Acta IMEKO, Title ACTA IMEKO ISSN: 2221-870X December 2017, Volume 6, Number 4, 54-60 ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |54 Estimation of the signal component from random equivalent and non-coherent sampling measurements Dušan Agrež University of Ljubljana, Faculty of Electrical Engineering, Trzaska 25, 1000 Ljubljana, Slovenia Section: RESEARCH PAPER Keywords: random equivalent time sampling; non-coherent condition; signal component; interpolated DFT; parametric estimation Citation: Dušan Agrež, Estimation of signal component from random equivalent and non-coherent sampling measurements, Acta IMEKO, vol. 6, no. 4, article 8, December 2017, identifier: IMEKO-ACTA-06 (2017)-04-08 Section Editor: Alexandru Salceanu, Technical University of Iasi, Romania Received May 5, 2017; In final form June 27, 2017; Published December, 2017 Copyright: © 2017 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Corresponding author: Dušan Agrež, e-mail: dusan.agrez@fe.uni-lj.si 1. INTRODUCTION High-speed digitizing of waveforms plays an important part in broadband measurements and one of the possibilities to increase the sampling frequency is the equivalent time sampling with reconstruction of the signal in the prolongation of the acquisition time [1]. In many applications of practical interest, we often wish to estimate signal parameters and reconstruct a signal from incomplete time samples, which can be attained non-uniformly and randomly as in the case of random equivalent sampling [2], [3]. Unlike the uniform sampling case, where the aliases are simply periodic replicas of the signal’s original spectrum when the sampling theorem is fulfilled, random sampling theory shows that sparsing and randomization of the sampling intervals shape the aliases into a noise floor in the sampled spectrum [4]. By sampling signals non-uniformly, the average sample rate can be more than a magnitude lower than the Nyquist rate, provided that these signals have a relatively low information content as measured by the sparsity of their spectrum. Random equivalent time sampling is a technique which is used for achieving high sample rates with reduction of the equivalent sampling interval eqS,t . It is applicable only for periodic signals. It is usually used in digital oscilloscopes and some other high-speed sampling applications [5], [6]. The general idea behind equivalent time sampling is shown in Figures 1 to 6. Figure 1 shows one sinusoidal signal component g, which can be generally expressed as ( ) ( )∑ −= +⋅ 1 0 eqS,eqS, 2sin M m mmm tnfAtnw ϕπ ( mf , mA , and mϕ are frequency, amplitude, and phase of the particular component). The signal is presented by randomly sampled points with low ‘duty ratio’ NND rsrs = , where rsN is the number of random samples from a total of N samples in the non-coherent measurement interval .eqS,M NtT = The basic raster of N samples is equidistant with the sampling interval eqS,t [7]. The whole acquired signal contains 3.8 cycles of the sine 8.3eqS, =⋅=∆= tNfff mmmq on 512=N points and samples are captured randomly to the triggering point (signal ABSTRACT Estimations of the signal component parameters in the case of random equivalent time sampling and under non-coherent sampling condition comprise two main error contributions: spectral leakage effect due to non-coherency and additional noise due to the randomization of sampling intervals. In the estimation procedure the non-parametric interpolated DFT approach has to be used first to estimate the component frequency and, after that, an iterative 4-parametric sine-fit algorithm should be used for other component parameters (amplitude and phase). Their estimations are possible when the duty ratio of random samples from the total samples in the non-coherent measurement interval is above 0.1. With these duty ratios of random samples it is possible to achieve error levels of 0.001 bins of the frequency estimations in relation to the estimation on full ensemble of points. ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |55 zero crossing with phase deg0 ) in two sweeps with fix sampling raster eqS,S 26 tt = . In Figure 1 the samples of the first sweep are represented by solid lines and the samples of the second sweep by dotted lines). In this figure the sampling theorem is fulfilled with a relative sampling frequency of mSeqS,SS 27.1926512 qq ⋅>===∆= ttNff . Although the samples are randomly captured (essentially the first sample in one sweep) the systematic relation between the signal frequency and sampling frequency is clearly visible in the spectrum of the equivalent sampled signal (Figure 2 - line b) where we have well-known replicas of signal components with frequency mq around integer values of the sampling frequency mk qqq ±=′ S ( ...,2,1=k ). With increasing the number of sweeps the duty ratio approaches 1rs →D and the component rises from the noise floor and replicas diminish (Figure 2 - line a). If the Analog-to-Digital conversion is relatively slow compared to the signal period, the sampling raster relatively increases and the sampling theorem could not be fulfilled (see Figure 3: →= eqS,S 85 tt mSeqS,S 20.685512 qq ⋅<=== ttN ). To fill the empty points by signal values more sweeps are needed for the same duty ratio (in Figure 3: six sweeps are needed for 07.0rs ≈D ) but the systematic relation between sampling frequency and component frequency is still recognized (Figure 4 - line b). If one reduces the number of samples of one sweep to 1, what is the limit case, we get ‘pure’ random sampling (Figure 5: →= eqS,S 512 tt mS 21512512 qq ⋅<<== ). Due to relative sampling frequency of 1S =q all amplitude DFT coefficients are contaminated by a systematic sampling effect mk qq ±=′ ( ...,2,1=k ) and beside noise of the acquisition signal we have also the ‘noise’ of random sampling (Figure 6 - line b). This fact has to be considered in estimations of the signal components. Estimations of the component signal parameters are distorted due to the missing samples in the time domain [8] or due to large random sampling noise in the frequency domain (Figure 6 - line b) where SNR for the investigated component is very low when rsD is low. Considering also the non-coherent sampling condition with leakage effect, where the measurement interval MT is not an integer multiple of the signal period mmmm iTT δq +== M , parameter estimations are not an easy task [9], [10]. The leakage signal energy additionally spreads Figure 1. Signals of random sampling: a–original signal, b–random sampled signal; 512=N , 7.19S =q , 8.3=mq , 1=A , deg0=ϕ , 07.0rs ≈D . Figure 3. Signals of random sampling: a – original signal, b – random sampled signal; 512=N , 0.6S =q , 8.3=mq , 1=A , deg0=ϕ , 07.0rs ≈D . Figure 4. Spectra of signals from Figure 3: a – original signal, b – random sampled signal, c – reconstructed signal; 0.6S =q , 8.3=mq , 07.0rs ≈D . Figure 5. Signals of random sampling: a – original signal, b – random sampled signal; 512=N , 1S =q , 8.3=mq , 1=A , deg0=ϕ , 07.0rs ≈D . Figure 2. Spectra of signals from Figure 1: a – original signal, b – random sampled signal, c – reconstructed signal; 7.19S =q , 8.3=mq , 07.0rs ≈D . ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |56 along the frequency axis and disturbs the DFT coefficients (Figure 6) [11], which cannot be removed as in the case of compressive sensing with the coherent sampling model [12], [13]. 2. ANALYSIS FOR PARAMETERS’ ESTIMATIONS Accurate measurement of a multi-sine waveform is a classic problem in spectral analysis. It is well known that non- parametric algorithms based on the discrete Fourier transform (DFT) have to contend with spectral leakage, which affects both amplitude estimation accuracy and frequency resolution [14]. These methods usually estimate the signal component frequency as key parameter by searching for the maximum in the spectrum by using DFT [15]. Advanced windowed functions and interpolation schemes have been applied to compensate for the effects of leakage caused by incoherent sampling and finite frequency resolution problems [16], respectively. It is also common knowledge that approaches based on a parametric signal model can achieve much better frequency resolution, but this is obtained at the price of greater complexity [17]. For estimations of the signal parameters from sparse sampled periodic signals two methods are investigated and compared: the non-parametric interpolated DFT and the sine- fit parametric approach [18]. In the first approach, one can choose the number of DFT coefficients in interpolation and the window function w(*) [16]. In the second approach, a number of parameters has to be defined: non-iterative 3- parametric sine-fit estimation when the component frequency is known or 4-parametric estimation when also the frequency has to be estimated in the iterative procedure [17]. In the first approach typically two or three largest DFT coefficients )1( −miG , )( miG , )1( +miG and Rife-Vincent windows class I with suitable order P are used [16]: - Two-point frequency estimation where s is a sign of displacement mδ : ( ) ( ) ( ) ( ) ( )siGiG iGPsiGP sii mm mm mmmm ++ ⋅−+⋅+ +=+= 1 2 δq (1) - Three-point frequency estimation: ( ) ( ) ( ) ( ) ( ) ( )121 11 13 +++− −−+ ⋅++=+= mmm mm mmmm iGiGiG iGiG Pii δq (2) - One-point amplitude estimation: ( ) ( ) ( ) ( )mPl m m m P m iGlP A ∏ = −⋅⋅= 1 22 2 1 πsin π !2 2 2 δ δ δ  (3) - Three-point amplitude estimation: ( ) ( ) ( ) ( ) ( ) ( )( )121 πsin π !22 2 2 1 1 22 2 3 +++−⋅ −⋅⋅ + = ∏ + = mmm P l m m m P m iGiGiG l P A δ δ δ  (4) - Three-point phase estimation with the arguments of the three largest local DFT coefficients ( )[ ]mi iGm arg=ϕ : ( ) ( ) 2 π 3 2 6 141 11 3 +− +++− = +− mimiimm a mmm δϕδϕϕδ ϕ (5) Since the sparse sampling considerably increases the noise in the frequency domain (Figure 6) it is not suitable to increase the number of interpolation DFT coefficients in interpolations due to their distortions. Figure 7 shows the mean values of the absolute relative errors of the amplitude DFT coefficients ( ( )miGe rs , ( )1rs +miGe , and ( )1rs −miGe ) around the local peak in relation to the original spectrum on 512=N samples normalized by the duty ratio ( ) ( ) rs * DiGiG ⋅= in (6) (see Figures 2,4,6 - line c). ( )( ) ( ) ( ) ( )* * rs rs iG iGiG iGe − = (6) Analyses of Figure 7 show that it is better to use two-point estimation for frequency to determine the position of the measurement component mmm i−=qδ between DFT Figure 6. Spectra of signals from Figure 5: a – original signal, b – random sampled signal, c – reconstructed signal; 1S =q , 8.3=mq , 07.0rs ≈D . I) 8.3=q , 2.0−=mδ . II) 4.4=q , 4.0=mδ . Figure 7. Mean values of the absolute relative errors of the amplitude DFT coefficients; a – ( )iGe rs , b – ( )1rs +iGe , c – ( )1rs −iGe , 1=A , deg0=ϕ , 100repeat =N trials. ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |57 coefficients )( miG and )1( +miG surrounding the component m as the key parameter, due specific distortion in sparse sampling. This conclusion can be supported also by analyzing the error characteristics using two and three- point estimations (Figure 8) [16]. Figure 8 shows collected minimal curves of the maximal bias errors of the frequency estimations due to phase changing with the two-point (IDFT-2p) and three-point (IDFT-3p) interpolations of the DFT using RV-I windows and the corresponding orders 6...1=P with which these minimal values have been obtained. It can be seen that bias errors are on the same level with two (1) or three-point (2) estimations but window order P has to be increased with two-point estimation. Whenever there is no information of frequency content of the signal, the component frequency estimation is the first step and the key to estimate other parameters (amplitude, phase, and sometimes also damping). This is valid for both the parametric and the non-parametric approach. Results in Figures 9 and 10 confirm above presumptions, where the frequency estimation error is defined as ( ) trueest. qqq −=E . It can be noticed that the relative number of random samples NND rsrs = has to increase at least to 0.1 when the frequency estimation errors step-like drop below 0.1 bin but this is still far from the target values when we have the whole ensemble of samples (Figures 9 - line a and 10 - line a). With increasing rsD errors slowly drop but the number of iterations in the process of randomly adding additional samples relatively substantially increase (for 512=N we typically need 2000 iterations to achieve 98.0rs ≈D ). Due to the randomness, some positions may be repeatedly sampled, whereas other positions are left blank. The parametric estimation approach (Figure 9 - line c: 4- parametric estimation with 9 iterations) gives better results than the non-parametric IDFT approach (Figures 9 - line b and 10 - line d) but the frequency has to be estimated first (typically by symmetric 3-point estimation using the Hann window 1=P and (2)). If we don’t have a good initial value of the component frequency the parametric approach gives useless results (Figure 10 - line c). After the point where the frequency component exceeds the random sampling noise (Figure 6 - line b: 074.0rs =D ) two- point IDFT estimation (Figures 9 - line d and 10 - line d) gives lower errors than the three-point estimation and especially in the case of large non-coherency (Figure 10 - line d: 4.0=mδ ). If the sampling frequency relatively increases to the signal frequency as in the case in Figures 1 and 2 ( 28.37.19mS >=qq ) the ‘random noise’ effect around the component diminishes and the non-parametric IDFT approach gives better results at lower values of rsD (Figure 11: lines b, d). The same behavior of the estimation errors can be noticed also for the amplitude estimations (Figures 12 and 14: ( ) 1trueest. −= AAAe ) and phase estimations (Figure 13: ( ) trueest. ϕϕϕ −=E ) where the same test conditions were used as in Figures 9 and 10. Errors drop very fast where rsD starts increasing and after the point around 1.0rs ≈D the decrease of the errors slowed down. Figure 9. Mean values of the absolute errors of the frequency estimation; a – original signal – IDFT(f-3p), b – random sampled signal – IDFT(f-3p), c – random sampled signal – 4-par. sine-fit (start with estimated f-3p), d – random sampled signal – IDFT(f-2p); 1S =q , 8.3=q , 1=A , deg0=ϕ , 100repeat =N trials. Figure 10. Mean values of the absolute errors of the frequency estimation; a – original signal – IDFT(f-3p), b – random sampled signal – IDFT(f-3p), c – random sampled signal – 4-par. sine-fit (start with 2=q ), d – random sampled signal – IDFT(f-2p) ; 512=N , 1S =q , 4.4=q , 1=A , deg0=ϕ , 100repeat =N trials. P 5 8 4 5 6 7 1 2 3 q )(max qE 2p -IDFT 3p -IDFT 2p -IDFT 3p -IDFT 1010− 010 210− 410− 610− 810− 1 3 Figure 8. Minimal curves of the maximal errors of the frequency estimations with the two- (IDFT-2p) and three- (IDFT-3p) interpolations of the DFT using RV-I windows and the corresponding orders 6...1=P . ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |58 The iterative 4-parametric estimation (Figures 12 - line c and 13 - line c) gives a better result than one-step 3-parametric estimation (Figure 13 - line d) when the sampling frequency is relatively low but the initial frequency has to be very well estimated. If we fix the value of the estimated frequency by IDFT the one step 3-parametric sine-fit estimation gives the error values as with the non-parametric approach (Figure 13 - line d). 3. EXPERIMENTAL RESULTS To demonstrate the behavior of estimations of the periodic signal parameters in the case of random sampling and under non-coherent sampling condition in a real measurement environment a digitizing oscilloscope (DO), Agilent 54501A [19] was used to randomly acquire signals. The signal was generated by a stable voltage generator, Keysight 33500B [20], with a nominal sine voltage of amplitude V900.0v =A . First, the signal frequency was fixed to MHz9.1v =f and with time constant of divns200time =c and 10 divisions in the x-axis, the DO acquires and shows 3.8 periods of the signal in the duration of μs2M =T ( 3.8μs2MHz9.1Mv =⋅=⋅= Tfq ). Since the DO display uses 500=N points to represent the signal in the x-axis the equivalent sampling frequency was MHz250MeqS, == TNf what is 25-times larger than the DO real sampling frequency MHz10S =f with 2025500eqS,SS ==⋅= ffNq . Sampling points were acquired with the multiple ‘randomly triggered’ sweeps to fill the points for display presentation [19]. To fulfil all 500=N points typically 150 sweeps were needed and this acquisition procedure was controlled remotely by SCPI commands: :TIM:MODE SING and :RUN [19]. Figures 15 (frequency estimation) and 16 (amplitude Figure 11. Mean values of the absolute errors of the frequency estimation; a – original signal – IDFT(f-3p), b – random sampled signal – IDFT(f-3p), c – random sampled signal – 4-par. sine-fit (start with estimated f-3p), d – random sampled signal – IDFT(f-2p) ; 512=N , 7.19S =q , 8.3=q , 1=A , deg0=ϕ , 100repeat =N trials. Figure 12. Mean values of the relative errors of the amplitude estimation; a – original signal – IDFT(f-3p, A-3p), b – random sampled signal – IDFT(f-3p, A-3p), c – random sampled signal – 4-par. sine-fit (start with estimated f- 3p), d – random sampled signal – 3-par. sine-fit (f-3p), e – random sampled signal – IDFT(f-2p, A-1p) ; 512=N , 1S =q , 8.3=q , 1=A , deg0=ϕ , 100repeat =N trials. Figure 13. Mean values of the absolute errors of the phase estimation; a – original signal – IDFT(f-3p, ph-3p), b – random sampled signal – IDFT(f-3p, ph-3p), c – random sampled signal – 4-par. sine-fit (start with estimated f- 3p), d – random sampled signal – 3-par. sine-fit (f-3p) ; 512=N , 1S =q , 8.3=q , 1=A , 100repeat =N trials. Figure 14. Mean values of the relative errors of the amplitude estimation; a – original signal – IDFT(f-3p, A-3p), b – random sampled signal – IDFT(f-3p, A-3p), c – random sampled signal – 4-par. sine-fit (start with 2=q ), d – random sampled signal – 3-par. sine-fit (f-3p), e – random sampled signal – IDFT(f-2p, A-1p); 512=N , 1S =q , 4.4=q , 1=A , deg0=ϕ , 100repeat =N trials. Figure 15. Mean values of the maximal absolute errors of the frequency estimation; a – target estimation error, b – random sampled signal – IDFT(f- 3p), c – random sampled signal – 4-par. sine-fit (start with estimated f-3p), d – random sampled signal – IDFT(f-2p); 500=N , 20S =q , 8.3=q , deg0=ϕ , 150sweeps =N . ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |59 estimation) show estimation errors. The reference value for the specific parameter is the one attained with estimation on the whole number of the filled points 1rs =D after 150 sweeps with 3-point estimations ((2) and (4)). Estimation errors are defined as ( ) 1,3p,est rsrsrs =−= DDDE qqq for the relative frequency (Figure 15) and as ( ) 11,3p,est rsrsrs −= =DDD AAAe for the amplitude (Figure 16). Estimation error behaviors in Figure 15 confirm simulation results from Figures 9 and 11 with a very fast decrease at low values of 1.0rs D errors drop slowly to the final estimation result with the whole ensemble of points at 1rs =D . The same estimation error behaviors can be noticed also in the case of the amplitude estimation (Figure 16). The best results are obtained by parametric sine-fit estimation procedures where the initial frequency was estimated first by the 3-point interpolated DFT approach (2). To analyze the estimation error behavior when the relative frequency is changing the testing conditions from Figure 15 were enhanced from one frequency of 8.3=q to a frequency range 64.1 ÷=q with frequency increments 05.0=∆q (Figure 17). For this testing the signal frequency was changed from MHz7.0minv, =f to MHz0.3maxv, =f with steps of kHz25v =∆f and at each frequency the estimation error was defined as difference between estimation value around 2.0rs =D and estimation value when 1rs =D (Figure 17: ( ) 1,est2.0,est1/2.0 rsrsrs === −= DDDE qqq ). The frequency estimation errors in Figure 17 show that by changing the relative frequency the relation of errors between estimations with low sampling duty cycle and estimations with the whole ensemble of points remain on the same level. 4. CONCLUSIONS Estimations of the signal component parameters from random equivalent and non-coherent sampling periodic signals comprise additional problems of random sampling noise. The non-coherency is more problematic due to frequency leakage than in the case of the coherent sampling where the problem of frequency estimation is much reduced like in approaches of compressive sensing. To achieve the lowest estimation errors the non-parametric IDFT estimation of frequency has to be used first for detection of the component frequency in both the parametric and non-parametric approach and after that the iterative parametric method should be used for other parameters when the sampling frequency is relatively lower than the Nyquist frequency and the sampling procedure becomes pure random. Simulations and experimental results show that the estimations of the component frequency, amplitude and phase are possible when the duty ratio of the random samples from the total number of samples in the non-coherent measurement interval is very low - around 0.1. REFERENCES [1] Y.-C. Jenq, Perfect reconstruction of digital spectrum from nonuniformly sampled signals, IEEE Trans. Instrum. Meas. 46(3) (1997) pp. 649-652. [2] Y. Zhao, Y. H. Hu, H. Wang, Enhanced random equivalent sampling based on compressed sensing, IEEE Trans. Instrum. Meas. 61(6) (2012) pp. 579–586. [3] EJ. Candes, 1. 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Rapuano, An extension to IEEE Std. 1241 sine fit for analog-to-information converters testing, In Proc. of I2MTC/2015 Pisa, Italy (2015) pp. 1933-1937. [9] M. Bertocco, G. Frigo, C. Narduzzi, High-accuracy frequency estimation in compressive sensing-plus-DFT spectral analysis, In Proc. of I2MTC/2015. Pisa, Italy (2015) pp. 1668-1671. [10] D. Bao, P. Daponte, L. DE Vito, S. Rapuano, Frequency- domain characterization of random demodulation analog-to- Figure 16. Mean values of the maximal relative errors of the amplitude estimation; a – target estimation error, b – random sampled signal – IDFT(f- 3p, A-3p), c – random sampled signal – 4-par. sine-fit (start with estimated f-3p), d – random sampled signal – 3-par. sine-fit (f-3p), e – random sampled signal – IDFT(f-2p, A-1p); 500=N , 20S =q , 8.3=q , deg0≈ϕ , 150sweeps =N . q ( )q1/2.0rs =DE 1 2 3 4 10-4 10-2 1 b 102 6 c a d 5 Figure 17. Mean values of the maximal absolute errors of the frequency estimation; a – target estimation error, b – random sampled signal – IDFT(f- 3p), c – random sampled signal – 4-par. sine-fit (start with estimated f-3p), d – random sampled signal – IDFT(f-2p) ; 500=N , deg0≡ϕ , 150sweeps =N . http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin= http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin= http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6638701&newsearch=true&queryText=random%20sampling ACTA IMEKO | www.imeko.org December 2017 | Volume 6 | Number 4 |60 information converters, ACTA IMEKO. 4(1) (2015) pp. 111- 120. [11] D. Agrež, Periodic signal parameters’ estimations from random sampling measurements under non-coherent sampling condition, in Proc. of the 21st Symposium IMEKO TC-4, Budapest, Hungary (2016) pp. 190-195. [12] E.J. Candes, M.B. Wakin, An introduction to compressive sampling. IEEE Signal Proc. Magazine 25 (2008) pp. 21–30. [13] R. 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[20] Keysight Technologies, 33500B Series Waveform Generators – Data Sheet, 5991- 0692EN (2015). 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Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice