Estimation of Exoplanetary Planet-to-Star Radius Ratio with Homomorphic Processing
Rodrigo Mahu, Patricio Rojo, Ali Dehghan Firoozabadi, Ismael Soto, Elyar Sedaghati, Nestor Becerra Yoma
aa r X i v : . [ a s t r o - ph . I M ] O c t Estimation of Exoplanetary Planet-to-Star Radius Ratio withHomomorphic Processing
Rodrigo Mahu a , Patricio Rojo b , Ali Dehghan Firoozabadi c , Ismael Soto c , ElyarSedaghati d , Nestor Becerra Yoma a, ∗ a Department of Electrical Engineering, Universidad de Chile, Av. Tupper 2007, Santiago, Chile b Departamento de Astronomia, Universidad de Chile, Camino el Observatorio 1515, Las Condes,Santiago, Chile c Electrical Engineering Dept., Universidad de Santiago de Chile, Ave. Ecuador 3519, Zip Code:9170124, Santiago, Chile d European Southern Observatory, Alonso de Cordova 3107, Santiago, Chile
Abstract
In this paper a homomorphic filtering scheme is proposed to improve the estimationof the planet/star radius ratio in astronomical transit signals. The idea is to reduce theeffect of the short-term earth atmosphere variations. A two-step method is presentedto compute the parameters of the transit curve from both the unfiltered and filtereddata. A Monte Carlo analysis is performed by using correlated and uncorrelated noiseto determine the parameters of the proposed FFT filter. The method is tested withobservations of WASP-19b and WASP-17b obtained with the FORS2 instrument at theVery Large Telescope (VLT). The multi parametric fitting and the associated errors areobtained with the JKTEBOP software. The results with the white light of the exo-planetdata mentioned above suggest that the homomorphic filtering can lead to substantialrelative reductions in the error bars as high as 45.5% and 76.9%, respectively. Theachieved reductions in the averaged error bars per channel were 48.4% with WASP-19band 63.6% with WASP-17b. Open source MATLAB code to run the method proposedhere can be downloaded from . This code was used to obtain theresults presented in this paper.
Keywords:
Exoplanet atmosphere detection, Homomorphic Processing, FFT filtering
1. Introduction
Astronomical observations have greatly benefited from the introduction of digitaldetectors. The migration from analogue photographic-plate detectors to digital array ofpixels such as the classic charge-coupled-devices (CCDs) happened towards the end of theprevious century. The much enhanced sensitivity and robustness of the digital devicespermitted giant leaps in observational astronomy. However, data analysis in astronomymay still greatly benefit from signal processing techniques. ∗ Corresponding author
Email address: [email protected] (Nestor Becerra Yoma)
Preprint submitted to Astronomy and Computing October 3, 2017 he first exoplanet around a main sequence star was detected two decades ago(Mayor and Queloz, 1995). In the short period since that discovery, the field of exo-planetary research has evolved beyond detection, where atmospheric and interior char-acterization of these alien worlds have been made. A multi-wavelength time-series ob-servation of an exoplanetary transit with sufficient baseline allows for the measurementof the transmission signature of stellar light’s tangential path along the planetary ter-minator; likewise, an equivalent observation of an exoplanetary occultation facilitatesthe measurement of the emission spectrum from planet’s dayside. For both complemen-tary methods, the depth of the feature in the lightcurve is the parameter that, afteradequate modeling, relates to the exoplanetary atmospheric properties like composition,temperature-pressure profiles e.g. (Madhusudhan and Seager, 2009; Fortney et al., 2010;Iro and Deming, 2010); in the case of transits, the depth is related to the planet-to-starradius ratio.Given the extra distortion by the Earth’s atmosphere (telluric effects), early exoat-mosphere characterization was possible only through space-based instrumentation, e.g.(Charbonneau et al., 2002; Vidal-Madjar et al., 2004, 2003) by detecting wavelength-dependent variations of transit depth in a time-series spectral observations as the planetcrosses in front of the host star. More troublesome became securing the first ground-basedobservations; after several null results e.g. (Richardson et al., 2003; Rojo, 2006) the firstbona-fide detections of exoatmospheres were only possible in 2008 (Redfield et al., 2008;Snellen et al., 2008). Those first detections observed the single target star correcting thetelluric effects by out-of-transit baselines, an approach that later kept yielding detectionof new atomic species as in (Astudillo-Defru and Rojo, 2013).However, simultaneously observing one or more reference stars together with the exo-planetary system has been the favored method that, complementing space-based mea-surements, have shown the large diversity of exoplanetary atmospheres (Bean et al., 2011;Mancini et al., 2014; Jord´an et al., 2013, among others). Other ground-based methodswith mixed success involve searching for the planetary spectral lines as they Doppler-wobble with respect to the telluric lines (Barnes et al., 2010; Rodler et al., 2012, 2013;Cubillos et al., 2011).In this paper, we will study the effect on the measurement of radius-ratio in anexoplanetary lightcurve after a filtered homomorphic process. Section 2 presents themodel, Section 3 applies the model to a real exoplanetary dataset for validation, Section4 presents the results, and Section 5 presents the concluding remarks.
2. Homomorphic Processing
The main motivation of homomorphic processing is to separate signals that have beencombined not additively but instead through, for instance, convolution or multiplication.This is achieved by mapping this original system onto a different space. For instance, ho-momorphic deconvolution and cepstral (frequency analysis in logarithmic space) analysishave been applied successfully in a variety of areas including speech and audio pro-cessing, geophysics, radar, medical imaging, and others (Oppenheim and Schafer, 2004;Oppenheim, 1969; Sreenivasan et al., 2015, among others).In this paper we apply homomorphic processing to separate the flux from the star ofthe target exoplanetary system and a simultaneously-observed flux from a reference starin the same field of view, and later filter out some of the telluric-induced high frequency2 igure 1: Block diagram of the proposed homomorphic filtering technique to improve the estimationaccuracy of the radius-ratio components from the data. By doing so, we seek to improve the accuracy of the exoplanetradius-ratio estimation. Figure 1 shows the proposed homomorphic filtering scheme.
The observed flux from the star with the transiting planet ( F T ) for each λ (wave-length) is modeled as follows: F T ( λ, t ) = F T O ( λ, t ) × A E ( λ, t ) + n T ( λ, t ) (1)and the corresponding observed flux for the reference ( F R ) star can expressed as: F R ( λ, t ) = F RO ( λ ) × A E ( λ, t ) + n R ( λ, t ) (2)where F T O ( λ, t ) and F RO ( λ ) denote the original flux without the telluric effects for thetarget and reference star, respectively; A E ( λ, t ) corresponds to the telluric atmosphereresponse; and, n T ( λ, t ) and n R ( λ, t ) indicate the observation noise for the target andreference stars. Also, the original flux from the reference star, F RO ( λ ), is independent oftime.The transit light curve for a given λ , F T O ( λ, t ), is described by the Mandel & Agolmodel (Mandel and Agol, 2002). The flux received from the target star depends on theradiated energy, which is a function of λ , and on the exoatmosphere absorption duringthe planetary transit that also depends on λ . In the case of the reference star we canassume that there is no time dependency.The telluric effect on the flux received from the target and reference stars, A E ( λ, t ),is the standard attenuation term ( ∝ e − τ ). This component is wavelength and timedependent as the atmosphere changes. Atmosphere perturbations can be classified aslong- or short-term. The air mass changes slowly in a few-hour observation window asthe star moves in the sky. In contrast, parameters such as density, pressure and amountof water may present rapid fluctuations on timescales of 10 or 15 minutes.The additive observation noise is composed of a white and a red noise component.Both types of noise present high frequency energies, although they are more significantin the white noise. 3 .2. Mapping We study homomorphic mapping into logarithmic space by first applying the loga-rithm operator to the observed light fluxes in (1) and (2) to transform the multiplicationto addition: log [ F T ( λ, t )] = log [ F T O ( λ, t )] + log [ A E ( λ, t )]+ log (cid:20) n T ( λ, t ) F T O ( λ, t ) A E ( λ, t ) (cid:21) (3)log [ F R ( λ, t )] = log [ F RO ( λ )] + log [ A E ( λ, t )]+ log (cid:20) n R ( λ, t ) F RO ( λ ) A E ( λ, t ) (cid:21) (4)By considering that F T O ( λ, t ) A E ( λ, t ) ≫ n T ( λ, t ) and F RO ( λ ) A E ( λ, t ) ≫ n R ( λ, t ),we can make use of the approximation log(1 + x ) ≈ x , if x ≪
1. The normalizationof F T ( λ, t ) with respect to F R ( λ, t ), i.e. the subtraction of (4) from (3), can then beexpressed as: log (cid:20) F T ( λ, t ) F R ( λ, t ) (cid:21) = log [ F T O ( λ, t )] − log [ F RO ( λ )]+ n T ( λ, t ) F T O ( λ, t ) A E ( λ, t ) − n R ( λ, t ) F RO ( λ ) A E ( λ, t ) (5)where the component log [ A E ( λ, t )] in (3) and (4) are canceled out since the target andthe reference star’s angular separation on the sky is negligible. The flux ratio (5) is thus processed with a low-pass Fast Fourier Transform (FFT)filter to suppress or reduce the high frequency components of the noise terms. Basically,the main idea is to reduce the distortion caused by the short term variations of the telluricresponse and of the observation noise. If the attenuated components are faster than thetransit time, then a more accurate estimation of the radius ratio could be achieved.The low-pass filter is applied in the frequency domain to prevent the addition ofdelay and phase distortion of the frequency components. The resulting filter is non-causal and requires processing the whole signal at once. This is achieved by multiplyingthe FFT spectrum with a frequency mask that attenuates all the components above a cutoff frequency. Figure 2 shows the low-pass filter mask defined by the cut-off frequency(COF) and the rejection band gain (RBG) in dB. Then, the inverse FFT is employed tocome back to the time domain. Finally, after low-pass filtering the log of the normalizedtransit flux curve, the inverse logarithm operator is applied.4 igure 2: FFT Low-pass filter mask defined by the cut-off frequency (COF) and the rejection band gain(RBG).
We follow the standard prescription of Mandel and Agol (2002) to find the best-fitparameters to the data. Since for characterizing extrasolar atmospheres, it is typicallyrequired to have a precision better than 10 − of the incoming flux. It is important toverify that new methods do not add unwanted systematic effects that could offset themeasurement. In fact, our early tests did show that a low-pass filtering distorts the shapeof the lightcurve for the estimation of some of the parameters (most greatly for the sumof radii and the inclination).Therefore, in order to counteract the above-mentioned distortion induced by thehomomorphic low-pass filter, and since the most relevant parameter for exoatmosphericmeasurements is the radius-ratio, we develop a two-step method (Fig. 3) that only triesto fit this parameter, while minimizing errors associated to possible degeneracies withother parameters. First, the observed transit curve is fitted with the Mandel & Agolmodel. In this paper this is achieved by using JKTEBOP (Southworth, 2013). As aresult the fitted parameters are estimated and the radius ratio coefficient is discarded.Second, the homomorphic low-filtering is applied and the resulting curve is also fittedwith the Mandel & Agol model. However, in contrast to the previous step, the Mandel& Agol fitting makes use of all the parameters previously estimated except radius ratio,which is now determined (Table 1).
3. Validation
We evaluated the proposed method with the transit time series corresponding toplanets WASP-19b (Sedaghati et al., 2015) and WASP-17b (Sedaghati et al., 2016). TheWASP-19b and WASP-17b data were obtained on the nights of Nov 16, 2014, and Jun18, 2015, respectively, using FORS2 at the VLT with Grism 600RI. In the case of WASP-19b, 30” ×
10” slits were used on six reference stars, in addition to WASP-19, and the5 igure 3: Diagram for the two-step method of estimation.Table 1: Parameters from the Mandel and Agol (2002) prescription that are fit during the two-stepmethod
Parameter Fit to curveName unfiltered filteredradius ratioradii suminclinationlinear limb darkeningquadratic limb darkeningperiodmid transit 6 ime (Days) R e l a t i v e F l u x WASP-19b
Time (Days) R e l a t i v e F l u x WASP-17b
Figure 4: White light data from WASP-19 (top) and WASP-17 (bottom). The curves were obtained bysumming all the spectral channels from 535nm to 837nm and from 570nm to 790nm for WASP-19b andWASP-17b, respectively. Finally, the curves were normalized by the flux average of the reference star. exposure time was 30s. For WASP-17b, 30” ×
15” slits were employed on five referencestars, in addition to WASP-17, and the exposure time was 35s (Fig. 4).
To estimate the parameters of the proposed homomorphic FFT filter, we generated aset of artificial 1000 transit noisy signals for each planet considered here, i.e. WASP-19band WASP-17b. This dataset was produced by adopting the following procedure: first,a synthetic transit curve was obtained for each planet with the Mandel & Agol model bymaking use of the estimated transit parameters of WASP-19b and WASP-17b accordingto (Sedaghati et al., 2015) and (Sedaghati et al., 2016), respectively; then, white noiseand red noise were added to the synthetic transit curves. The red noise used here presentsa spectral density distribution that decreases at a rate of f − (Carter and Winn, 2009).The red and white noise power were the same. The resulting SNR after the additionof both types of noise was equal to the SNR of the white light curves shown in Figure4. Figure 5 depicts example of the noisy transit curves generated for WASP-19b andWASP-17b. The grid search described in Subsection 3.3 was employed to estimate thehomomorphic FFT filter parameters, i.e. COF and RBG.Figure 6 shows the Monte Carlo-based estimation method. The 1000 noisy signalswere filtered with the homomorphic low-pass filter. Then JKTEBOP was used to es-7 ime (Days) R e l a t i v e F l u x WASP-19b
CleanNoisy
Time (Days) R e l a t i v e F l u x WASP-17b
CleanNoisy
Figure 5: Examples of artificially generated noisy transit curves employed in the Monte Carlo simulations:WASP-19b (top) and WASP-17b (bottom). igure 6: Monte Carlo simulation to estimate the parameters of the homomorphic FFT filter. timate the transit parameters for the unfiltered and filtered signals. After estimatingthe parameters with the original 1000 unfiltered and filtered signals, we calculated theaverage estimation of radius ratio and corresponding error bar considering an interval of68% of confidence that corresponds to one gaussian standard deviation.As can be seen in Figure 7, in the early test (i.e. fitting all parameters after thefilter), the homomorphic low-pass filter led to a reduction of 45.4% in the estimation oferror bar for the radius ratio with WASP-19b. However, the method is unable to findthe correct estimation for inclination and radii sum, and the error bar of the mid transitincreased, because of the distortion incorporated by the homomorphic low-pass filter inthe transit curve transition.As can be seen in Fig. 8, the early test with WASP-17b leads to a low error reductionof 1.69 %, which is much lower than the error reduction achieved with WASP-19b. Thisresult must be due to the fact that, according to Table 2, the white light curve of WASP-17b provides an SNR that is 77% higher than the one observed in the WASP-19b lightcurve when the noise signal is obtained as the residual of the JKTEBOP estimation. Incontrast, Table 2 also shows that the SNR averaged across all the channels is just 20%higher in WASP-17b than in WASP-19b. This apparent contradiction is explained asfollows. According to Table 4, the noise in WASP-17b is much less correlated betweenchannels than in WASP-19b. Consequently, when the residual noise signals are summedover all the channels, the resulting noise energy is much lower with WASP-17b thanwith WASP-19b. Coefficient Q in Table 4 is defined as the quotient of the energy of thesignal resulting from the summation of all the channel noises by the summation of theindividual channel noise energies as in Eq. 6 (see Table 3): Q = P t [ N oise ( t )] P λ P t [ N oise ( t, λ )] (6)where N oise ( t, λ ) denote sample t of the noise signal at channel λ and, N oise ( t ) = X λ N oise ( t, λ ) (7)Accordingly, WASP-17b delivers a Q value that is 3.5 times lower than the one obtainedwith WASP-19b. 9 adius Ratio N Unfiltered
Radius Ratio N Filtered
Figure 7: Histograms of the estimated radius-ratio for WASP-19b with the early test: unfiltered (top)and filtered (bottom) data. The red continuous lines, at the outer sides of the histogram, bound the68% confidence interval. The blue and dashed red lines, in the middle of the histogram, indicate thereference radius-ratio and the average estimation, respectively. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.) adius Ratio N Unfiltered
Radius Ratio N Filtered
Figure 8: Histograms of the estimated radius-ratio for WASP-17b with the early test: unfiltered (top)and filtered (bottom) data. The red continuous lines, at the outer sides of the histogram, bound the68% confidence interval. The blue and dashed red lines, in the middle of the histogram, indicate thereference radius-ratio and the average estimation, respectively. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.) adius Ratio N Unfiltered
Radius Ratio N Filtered
Figure 9: Histograms of the estimated radius ratio for WASP-19 with the proposed estimation method:unfiltered (top) and filtered (bottom) data. The red continuous lines, at the outer sides of the histogram,bound the 68% confidence interval. The blue and dashed red lines, in the middle of the histogram,indicate the reference radius-ratio and the average estimation, respectively. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)
Results are shown in Fig. 9 for the proposed two-step method. The reduction in theestimation error bar for WASP-19 was as high as 28.4% without additional distortion.Fig. 10 shows an improvement of 0.66% for the case of WASP-17.
The homomorphic FFT low-pass filter defined in Figs. 1 and 2 has two parametersthat need to be tuned, COF and RBG. These parameters were optimized by making useof a two-dimensional grid search methodology. In the case of WASP-19b, 153 samples,the 256 FFT was obtained and the optimal COF was searched in the integer interval[1, 127]. The optimal RBG is found within the set [10dB, 20dB, 30dB, 40dB, 50dB,60dB, 70dB, 80dB]. For each pair (COF, RBG) the Monte Carlo simulation, described inSubsection 3.2 , was performed and the optimal pair (COF, RBG) was determined withrespect to the lowest interval of 68% confidence that includes the correct radius ratio.Accordingly, the optimal filter parameters for WASP-19b are: for the early test, COFequal to five FFT samples and RBG equal to 10dB; and, for the proposed method, COFcorresponds to 14 FFT samples and RBG is equal to 30dB. The WASP-17b data contains477 samples and the 512 FFT was computed. To adopt a similar resolution to WASP-19b, the homomorphic FFT low-pass filter for WASP-17b was optimized by searching12 adius Ratio N Unfiltered
Radius Ratio N Filtered
Figure 10: Histograms of the estimated radius ratio for WASP-17 with the proposed estimation method:unfiltered (top) and filtered (bottom) data. The red continuous lines, at the outer sides of the histogram,bound the 68% confidence interval. The blue and dashed red lines, in the middle of the histogram,indicate the reference radius-ratio and the average estimation, respectively. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)Table 2: SNR in the white light curves and the SNR averaged over all the channels with WASP-17b andWASP-19b. The noise is estimated as the residual between the real signal and the JKTEBOP estimationof the transit curve.
Planet White light curve SNR averaged overName SNR all the channelsWASP-17b 147.99 74.75WASP-19b 83.45 62.3013 able 3: SNR at each channel with WASP-17b and WASP-19b. The noise is estimated as the residualbetween the real signal and the JKTEBOP estimation of the transit curve.
WASP 17b WASP 19bChannel SNR Channel SNR560.00 nm 57.02570.00 nm 12.25 570.00 nm 52.57580.00 nm 27.07 581.00 nm 53.21590.00 nm 63.05 590.00 nm 58.70600.00 nm 83.91 601.00 nm 61.17610.00 nm 94.11 610.00 nm 65.24620.00 nm 82.24 620.00 nm 67.52630.00 nm 79.32 630.00 nm 64.09640.00 nm 85.83 640.50 nm 60.31650.00 nm 95.70 650.00 nm 61.87660.00 nm 99.10 660.50 nm 62.94670.00 nm 111.27 670.00 nm 63.95680.00 nm 118.00 681.25 nm 67.56690.00 nm 99.97 690.00 nm 69.80700.00 nm 93.96 701.25 nm 64.27710.00 nm 98.89 711.25 nm 57.97720.00 nm 83.70 720.00 nm 57.44730.00 nm 81.65 731.25 nm 49.19740.00 nm 84.22 740.00 nm 59.50750.00 nm 60.62 748.00 nm 73.07760.00 nm 34.58 760.00 nm 45.15770.00 nm 34.16 768.00 nm 51.08780.00 nm 47.20 780.00 nm 74.46790.00 nm 48.41 790.00 nm 72.17800.00 nm 73.82810.00 nm 71.10820.00 nm 66.99
Table 4: Correlation coefficient between channel noises averaged across all the channels and metric Qaccording to (6) with WASP-17b and WASP-19b.
Comparison Planet NameCriteria WASP-17b WASP-19bAverage correlation coefficientof the noise between channels. 0.4521 0.7309Q, as defined in (6). 0.0431 0.151314
COF (index)
RBG (dB) R ad i u s R a t i o % Figure 11: Evaluation of the distortion introduced in the original clean transit curve by the proposedmethod. The error curves were obtained when radius-ratio was made equal to 0.1416. The filteredradius-ratios were estimated with JKTBOP. The optimal filter (COF=14 FFT samples and RBG=30dB) is indicated by the red lines.(For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)
COF in the interval [1, 255] and discarding one every two FFT samples. Accordingly,the optimal filter parameters for WASP-17b are: for the early test, COF equal to 159FFT samples and RBG equal to 40dB; and, for the proposed method, COF correspondsto 15 FFT samples and RBG is equal to 10dB.
Distortion analysis
The distortion introduced by the proposed method and its dependence on filter pa-rameters was evaluated by processing the clean transit curves with the scheme shownin Fig.1 by applying the optimal FFT low-pass filter with COF=14 FFT samples andRBG=30dB. The clean transit curves were generated by using the parameters fromWASP-19b and varying radius-ratio. The procedure was repeated with six radius-ratiosrepresentative of real exoatmospheric modulation along a typical spectrum: 0.136, 0.138,0.14, 0.1416, 0.142 & 0.144. Then, the error between the radius-ratios in the originalclean and filtered transit curves is computed. Figure 11 shows the error curve whenradius-ratio was made equal to 0.1416. As can be seen in Fig. 11, there is a wide rangeof values for COF and RBG, including the optimal ones, where the error is lower than10 − . The same behavior is obtained with the other five reference radius-ratios. Thisresult strongly suggests that the proposed homomorphic filtering does not introduce anysignificant distortion into the observed original transit curves
4. Results
We used JKTEBOP with the white light from WASP-19b and WASP-17b to com-pare the ordinary estimation of the transit curve parameters (baseline) with the one15 nfiltered Proposed Method R ad i u s R a t i o WASP-19b
Unfiltered Proposed Method R ad i u s R a t i o WASP-17b
Figure 12: Estimation of radius ratio on white light data: black, unfiltered; and, red, with the proposedmethod. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)
Wavelength
550 600 650 700 750 800 850 R ad i u s R a t i o Unfiltered Proposed
Wavelength
550 600 650 700 750 800 850 E rr o r B a r Figure 13: Estimations of radius-ratio with the corresponding confidence intervals by employing JKTE-BOP: black, unfiltered data; red, proposed homomorphic filtering method. The wavelength channels aredefined in (Sedaghati et al., 2015) for WASP-19b. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.) avelength
550 600 650 700 750 800 R ad i u s R a t i o Unfiltered Proposed
Wavelength
550 600 650 700 750 800 E rr o r B a r Figure 14: Estimations of radius-ratio with the corresponding confidence intervals by employing JKTE-BOP: black, unfiltered data; red, proposed homomorphic filtering method. The wavelength channels aredefined in (Sedaghati et al., 2016) for WASP-17b. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)
5. Conclusions
The homomorphic filtering technique proposed here is applied to improve the estima-tion of the radius ratio in astronomical transit signals. The main motivation is to reducethe interference from the short-term earth atmosphere variations. The parameters of thehomomorphic FFT filter are estimated with a Monte Carlo-based scheme. The proposedmethod was tested with the white light of WASP-19b and WASP-17b real data usingJKTEBOP, and was able to lead to dramatic reductions of 45.5% and 76.9% in the er-ror bars, respectively. Similar reductions in the averaged error bars per channel wereachieved with WASP-19b and WASP-17b: 48.4% and 63.6%, respectively. Moreover, thesensibility analysis shows that the distortion caused by the technique is lower than thetarget precision, i.e. 0.1%. Finally, the application of the homomorphic filtering schemeto additional exo-planet real data is proposed as a future research.
Acknowledgements
The research leading to these results was funded by CONICYT-ANILLO project ACT1120, project POSTDOC DICYT (No.041613DA POSTDOC), Universidad de Santiagode Chile (USACH).
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