Wave Propagation 2010 Part 19 ppt

Vianna2and Helio Takai3 1Federal University of Rio de Janeiro/Signal Processing Laboratory/COPPE-Poli 2Federal University of Rio de Janeiro/Physics Institute 3Brookhaven National Labarat

positional angle θ is formed by the union of elementary scatterer radar cross section (4 items) plus 6 cosine oscillations It is not difficult to see that every cosine functions are caused by the interference effect between the fields scattered by a pair of elementary scatterers forming the RCRO The number of this pairs can be found with the use binomial coefficient

N M

C =M ⎡⎣N M N− ⎤⎦ , where M is a number of values, N is a number elements in the combination In the case

when 4M = , 2 N = , we have C = So, the angular response function of the complex 24 6radar object considered will include 6 space harmonic functions as the interference result summarize how it follows from the expression (48) where the values d12=X1−X2;

13 1 3; 14 1 4; 23 2 3; 24 2 4; 34 3 4

d =X −X d =X −X d =X −X d =X −X d =X −X are the space diversity

of scattered elements for every interference pair The space harmonic function

cos 2

i k kd ik

σ σ θ corresponds to the definition that was done in (Kobak, 1975), (Tatarinov

et al, 2007) In accordance with this definition, the harmonic oscillation in the space having the type cos 2kdθ( ) is defined by the full phase ψ θ( )=2kdθ=(2 /π λ)2dθ, the derivative from which is the space frequency f SP=2 /d λ having the dimension Rad−1 The period

T = f =λ dhas the dimension Rad , which corresponds to this frequency

So, a full power distribution of the field, scattered by complex radar object, is an union of the interference pictures, which are formed by a collection of elementary two-points interferometers

Thus, we can write a scattered power random angular representation, depending on the positional angle, in the form

C C= is combinations number, M is a full number of RCRO elementary scatterers

It was demonstrated above that the electromagnetic field Stokes parameter S S0, 3 angular distribution at the scattering by two-point distributed object has the form

C C= is combinations number An amplitude of every space harmonics and initial

space phases of these harmonics will be stochastic values and the further analysis must be

statistical First of all we will find a theoretical form of scattered field Stokes parameter S3

angular distribution autocorrelation function As far as we would like to find the

autocorrelation function (not covariance function!), we must eliminate a random constant

∑ from the stochastic function S3( )θ for the guarantee of zero mean value

Taking into account that the value 3

1

M m m

C

S θ =∑ D kd θ η+Its autocorrelation function can be found as

2 1

Here amplitudes D and space initial phase ηof space harmonics are random values,

which can be characterized by two-dimensional probability distribution densityW2( D,η),

and Δ =θ θ1−θ2 We will suppose that random amplitudes and phases are independent

variables For this case two-dimensional probability distribution can be presented as two

one-dimensional distributions densities product

Let’s suppose also that random phase has the uniform probability distribution density on

the interval (−π π, ) i.e W( )η =1 / 2π A probability distribution density for the random

amplitude D can be preassigned, however for all cases it will be one-sided After the

integration we obtain the value of double integral in the form

1 0

where <D N > is the polarization distance mean value, which was found by the average

along the statistical ensemble of random values D N for all space harmonics having the

space frequency f SP N =2d N/λ Thus, we can write the theoretical form of scattered field

Stokes parameter angular distribution autocorrelation function in the form

an isolated space harmonic oscillation S N( )θ = D Ncos 2( kd Nθ η+ N) having random

amplitude D N and random initial space phase ηN, i.e

it is not difficult to see that the autocorrelation function of the Stokes parameter stochastic

realization is the union of individual autocorrelation functions of all space harmonics:

Let’s now to find a complex radar object averaged space spectra using the expressions (8) for

polarization-angular response autocorrelation function The power spectra for the case of

isolated space harmonic can be found as the Fourier transformation above the

power spectra lines is determined by polarization distance between polarization states of

two scatterers forming the radar object

The full space spectra of stochastic polarization-angular response, i.e Fourier

transformation of the autocorrelation function (53) is:

It is necessary to indicate here that a connection between scattered (diffracted) field

polarization parameters and polarization parameters distribution along a scattering

(diffracting) object in the form of Fourier transformation pair is established in the first time

However, this connection is correct for fourth statistical moments: scattered field intensity

correlations (include mutual intensity) and polarization proximity (distance) distribution

along a scattering (diffracting) object

In the conclusion we consider some results of scattered field polarization parameters

investigation at the scattering by random distributed object having a lot of scattering centers

– “bright” points It follows also both from theoretical and experimental investigations

results that polarization-angular response function of a RCRO in the form of the 3-rd Stokes

parameter angular dependence corresponds to a narrow-band random process The

experimental realization of this parameter has shown on the fig.7 The angular interval for

this dependence is ±200 The rotated caterpillar vehicle (the sizes 5,5x2,5x1,5 m) placed on

the distance 2 km was used as complex radar object The autocorrelation functions (ACF) of

this object response S3( )Δθ are shown on the fig.14 The ACF on the angular interval ±200concerning the direction to the object board is designated by dotted line and the ACF into the same interval in direction to the stern of the object is continue line The measurements in these directions allow us to take into account the difference in the radar object space spectra band at its observation in areas of perpendiculars to the board and to the stern of the object

On the fig 15 RDRO mean power spectra are shown Dotted line is corresponding to direction to the object board and continue line corresponds to object stern

Fig 14 Autocorrelation functions of RDRO Fig.15 Mean power space spectra of RDRO stochastic polarization-angular response

6 Conclusion

In the conclusion we can to indicate that in the Chapter proposed a new statistical theory of distributed object polarization speckles (coherent images) has been developed The use of fourth statistical moments and emergence principle allow us to find the answers for a series

of problems which are having the place at the electromagnetic waves coherent scattering by distributed (complex) radar objects

7 References

Proceedings of the IEEE (1965) Special issue Vol 53., No.8, (August 1965)

Ufimtsev, P (1963) A Method of Edge Waves in Physical Diffraction Theory, Soviet Radio Pub

House, Moscow, Russia

Proceedings of the IEEE (1989) Special issue Vol 77., No.5, (May 1965)

IEEE Transaction on Antennas and Propagation (1989) Special issue No.5, (May 1965)

Ostrovitjanov, R & Basalov F (1982) A Statistical Theory of Distributed Objects Radar, Radio

and Communication Pub House, Moscow, Russia

Shtager, E (1986) Waves Scattering by Complicated Radar Objects, Radio and Communication

Pub House, Moscow, Russia

Kell, R (1965) On the derivation of bistatic RCS from monostatic measurements Proceedings

of the IEEE, Vol 53, No 5, (May 1965), pp 983-988

Stratton, J & Chu, L (1939) Diffraction theory of electromagnetic waves Phys Rev., Vol 56,

pp 308-316

Tatarinov, V ; Tatarinov S & Ligthart L (2006) An Introduction to Radar Signals Polarization

Modern Theory (Vol 1 : Plane Electromagnetic Waves Polarization and its Transformations), Tomsk State University Publ House, ISBN 5-7511-1995-5, Tomsk,

Russia

Foreign Radioelectronics, No 4-5, (May 1994), pp 22-40, Russia

Steinberg, B (1989) Experimental localized radar cross section of aircraft Proceedings of the

IEEE, Vol 77, No 5, (May 1989), pp 663-669

Kobak, V (1975) Radar Reflectors, Soviet Radio Pub House, Moscow, Russia

Kanareikin, D.; Pavlov, N & Potekchin V (1966) Radar Signals Polarization, Soviet Radio

Pub House, Moscow, Russia

Pozdniak, S & Melititsky V (1974) An Introduction to Radio Waves Polarization Statistical

Theory, Soviet Radio Pub House, Moscow, Russia

Franson, M (1980) Optic of Speckles Nauka Pub House, Moscow, Russia

Peregudov, F & Tarasenko, F (2001) The Principles of Systems Analysis, Tomsk State

University Publ House, Tomsk, Russia

Azzam, R & Bashara, N (1977) The Ellipsometry and Polarized Light, North Holland Pub

House, New York-Toronto-London

Tatarinov, V ; Tatarinov, S & van Genderen P (2004) A Generalized Theory on Radar

Signals Polarization in Space, Frequency and Time Domains for Scattering by Random Complex Objects Report of IRCTR-S-004-04, Delft Technology University, the Netherlands

Born, M & Wolf, E (1959) Principles of Optics Pergamon Press, New-York-Toronto-London Potekchin, V & Tatarinov, V (1978) The Coherence Theory of Electromagnetic Fielg, Svjaz Pub

House, Moskow, Russia

Tatarinov, V ; Tatarinov S & Kozlov, A (2007) An Introduction to Radar Signals Polarization

Modern Theory (Vol 2: A Statistical Theory of Electromagnetic Field ), Tomsk State

University Publ House, ISBN 978-5-86889-476-3, Tomsk, Russia

1 Introduction

In the solar system, debris whose mass ranges from a few micrograms to kilograms are calledmeteoroids By penetrating into the atmosphere, a meteoroid gives rise to a meteor, whichvaporizes by sputtering, causing a bright and ionized trail that is able to scatter forward VeryHigh Frequency (VHF) electromagnetic waves This fact inspired the Radio Meteor Scatter(RMS) technique (McKinley, 1961) This technique has many advantages over other meteordetection methods (see Section 2.1): it works also during the day, regardless of weatherconditions, covers large areas at low cost, is able to detect small meteors (starting frommicrograms) and can acquire data continuously Not only meteors trails, but also many otheratmospheric phenomena can scatter VHF waves and may be detected, such as lightning ande-clouds

The principle of RMS detection consists in using analog TV stations, which are constantlyswitched on and broadcasting VHF radio waves, as transmitters of opportunity in order tobuild a passive bistatic radar system (Willis, 2008) The receiver station is positioned far awayfrom the transmitter, sufficiently to be bellow the horizon line, so that signal cannot be directlydetected as the ionosphere does not usually reflect electromagnetic waves in VHF range (30 -

300 MHz)(Damazio & Takai, 2004) The penetration of a meteor on Earth’s atmosphere createsthis ionized trail, which is able to produce the forward scattering of the radio waves and thescattered signals eventually reach the receiver station

Due to continuous acquisition, a great amount of data is generated (about 7.5 GB, each day)

In order to reduce the storage requirement, algorithms for online filtering are proposed in bothtime and frequency domains In time-domain the matched filter is applied, which is optimal

in the sense of the signal-to-noise ratio when the additive noise that corrupts the receivedsignal is white In frequency-domain, an analysis of the power spectrum is applied

The chapter is organized as it follows The next section presents the meteor characteristics,and briefly introduces the several detection techniques Section 3 describes the meteorradar detection and the experimental setup Section 4 shows the online triggering algorithmperformance for real data Finally, conclusions and perspectives are addressed in Section 5

Eric V C Leite1, Gustavo de O e Alves1, Jos´e M de Seixas1, Fernando Marroquim2, Cristina S Vianna2and Helio Takai3

1Federal University of Rio de Janeiro/Signal Processing Laboratory/COPPE-Poli

2Federal University of Rio de Janeiro/Physics Institute

3Brookhaven National Labaratory

1,2Brazil

3USA

Radar Meteor Detection: Concept, Data

Acquisition and Online Triggering

25

2 Meteors

Meteoroids are mostly debris in the Solar System The visible path of a meteoroid that entersEarth’s (or another body’s) atmosphere is called a meteor (see Fig.??) If a meteor reaches the

ground and survives impact, then it is called a meteorite Many meteors appearing seconds

or minutes apart are called a meteor shower The root word meteor comes from the Greek

μτωρoν, meaning ”high in the air” Very small meteoroids are known as micrometeoroids,

1g or less

Many of meteoroid characteristics can be determined as they pass through Earth’s atmospherefrom their trajectories, position, mass loss, deceleration, the light spectra, etc of the resultingmeteor Their effects on radio signals also give information, especially useful for daytimemeteor, cloudy days and full moon nights, which are otherwise very difficult to observe.From these trajectory measurements, meteoroids have been found to have many differentorbits, some clustering in streams often associated with a parent comet, others apparentlysporadic Debris from meteoroid streams may eventually be scattered into other orbits Thelight spectra, combined with trajectory and light curve measurements, have yielded variousmeteoroid compositions and densities Some meteoroids are fragments from extraterrestrialbodies These meteoroids are produced when these are hit by meteoroids and there is materialejected from these bodies

Most meteoroids are bound to the Sun in a variety of orbits and at various velocities Thefastest ones move at about 42 km/s with respect to the Sun since this is the escape velocityfor the solar system The Earth travels at about 30 km/s with respect to the Sun Thus, whenmeteoroids meet the Earth’s atmosphere head-on, the combined speed may reach about 72km/s

A meteor is the visible streak of light that occurs when a meteoroid enters the Earth’satmosphere Meteors typically occur in the mesosphere, and most range in altitude from 75 to

Fig 1 Debris left by a comet may enter on Earth’s atmosphere and give rise to a meteor

Radar Meteor Detection: Concept, Data Acquisition and Online Triggering 3

100 km Millions of meteors occur in the Earth’s atmosphere every day Most meteoroids thatcause meteors are about the size of a pebble They become visible in a range about 65 and 120

km above the Earth They disintegrate at altitudes of 50 to 95 km Most meteors are, however,observed at night as low light conditions allow fainter meteors to be observed

During the entry of a meteoroid or asteroid into the upper atmosphere, an ionization trail

is created, where the molecules in the upper atmosphere are ionized by the passage of themeteor (Int Meteor Org., 2010) Such ionization trails can last up to 45 minutes at a time.Small, sand-grain sized meteoroids are entering the atmosphere constantly, essentially everyfew seconds in any given region of the atmosphere, and thus ionization trails can be found inthe upper atmosphere more or less continuously

Radio waves are bounced off these trails Meteor radars can measure also atmospheric density,ozone density and winds at very high altitudes by measuring the decay rate and Doppler shift

of a meteor trail The great advantage of the meteor radar is that it takes data continuously, dayand night, without weather restrictions The visible light produced by a meteor may take onvarious hues, depending on the chemical composition of the meteoroid, and its speed throughthe atmosphere This is possible to determine all important meteor parameters such as time,position, brightness, light spectra and velocity Furthermore it is possible also to obtain lightcurves, meteor spectra and other special features.The radiant and velocity of a meteoroid yieldits heliocentric orbit This allows to associate meteoroid streams with parent comets Thedeceleration gives information regarding the composition of the meteoroids From statisticalsamples of meteor heights several distinct groups with different genetic origins have beendeduced

2.1 Meteor observation methods

There are many ways to observe meteors:

– Visual Meteor Observation - Monitoring meteor activity by the naked eye Least accurate

method but easy to carry out in special by amateur astronomers Large numbers ofobservations allow statistically significant results Visual observations are used to monitormajor meteor showers, sporadic activity and minor showers down to a zenithal hourlyrate (ZHR) of 2 The observer can count and estimate the meteor magnitude using a taperecorder for later to plot a frequency histogram The visual method is very limited sincethe observer cannot work during the day or cloudy nights Such an observation can bequite unreliable when the total meteor activity is high e.g more than 50 meteors per hour.The naked eye is able to detect meteors down to approximately +7mag under excellentcircumstances in the vicinity of the center of the field of view (absolute magnitude - mag - isthe stellar magnitude any meteor would have if placed in the observer’s zenith at a height

of 100 km A 5th magnitude meteor is on the limit of naked eye visibility The higher thepositive magnitude, the fainter the meteor, and the lower the positive or negative number,the brighter the meteor)

– Photographic Observations - The meteors are captured on a photographic film or

plate (Hirose & Tomita, 1950) The accuracy of the derived meteor coordinates is veryhigh Normal-lens photography is restricted to meteors brighter than about +1mag.Multiple-station photography allows the determination of precise meteoroid orbits.Photographic methods can hardly compete with video advanced techniques The effort to

be spent for the observation equipment is much lower than for video systems For thisreason photographic observations is widely used by amateur astronomers On the otherhand, the photograph methods allow to obtain very important meteor parameters: accurate

539Radar Meteor Detection: Concept, Data Acquisition and Online Triggering

position, height, velocity, etc The sensitivity of the films must be considered There is nowvery sensitive digital cameras with high resolution for affordable prices, which produce agreat impact to this technique This method is restricted also to clear nights.

– Video Observations - This technique uses a video camera coupled with an image

intensifier to record meteors (Guang-jie & Zhou-sheng, 2004) The positional accuracy isalmost as high as that of photographic observations and the faintest meteor magnitudes arecomparable to visual or telescopic observations depending on the used lens Meteor showeractivity as well as radiant positions can be determined Multiple-station video observationsallow the determination of meteoroid orbits

Advanced video techniques permit detection of meteors up to +8mag Video observation

is the youngest and one of the most advanced observing techniques for meteor detection.Professional astronomers started to use video equipment at the beginning of the seventies

of the last century Currently the major disadvantage is the considerable price of a videosystem

– Telescopic Observations - This comprises monitoring meteor activity by a telescope,

preferably binoculars This technique is used to determine radiant positions of majorand minor showers, to study meteors much fainter than those seen in visual observationsones, which may reach +11mag Although the narrower field, the measurements are moreprecise

– Radio Observations - Two main methods are used, forward scatter observations and radar

observations The first method is easy to carry out, but delivers only data on the generalmeteor activity The last is carried out by professional astronomers Meteor radiants andmeteoroid orbits can be determined Radar meteors as well as telescopic ones may be asfaint as +11mag

Radio meteor scatter is an ideal technique for observing meteors continuously, day and nightand even in cloudy days Meteor trails can reflect radio waves from distant transmittersback to Earth, so that when a meteor appears one can sometimes receive small portions ofbroadcasts from radio stations up to 2,000 km away from the observing site

The technique is strongly growing in popularity amongst meteor amateur astronomers In therecent years, some groups started automating the radio observations by monitoring the signalfrom the radio receiver with a computer and even in cloudy days (see Fig 2) Even for suchhigh performance, the interpretation of the observations is difficult A good understanding ofthe phenomenon is mandatory

3 Meteor radio detection

Measurements performed by Lovell in 1947 using radar technology of the time showed thatsome returned signals were from meteor trails This was the start of a technique known today

as RMS, which was intensely developed in the 50’s and 60’s Both experimental and theoreticalwork have been developed Today, radio meteor scatter can be easily implemented having inhands an antenna, a good radio receiver and a personal computer

There are two basic radar arrangements: backscattering and forward scattering Backscattering is the traditional radar, where the transmitting station is near the receiving antenna.Forward scattering is used when the transmitter is located far from the receiver Botharrangements are used in the detection of meteors Back scatter radar tends to be pulsed andforward scatter continuous wave (CW) Forward scatter radar shows an increase in sensitivity

Radar Meteor Detection: Concept, Data Acquisition and Online Triggering 5

Fig 2 RMS detection principle

for the detection of small trails when compared to a same power backscatter due to thedifferences in aspect ratios Forward scattering also avoids possible confusion of echos bythe ionosphere as discussed by (Matano et al., 1968)

One of the main challenges to estimate the signal return power and its duration lies in abetter understanding of the lower atmosphere chemical properties At higher altitudes wheremeteors produce ionization trails, 80 to 120 km, the return signal duration only depends onthe hot plasma diffusion rate At lower altitudes, electron attachment to molecular oxygenlimits the signal duration for their detection In addition, the shorter mean free path causesthe electron to scatter while radiating and therefore dampening the return power An energy

of 1eV electron will roughly undergo 109 collisions per second, or 10 collisions in a onewavelength at 100 MHz The formalism to evaluate both signal duration and reflected power

is well understood for meteor trails

A specular reflection from an electron cloud only happens when a minimum free electrondensity is reached This is known from plasma physics and is given by:

ν p=



n e e2

where n e , e and m eare the electron density, charge and mass, respectively, which takes a value

of ne=3.8×1013m −1 for f=55.24MHz (channel 2) and ne=5.6×1013m −1 for f=67.26MHz

(channel 4) Below this critical density the reflection is partial and decreases with decreasingelectron density A total reflection happens because the electron density is high enough so thatelectrons reradiate energy from its neighbors This happens in meteor trails that are usuallycalled to be in an overdense scattering regime The converse is the underdense, for which thedensity is lower and there is no re-radiation by electrons in the cloud Both regimes are well

541Radar Meteor Detection: Concept, Data Acquisition and Online Triggering

Fig 3 Experimental setup for radar signal reception.

known from the radio meteor scatter science Meteor ionization is produced at altitudes above

80 km where the atmosphere is rarefied and gases are from the meteor elements itself Thelifetime of the ionization trail produced by a meteor is a function of diffusion that cools the hottrail and recombination of electrons to the positive ions Because of the elevated temperaturethe ionization lasts typically from 0.2 to 0.5 seconds

The formalism to calculate the reflected power by a meteor trail is well developed both as amodel and numerical integration Models provide good means to understand the underlyingprocesses and for the case of meteors they have been perfected over decades to providereliable values for power at the receiver Development of these models is driven by theapplication known as meteor burst communication where the ionization trails are used tobounce VHF for distances over 2,000 km

3.1 Experimental setup

As an example, the setup for experimental data acquisition used here to quote theperformance of the online detection algorithms is shown in Fig 3 It includes a doubledipole ( in ”V”, inverted) antenna (Damazio & Takai, 2004) for a nearly vertical detection,

a computer controlled radio receiver tuned to video carrier of an analog TV channel and apersonal computer equipped with an off-board sound card, able to perform sampling rates

up to 96 kHz Due to continuous operation, a hard disk of high capacity is required

4 Online triggering

The continuous acquisition is an inherent characteristic to radar technique Acquiring datacontinuously means generating a great amount of data, which must be stored for a posterioranalysis, or processed online for the extraction of the relevant information Moreover, most ofthe data are from background noise events, which makes it difficult the detection of interestingevents due to the data volume If the online trigger is not implemented, the full data storagerequires a more complex storage system, which increases the final cost of the experimental

Radar Meteor Detection: Concept, Data Acquisition and Online Triggering 7

setup In other hand, if the data are processed online, only what is judged interesting will beretained, which translates into a significant reduction on the data volume to be stored

In order to obtain an efficient detection and classification of received signals, online algorithmsare designed in both time and frequency domains In time-domain, the matched filter isapplied In frequency-domain, an analysis of the cumulative spectral power is applied Thenext subsections provide a brief description of such techniques

4.1 Signal detection

When the RMS (Radio Meteor Scatter) technique is considered, the signal detection problemcan be formulated as the observation of a block of received data for decision among two

hypotheses (Shamugan & Breipohl, 1998): H0, also called the null hypothesis, which states

that only noise is present, and H1, also called the alternate hypothesis, which states that theblock contains meteor signal masked by additive noise In a simpler case, the signal to bedetected may be known a priori (deterministic signal detection), and samples are corrupted

by noise Due to natural randomness of the occurrence of meteor events, the signal generated

by them is considered as a stochastic process (Papoulis, 1965) Thus, from an observation

Y of the incoming signal, P(H i | y)with i=0, 1 represents the probability, given a particular

value Y=y, that H iis true The decision in favor of each hypothesis considers the largest

probability: if P(H1| y ) > P(H0| y)choose H1, or if P(H0| y ) > P(H1| y)choose H0:

The ratio at the left in Equation 5 is called the likelihood ratio, and the constant

P(H0)/P(H1) =γ is the decision threshold.

Due to noise interference and other practical issues, the detection system may commitmistakes The meteor signal detection system performs a binary detection, so that two types

of errors may occur:

– Type-I: Accept H1 when H0 is true (which means taking noise as a meteor signal andproduce a false alarm)

– Type-II: Accept H0when H1is true (which means to miss a target signal)

543Radar Meteor Detection: Concept, Data Acquisition and Online Triggering

The probability to commit a type-I error is called false alarm probability, denoted as P F, and

the probability of type-II error is called probability of a miss (P M) (Shamugan & Breipohl,

1998) In addition, we can define P D=1− P M, which is called the detection probability Thedecision threshold can be handled to produce acceptable values for both detection and falsealarm probabilities If the decision threshold is varied, the Receiver Operating Characteristics

(ROC) curve can be constructed (Fawcett, 2006) This means to plot P D versus P F Asthe signal-to-noise ratio (SNR) decreases, detection efficiency deteriorates, which translates

into ROC curves near the diagonal and for a given fixed P D, the false alarm probability

increases Therefore, the detection system can be designed by establishing the desired P D and minimizing P F, which is known as the Neyman-Parson detector (Trees - Part I, 2001).Another useful performance index is the sum-product (Anjos, 2006), which is defined as

SP= (P D+1− P F)

By maximizing the SP index, a balanced detection efficiency is achieved for both hypotheses

H0and H1

4.2 The matched filter

In the case the signal to be detected is known (deterministic), from Equation 5, considering

that the block of received data s[n]comprises N samples and the additive noise is white (Trees

Radar Meteor Detection: Concept, Data Acquisition and Online Triggering 9

x 10−6

Fig 4 Covariance matrix for the noise process (development set)

4.2.1 Noise whitening

If the additive noise is a colored noise, is desirable to apply a whitening filter (Whalen, 1995)

on a preprocessing phase When a given zero-mean signaly is said white, its samples are

uncorrelated and the corresponding variance is unitary (Hyv¨arinen, 2001) As a consequence,its covariance matrix equals de identity matrix:

It is possible to obtain a linear transformation that applied to a processy produces a new signal

processv that is white A common way to obtain the whitening transformation is through the

decomposition of the covariance matrix into its eigenvalues and eigenvectors (Hyv¨arinen,2001):

where E is the orthogonal matrix of eigenvectors and D is the diagonal matrix of the

eigenvalues The whitening matrix is then obtained through (Hyv¨arinen, 2001):

And the signal transformation

obtains the white signal processv.

The covariance matrix for the raw data (see Section 4.2.3 next) is shown in Fig 4 Thecovariance matrix exhibits crosstalks, which point out a deviation from a fully white noise

545Radar Meteor Detection: Concept, Data Acquisition and Online Triggering