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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;
[email protected]
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:[email protected]://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
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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
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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
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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
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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
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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
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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
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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
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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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International Journal of Optics 11
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