) is the No.iIMF in the echo signal, which characterizes the finest or periodic shortest component of the echo signal.Rn(t) means that the echo signal is decomposed into residual signals afternIMF, also called residual signal. When not considering the effects of noise, the time-frequency distribution of multi-component echo signals is composed of multiple ridge peaks, and each ridge has its own phaseφi(t) and IFfi(t),

(3)

Representation of the exterior ballistic characteristic points echo signal, at any time, may contain more than one oscillation mode. That is, when there are multiple feature points in the radar beam, if there are velocity differences between these targets, and the targets moving speed can be resolved by radar. The radar can handle each part of feature points’ information.

Thus, the multiple component signals of characteristic points can be described by AM-FM radar echo signal with the amplitude ofA(t),

r(t)=A(t)ej(t)

(4)

Then according to Eq.(2), there is

(5)

In the formula, the residual functionRn(t) is omitted, and Re represents the real part. Thus,r(t) signal of the HT signal is

(6)

Thus, the amplitude spectrum of the echo function of the exterior ballistic characteristic points can be calculated correspondingly, and the IF is given at last.

In general, the original echo data of radar interval is 5-10 μs, the highest up to 2 μs. Theoretically the accuracy of the characteristic time for the device can distinguish the smallest time unit half. Therefore the original echo signal were observed. The theoretical time accuracy is microsecond or ten microseconds, which is actually not achievable: the original echo signal mixed with a lot of noise, so useful signal cannot be directly resolved. We must take a signal processing approach. The data interval after signal processing determines the highest characteristic time accuracy, which can be given by the IF.

2 Characteristic Points Detection and EMD Extraction of Echo Aliasing Signals

In order to accurately describe the exterior ballistic feature points’ frequency varing with time, the echo signal mutation should be detected, processed and resolved.And HHT is an adaptive IMF generated by the echo signal itself. It is shown as a scale band-pass filter. It filters the signal. Thus, the signals within a certain range will be separated out to form a limited bandwidth of the inherent mode of single-component signal. Therefore, HHT has good local adaptability to the signal, which can effectively separate the mutation position of the echo signal and determine the mutational points’ position. That can be used to detect the mutation problem of the echo signal.

2.1 Detection of exterior ballistic mutational points

The mutation of the ballistic echo signal shows a change in the peak in the Hilbert spectrum, so the HT ofr(t) is

(7)

Shown by analytic signals,

z(t)=r(t)+jH[r(t)]

(8)

Then the corresponding amplitude and phase are

(9)

Thus, when the two noise mutation signalsn(t) are superimposed in ther(t) signal, the echo signal amplitude and the IF corresponding to the echo signal are shown in Fig.1.

Fig.1 Echo mutation signal and its analytical signal

In Fig.1, there is a significant correspondence between the signalr(t) with two mutational points and the corresponding analytic signal amplitudeA(t) and the IFfi(t).That is, at the mutational point of the signalr(t), we use the simulation to solve the HHT. In the amplitude and IF curve, there are corresponding changes in the regular time, through the peak of the HT spectrum to detect the echo signal mutational point moment, and then the mutational points’ position is determined.

Further, the IF of the power spectrum of the echo mutation signal is estimated using HHT, as shown in Fig.2.

Fig.2 IF of characteristic point echo signal

The IF of the echo mutation signal can be accurately estimated in Fig.2, and the IF changes continuously with the mutation position. It corresponds to the only mutation position and the IF at each moment.

2.2 EMD extraction of echo aliasing signals

During radar radiation source encountering target, target mutates due to the event. The echo signal will make echo signal produce parasitic modulation, aliasing or embedded in radar echo, thus forming the aliasing modulation phenomenon.

According to the nonlinear system, the sine function mathematical model of the echo aliasing signalr(t) can be established:

r(t)=sin (ωt+mεcosωt)

(10)

mεis the aliasing modulation factor. Then the time domain waveform ofr(t) is shown in Fig.3.

Fig.3 Time domain waveform of echo aliasing modulation

From Fig.3, aliasing modulation is similar to the harmonic distortion phenomenon. There are a more sharp bottom and a smooth peak. When the cosine function is used as the model, its time domain waveform appears to have a higher tip and a smooth bottom. This is often seen as a nonlinear distortion caused by harmonic interference in the traditional signal analysis. In the HHT analysis,r(t) can be developed directly after the HHT.

r(t)=sinωtcos (mεcosωt)+
cosωtsin (mεcosωt)

(11)

Usuallymεis less, therefore,

sin (mεcosωt)≈mεcosωt,cos (mεcosωt)≈1

(12)

It can be seen that the aliasing modulation is composed of two or more waveforms whose IF can be seen as being composed of two sine waves:

r(t)=r1(t)+r2(t)=A1ejω1t+A2ejω2t

(13)

Whenω1,ω2>0，the signal is parsed. At this point, the IMF condition is satisfied. The echo phaseφ(t) and the IFfi(t) are

(14)

(15)

Thus the IF waveform of the echo mixing signal is shown in Fig.4.

Fig.4 IF of mixed frequency modulated wave

Obviously, when the mutation signal is submerged in the noisy environment or the mutation is slow and aliasing, it is difficult to determine the mutation time from the radar original measurement data. It is difficult to distinguish the echo pattern of the echo signal from the HHT find the IF, which can effectively analyze algebraic modulation of the feature points’ echo signal.

3 Experimental Verification of Parameter Extraction of Ballistic Characteristic Points

The speed is the primary factor in the continuous wave radar tracking measurement for the targets. As long as the speeds of the different targets are different, the echo signal to noise ratio exceeds the tracking threshold, and the radar can track the measurement. In the measurement of a rocket bullet separation and opening time test after the main body of the rocket is separated from the bullet and the cabin is opened, there are many targets in the radar beam. Subsequently, the signal-to-noise ratio (SNR) of the radar signals will decrease, which has both mutation and aliasing, the radar echo raw data and its spectrum are shown in Fig.5 and Fig.6.

In Fig.5 and Fig.6, both from the radar echo data and spectrum, it is difficult to directly use the signal to noise ratio to determine or roughly determine the separation point and the open-cabin-point and other ballistic characteristics of the parameters, it can be observed that the signal to noise ratio of two declined. And the rocket separation of the bullet cartridge by the explosive force to push forward, forward bullets, in the speed curve, in principle, can clearly see the separation of two speed curve(multi-target). And from speed of waterfall, a rough judge for the separation point of time can be made, then, in Fig.5 for post-processing, there are velocity cascades in the vicinity of separation and mutation near the open-cabin-point, as shown in Fig.7.

Fig.5 Radar raw data

Fig.6 Echo spectrum

Fig.7 Rocket separation and open cabin near waterfall plot

From Fig.7, it is estimated that there are at least two changes in the velocity. The separation point appears near 102.8 s. The second is that the open-cabin-point may appear near 107 s. Since the most obvious representation of the ballistic mutation is the acceleration mutation, the 3 748 post-processing data points of 101 s to 118 s are differentiated according to Fig.5 and Fig.7, and the radar radial acceleration curve is shown in Fig.8.

Fig.8 Radar radial acceleration curve

Fig.8 can be used to determine the acceleration. As it is shown, relative to the rocket main

acceleration, the first decline in the vicinity of 102.85 s. The second acceleration drop position is not noticeable. Considering the acceleration of the original data is relatively smooth, the open-cabin-point may be at 107.1 s.

In order to judge the characteristic point more accurately, the 3 748 groups of data (including the separation point and the capsule information) obtained by post-processing are decomposed into a fourth-order IMF using EMD, as shown in Fig.9.

Fig.9 EMD decomposition of feature point echo

Fig.9 shows the IMF1-IMF4 hierarchical decomposition of the original data EMD, essentially a gradual refinement on a time scale. That is, to keep the original signal of the IMF characteristics of the same time, in the time domain to compress, the frequency spectrum will be broadened to reach the high resolution on the frequency domain. The corresponding time-scale transformation between the two levels of IMF is

ni=[(ni-1-1)/2]+N,(i=1,2,3,4)

(16)

whereniis the length of data segment; [] is the rounded symbol;Nis half the length of the filter. Thus, the characteristic point judged by the EMD decomposition diagram 9 can be reduced to the original signal. That is to say, the number of mutation points on the IMF will be converted to the corresponding time in the time domain. High resolution is achieved in the time domain.

From the IMF4 in Fig.9, the first mutation point corresponds to the 46th point data, that is, the separation point. The second mutation point corresponds to the 124th data point, which is the open-cabin-point. Using Eq.(16) with 46 points and 124 points step by step back to the original signal, you can determine that the separation point time is 102.95 s and the open-cabin-point is 107.21 s. It can be seen that the time of separation point and open-cabin-point time lag by 100 ms and 101 ms respectively. This also conforms to the essential characteristics of the feature quantities used by the two discriminants, since the differential effect from velocity to acceleration is of an advanced and predictive characteristic.

4 Conclusion

The exterior ballistic parameters and the parameters of the flying arrows’ attitude characteristics directly affect the combat performance of the weapon system. Therefore, it can bring practical significance to improve the accuracy of ballistic tracking data, correct system error, analyze error, cause of failure and so on by effective and high-precision measurements. In the exterior ballistic parameter measurement, continuous wave radar has the characteristics of far distance, all-weather operation, good space and velocity resolution and so on. Thus, it is possible to quickly and accurately detect the exterior ballistic characteristic points from a large number of trajectories to track the original data by using a kind of time-frequency analysis such as EMD. It determines the accuracy of the ballistic feature points, which can effectively eliminate the acceleration error caused by the mutation problem and provide reliable test measurement data for weapon identification.