2008 International Conference on Electronic Design December 1-3,2008, Penang, Malaysia
ApPLICATION OF COMPLEX WAVELETS IN RADAR SIGNAL PROCESSING
S.Thenappanl ProfM N Giriprasad2 Dr S.Vara~ajan3 TSreenivasuluReddl. .1 Research Scholar, JNTU, 2 JNTU, Pulivendla S V U College ofEngg., T,rupat"4 Research Scholar, S V U ,. MailID : [email protected]
Abstract
The present paper discusses the Complexwavelet based signal processing techniques for themeteor detection. Firstly the signal containing meteoris processed with the wavelet based noise levelestimation technique, which will improve SNR of thesignal and improves the probability of detection ofWeak Meteors. Secondly, Complex wavelets(constructing filterbanks based on Hilbert transformconcept) that allow detecting certain parametersrelated to different Doppler frequency components.When the parameters exceed a certain threshold, itindicates that the meteor is detected. Meteor DetectionFunction (MDF) is defined using the above statedtechniques. MDF gives the Meteor count, Time ofoccurrence, Meteor scan number. MDF is a uniquefunction and it is very useful in Astronomical studies.The advantages of this method are directly derivedfrom the limitations of the existing methods. A greatimprovement in the characteristics ofthe meteor eventsis observed when processed with complex waveletbased signal processing techniques. Experimentalresults showed an increase in the detection of meteorevents compared to the traditional method - whichuses the spectrum(frequency domain) data. No need ofintense observation on the data for threshold setting,the effect of 'Noise' and 'Interference' is also reducedto a great extent.
KEYWORDSMST Radar (Mesosphere,Stratosphere,Troposphere)Complex wavelets, Hilbert Transfonn,SNR,MeteorDetection Function (MDF), Head echo.
Radar is a high power pulse coded phase coherent VHFRadar, employing advanced signal and data processingtechniques. A major MST Radar facility intendedprimarily for studies on low latitude middleatmosphere has been established at Gadanki, India in1993. The operating frequency of the MST Radar is
53MHz with a peak power aperture product of 3xl01O
Wm2. Atmospheric radar signal means the signal
received by the Radar due to the back scatteringproperty of the atmospheric layers. Generally thereceived back scatter signals otherwise called as Radarreturns are very much associated with Gaussian noise.The noise dominates the signal as the distance betweenthe radar and target increases and this led to decreasein Signal to Noise Ratio.
Meteor detection meant identifying and separating thescans that contain meteors from the nonnal ones. Theoccurrence of the radar meteor echoes is a sporadicphenomenon. Meteor concept is a rando~
phenomenon. i.e., the day, month and year of theIroccurrence can be stated precisely but at whichparticular time in a day they occur can't be said. Eventhough it has been observed for several years, the 'Peakevent' pinpoint time can't be stated accurately. Only the'Day of Peak Event' (DPE) can be stated. The abovestatements strongly describe the Degree ofRandomness of the phenomenon. Different methodsare designed for meteor detection in the past. All ofthem have got their own limitations. In this paper, afundamental signal processing technique is used todetect meteors, which provide very precise detectioneven for very low signal-to-noise ratio (SNR) meteorreturn signals.
2. METEOR RETURN SIGNAL MODEL
Given the transmission of a T-JlS pulse, themeteor return signal is also T-Jls along with acorresponding Doppler frequency. The mathematicalmodel of the sampled, noise-free return signal as:
m[n] =Aexp{j(OJd + t/J)}~[n -1m ], n = 1,2. N1PP
______________(1)
where A is the amplitude,
OJd is Doppler frequency,
t/J is the phase,
1m is the location of the meteor,N1PP is the number of samples in one IPP,T is time period of the transmitted pulse anda[n] is expressed as
a[n] = u[n] - u[n-(T-1)] -------------- (2)
5
3.1 TRADITIONAL METHOD
Though this method appears to be simple in concept,the real difficulty lies in its execution. The two majordisadvantages are1. A huge amount of data has to be observed beforecoming to a conclusion about the Threshold value.Moreover, the extent to which observations has to bemade is Unknown. The major reason for this is "TheMeteor occurring is a random event".2. This method cannot detect the weak meteorsefficiently. The Complex wavelet based methodovercomes these limitations. Besides intensifying thedetection process this method also improves the meteorcharacteristics greatly.
3.2.1 Need for Complex Wavelets1,2
Complex Wavelets Transforms (CWT) usecomplex-valued filtering (analytic filter) thatdecomposes the real/complex signals into real andimaginary parts in transform domain. The real andimaginary coefficients are used to compute amplitudeand phase information, just the type of informationneeded to accurately describe the energy localization ofoscillating functions (wavelet basis). The proposedalgorithm uses wavelet based noise estimationtechnique to remove undesirable components andenhance the detection of signal (Meteor in our case)the explanation ofwhich is given below:
3.2 COMPLEX WAVELET BASED METEORDETECTION
3.2.1WAVELETS BASED NOISELEVELESTIMATION
In wavelet denoising the thresholdingmethods can be grouped into two categories, namelyglobal thresholds and level dependent threshold. Theformer means that we choose a single value forthreshold A to be applied globally to all empiricalwavelet coefficients while latter means that a possiblydifferent threshold value A( j ) is chosen for eachresolution level j. In what follows, we consider bothglobal and level dependent thresholds. Thesethresholds require an estimate of the noise level cr.Dohono and Johnstone considered estimating cr in thewavelet domain and suggested a robust estimate that isbased only on the empirical wavelet coefficients at thefinest resolution level5
• The reason for consideringonly finest level is that the corresponding empiricalwavelet coefficients tend to consist mostly of noise.The noise level cr (based on median absolute deviation)is given by
4.51.5 2 2.5 3 3.5Time(sec)
0.5
Figure l(b) Range bin without meteor
2
where u[n] is the unit step function.
In the Figure 1(b), the distribution of the amplitude in anon-meteor Range bin is shown. Here the differencebetween the peak and mean amplitude is very lowsuggesting no strange event has occurred whereas ithappened in the meteor case as shown in the Figurel(a).A threshold has to be set on observing the peaks ofsuch events. This is not a simple event as all the peaksare not of equal value. A high threshold can detectstrong meteors quite easily but weak meteors aretotally missed. Likewise, a low threshold may detectnon-meteor scans as meteor-scans besides the correctdetection of the meteor scans. One way to solve theproblem is to choose an optimum threshold value, butthis yields unsatisfactory results.
~j 0.015
The meteor detection technique based on theTraditional method is just setting a threshold onobserving some scans containing meteors.In this method the data from the back scattered signalis plotted for all scans.
3. METEOR DETECTION
Firstly, the scans containing the meteors are separatedmanually and an intense observation is made on thesemeteor scans. A sudden drastic jump in the amplitude(power) occurs in the range bin in which the meteor ispresent as shown in the Figure 1.
Figure 2. Simulated Radar signal (a) without noise,(b) with noise of -15 db SNR,
Fig.2 (c) After denoising using wavelet, (d) Denoisedsignalwith adaptive window, (e) Echo position.
Here w (k) is detail coefficients at the finest level. Ifwe have started with n sampled data, after first level offiltering, we will have n/2 detail coefficients8
• Noiselevel threshold (0) is calculated based on the equation3.
the Fig 2e. Therefore proposed method based onwavelet is very potential and efficient one.
Complex wavelets (constructing filterbanks based onHilbert transform3
,4 concept) that allow detectingcertain parameters related to different Dopplerfrequency components. When the parameters exceed acertain threshold, it indicates that the meteor isdetected.Gabor introduced the Hilbert transform into signaltheory, by defining a complex extension of a realsignal f(t) as:
x(t) = f(t) + j g(t) ------------------------ (4)where, g(t) is the Hilbert transform of f(t) and denoted
1/2
as H{f(t)} and j = (-1) .
The signal g(t) is the 90 shifted version of f(t) asshown in figure (3.1 a).The real part f(t) and imaginarypart g(t) of the analytic signal x(t) are also termed asthe 'Hardy Space' projections of original real signalf(t) in Hilbert space. Signal g(t) is orthogonal to f(t). Inthe time domain, g(t) can be represented as 1,6:
g(t)=H{f(t)}=.!. } J(t) dT= J(t)*~------(5)J( -00 t -r ret
This analytic extension provides the estimate ofinstantaneous frequency and amplitude of the givensignal x(t) as:
Magnitude ofx(t) = ~J(t)2 + g(t)2-1
Angle ofx(t) = tan [g(t)/ f(t) ] ---------(6)The same concept of analytic or quadratureformulation is applied to the filterbank structure ofstandard DWT to produce complex solutions and intum the COMPLEX WAVELET TRANSFORM.Figure 3 illustrates the process of meteor detectionusing complex wavelets. The description of each blockis as follows.
(e)0.25
0.20
0.15
0.00 -I---.A..l----l
0.10
o.as
.. ... ~ 0 2 4. ~ ~ 0 1 2
2.0
2.5
0.5
1.5
1.0
..... ·2024.
2.5
3.5
2.0
.. .. ·2 0 2 • 6
0.5
3.0
1.5
1.0
""",,1•• /..,,\(c) ,....--.-,..-------, (d)
(8) (b) 1.
1
12
105
-I:::I=•A.EC 3
A radar signal for one bin is simulated and isartificially corrupted with Gaussian noise of differentSNR's. As a case in a point, Figure 2a shows thespectrum of simulated radar signal. From that it is clearthat the Doppler is at -0.24 Hz. Fig 2b shows thespectrum of radar signal corrupted by noise with SNRof -15 db. Addition of noise has resulted in generationof multiple echoes around genuine echo. Fig 2c.shows spectrum of wavelet based noise level estimatedsignal after denoising. Fig 2d. shows spectrum ofadaptive window wavelet based noise level estimatedsignal. Within this window there is more than one echowhich resembles the Doppler. The genuine echo isfound on the basis of maximum slope7
,9 as shown in
3.2.2 Data converter and complex waveletProcessor.
Processing on raw data is very difficult. So, it is firstconverted into desired form and this is achieved withthe Data converter section. The output data from theConverter is in the complex form. Let it be x(t) and isgiven by the following equationx(t) = I(t) +j Q(t)I(t) = A cos(2 J( fdt)
Q(t) = A sin(2 J( fdt) -------------- ----(7)whereI(t) - In phase component ofx(t)Q(t) - Quadrature phase component of x(t)A - Amplitude of the received echo in volts
fd - Mean doppler frequency in HzThe ith component of the in-phase and quadrature phasecomponents can bewritten asx(i) = I(i) + jQ(i)I(i) = A cos(21t fd i~t)
Q(i) = A sin(2 1t fd i ~t ) ---------------(8)where i = 0, 1,2,....N-l~t is the sampling time.
RAWDATA
Complex wavelet Processor
Parameter Extractor
Threshold selector
OutputFigure 3: Flow Chart of Complex wavelet based
meteor detection
The complex wavelet processor consists of a set offilter banks, designed one each for real and imaginarypart of the complex input signal6
• The basic operationthat takes place in this processor is Signaldecomposition, Altering or manipulating the filtercoefficients obtained from the decomposition,Thresholding5 these coefficients (Soft thresholding)and Reconstructing the signal from those modifiedcoefficients. Each of these stages has got their ownimportance and none of them can be neglected. Thetwo basic thresholding techniques available in thisprocessor are Hard and Soft Thresholding. (as shownin figure 4)The functions for soft and hard thresholding aredefined by
{(II - }..,".""I)" lui ~ A
9$(u) :=0, lui < it.
lui::: Alui <A
------- (9)Here, A is an adequate threshold.
~ o·
Figure 4: Soft and Hard thresholding respectively
3.2.3 Parameter Extractor"SA set of parameters are defined from the results of'Before processing' and 'After processing'. Thedescriptions of the parameters are as follows.
1. Line of Control (LoC): It is created fromthe data corresponding to the height at whichthe maximum occurs. This is done on bothunprocessed data and the processed data.
2. PARR: It is created from the data obtainedfrom the real tree and hence it is named so. Itis defined below.
max(Real tree data)PARR =------
mean(Real b'ee data) (10)
3. PARI: It is created from the data obtainedfrom the Imaginary tree (see chapter 3) andhence it is named so. It is defined below.
max(lmagincu"y D'ee data)PARI =
mean(lmagincuy D'ee data)
----(11 )
4. POSR: It is created from the datacorresponding to the position (address) of thereal tree data elements.
5. POSI: It is created from the datacorresponding to the position (address) of theimaginary tree data elements.
These 'Parameters' along with the 'Thresholds' set forthese parameters in the later stage directly indicate themeasure of 'Accuracy of the Complex wavelet basedmethod'. 'LoC is very important of all theseparameters. It reduces 75% of the ambiguity in thethreshold selection process.
3.2.4 Threshold selector
A separate threshold has been selected for each of theparameters. Although threshold selection is inevitable,the degree of ambiguity is very less compared to thatof first method.
NARL Meteor MST2005-08-11-19-13-56E
6.1STRONG METEOR
6. RESULTS
TIME(sec)
TIME(sec)so
Before Processing - Strong Meteor
HEIGHT(km)
HEIGHT(km)
~ 0.8_·
fa 06
i 04-·
~
* POWER, is taken as a term proportional tothe square of the amplitude.
NARL Meteor MST2005-08-11-19-13-56E
Figure 5: Mesh plot of a scan giving threedimensional view of a strong meteor withTraditional and Complex wavelet basedmethod respectively
5.2 Advantages of complex wavelet based method
5. ADVANTAGES
4. METEOR DETECTION FUNCTION (MDF)
Limited redundancy with very goodproperties of shift-invarianceImproved directionality and availability of
phase information, which are not present instandard DWT.
The threshold selection is a two step process.1. Loe remains almost same for both the
'Unprocessed' and 'Processed data'(meteorcase). Hence there is no ambiguity in settingthreshold for this parameter. POSR, POS! arealso conditioned in this step itself.
2. PARR and PARI are conditioned in this step.As these two parameters are just ratios ofmaximum to mean, they posses large valuesfor the meteor cases.
5.1 Advantages of using complex waveletsThe Complex wavelets have
Meteor Detection Function (MDF) is definedusing the above stated techniques. MDF gives theMeteor count, Time of occurrence, Meteor scansnumber. MDF is a unique function and it is very usefulin Astronomical studies.
Sefor. proo•••jng
20% More meteor detection is reported.Weak meteors can be efficiently detected.No need of intense observation on the data forthreshold setting.The effect of 'Noise' and 'Interference' is alsoreduced to a greater extent.Improvement in Meteor Characteristics
5.3 Meteor Label
Upon detection, each meteor event is assigned a unique'Laber. The sequence in the label is Research center,Radar name, year, month, day number, hour, minute,seconds and beam direction.
'E'~ 105
~ 100
'E'~ 105
~100
:2TIME(eec)
Figure 6: The altitude/height (km) at which themeteor is detected and the meteor's 'Time ofPersistence' with Traditional and Complexwavelet based methods respectively. Themeteor occurred at a height of 101km.
Figure 8: Spectrum of the 'Meteor event' withTraditional and Complex wavelet methodsrespectively
0.5 2TIME(sec)
Figure 10: The altitude/height (km) at whichthe meteor is detected and the meteor's 'Time ofPersistence' with Traditional and Complexwavelet based methods respectively. Themeteor occurred at a height of 106km
NARL Meteor MST2005-08-12-06-14-41E
6.3 Meteor Echoes
NARL Meteor MST2005-08-12-01-37-17E
12001000400 600 800No. of 1ft points
200
-0.01
Figure 13: Underdense echo. The figure isplotted only for the Range bin in which themeteor occurred
NARL Meteor MST2005-08-12-01-34-13E
Overdense Echo
NARL Meteor MST2005-08-12-04-28-27E
Underdense Echo
Figure 14: Overdense echo. The figure isplotted only for the Range bin in which themeteor occurred.
-0.005
0.005
0.015.---------.--------.----.---~--,...---____.
0.01
I
85 90 95 100 105 110 115 120 125HEIGHT(km)
NARL Meteor MST2005-08-12-06-14-41E
Figure 11: The distribution ofPower Vsaltitude/height (kIn) of the meteor event.
Before proceesing
j 0.035
ii 0.03
~ 0.025
J 0.02
j o~o::
60 80F·requency(Hz)
Figure 12: Spectrum of the 'Meteor event'with Traditional and Complex waveletmethods respectively
110
1105
~100
2TIME(sec)
Figure 15: Two meteors are detected in asingle scan. The second one (Bunch likestructure) is because ofHead echo.
* Head echo is from the meteor itself. i.e.,spatially localized and moving along with themeteor.
1. Meteor phenomenon disobeys Gaussian distribution.(See 'Results').2. Frequency of the meteor signal varies at about andaround 4Hz.3. Maximum meteor trails are observed at about andaround 100km altitudes.4. Echoes with lesser duration are larger in numbercompared to long duration echoes Most of the echoesfrom the meteor trails last typically for less thanlOOms. i.e., meteor trail echo is a short livedphenomenon.5. Meteors start out as both 'overdense' and'Underdense' for VHF and UHF radars (MST Radars),
depending on the altitude, mass, velocity of the meteor.Most of the meteor (observed in this work) are'Underdense'.Using the proposed signal processing techniques themeteor detection and the interference removal candetect very weak meteor events. Experimental resultsshow that it is possible to detect about 200/0 moremeteor events compared to the traditional method.Also the whole meteor detection and interferenceremoval processes are done by automated fashion,which saves a lot ofprocessing time.
ACKNOWLEDGMENT
The authors gratefully acknowledge Prof. D.N.Rao,Director, NARL, for his encouragement and supportthrough out this work. The authors would like toexpress their sincere thanks to Dr. V.K.Anandan,NARL, for his valuable suggestions.
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[3] Donoho et. al Density Estimation for Waveletthresholding, Preprint, Dept. of Statistics,Stanford University 1993.
[4] Daubechies, I., Ten Lecturers on Wavelets,SIAM, Philadelphia, 1992.
[5] Donoho D.L., Denoising by soft thresholding.IEEE Transaction on Information Theory Vol.41
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