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Atmospheric Research 129–130 (2013) 58–66
Contents lists available at SciVerse ScienceDirect
Atmospheric Research
journal homepage: www.elsevier.com/locate/atmos
Lightning VHF radiation location system based on short-baseline TDOA
technique — Validation in rocket-triggered lightning
Zhuling Sun a, b, Xiushu Qie a,⁎, Mingyuan Liu a, Dongjie Cao a, Dongfang Wang a
a
Key Laboratory of Middle Atmosphere and Global Environment Observation (LAGEO), Institute of Atmospheric Physics, Chinese Academy of Sciences,
Beijing 100029, China
b
University of Chinese Academy of Sciences, Beijing 100049, China
a r t i c l e
i n f o
Article history:
Received 29 February 2012
Received in revised form 23 October 2012
Accepted 21 November 2012
Keywords:
Short-baseline
Time-difference of arrival
VHF radiation
Discharge process
Rocket-triggered lightning
a b s t r a c t
A lightning VHF radiation location system based on short-baseline time-difference of arrival
(TDOA) technology is newly developed. Based on the orthogonal 10 m-baseline antenna array
with four identical broadband flat plane antennas, this system receives the lightning broadband
VHF radiation signals and calculates TDOA between antennas in order to determine the location of
lightning radiation sources in two dimensions (elevation and azimuth). To reduce noise and
improve estimation accuracy of time delay, a general correlation time delay estimation algorithm
based on direct correlation method and wavelet transform is proposed. Moreover, parabolic
interpolation algorithm is used in the fractional delay estimation to improve the time resolution of
the positioning system. In this paper, a rocket-triggered lightning discharge and a cloud lightning
discharge are analyzed respectively, combining with simultaneous observations of high-speed
camera and fast/slow electric field changes. The results indicate that the TDOA location system
could effectively map the lightning radiation sources in 2 dimensions and are in a good agreement
with the high-speed video camera images.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Lightning discharges emit electromagnetic radiation in
broadband frequency ranges covering from ULF/VLF, VHF/
UHF, and even up to Gamma-ray. Observations have shown
that radiation of different frequency spectrum corresponds to
different stage of the lightning discharge, so the development
process and characteristics of the lightning discharge can be
determined based on the lightning detection and location
technique at different frequencies.
VLF/LF signals generally correspond to larger-scale
breakdown processes in lightning discharges and can't be
used to describe entire development of the lightning
channel, while VHF/UHF is emitted during smaller-scale
breakdowns, so VHF/UHF sources can be located to image the
lightning channel development and to realize the detection of
both cloud-to-ground lightning and intra-cloud lightning (e.g.,
⁎ Corresponding author. Tel.: +86 10 82995091; fax: +86 10 82995073.
E-mail address: qiex@mail.iap.ac.cn (X. Qie).
0169-8095/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.atmosres.2012.11.010
Shao et al., 1995; Rison et al., 1999; Thomas et al., 2001; Zhang
et al., 2010).
For the short-baseline VHF band location, two main
lightning location techniques have been proposed and applied,
interferometer and time-difference of arrival (TDOA). The basic
principles are measuring the phase or time-difference of
arrivals between two antennas in order to map lightning
radiation sources, respectively.
In recent years, the interferometer location technology has
been developed quickly. Narrowband interferometer uses
single station location technology to reconstruct the temporal
and spatial development of lightning discharge visually (e.g.
Richard and Auffray, 1985; Rhodes et al., 1994; Shao et al.,
1995; Zhang et al., 2008). Shao et al. (1996) first proposed
broadband interferometer which located the lightning radiation by broadband lightning radiation signals, and it has been
improved and developed quickly afterward (e.g. Ushio et al.,
1997; Kawasaki et al., 2000; Dong et al., 2001). With the
synchronic observation of the electric field changes, the
direction and speed of the space charge movement could be
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Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
indicated, moreover, the distribution characteristics of electric
charge inside the thunderstorm could be estimated. More
recently, two sets of broadband interferometers were
employed synchronously to locate lightning radiation sources
in three dimensions (e.g. Morimoto et al., 2005; Akita et al.,
2010; Yoshida et al., 2012).
The impact of lightning signal waveform and amplitude
on the interferometer location technology is small; however,
phase ambiguity problem affects the reliability of location. In
contrast, time-difference of arrival technique can avoid this
problem with higher location accuracy. Tayloy (1978) first
used this technique operated at a bandwidth from 20 to
80 MHz and a 10 m baseline length to estimate the elevation
and azimuth of radiation sources, but could only distinguish
independent pulse events of lightning. Zhang et al. (2003)
designed a 10 m short baseline time-of-arrival lightning
radiation detection system with a center frequency of
280 MHz and a bandwidth of 10 MHz. It was used in the
observation on lightning physics and mapped the lightning
VHF sources in two dimensions, while the system detection
efficiency was relatively low, and error for elevation was
relatively large. Cao et al. (2012) developed a new short
baseline lightning location system but with the bandwidth
from 125 to 200 MHz to locate CG lightning discharge
processes in two dimensions.
The traditional time-of-arrival location technology considers the peak of radiation signal waveform as arrival
instant to solve time-difference between two antennas,
resulting in this technology being effective to just isolate
short pulses. Because the peak in continuous radiation signal
waveform is hard to be identified at the high frequency band
and easily polluted with noise, the time-difference of arrival
method can't have a good performance. In this paper, based
on the short baseline system mentioned by Cao et al. (2012),
a scheme using generalized cross correlation time delay
estimation algorithm together with wavelet transform is
raised to reduce the noise and improve the system location
accuracy. Radiation source location results and characteristics of electric field change have been analyzed for a
triggered lightning discharge and a cloud lightning discharge, and the radiation source locations are compared
with the synchronization optical observation.
59
Fig. 1. Schematic diagram of TDOA lightning radiation location with two
antennas A and B separated horizontally with a distance of d.
where c is the speed of light in vacuum. Formula (1) shows that,
as long as Δt is determined, the direction of radiation signals can
be obtained. The radiation sources can be mapped in
two-dimensional (2-D) azimuth and elevation format with
three or more non-collinear antennas and two independent
non-collinear baselines.
2.2. Radiation sources location method in 2-D
Due to the simple spatial geometry relationship, the
orthogonal baseline configuration is widely applied in the
short-baseline location system, as illustrated in Fig. 2. Four
antennas A, B, C, and D are arranged at the vertices of a square,
and its two sides CD and CB correspond to the direction of
south and east, respectively. If an arbitrary space radiation
source P occurred in the quadrant BCD, using the law of cosines
in space right-angle triangle, time-differences of arrival of a
signal on two mutually orthogonal baselines can be expressed
as:
cosPCD ¼ cosðAZ Þ cosðELÞ
cosPCB ¼ cosð90B−AZ Þ cosðELÞ
¼ sinðAZ Þ cosðELÞ:
ð2Þ
2. Method of TDOA lightning location
Azimuth angle AZ and elevation angle EL can be determined as follows:
2.1. Basic principle
AZ ¼ tan
The principle of the location technology based on the TDOA
technique is shown in Fig. 1. Two identical receiving antennas A
and B are separated with a distance d in a horizontal plane.
Since the distance between lightning radiation source and each
receiving antenna is extremely larger than that between two
antennas, we can consider high frequency electromagnetic
waves of radiation sources to be approximately plane wave
when reaching the receiver, with the incident angle θ against to
the baseline. If the time-difference of arrival between the couple
of antennas is assumed to be Δt, the relation between Δt and θ
can be expressed by the following equation:
cosθ ¼
cΔt
d
cosPCB
−1 Δt CB
¼ tan
Δt CD
cosPCD
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−1
2
2
EL ¼ cos
cos PCB þ cos PCD
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−1 c
¼ cos
Δt CB 2 þ Δt CD 2 :
d
−1
ð1Þ
ð3Þ
The TDOA method has some advantages over other lightning
location techniques. With a very short baseline, this method does
not require knowledge of the absolute arriving time of two
signals what a long-baseline time of arrival method needs, and
the experiment field selection could have more flexibility in
scale. To determine the location of a radiation source, the TDOA
method only needs to estimate the time delay in time domain,
while the interferometer location technology must consider the
phase ambiguity problem of phase difference estimations in
frequency domain, which might be difficult to determine in
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2.4. The TDOA estimation technique
Fig. 2. An orthogonal baseline model for two-dimensional location using
time-difference of arrival method.
practice. In this respect, the TDOA method has its advantages
comparing to the interferometer location technology.
The generalized cross correlation (GCC) technique is one of
the most classic time delay estimation methods, improving the
computational complexity, convergence rates and estimation
accuracy. In the GCC technique, the time delay is estimated to
be the time corresponding to the maximum peak of the
cross-correlation function of two signals. In order to improve
the signal to noise ratio (SNR) and weaken the effect on the
peak solution of cross-correlation by multiple peaks or false
peak errors, the wavelet transform is actually adopted in the
construction of the weighting function, since it performs very
well in the scale decomposition and frequency filtering.
Appendix A gives more details of the GCC technique and
wavelet transform in time delay estimation.
The result of general cross-correlation time delay estimation has an integer time delay of sampling interval accurately,
depended on the sampling rate. If the actual value of the time
delay isn't integer multiples of the sample period, the peak of
cross-correlation is supposed to be unsampled. In order to
achieve a high-precision estimation of the time delay, interpolation method is applied to the cross-correlation function and
discussed in detail in Appendix B.
3. Hardware of the TDOA location system and the
installation in SHATLE
2.3. Location error analysis
Calculating the difference of both sides of the Eq. (1), the
uncertainty caused by the error in time-difference measurement can be estimated, as shown in Eq. (4).
dθ
c
¼
:
dΔt d sinθ
ð4Þ
In the case that both the length of baseline and the error
in time-difference measurement are certain, the measurement error depends on the incident angle. The error is the
greatest when the incident direction parallels to the baseline,
therefore, other sensors should be selected to the calculation
when the radiation source is within 25° angle to the baseline.
From the Eq. (4) it is shown that the accuracy of the TDOA
location system depends on that of the time-difference
measurement between antennas, so it is very crucial to
calculate the time-difference of arrival. Usually the baseline is
very short, with a length of only a dozen meters; as a result,
the value of the time-difference is in the order of only a dozen
nanoseconds to a few tens of nanoseconds, which imposes
the technical difficulty of measuring.
Zhang et al. (2003) used narrow-band signal peak-seeking
technique to calculate the time-difference. However the
wide-band signal, especially at high frequencies, is very sensitive
and susceptible to being interfered with the noise, so there will
be great disparity between the results of peak-seeking and the
reality. Qiu et al. (2009) utilized a correlation algorithm to
determine the difference in time between waves arriving from
the same signal which reduced impact noise effectively. In this
paper we propose a modified cross correlation method to solve
problems in the time-difference of arrival calculation and to
improve the location accuracy directly.
Block diagram of the TDOA location system is shown in
Fig. 3. The TDOA location system used four broadband flat
antennas to receive lightning VHF radiation signals. Each one is
arranged at a vertex of a square of side 10 m, and two adjacent
sides constitute two orthogonal baselines with east-westward
and north-southward, respectively. Each antenna signal is
sequentially passed through preamplifier, band-pass filter and
coaxial cable with the same length and frequency response. A
LeCroy oscilloscope is used for the data acquisition. The
bandwidth of the band-pass filter is about 125–200 MHz. The
sample frequency of oscilloscope is 1 GS/s and vertical
resolution is 8 bits. Sequential record mode is used, and each
channel can record up to 4000 segments with 2002 samples for
each segment. The fast antenna and slow antenna are used for
electric field change measurement with a sampling rate of
Fig. 3. Block diagram of the TDOA location system components.
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Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
5 MHz. Their bandwidth is 5 MHz and 2 MHz, and time
constants are 3 s and 1 ms, respectively.
In the summer of 2010, the lightning VHF radiation location
system based on short-baseline TDOA technology was used in
the SHandong Artificial Triggering Lightning Experiment
(SHATLE) (Qie et al., 2007, 2009). Fig. 4 shows a diagram of
STATLE installation. The TDOA location system and the high
speed Camera were installed about 700 m and 670 m away
from the rocket launcher, respectively. The system uses a
high-time accuracy GPS for time synchronization of the TDOA
location system and the electric field changes measurement
system, maintaining accuracy to within 50 ns of absolute time.
The remove of negative charge in the cloud is defined to
produce a positive electric field change in this paper. Fast and
slow electric field changes can assist to judge the development
characteristics of radiation sources located by the TDOA in
different lightning discharge phases.
4. Validation of the TDOA location system
During the experiment, a great deal of data was achieved.
In this paper, a triggered lightning and an intra-cloud
lightning discharge are analyzed to validate the hardware
and the algorithm of the TDOA location system by comparing
the VHF location results with the high-speed video images.
4.1. Location results of a rocket-triggered lightning flash
The rocket-triggered lightning occurred at 23:01:32, Aug
20, 2010, and its entire discharge lasted about 0.5 s. Fig. 5(a)
shows the discharge current waveform measured at channel
base of the triggered lightning. Fig. 5(b) and (c) shows the
corresponding electric field changes of the lightning
recorded by the fast and slow antennas. Time zero just
represents the triggering moment of the recorded signal,
and bar positions on the time-axis are coincident with times
when radiation sources were detected by the TDOA location
system. Although the waveform of fast electric field changes
61
was saturated, it can still be recognized clearly that this
lightning process contained a total of 9 return strokes. Just
60 radiation segments were recorded during this flash, due
to the preset high triggering level. Radiation sources were
emitted intermittently in different phases of lightning
discharge process, as can be seen from the density of bar
positions in Fig. 5. Most radiations concentrated near the
times of return strokes, corresponding well to electric field
changes in the time course. It's worth mentioning that in this
lightning discharge the number of radiation sources had a
great relation to the stroke intensity, and most radiation
signals were acquired near time of the 6th return stroke,
with the maximum return stroke electric field peak and
maximum current peak.
Fig. 6(a) shows the image of the 6th return stroke captured
by high-speed video camera located 670 m away from the
lightning channel. The straight part of the lightning channel
was the trajectory of the metal wire after evaporation, and
elevation of its top end was about 20.8° with a height of 256 m.
The upper curved channel was corresponding to the natural
discharge. The top of the visual channel outside the thundercloud corresponds to an elevation of approximately 37°.
Fig. 6(b) shows the radiation source locations for the whole
triggered lightning in two-dimensions. The north direction is
the reference azimuth with increasing clockwise. Color
changes with time from blue to red. It can be seen that the
channel from the bottom to the 21° elevation was straight
approximately, corresponding to the trajectory of metal wire.
Up to the 35° elevation, the shape and the height of the channel
showed certain consistency with the optical observation. The
high speed camera couldn't capture the channel with elevation
from 35° to 76° for cloud shielding effect, while VHF radiation
source locations show advantage of depicting in-cloud lightning channel.
Fig. 7 shows the radiation source locations of the 6th
subsequent dart leader, which occurred about 99.74 ms
after the electric field changes reaching its trigger level. In
the dart leader phase, the waveform of fast electric field
Fig. 4. Installation diagrams of the TDOA location system and high-speed video camera in SHATLE2010.
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Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
Fig. 5. The discharge current of a triggered lightning measured at channel base (a), normalized electric field changes recorded by fast antenna (b) and slow antenna (c).
change is saturated. The VHF radiation lasted for 68 μs. As
the record interval equals to the dead time of oscilloscope
recording, there could be a continuing development of dart
leader radiation sources.
The starting position of radiation sources was in the
higher altitude inside the thunderstorm at 76° elevation and
336° azimuth. Since then the general trend of elevation was
downward going along a previous well-defined channel,
with small changes in azimuth range. The average two
dimensional speed of the radiation source progression is
estimated to be 6.6 × 10 6 m/s. Zhang et al. (2008) reported a
progression speed of about 4.1 × 10 6 m/s for dart leaders in
natural lightning. Qiu et al. (2009) presented 2.5 × 10 6 m/s
for dart leaders in rocket-triggered lightning. The result here
agrees with the results above.
4.2. Location results of an intra-cloud lightning flash
This intra-cloud lightning flash occurred at 00:52:55, July
08, 2010, and the TDOA location system depicted the
development of lightning radiation sources in 2-D during a
period of 0.6 s. Unfortunately, fast and slow antennas didn't
well record the corresponding electric field changes. According
to the observation records, the time interval between the
perception of lightning and the first sound of thunder was
Fig. 6. Discharge channel of the triggered lightning flash. (a) One frame from high speed video images, (b) 2-D VHF radiation source locations.
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Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
63
Fig. 7. Radiation sources of the dart leader before the 6th subsequent return stroke in the triggered lightning. (a)Azimuth-elevation display, (b) elevation versus
time, (c) azimuth versus time, and (d) electric field changes recorded by the fast and slow antennas.
about 15 s, and the lightning was estimated to be 5 km away
from the TDOA location system.
Fig. 8(a) and (b) shows the location result of the radiation
sources and one frame of the high-speed video images,
respectively. For a better description of the lightning, the
reference azimuth in Fig. 8(a) is performed to be the south
direction. This lightning happened in the north of the
observation station at 15° elevation, estimating that the
distance from the ground to the start region was about
1.3 km. Comparison of Fig. 8(a) and (b) illustrates that there
is a good consistency between the VHF radiation channels
and optical images, and the TDOA location system could
definitely depict lightning branches of the cloud flashes.
The propagation paths of lightning radiation sources were
branched extensively for this intra-cloud. As indicated by
arrows b1, b2 and b3 in Fig. 8(a), there are 3 main branches
developing in the space towards the west. The lowest branch
b1 was first developed at 15° elevation and 183° azimuth,
followed by the branches b2 and b3 starting from the same
region roughly. The elevation direction of the branches b1 and
b2 was downward-moving, and the azimuth extension was
from 135° to 185°. The average velocities of the branches b1
and b2 were 5.6 × 105 m/s and 4.6 × 105 m/s, respectively.
These results are similar to that of Shao et al. (1996) on the
intra-cloud lightning discharge. Radiation sources of branch b3
were highly branched and expanded widely from 50° to 250°
azimuth. Propagation direction of b3 was mainly upward with
an average speed of about 2.5× 105 m/s.
As the development of the lightning main branches, there
were radiation sources developing continuously as A, B, C, D
and E in Fig. 8(b). Discharge A occurred during the expansion of
the middle branch b2. The TDOA location system detected
Fig. 8. Similar to Fig. 6, but for an intra-cloud lightning flash at 00:52:55, July 08, 2010.
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Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
eastward development horizontally of the lightning radiation
sources around 206° azimuth. Progression time of the path was
approximately 18 μs with a speed of about 6.8 × 106 m/s. Then
during the extension of the branch b3, there was a downward
discharge B propagating around the same region, traversing
the path established by previous discharge A with a speed of
about 6.3 × 106 m/s during about 800 μs. Thereafter, at 31°
elevation and 150° azimuth, discharge C sloped down toward
branch b2 and the velocity was about 8.2× 106 m/s within
230 μs. Similarly, both discharges D and E were progressed
gradually closer to the branch b3 inside the thundercloud. The
velocities of discharges D and E were 1.2 × 107 m/s and
4.7 × 107 m/s, respectively. Closely following discharges C, D
and E, there were lightning radiation sources detected ahead of
branches b2 and b3. Though there was no electric field change
corresponded, it still can be inferred that the existence of the
negative recoil streamer, which progressed through the
existing channel by the previous positive breakdown in a
retrogressive manner, transferred negative charges from air to
the lightning channel and caused extension of the channel.
Velocities of breakdown discharges C, D and E agreed well with
the previous result reported by Shao et al. (1996) within an
order of magnitude (106–107 m/s).
5. Conclusion and discussion
As a lightning radiation source location technology,
short-baseline TDOA VHF lightning radiation source location
system is now still in the improving and perfecting stage. This
study describes a short baseline lightning location system using
relevant knowledge of signal processing, utilizing generalized
cross correlation time delay estimation, wavelet analysis
algorithm technique to lightning radiation signal processing,
applying parabolic interpolation method to estimate effectively
signal fractional delay, accurately calculating the difference of
time delay arriving at the receiving antennas, realizing the
location of VHF lightning radiation sources in two dimensions,
and finally observing the progression of lightning initiating and
progressing in time and space.
The discharge channel determined by radiation source
locations, for a rocket-triggered lightning and an intra-cloud
lightning, was generally in good agreement with the
high-speed video camera observations. The short-baseline
TDOA VHF lightning radiation source location system effectively reconstructs the lightning discharge channel, visually
reproduced the formation and development of lightning
occurring inside the cloud, and successfully described the
temporal and spatial development characteristics of lightning
discharge channels. It can be concluded that the algorithm
weakens the influence of noise on the time delay estimation,
providing better positioning accuracy in a certain extent.
However, because of shortcomings of the system itself in
location method and antenna arrangement, there are certain
disadvantages, such as the relatively large error for azimuth, no
distance detection of the radiation sources. Therefore, several
improvements are still required for the system in the future,
such as adding additional baseline at vertical direction and
adopting the cooperation of multiple antennas, in order to
further improve the accuracy and availability of the system.
According to the comparison of VHF radiation location
results between the rocket-triggered lightning and the
intra-cloud lightning, it is clear that the channel of intra-cloud
lightning discharge is very complex, having various kinds of
discharge process and widespread branches both in the
horizontal and vertical directions. On the contrary, the
rocket-triggered lightning here has just one channel without
any branch, and presents more coherent and compact radiation
sources location result. More different lightning case studies
are needed to reveal similarities and differences in various
kinds of lightning discharge.
Acknowledgments
The research was supported by the National Natural
Science Foundation of China (grant nos. 41175002 and
40930949) and the National Science and Technology Support
Projects (grant no. 2008 BAC36B03).
Appendix A. Wavelet-based generalized cross correlation
(GCC) time delay estimation
Two received signals by the two antennas can be modeled
by:
s1 ðt Þ ¼ sðt Þ þ e1 ðt Þ
s2 ðt Þ ¼ Asðt−τÞ þ e2 ðt Þ
ðA:1Þ
where s1(t) and s2(t) are received signals by two spatially
separated antennas, s(t) is the lightning radiation source signal,
e1(t) and e2(t) represent the additive noises, τ yields the time
delay between the two received signals, and A denotes the
attenuation coefficient for propagation. Cross-correlation reflects the level of similarity between two related time series at a
different time, and it can be expressed mathematically by:
Rs1 s2 ðτ^ Þ ¼ E½s1 ðt Þs2 ðt þ τ^ Þ
¼ ARss ðτ^ −τ Þ þ ARse1 ðτ^ −τÞ þ Rse2 ðτ^ Þ þ Re1 e2 ðτ^ Þ:
ðA:2Þ
Noises are assumed to be random processes and uncorrelated
with the signal s(t). Both the cross-correlation between two
noises and that between noise and thunder signal tend to be
zero, so the cross-correlation function becomes:
Rs1 s2 ðτ^ Þ≈ARss ðτ^ −τ Þ ¼
N
1X
sðnÞsðn þ τ^ −τÞ:
N n¼0
ðA:3Þ
This method is simple in principle and easy to implement,
and it also weakens the noise influence on the time delay
estimation. However, the cross-correlation method also faces
some practical problems, such as, large amounts of multiply–
accumulate computation, time consuming and failure to meet
the fast processing requirement. The Wiener–Khinchin theorem states that the time domain correlation function is the
inverse Fourier transform of the power spectral density, so we
can use the Fourier transform algorithm to calculate the
correlation in the frequency domain, which will speed up
calculations and reduce amount of computation greatly. First,
the received signals are transformed by fast Fourier transform
into frequency domain, and then the conjugate convolution is
calculated and inversely transformed to the cross-correlation.
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Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
65
Fig. A.1. Flow chart of cross-correlation algorithm.
A block diagram of a cross-correlation processor is shown in
Fig. A.1.
In the generalized cross correlation (GCC) approach,
each received signal is linear pre-filtered before taking
cross-correlation in order to improve the signal to noise ratio
(SNR). This scheme is actually weighting cross power spectral
density before doing the inverse Fourier transformation to get
the cross-correlation function, and the pre-filter function is
called the weighting function, as H(ω) in Eq. (A.4).
′
Gs1s2 ðωÞ ¼ HðωÞGs1s2 ðωÞ:
ðA:4Þ
In order to obtain accurate delay estimation, the weight
function should be selected appropriately. Using the wavelet
transform in the construction of the weighting function,
signal resolutions are different at different positions in the
time-frequency plane, and mutation signal can be effectively
distinguished from noise in the coarse-to-fine analytic
process. The wavelet transform of a limited energy signal
x(t), based on the basis wavelet function ψ ab(t), can be
expressed as follows:
þ∞
W ða; bÞ ¼ ∫ xðt ÞΨa;b ðt Þdt
ðA:5Þ
−∞
Ψa;b ðt Þ ¼ jaj
−1=2
Ψ
t−b
; a; b∈R; a≠0
a
ðA:6Þ
where ψ (t) is a mother wavelet function, which can produce
different ψ ab(t) by time shifting and scale stretching. The
asterisk is referred to the complex conjugate of a variable.
Letters a and b stand for the scale parameter and the time
displacement parameter, respectively.
Using the related frequency domain expression shown in
Eq. (A.7), Eq. (A.4) can be described like Eq. (A.8).
pffiffiffi þ∞
a
jωτ
WT ða; bÞ ¼
∫ X ðωÞΨa;ωn ðaωÞe dω
2π −∞
′
Gs1s2 ðωÞ ¼
pffiffiffi aΨ ðaωÞGs1s2 ðωÞ
ðA:7Þ
ðA:8Þ
where X(ω), Ψa;ωn ðaωÞ denote Fourier transforms of x(t) and
ψab(t), respectively.
This operation equates to a set of band pass filters with
relative frequency characteristics Ψa;ωn ðaωÞ. The energy of the
lightning signal is concentrated in the wavelet domain, and
corresponding wavelet coefficients are large. On the contrary,
the energy of noise signal has small amplitudes, with coefficients uniformly distributed. After multiple-level wavelet
analysis, wavelet coefficients with larger amplitude generally
stand for the useful signal, while the smaller stand for noise
with a high probability. Therefore, for noise reduction, a
suitable threshold should be used to retain or contract wavelet
coefficients above the threshold, and to zero wavelet coefficients below the threshold.
In practical, to meet the needs of data processing and
analysis, discrete wavelet transform is used, where a and b in
Eq. (A.7) are fixed to integers. The process of transform is
mainly divided into four stages:
(1) Selection of suitable wavelet basis to the wavelet
decomposition
Use Dmey wavelet as a wavelet basis function in the
TDOA location system, which is defined in frequency
domain and has symmetry and excellent multiple
resolution characteristics. At the meantime, Dmey
wavelet can effectively avoid the signal distortion by
phase distortion in the signal processing, and has a
better performance in the cutoff characteristics of
low-pass and high-pass filters than the others.
(2) Determination of the denoising threshold
Usually SNR of the primary signal is the important
evaluation to appropriate values for the threshold. We
apply heuristic SURE thresholding method in the
system, which selects the optimal one from either
the FIXTHRES threshold or the SURE threshold. When
the SNR is very small, and SURE estimation will have
large error, so the FIXTHRES one should be chosen, the
opposite is also true.
(3) Choice of an appropriate threshold rule
In this system, we use the soft-thresholding selection
function and the function is formulated as Eq. (A.9),
f ðdi Þ ¼
sgnðdi Þðjdi j−λÞ jdi j≥λ
0
jdi jbλ
ðA:9Þ
where λ is the median threshold value based on wavelet
coefficients, di is the wavelet coefficient at each wavelet
decomposition level, and f(di) is the coefficient after
thresholding. Using this rule, wavelet coefficients with
absolute value smaller than the median threshold are
zeroed as noises, and the remainders are retained as a
useful signal, with value contracted.
(4) Signal reconstruction
With the discrete wavelet transform, the signal denoised
can be reconstructed combining with all high frequency
wavelet coefficients and low frequency wavelet coefficients at bottom levels, then further cross power spectral
density weighted by the wavelet can be offered.
Generally, the wavelet-based generalized correlation for
time delay estimation, can not only reduce noise
effectively, but also weakens the effect on the peak
solution of cross-correlation by multiple peaks or false
peak errors.
Author's personal copy
66
Z. Sun et al. / Atmospheric Research 129–130 (2013) 58–66
Fig. B.1. The parabolic interpolation in cross-correlation function.
Appendix B. The non-integer time delay estimation
Analyzing the characteristics of correlation function
waveform, it is known that the auto-correlation functions of
periodic signals are still periodic with the same period, which
can be written in cosine form; while that of broad-band
random signals attenuate rapidly, and can be written in sinc
form. Sine Taylor-series expansions of both cosine and sinc
functions can be represented by the parabolic form, then the
parabolic interpolation can be regarded to be a proper
method to get the non-integer time delay estimation. This
method is schematically shown in Fig. B.1.
Assuming that the cross-correlation function around the
peak can be expressed as a parabola function, the peak of
cross-correlation is R(τ), and values on the left and right sides
of the peak respectively are R(τ − 1), R(τ + 1), respectively. The
center location of the parabola function, and the actual time
delay estimation including both integer and non-integer, can
be expressed as:
τ^ ¼ τ þ
Rðτ−1Þ þ Rðτ þ 1Þ
:
2ðRðτ−1Þ−2RðτÞ þ Rðτ þ 1ÞÞ
ðB:1Þ
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