Research ArticleLaser Doppler Signal Denoising Based on Wavelet PacketThresholding Method

Da Zhang 1 and Ranglei Liu2

1College of Automation & Electronic Engineering, Qingdao University of Science and Technology, Qingdao 260061, China2College of Electromechanical Engineering, Qingdao University of Science and Technology, Qingdao 260061, China

Correspondence should be addressed to Da Zhang; qdzd721@163.com

Received 30 August 2019; Revised 17 October 2019; Accepted 31 October 2019; Published 14 November 2019

Academic Editor: Wonho Jhe

Copyright © 2019 Da Zhang and Ranglei Liu. ,is is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In laser Doppler velocimeter (LDV), calculation precision of Doppler shift is affected by noise contained in Doppler signal. Inorder to restrain the noise interference and improve the precision of signal processing, wavelet packet threshold denoisingmethods are proposed. Based on the analysis of Doppler signal, appropriate threshold function and decomposition layer numberare selected. Heursure, sqtwolog, rigrsure, and minimaxi rules are adopted to get the thresholds. Processing results indicate thatsignal-to-noise ratio (SNR) and root mean square error (RMSE) of simulated signals with original SNR of 0 dB, 5 dB, and 10 dB inboth low- and high-frequency ranges are significantly improved by wavelet packet threshold denoising. A double-beam anddouble-scattering LDV system is built in our laboratory. For measured signals obtained from the experimental system, theminimum relative error of denoised signal is only 0.079% (using minimaxi rule). ,e denoised waveforms of simulated andexperimental signals are much more smooth and clear than that of original signals. Generally speaking, denoising effects ofminimaxi and saqtwolog rules are better than those of heursure and rigrsure rules. As shown in the processing and analysis ofsimulated and experimental signals, denoising methods based on wavelet packet threshold have ability to depress the noise in laserDoppler signal and improve the precision of signal processing. Owing to its effectiveness and practicability, wavelet packetthreshold denoising is a practical method for LDV signal processing.

1. Introduction

As having advantages as high sensitivity, high precision,wide measuring range, and noncontact measurement, thelaser Doppler velocimeter (LDV) has widespread applica-tions in industry and scientific research [1–6]. Research anddesign of LDV have drawn a lot of attention from re-searchers [7–11]. Double-beam and double-scattering op-tical structure adopted in our research is mainly used for thecontactless velocity measurement of solid and fluid. Forexample, in high-speed mill, laser Doppler velocimeters areutilized to measure the accurate speed parameters of therolling strip for the precise control of the strip rollingthickness, which has important significance for the pro-duction of high-quality rolling products. And in some sci-entific research studies, LDV is adopted for the velocity

measurements of fluid, gas fluid, and flame (such as exhaustflame of rocket engine). Applications of LDV put forwardrequirement for the real-time performance of signal pro-cessing. Meanwhile, for commercial applications such ashigh-speed mill and fluid velocity measurement, the cost ofsignal processing system is limited. ,erefore, a complexdenoising method is not applicable for LDV. ,e aim of ourresearch is to find a practical, stable, reliable, and less-complicated noise reduction method for the laser Dopplervelocimeter. As the LDV has a wide measuring range, fromstatics to hypersonics, the distribution range of Dopplerfrequency is wide. Meanwhile, because of the complexity ofthe surface morphology of tested solid and the changeof diameter of measured particles in fluid, the broadeningof the peak spectrum of Doppler signal often occurs inpractical applications. ,e above reasons lead to the poor

HindawiInternational Journal of OpticsVolume 2019, Article ID 1097292, 11 pageshttps://doi.org/10.1155/2019/1097292

mailto:qdzd721@163.comhttps://orcid.org/0000-0002-0805-7066https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/1097292

performance of denoising methods based on frequency-domain analysis in LDV. �e time-frequency analysismethod is more applicable for LDV signal denoising.

Compared with wavelet denoising, wavelet packetdenoising subdivides the high-frequency signal and cananalyze the signal better. In this paper, the selection ofparameters, such as the basis function, the number ofdecomposition layers, and the threshold determinationmethod, which directly aect the noise reduction eect, isstudied in detail through the analysis of experimental signaland the comparison of SNR and RMSE. And the optimalparameters are determined. �e main novelty of the pro-posed work can be summarized as follows: (1) the study ofthe applicability of wavelet packet denoising method toLDV signal; (2) the construction of double-beam anddouble-scattering experiment system; (3) the optimizationof the parameters of wavelet packet threshold denoisingmethod applied in LDV signal. �e study provides apractical and reliable method for LDV signal denoising inapplication.

2. Principle of Laser Doppler Velocimeter

In this experiment, the double-beam and double-scatteringmodel in optical heterodyne detection mode was selected forthe detection of Doppler shift. �e Doppler shift in thedouble-beam and double-scattering model is independent ofthe scattered light and is only related to the directions of thetwo incident light beams. In an actual measurement, twobeamsW1 andW2 are incident on the surface of the movingobject, forming a very small spot on the surface of the objectmeasured. Both incident beams are scattered. �e basicmodel is shown in Figure 1.

�e relationship between Doppler shift and velocity is asfollows:

V �λ

2 sin θΔf, (1)

where Δf is the Doppler shift, θ is the angle between theincident beam and the intersection bisector, and λ is the laserwavelength. From equation (1), we know that Δf is a linearfunction of velocityV, so in the actual speed measurement, ifwe can accurately determine the Doppler shift, we are able toquickly calculate the corresponding speed.

3. Selection and Principle of Wavelet PacketDenoising Method

3.1. Selection of Wavelet Packet Denoising Method. Owing tothe complexity of noise components of Doppler velocitymeasurement signals, the noise signal may be distributed indierent time-frequency subspaces. �e wavelet-based signalanalysis is not able to redecompose the signal in the high-frequency region, and the high-frequency noise rejection is notideal. To deal with this shortcoming, wavelet packets are ap-plied to process Doppler signals, which, comparedwith wavelettransforms, can further decompose the high-frequency com-ponents that are not subdivided in the multiresolution analysis

and can thus readily eliminate the noises in each spectrumrange of the signal [12, 13].

Compared with other wavelet packet denoising methodssuch as modulus maxima reconstruction denoising based onsignal singularity and spatial correlation denoising based onsignal correlation, threshold denoising has the characteris-tics of low computation complexity, good denoising per-formance, and applicability to low signal-to-noise-ratio(SNR) signal processing. As the process of LDV signal re-quires a high real-time response, threshold denoising is usedin our research.

Wavelet packet thresholding denoising is divided intothree steps: wavelet packet decomposition, thresholdquantization on the coecients created by the de-composition, and signal reconstruction. �e denoisingperformance depends on the following factors: selection ofthe wavelet packet basis, determination of the number ofpacket decomposition layers, and selection of the thresholdfunction and the threshold estimation method.

3.2. Principle of Wavelet Packet Transform. In the multi-resolution analysis of orthogonal wavelet basis, the scalesubspace and wavelet subspace are dened as Vj and Wj,respectively, where j is the scale factor (j ∈ Z). �e or-thogonal decomposition of the Hilbert space L2(R) can beexpressed as Vj− 1 � Vj ∪Wj. �e two-scale equation of thescaling function φ(t) is [13]

ϕ(t) ��2

√∑k∈Z

hkϕ(2t − k),

Ψ(t) ��2

√∑k∈Z

gkϕ(2t − k),

(2)

where hn{ }n∈Z and gn{ }n∈Z are, respectively, the low- andhigh-pass representations of a set of conjugate image lters,gn � (− 1)

nh1− n. We introduce new notations μ0(t) � φ(t)and μ1(t) � ψ(t), and then, μ0(t) and μ1(t) satises thefollowing equations:

μ0(t) ��2

√∑k∈Z

hkμ0(2t − k),

μ1(t) ��2

√∑k∈Z

gkμ1(2t − k).

(3)

Using μ0, μ1, g, and h, a set of wavelet packet functions ata certain scale are dened as μn, n� 0, 1, 2, . . .:

W1 W2

V

θ

Solid surface

Figure 1: Double-beam and double-scattering model.

2 International Journal of Optics

μ2n(t) ��2

√k∈Z

hkμn(2t − k),

μ2n+1(t) ��2

√k∈Z

gkμn(2t − k).

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(4)

,e function μn is referred to as the wavelet packetdetermined by function μ0 � ϕ. ,e discrete wavelet packetdecomposition function is

d2nj [k] � l∈Z

hl− 2kdnj+1[l],

d2n+1j [k] � l∈Z

gl− 2kdnj+1[l].

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(5)

,e discrete wavelet packet reconstruction function is

dnj+1[k] �

l∈Zhk− 2ld

2nj [l] +

l∈Zgk− 2ld

2n+1j [l]. (6)

3.3. Wavelet Packet Global 2reshold Denoising. ,e appli-cation of the wavelet packet function to signal denoisingmainly has twomethods: one is wavelet packet multithresholddenoising, and the other is wavelet packet global thresholddenoising. ,e main difference between them is that waveletpacket multithreshold denoising is used to threshold thedecomposition coefficients, the principle of which is to selectan appropriate threshold for each wavelet packet de-composition coefficient for threshold quantization, whilewavelet packet global threshold denoising is used to select anappropriate threshold for all wavelet packet decompositioncoefficients for denoising processing. Althoughwavelet packetmultithreshold denoising is a more accurate denoisingmethod, the computation in this method is high in complexityand is slow in speed. For the characteristics of Doppler speedmeasurement signal (a stringent real-time requirement), wechoose the wavelet packet global threshold denoising methodwith faster computation speed.

3.4. Selection of 2reshold Function. ,e difference inthresholding functions reflects the difference of coefficientprocessing rules; the two commonly used thresholdingfunctions are the soft thresholding function and hardthresholding function. ,e hard thresholding function is[13]

wi,j �wj,k, wj,k

≥ λ,

0, wj,k

< λ,

⎧⎪⎨

⎪⎩(7)

and the soft thresholding function is

wi,j �sign wj,k wj,k

− λ , wj,k

≥ λ,

0, wj,k

< λ,

⎧⎪⎪⎨

⎪⎪⎩(8)

where wj,k and wi,j are the wavelet coefficients before andafter the denoising process, respectively; λ is the definedthreshold; and the two thresholding functions are used toremove the small wavelet coefficients and to shrink or to

retain the large wavelet coefficients. On the basis of hardthresholding, soft thresholding shrinks the continuouspoints to zero on the declining boundaries, which can ef-fectively avoid the discontinuity, making the reconstructedsignals smoother. However, in the soft thresholding method,wj,k and wi,j always have a constant deviation and lead toloss of certain high-frequency information. However, thehigh-frequency information in the Doppler speed mea-surement signals contains less useful signals, so the softthresholding function is used in this experiment.

3.5. Estimation Method of 2reshold. ,ere are usually fourthreshold estimation methods available: heuristic thresh-olding (heursure), fixed thresholding (sqtwolog), unbiasedrisk estimation thresholding (rigrsure), and minimaxithresholding (minimaxi) [13]. ,e rules for the minimaxithresholding and the rigrsure thresholding are conservative.When a very small amount of the high-frequency in-formation of the noisy signal exists in the noise range, thesetwo thresholdings are very useful and able to extract theweak signals. ,e sqtwolog and heursure rules are morecomplete and effective in denoising; however, they tend tomistake useful high-frequency signals as noise and removethem. ,e four threshold rules are as follows [14].

① Heuristic threshold (heursure):

c �

����������

1N

lnNln 2

3

,

e �

Ni�1 Si

2

− N

N,

(9)

where, N is the length of signal Si. If e< c, a fixedthreshold (sqtwolog) should be selected. Otherwise,the smaller threshold obtained from the fixedthreshold (sqtwolog) and the heuristic threshold(heursure) should be selected by this criterion.

② Fixed threshold (sqtwolog):

λ ���������

2 log(N)

, (10)

where N is the length of signal sequence Si.③ Unbiased risk estimation threshold (rigrsure):

Each element of the signal Si is taken as an absolutevalue and then sorted from small to large. After that,each element is squared to obtain a new sequence:

f(k) � sort Si

2, k � 0, 1, 2, . . . , N − 1. (11)

If the threshold is the square root of the kth element off(k), as shown in equation (12), then

λk �����

f(k)

. (12)

,e risk generated by the threshold is

International Journal of Optics 3

Risk(k) � N − 2k +∑ki�1f(i) +(N − k)f(N − k)

N[ ].

(13)

In the risk curve Risk(k), the value of k correspondingto the minimum risk point is denoted as kmin. And thethreshold of rigrsure rule is dened as

λ ��������f kmin( )√

. (14)

④ Minimaxi threshold (minimaxi):

λ �0.3936 + 0.1829

lnNln 2

( ), N> 32,

0, N≤ 32.

(15)

4. Wavelet Packet Denoising ofSimulated Signals

4.1. Simulation of LaserDoppler Signal. In order to verify theeectiveness of wavelet packet threshold denoising, simu-lated laser Doppler signals were processed by heursure,sqtwolog, rigrsure, and minimaxi rules separately. �e ideallaser Doppler signal is a continuous Gauss-distributed si-nusoidal waveform. �e Gauss-distributed base is caused byintensity variety of incidence light. And the sinusoidal signalmodulated on base signal is caused by the light interferenceon the surface of test object. As the actual signal containshigh intensity noises, including stray light noise, shot noise,and Johnson noise, Gaussian white noise was selected as thesynthetic noise. A sequence of simulated LDV signal withSNR of 0 dB is shown in Figure 2.

4.2. SelectionofDecompositionLayerNumber. As the waveletpacket function and the number of decomposition layers willaect the performance of wavelet packet thresholdingdenoising, it is important to choose the optimal waveletpacket function and the optimal number of the waveletpacket decomposition layers. In this experiment, the sym8function in the symN function system of the wavelet packetfunction is used because the symN function is an approx-imate symmetric wavelet packet function, and its waveformis similar to the waveform of the Doppler signal. In addition,the sym8 function is the wavelet packet function with thehighest symmetry and the highest similarity in the symNfunction system [15]. In the process of selecting the numberof the wavelet packet decomposition layers, we use the sym8wavelet packet function and thresholding for the denoisingof noise-added simulation signals and determine thenumber of the decomposition layers by analyzing andcomparing the denoising eect. Taking sqtwolog rule, forexample, ten groups of signals with original SNR of 0 dB aredenoised by one to ve layer decomposition. Table 1 showsthe average SNR of each layer. �e calculation formula ofSNR is shown in the following equation [14]:

SNR � 10 lg PsPn, (16)

where Ps is the power of the original signal and Pn is thepower of noise. �e denoising eect is proportional to SNR[16].

Figure 3 shows the denoised waveforms of the signalgiven by Figure 2. As shown in Figure 3, compared withother three denoised signals, four-layer decompositiondenoised signal has the clearest and smoothest waveform.

It can be concluded from Table 1 that in the denoisingprocess performed on the noise-added simulation signals, asthe number of decomposition layers increases, the denoisingeect improves. However, after the number of de-composition layers reaches four layers, the increasing of thedenoising eect is not obvious and meanwhile the com-putation complexity is increased. By performing multipledenoising comparisons on the noise-added simulation sig-nals, it was decided to use four-layer decomposition forwavelet packet decomposition.

4.3. Processing of Simulated Signals. Wavelet thresholddenoising of heursure, sqtwolog, rigrsure, and minimaxirules was utilized to process simulated signals in both low-frequency range (100 kHz∼1000 kHz, corresponding to lowspeed) and high-frequency range (100MHz∼9000MHz,corresponding to high speed). �e sampling frequency was5MHz in low-frequency range and 5GHz in high-frequencyrange. �e simulated signals were interfered by Gaussian-distributed white noise with SNR of 0 dB, 5 dB, and 10 dB.SNR comparisons of four threshold rules in low- and high-frequency ranges are shown in Figures 4 and 5.

As shown in Figures 4 and 5, all four kinds of thresholdrules have signicantly improved the SNRs of simulatedsignals in both low- and high-frequency bands. �emaximum dierence between denoised signal and originalsignal is 12.52 dB (1000 kHz signal with an original SNR of0 dB denoised by minimaxi rule). Taking SNR as an in-dicator, the denoising eects of minimaxi and sqtwologrules are more obvious than that of heursure and rigrsurerules.

Along with SNR, root mean square error (RSME) isanother parameter for the evaluating of denoising eect.�eformula of RMSE is as follows [14]:

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

Figure 2: Simulated signal with SNR of 0 dB.

Table 1: SNR comparison of dierent layers.

Layer number 1 2 3 4 5SNR 3.41 6.70 8.92 11.94 9.36

4 International Journal of Optics

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(a)

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(b)

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(c)

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(d)

Figure 3: Denoising eects of dierent decomposition layers. (a) One-layer decomposition denoising. (b) Two-layer decompositiondenoising. (c) �ree-layer decomposition denoising. (d) Four-layer decomposition denoising.

0 200 400 600 800 10008

9

10

11

12

13

Signal frequency (kHz)

SNR

(dB)

HeursureSqtwolog

RigrsureMinimaxi

(a)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 100011

12

13

14

15

16

Signal frequency (kHz)

SNR

(dB)

(b)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 100015

16

17

18

19

20

Signal frequency (kHz)

SNR

(dB)

(c)

Figure 4: SNR comparison of four threshold rules in low-frequency range. (a) SNR of original signal is 0 dB. (b) SNR of original signal is5 dB. (c) SNR of original signal is 10 dB.

International Journal of Optics 5

RMSE �������������������1N

N

x(n) − xs(n)

2

, (17)

where x(n) is the original signal, xs(n) is the filtered signal,and N is the length of the signal. ,e smaller the RMSE is,the better the denoising effect is. ,e comparisons of RMSEof signals in both low- and high-frequency bands are shownin Figures 6 and 7.

As shown in Figures 6 and 7, all four kinds of thresholdrules have satisfactory RMSEs for simulated signals in bothlow- and high-frequency bands. Taking RMSE as an in-dicator, the denoising effects of minimaxi and sqtwolog rulesare better than that of heursure and rigrsure rules.

In Figure 8, the waveform of an original simulated signal(SNR� 0 dB) is shown.,e denoised signals using four kindsof threshold rules are shown in Figure 9.

As shown in Figure 9, the noise contained in originalsignal can be reduced significantly by all four threshold rules.

Compared with original signal shown in Figure 8, wave-forms of the denoised signals are more smooth and clear.Generally speaking, minimaxi and sqtwolog rules havebetter performance than heursure and rigrsure rules.

5. Wavelet Packet Denoising ofExperimental Signals

An experimental LDV system with double-beam and dou-ble-scattering structure was built in our laboratory. ,estructure of this experimental system is shown in Figure 10.An optical fiber splitter (TN632R5F2, ,orlabs) was utilizedto equally divide the output laser into two beams. ,e twobeams were emitted from two GRIN single-mode fibercollimators (50-630-FC, ,orlabs) parallely to a focusinglens and focused on the surface of a 150mm diameter metaldisk driven by using a DC motor. ,e light scattered frommetal disk surface was collected and focused on another

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10008

9

10

11

12

13

Signal frequency (MHz)

SNR

(dB)

(a)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 100011

12

13

14

15

16

Signal frequency (MHz)

SNR

(dB)

(b)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 100015

16

17

18

19

20

Signal frequency (MHz)

SNR

(dB)

(c)

Figure 5: SNR comparison of four threshold rules in high-frequency range. (a) SNR of original signal is 0 dB. (b) SNR of original signal is5 dB. (c) SNR of original signal is 10 dB.

6 International Journal of Optics

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10000.001

0.0015

0.002

0.0025

Signal frequency (kHz)

RMSE

(a)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10000.001

0.0015

0.002

0.0025

Signal frequency (kHz)

RMSE

(b)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10000.001

0.0015

0.002

0.0025

Signal frequency (kHz)

RMSE

(c)

Figure 6: RMSE comparison of four threshold rules in low-frequency range. (a) SNR of original signal is 0 dB. (b) SNR of original signal is5 dB. (c) SNR of original signal is 10 dB.

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10000.001

0.0015

0.002

0.0025

Signal frequency (MHz)

RMSE

(a)

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10000.001

0.0015

0.002

0.0025

Signal frequency (MHz)

RMSE

(b)

Figure 7: Continued.

International Journal of Optics 7

collimator by collecting lens. And then, the light wastransmitted to APD detector through an optical ber. �reecollimators were xed in an aluminium support.

�e He-Ne laser source (HNLS008R-EC, �orlabs) usedin our experiments has 0.8mW output power and 632.8 nmwavelength.�e APD detector (APD 430A/M,�orlabs) has400–1000 nm detection wavelength range, 8–80 μm opticalpower, and 400MHz output bandwidth. �e rotating rate of

the metal disk was measured by using a photoelectric en-coder (UCD-IPH00-L100-ARW, Posital). And the linearvelocity of the disk was calculated out by rotating rate anddisk diameter. Half angle of two incident beams was set as17°. �e experimental LDV system is shown in Figure 11.

Ten groups of experimental signals were obtained fromthe LDV system shown in Figure 11. And then, these signalswere processed by wavelet packet threshold denoising using

HeursureSqtwolog

RigrsureMinimaxi

0 200 400 600 800 10000.001

0.0015

0.002

0.0025

Signal frequency (MHz)

RMSE

(c)

Figure 7: RMSE comparison of four threshold rules in high-frequency range. (a) SNR of original signal is 0 dB. (b) SNR of original signal is5 dB. (c) SNR of original signal is 10 dB.

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

Figure 8: Original signal with SNR� 0 dB.

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(a)

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(b)

0 500 1000 1500 2000 2500 3000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

(c)

Sampling points0 500 1000 1500 2000 2500 3000

–0.02

0

0.02

0.04

Am

plitu

de (V

)

(d)

Figure 9: Denoising eects of four threshold rules. (a) Heursure threshold rule denoising. (b) Sqtwolog threshold rule denoising. (c)Rigrsure threshold rule denoising. (d) Minimaxi threshold rule denoising.

8 International Journal of Optics

heursure, sqtwolog, rigrsure, and minimaxi rules separately.�e wavelet packet function is sym8. �e decompositionscale is 4. �e sampling frequency was set as 10MHz. Afterthe threshold denoising, 1024 points fast Fourier transform(FFT) was adopted to get the Doppler frequency shift Δf.�en, equation (1) was used to calculate the rotating linearvelocity of metal disk. Table 2 shows the accuracy com-parison of four kinds of threshold rule applied on ten ex-perimental signals.

As shown in Table 2, the threshold denoising methodshave ability to provide satisfactory process eects. Overallyspeaking, minimaxi and sqtwolog rules have higher pre-cision. �e minimum relative error is 0.079% (No. 8 signalusing minimaxi rule). Taking a set of signal with a calibratedspeed of 0.2705m/s, for example, the signal waveform isshown in Figure 12. And the denoised signals using fourthreshold rules are shown in Figure 13.

As shown in Figure 13, the noise interferences werereduced signicantly by four threshold rules. Minimaxi andsqtwolog rules denoised signal has more smooth and clearwaveform than other denoised signals. �reshold denoisingmethods based on wavelet packet are able to depress thenoise in experimental signal and are satisfactory for practicalapplications. In circumstance of Intel Core i3-3240 (CPU

V

Rotatingdisk

Focusinglens

Collectinglens

Fiberheads Fiber

splitter

APDdetector

He-Ne lasersource

Figure 10: Structure of the experimental LDV system.

Aluminiumsupport

Laser source

Emergentfiber heads

APDdetector

Collectinglens

Focusinglens

Metal disk

Fibersplitter

Incidentfiber heads

Figure 11: Experimental LDV system.

Table 2: Processing accuracy of experimental signal.

No. Reference speed(m/s)Error (m/s)

Heursure Sqtwolog Rigrsure Minimaxi1 0.2705 0.0007 0.0006 0.0010 0.00032 0.3070 0.0007 0.0005 0.0011 0.00043 0.3852 0.0007 0.0006 0.0011 0.00034 0.4315 0.0008 0.0007 0.0012 0.00045 0.4861 0.0008 0.0006 0.0011 0.00056 0.5307 0.0008 0.0008 0.0013 0.00057 0.5725 0.0009 0.0007 0.0013 0.00068 0.6356 0.0010 0.0008 0.0012 0.00059 0.7012 0.0010 0.0009 0.0014 0.000610 0.7358 0.0009 0.0008 0.0013 0.0007

0 2000 4000 6000 8000 10000 12000–0.02

0

0.02

0.04

Sampling points

Am

plitu

de (V

)

Figure 12: Experimental signal.

International Journal of Optics 9

frequency of 3.4GHz, memory of 4GB), the average uptimeof above 10 signal sequences is 0.00010 s. Taking the trackingltering method adopted in [17] as a comparison, the av-erage uptime of above 10 signal sequences is 0.00052 s. �epassband width of tracking lter is set as 100 kHz, the sameas [17]. It can be seen that the proposed method has a higheroperation speed, which is important for high-speedmeasurement.

6. Conclusion

A signal denoising method based on wavelet packet isproposed in this research. Wavelet packet thresholddenoising using heursure, sqtwolog, rigrsure, and mini-maxi rules is used to process both simulated and measuredsignals. Features such as signal waveform, SNR, andRMSE of simulated signals with dierent original SNRs inlow- and high-frequency ranges are signicantly im-proved after the denoising. Furthermore, the validity ofproposed methods on measured signals is proved by thedenoising results of signals obtained from double-beamand double-scattering experimental system built-in lab-oratory. In general, the performances of thresholddenoising using minimaxi and sqtwolog rules are betterthan those of heursure and rigrsure rules. As indicated inabove research studies, the wavelet packet thresholddenoising methods are eective and applicable for LDVsignal.

Data Availability

�e data used to support the ndings of this study have notbeen made available because the data also form critical partof ongoing study.

Conflicts of Interest

�e authors declare that there are no con©icts of interestregarding the publication of this paper.

Acknowledgments

�is research was funded by the National Natural ScienceFoundation of China (61803219).

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Figure 13: Denoised experimental signal. (a) Heursure threshold rule denoising. (b) Sqtwolog threshold rule denoising. (c) Rigrsurethreshold rule denoising. (d) Minimaxi threshold rule denoising.

10 International Journal of Optics

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