Nonlinear adaptive noise-induced algorithm and its application in penetration signal

Measurement 58 (2014) 556–565

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Nonlinear adaptive noise-induced algorithm and its application in penetration signal Zongbao Liu a,b,⇑, Shiqiao Gao a, Haipeng Liu a, Dongmei Zhang a, Lei He b a b

State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China Department of Electrical Engineering, University of California, Los Angeles, CA 90095, USA

a r t i c l e

i n f o

Article history: Received 12 November 2013 Received in revised form 20 June 2014 Accepted 14 August 2014 Available online 4 September 2014 Keywords: Nonlinear weight Adaptive signal algorithm Moving window autocorrelation Penetration Polynomial fitting

a b s t r a c t A nonlinear adaptive noise induced algorithm with nonlinear weights was proposed to extract rigid body deceleration during penetration events; it has 3rd-order nonlinear weight, which ensures deceleration curve is smooth everywhere (not only continuous) and avoids sharp points (crucial for targets detection). In addition, an autocorrelation algorithm was improved by applying moving window method to be compared with the proposed nonlinear adaptive algorithm. By calculating penetration depth and Power Spectrum Density (PSD) of 4 deceleration time series, we show that the nonlinear adaptive algorithm more effectively reduces noise in deceleration for striking velocities between 538 and 800 m/s compared with Adaptive Paqta Criterion, moving window autocorrelation and wavelet algorithms. It is further shown that the proposed adaptive algorithm is of the same order as the other 3 methods in terms of computational complexity. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The objective of penetration signal processing is to extract the characters of deceleration during penetration events for the optimization of the projectiles’ burst points and determination of the materials impacted (concrete, soil, sand, etc.), which could enhance the weapons’ performance. From Refs. [1–4], penetration deceleration has mainly 3 physical components: rigid body deceleration, high frequency vibrations from the accelerometer and projectile and high frequency noise. Filtering is the first step to characterize deceleration, detect multi-layer targets and explode the projectile precisely. In a broad class of filtering methods for projectile deceleration we can identify three major subclasses: filter circuits, frequency based methods and time-frequency based methods [3–5]. ⇑ Corresponding author at: State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China. E-mail address: [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.measurement.2014.08.025 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.

Filter circuits inevitably increase the size of earth penetrator instrumentation, which is crucial for the miniaturization of penetrator [5–7]. Because the rigid body deceleration of a projectile is a nonstationary random signal (overlapping frequency bands), frequency based implementation fails to reveal the time-varying characteristics of the signal. The averaging process inherent in the frequency methods (mostly Fourier Transform) smears the time varying spectral features [8]. To deal with the nonstationary nature of penetration signals, the customary practice is to invoke the use of adaptive filtering [9]. The adaptive filters (nonlinear systems) are time varying since their parameters are continually changing in order to meet a performance requirement. In this paper, an adaptive algorithm (adaptive filter) with nonlinear weight is introduced to remove noise in penetration deceleration. Additionally, a moving window autocorrelation algorithm is proposed to reduce noise effectively. Comparison of the two proposed algorithms and two previous algorithms are performed for striking velocities between 538 and 800 m/s and shows better

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Nomenclature Notation f ðxÞ K ai dx Q Y F e A a x1 ; x2 yðiÞ

yðiþ1Þ yðcÞ k k0 kw Rxx x0 ðnÞ

ði þ 1Þth segment of time series overlapped region of time series adaptive Paqta Criterion coefficient initial value of k gradient of k autocorrelation of time series filtered data by moving window autocorrelation M length of time series Wn; N; w filter window size v0 striking velocity of penetration e repetition time of moving window autocorrelation

fitted polynomial of time series order of polynomial coefficients of polynomial sampling interval residual error of fitting matrix of yi matrix of f ðiÞ matrix of error Vandermonde matrix matrix of ai normalized weights ith segment of time series

n X

filtering effectiveness and computational complexity of the nonlinear adaptive algorithm.

Q¼

2. Nonlinear adaptive algorithm [10]

Coefficients a0 ; a1 ; a2 ; . . . ; aK can be derived from polynomial regression Y ¼ F þ e ¼ Aa þ e

Assume the time series fx1 ; x2 ; . . . ; xN g is partitioned into several segments with window size w ¼ 2n þ 1, where the adjacent segments have n þ 1 overlapped points. For each segment the time series is fitted by Kth-order polynomial, and the polynomial of the ith and ði þ 1Þth segments are denoted as yi ðl1 Þ; yiþ1 ðl2 Þ; l1 ; l2 ¼ 1; . . . ; 2n þ 1, while the last segment may be less than 2n þ 1 [11,12]. Let the corresponding curve be

i ¼ 1; . . . ; w ¼ 2n þ 1

the least square polynomial fit is ðxi ; f i Þ; i ¼ 1; . . . ; w, where the fitted polynomial is

f ðxÞ ¼ b0 þ b1 x þ b2 x2 þ    þ bK xK

ð1Þ

where K is the order of the polynomial [12]. Suppose the sampling interval is dx, then xi can be written as [13,14]:

xi ¼ idx þ a

ð2Þ

We can always find a, which makes i ¼ n; n þ 1; . . . ; 0; 1; 2; . . . ; n  1; n, then (1) can be written as 2

f ðiÞ ¼ a0 þ a1 i þ a2 i þ    þ aK i

2

2

2.1. Principle of the algorithm

ðxi ; yi Þ;

½ f ðiÞ  yi 

I¼n

K

ð3Þ

y1 6y 6 2 Where Y ¼ 6 6 .. w1 4.

3 7 7 7; 7 5

yw 2

3 a0 6a 7 6 17 7 a ¼6 6 .. 7; ðKþ1Þ1 4. 5 aK 2

i1

i1

2



i1

i2 .. .

i2 .. .

2

 .. .

i2 .. .

1 iw

iw

1 6 61 6 A ¼6. wðKþ1Þ 6. 4.

2

K K

K

3 e1 6e 7 6 27 7 e ¼6 6 .. 7; K1 4. 5 eK 2

3 7 7 7 7 7 5

ð4Þ

   iw j

Note that A is a Vandermonde matrix, Aij ¼ i ; i ¼ n; n þ 1; . . . ; 0; 1; 2; . . . ; n  1; n; j ¼ 0; 1; . . . ; K The residual error of the fitted polynomial [15] is Q ¼ eT e ¼ ðY  AaÞT ðY  AaÞ ¼ YT Y þ aT AT Aa  2aT AT Y ) min ð5Þ

Apply the matrix differential rules from [16]:

@Q ¼ 2AT Aa  2AT Y ¼ 0 @a and the solution is given by

 1 AT Aa ¼ AT Y ) a ¼ AT A AT Y

The residual error of the fitted polynomial is

Fig. 1. Schematic diagram of the non-linear adaptive algorithm.

ð6Þ

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When fitted curve is regarded as the desired signal, the filter is a low-pass filter; when residual data is regarded as the desired signal and the fitted curve is the trend signal, it is a high-pass filter; for two different window sizes w1 < w2 , if the difference of the two fitted curves is regarded as the desired signal, it becomes a band-pass filter [11]. The fitted curve of the overlapped region is

yðcÞ ðlÞ ¼ x1 yi ðl þ nÞ þ x2 yiþ1 ðlÞ;

l ¼ 1; 2; . . . ; n þ 1

ð7Þ

where x1, x2 are weights of the overlapped region.

then N ðminÞ ¼ 3 When N ¼ 3, we can conclude that the 3rd-order weights are

8 > > > > x ¼ a; > > <

(

b1 þ 2b2 x þ 3b3 x2 ¼ 0 b0 þ b1 x þ b2 x2 þ b3 x3 ¼ 1

( > > b1 þ 2b2 x þ 3b3 x2 ¼ 0 > > > x ¼ b; > : b0 þ b1 x þ b2 x2 þ b3 x3 ¼ 0

2.2. Nonlinear weight [17,18]

(

The partitioned fitted curve is (see Fig. 1)

With

Left : yðiÞ ¼ a0 þ a1 x þ    þ aK xK

a¼1 b¼nþ1

; we have

Right : yðiþ1Þ ¼ b0 þ b1 x þ    þ bK xK Ov erlapped : yðcÞ ¼ x1 yðiÞ þ x2 yðiþ1Þ where



Suppose the overlapped segments start with x ¼ a and end with x ¼ b, then when x ¼ a, !  K  X  iai xi1   i¼1

i¼1



i¼0

þ

i¼0

K K X X iai xi1  ibi xi1 i¼1

x¼a

!

N X

x¼a

!

i

bi x  1

¼ 0;

x¼a

ð8Þ

i¼1

To let Eq. (8) always be correct, it must be

8 N P > i1 > > < ibi x ¼ 0

2 þ 3n 6ðn þ 1Þ þ n3 n3

3ðn þ 2Þ 2 2 3 l þ 3l n3 n

2 þ 3n 6ðn þ 1Þ 3ðn þ 2Þ 2 2 3  lþ l  3l n3 n3 n3 n

The fitted curve of the overlapped region is

i¼0

!

l

x2 ¼

To ensure the fitted time series is continuous every ðiÞ  ðcÞ  where, we have @y@x  ¼ @y@x  , then x¼a

x1 ¼ b0 þ b1 x þ b2 x2 þ b3 x3 ¼ 1 

: x¼a

Right-hand derivative: "  N K N K X X X X @yðcÞ  ¼ ibi xi1 ai xi þ bi xi iai xi1  @x x¼a i¼1 i¼0 i¼0 i¼1 ! ! # N K N K  X X X X  i1 i þ  ibi x bi x þ 1  bi xi ibi xi1   i¼1 i¼0 i¼0 i¼1

N K K X X X ibi xi1 ai xi  b i xi

Þ > > b2 ¼ 3ðnnþ2 > 3 > > > : b3 ¼ n23

Then we have

x1 þ x2 ¼ 1 ðnormalized weightsÞ x1 ¼ b0 þ b1 x þ    þ bN xN

@yðiÞ  Left-hand derivative : ¼ @x x¼a

8 b0 ¼ 1  2þ3n > n3 > > > > 6 ð nþ1 Þ > < b1 ¼ 3 n

  2 þ 3n 6ðn þ 1Þ 3ðn þ 2Þ 2 2 3 yðcÞ ðlÞ ¼ 1  þ l l þ 3l 3 3 3 n n n n   2 þ 3n 6 ð n þ 1 Þ 3 ð n þ 2Þ 2 2 3  lþ l  3l  yi ðl þ nÞ þ 3 3 3 n n n n  yiþ1 ðlÞ l ¼ 1; 2;. . . ; n þ 1

Comparing linear, 2nd-order, 3rd-order weights, as shown in Fig. 2, we conclude that the comprehensive evaluation of yðiÞ and yðiþ1Þ is more accurate when using a 3rdorder weight, where. Linear weight:     l1 i l  1 iþ1 yðcÞ ðlÞ ¼ 1  y ð l þ nÞ þ y ðlÞ; l ¼ 1; 2; . . . ; n þ 1 n n

i¼1

N > P > > : bi xi  1 ¼ 0 i¼0

therefore N ðminÞ should be 3. Similarly, when x ¼ b N K K X X X ibi xi1 ai xi  b i xi i¼1

i¼0

!

i¼0

N K K X X X bi xi iai xi1  ibi xi1 þ i¼0

i¼1

! ¼ 0;

x¼b

i¼1

To let the equation above always be correct, we have

8 N P > i1 > > < ibi x ¼ 0 i¼1

N > P > > : bi xi ¼ 0 i¼0

Fig. 2. Linear and nonlinear weights.

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2nd-order weight (1):

sharp points. Fig. 3 describes the curves filtered by linear and 3rd-order weights respectively for striking velocities between 538 and 800 m/s. A close-up view shows that the fitted curves using nonlinear weight are smoother without sharp points, which is crucial for the detection of the points where the target composition changes and the identification of materials impacted.



 1 2 1 yi ðl þ nÞ y ðlÞ ¼ þ þ 1 l n2 þ 2n n2 þ 2n   1 1 2 yiþ1 ðlÞ; l ¼ 1; 2; . . . ; n þ 1 l  2 þ 2 n þ 2n n þ 2n ðcÞ

2nd-order weight (2):

! 2 l 2ðn þ 1Þ 1 y ðlÞ ¼  þ 1 y i ðl þ n Þ lþ 2 n þ 2n n2 þ 2n n2 þ 2n ! 2 l 2ðn þ 1Þ 1 þ yiþ1 ðlÞ; þ l 2 n þ 2n n2 þ 2n n2 þ 2n

2.3. Window size w

ðcÞ

Window size w cannot exceed the local period of the variation. By comparing the deceleration curves with different window sizes as shown in Fig. 4, we conclude that the curve is smooth enough with minimum penetration depth error when w ¼ 105, while the deceleration curve distorts (rigid body deceleration is removed) significantly when w > 805 and cannot be used in this filtering algorithm [19,20].

l ¼ 1; 2; . . . ; n þ 1 3rd-order weight:   2 þ 3n 6ðn þ 1Þ 3ðn þ 2Þ 2 2 3 i yðcÞ ðlÞ ¼ 1  þ l l þ 3l y 3 3 3 n n n n   2 þ 3n 6ðn þ 1Þ 3ðn þ 2Þ 2 2 3 iþ1  ðl þ nÞ þ  l þ l  l y ðlÞ; n3 n3 n3 n3

˘ ta Criterion algorithm 3. Pau 3.1. Adaptive Pau˘ta Criterion [21]

l ¼ 1; 2; . . . ; n þ 1

With fixed criterion jv i j > 3r, Paqta Criterion processes data within a moving window with same confidence level, which has lower speed and less flexibility. Our

∧

Deceleration(m/s 2)

∧

Deceleration(m/s 2)

Nonlinear weight ensures the curve is smooth everywhere (not only continuous, as shown in [8]) and avoids

Time (ms)

∧

∧

Deceleration(m/s 2)

Deceleration(m/s 2)

Time (ms)

Time (ms)

Time (ms)

Fig. 3. Filtered deceleration with linear and nonlinear weights.

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Fig. 4. Deceleration versus time with different window sizes.

improvements are as follows: replace 3r with kr, and put k ¼ k0 þ kw , where k0 is a constant initial value and kw is the gradient of k. When k is large, some gross errors cannot be removed; when k is small, the useful information in the signal may be rejected. To overcome this problem, the gradient kw is introduced to adaptively remove gross errors. As presented in [22], k 2 ½1:3; 5 in general, then the adaptive criterion may be expressed as

jv i j > kr

ð9Þ

3.2. Adaptive Pau˘ta Criterion with moving window Consider a time series x1 ; x2 ; . . . ; xn ; xnþ1 ; xnþ2 ; . . . ; xM , firstly we remove gross errors of n (window size) points x1 ; x2 ; . . . ; xn by applying the proposed adaptive Paqta Criterion, then repeat the process by moving window to xL ðxL ; xLþ1 ; . . . ; xLþn Þ until the end of the time series. The process is regarded as one order. Note that each group of n data points is defined as window size, the number of points for each movement as step Lð1 6 L 6 nÞ, and the executing times as order e, which is shown in Fig. 5.

4. Moving window autocorrelation algorithm 4.1. Autocorrelation function [22] Noise is uncorrelated with deterministic signal and itself, so its correlation function is almost a delta function. These properties are exploited in many practical schemes for detecting signals in noise. For a time series xðkÞ, its autocorrelation function is defined as

Rxx ðsÞ ¼

N1 1X xðkÞxðk þ sÞ N k¼0

ð10Þ

where s ¼ 0; 1; 2; . . . ; M is the delay, xðkÞ is the time series and N is the window size of xðkÞ.

4.2. Moving window autocorrelation algorithm The moving-window method has been widely used in the past for non-stationary signal analysis [23].

Z. Liu et al. / Measurement 58 (2014) 556–565

Fig. 5. Adaptive Paqta Criterion with moving window.

Fig. 6. Principle of moving window filtering.

Fig. 7. Deceleration versus time with different window sizes.

Fig. 8. Displacement versus time with different window sizes.

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Fig. 9. Local details.

Applying the moving window method, we proposed a new filtering algorithm which we defined as moving window autocorrelation algorithm. It works as follows: Let xð1Þ; xð2Þ; . . . ; xðkÞ be the time series, s is delay, N is the

window size of xðkÞ. When we calculate the autocorrelation function, the first data length will be N 0 ¼ N þ s. Let the autocorrelation value of the N points be the value of point xððN þ sÞ=2ÞÞ, that is x0 ððN þ sÞ=2ÞÞ. Similarly we move the window to the next point and the result of it will be the value of point xððN þ sÞ=2 þ 1Þ, that is x0 ððN þ sÞ=2 þ 1Þ. Finally, we obtain a new time series x0 ððN þ sÞ=2Þ; 0 x0 ððN þ sÞ=2 þ 1Þ; x0 ððN þ sÞ=2 þ 2ÞÞ; . . . ; x0 ððN þ sÞ=2 þ k Þ. However, since the processed time series is shorter than the original one, we can set x0 ð1Þ ¼ xð1Þ; x0 ð2Þ ¼ xð2Þ; . . . ; x0 ðN þ s  1Þ ¼ xðN þs  1Þ when s, s þ 1Þ s þ 1Þ; and x0 ðM  Nþ ¼ xðM  Nþ n < Nþ 2 2 2 Nþs Nþs 0 0 x ðM  2 þ 2Þ ¼ xðM  2 þ 2Þ; . . . ; x ðMÞ ¼ xðMÞ, that is N þ s  1 points are not filtered. Moreover, the whole process can be repeated several times, and the repeated time is defined as order e, as shown in Fig. 6. To fully use the adjacent data points (10) can be rewritten as

Rxx ðsÞ ¼

sþN 1 X xðk  sÞxðkÞ N k¼sþ1

Fig. 10. Filtered deceleration versus time by autocorrelation algorithm.

Z. Liu et al. / Measurement 58 (2014) 556–565

then we have  Rxx

s;

 sþi X Nþs 1 Nþ þi ¼ xðk  sÞxðkÞ i ¼ 0; 1; 2 . . . ; M  ðs þ N Þ N k¼sþ1þi 2

the filtered time series is

   8 0 x ðnÞ ¼ x0 Nþ2 s þ i ¼ f Rxx s; Nþ2 s þ i ¼ > > > ( ) > < Nþ sþi P xðk  sÞxðkÞ ; Nþ2 s 6 n 6 M  Nþ2 s f N1 > k¼sþ1þi > > > : 0 x ðnÞ ¼ xðnÞ; 0 < n < sþN ; M  Nþ2 s 6 n 6 M 2 where f is data transformer operator (square-root function). In this algorithm, when

8 s ¼ 0; N ¼ 1; original signal > > > > N1 > X > > < s ¼ 0; N > 1; Rxx ðs; iÞ ¼ N1 x2 ðkÞ k¼0

> > Nþ sþi > X >  > > xðk  sÞxðkÞ > s > 0; N > 1; Rxx s; Nþ2 s þ i ¼ N1 : k¼sþ1þi

563

Only when the window size of the filter N 0 ¼ N þ s P 10, can the algorithm effectively remove noise. For larger N 0 , it will lead to longer time, and thus affect the real-time process. Comparing the filtered deceleration and displacement (double integration of rigid body deceleration), we find that the algorithm is optimal when N ¼ 16; s ¼ 2. Order e is limited by real-time speed of the process, usually it should be no more than 5, as larger e will result in longer time. Fig. 7–9 show the filtered deceleration with different window sizes and corresponding displacements (double integration of rigid body deceleration), it can be seen that deceleration curve is smooth enough with minimum penetration depth error while N ¼ 16; s ¼ 2. It is further shown from Fig. 10 that N ¼ 16; s ¼ 2 are the optimal parameters for filtered deceleration with different striking velocities. 5. Comparison of the four algorithms To appreciate the effectiveness and limitations of the 4 algorithms, we compared their denoising performance in

Fig. 11. Deceleration versus time with 4 filtering algorithms.

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Table 1 Comparison of penetration depth. Velocity &depth

538 m/s 0.49735 m 700 m/s 0.82532 m 763 m/s 0.91287 m 800 m/s 0.97745 m

Algorithm Paqta Criterion algorithm

Autocorrelation algorithm

Wavelet algorithm

Adaptive algorithm

0.40531 m 18.506% 0.89381 m 8.299% 1.12018 m 22.71% 1.20311 m 23.08%

0.4458 m 10.365% 0.7541 m 8.629% 0.8109 m 11.170% 0.8384 m 14.226%

0.5553 m 11.64% 0.9188 m 11.327% 1.026 m 13.576% 1.009 m 3.228%

0.4963 m 0.22% 0.8209 m 0.536% 0.9199 m 0.77% 0.9711 m 0.649%

time domain, power spectrum (frequency domain) and computation complexity. All the parameters are denoted below: Adaptive Paqta Criterion algorithm: coefficient k ¼ 1:8, step L ¼ 3, window size Wn ¼ 50. Moving window autocorrelation algorithm: delay s ¼ 2, window size N ¼ 16, order e ¼ 2. Wavelet algorithm: db6. Proposed nonlinear adaptive algorithm: order of fitted polynomial K ¼ 2, 3rd-order weight, window size w ¼ 105.

5.1. Filtered signal From Fig. 11 we observe that deceleration filtered by the Paqta Criterion and autocorrelation algorithms still fluctuate a lot, while the wavelet and adaptive algorithms are smoother. However there is peak’s delay between the measured and filtered deceleration for the wavelet algorithm. Table 1 illustrates that compared to the Paqta Criterion and autocorrelation algorithms, penetration depth derived from wavelet and adaptive algorithms have smaller errors

Fig. 12. Power Spectrum Density (PSD) of filtered deceleration.

Z. Liu et al. / Measurement 58 (2014) 556–565 Table 2 Comparison of algorithms’ performance. Algorithm

Effectiveness

Accuracy

Execution times

Paqta Criterion algorithm Autocorrelation algorithm Wavelet algorithm Adaptive algorithm

Good

Good

254,904

Good

Good

254,384

Excellent Excellent

Excellent Excellent

538,056 339,230

and depth error of adaptive algorithm is the minimum among the 4 methods. 5.2. Power Spectrum Density (PSD) [3] Analysis in the frequency domain complements the time domain analysis by giving us information about the spectral contents of the projectile deceleration. Fig. 12 shows that the filtering effect of the wavelet and adaptive algorithms at high frequency points perform well while adaptive algorithm is best because it can also keep the signal’s low frequency parts. 5.3. Computational complexity and real-time performance [24] Computational complexity is calculated when N ¼ 2502 and window size n ¼ 52. Comparison is set as the basic operation for Paqta Criterion algorithm and multiplication is set as the basic operation for autocorrelation, wavelet and adaptive algorithms. Table 2 illustrates the result, which shows that at the same order, Paqta Criterion and adaptive algorithms are computationally more efficient than the other two methods. 6. Conclusion Paqta Criterion algorithm only removes gross errors in deceleration time series. For signals with strong noise, moving window autocorrelation method should be executed several times to extract rigid body deceleration. The proposed nonlinear adaptive algorithm with 3rd-order weight ensures deceleration curves are smooth everywhere (not only continuous) and avoids sharp points (which is crucial for targets’ composition detection and material identification during penetration events), thus effectively reduces noise without priori knowledge of penetration dynamic process. Though Paqta Criterion and autocorrelation algorithms have lower computational complexity, nonlinear adaptive algorithm is of the same order as them. Therefore the proposed nonlinear adaptive algorithm is the best candidate to be used in the filtering of deceleration time series. Acknowledgments This work is funded by State Key Laboratory of Explosion Science and Technology, Beijing, China and China

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Scholarship Council. The authors would like to thank Rahul Krishnan from Design Automation Lab at University of California, Los Angeles for his help in writing this paper. We also thank the anonymous reviewers and the editors for their helpful suggestions, which have greatly improved the presentation of this paper. References [1] R.G. Lundgren, A strain gage based projectile health monitor and salvage indicating circuit for kinetic energy penetrating projectiles, in: 53th NDIA Fuze Conference, 2009. [2] R.G. Lundgren, Signal processing means for detecting and discriminating between structural configurations and geological gradients encountered by kinetic energy penetrating projectiles, in: 52th NDIA Fuze Conference, 2008. [3] J. Pabio, Analysis of the Deceleration Signal of a Missile During a Hard-Target Attack, University of Laval, Canada, 1997. [4] Sh.H. Wang, Study on the Acceleration Signal of the Hard Target Penetration, Beijing Institute of Technology, Beijing, 2010. [5] W.D. Zhang, L.J. Chen, J.J. Xiong, et al., Ultra-high g deceleration-time measurement for the penetration into steel target, Int. J. Impact Eng. 34 (2007) 436–447. [6] R.J. Franco, M.R. Platzbecker, Miniature Penetrator (MinPen) Acceleration Recorder Development and Test, Sandia National Laboratories, Albuquerque, NM 87185: SAND98-1172C, 1998. [7] T.A. Rohwer, Miniature, Single Channel, Memory-Based, High-G Acceleration Recorder (MilliPen), Sandia National Laboratories, Albuquerque, NM 87185: SAND99-1392C, 1999. [8] Z.M. Hussain, A.Z. Sadik, P. O’Shea, Digital Signal Processing: An Introduction with MATLAB and Application, Springer, Berlin, 2011. [9] S. Haykin, L. Li, Nonlinear adaptive prediction of nonstationary signals, IEEE Trans. Signal Process. 43 (2) (1995). [10] H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, second ed., Cambridge University Press, Cambridge, 2004. [11] W.W. Tung, J.B. Gao, J. Hu, L. Yang, Detecting chaos in heavy noise environments, Phys. Rev. E 83 (2011) 046210. [12] T. Ivancevic, L. Jain, J. Pattison, et al., Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis, Nonlinear Dynam. 56 (2009) 23–44. [13] J.B. Gao, J. Hu, W.W. Tung, Entropy measures for biological signal analysis, Nonlinear Dynam. 68 (2012) 431–444. [14] J.B. Gao, Y.H. Cao, W.W. Tung, J. Hu, Multiscale Analysis of Complex Time Series—Integration of Chaos and Random Fractal Theory, and Beyond, Wiley, New York, 2007. [15] H.H. Lin, Data Processing of Dynamic Measurement, Beijing Institute of Press, Beijing, 1995. [16] J. Polking, Differential Equations, second ed., Pearson, New Jersey, 2005. [17] Sh.Q. Gao, H.P. Liu, et al., A fuzzy model of the penetration resistance of concrete targets, Int. J. Impact Eng. 36 (2008) 644–649. [18] J.B. Gao, H. Sultan, J. Hu, W.W. Tung, Denoising non-linear time series by adaptive filtering and wavelet shrinkage: a comparison, IEEE Signal Process. Lett. 17 (2010) 237–240. [19] B.P. Lathi, Linear Systems and Signals, second ed., Oxford University Press, Oxford, 2004. [20] H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, second ed., Cambridge University Press, Cambridge, 2004. [21] Z.B. Liu, Sh.Q. Gao, An algorithm for measurement data eliminating gross error and smoothing, in: Presented at the 31st Chinese Control Conference. . [22] M.D. Ortigueira, C.J.C. Matos, M.S. Piedade, Fractional discrete-time signal processing: scale conversion and linear prediction, Nonlinear Dynam. 29 (2002) 173–190. [23] K. Kodera, C. Villedary, R. Gendrin, A new method for the numerical analysis of non-stationary signals, Phys. Earth Planet. Inter. 12 (1976) 142–150. [24] T.H. Cormen, C.E. Leiserson, R.L. Rivest, et al., Introduction to Algorithms, third ed., MIT Press, Cambridge, 2009.