A New Algorithm of Compensation of the Time Interval Error GPS ...

Sensors & Transducers Volume 112, Issue 1 January 2010

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ISSN 1726-5479

Editors-in-Chief: professor Sergey Y. Yurish, Phone: +34 696067716, fax: +34 93 4011989, e-mail: [email protected] Editors for Western Europe Meijer, Gerard C.M., Delft University of Technology, The Netherlands Ferrari, Vittorio, Universitá di Brescia, Italy Editor South America Costa-Felix, Rodrigo, Inmetro, Brazil

Editors for North America Datskos, Panos G., Oak Ridge National Laboratory, USA Fabien, J. Josse, Marquette University, USA Katz, Evgeny, Clarkson University, USA Editor for Asia Ohyama, Shinji, Tokyo Institute of Technology, Japan

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Djordjevich, Alexandar, City University of Hong Kong, Hong Kong Donato, Nicola, University of Messina, Italy Donato, Patricio, Universidad de Mar del Plata, Argentina Dong, Feng, Tianjin University, China Drljaca, Predrag, Instersema Sensoric SA, Switzerland Dubey, Venketesh, Bournemouth University, UK Enderle, Stefan, Univ.of Ulm and KTB Mechatronics GmbH, Germany Erdem, Gursan K. Arzum, Ege University, Turkey Erkmen, Aydan M., Middle East Technical University, Turkey Estelle, Patrice, Insa Rennes, France Estrada, Horacio, University of North Carolina, USA Faiz, Adil, INSA Lyon, France Fericean, Sorin, Balluff GmbH, Germany Fernandes, Joana M., University of Porto, Portugal Francioso, Luca, CNR-IMM Institute for Microelectronics and Microsystems, Italy Francis, Laurent, University Catholique de Louvain, Belgium Fu, Weiling, South-Western Hospital, Chongqing, China Gaura, Elena, Coventry University, UK Geng, Yanfeng, China University of Petroleum, China Gole, James, Georgia Institute of Technology, USA Gong, Hao, National University of Singapore, Singapore Gonzalez de la Rosa, Juan Jose, University of Cadiz, Spain Granel, Annette, Goteborg University, Sweden Graff, Mason, The University of Texas at Arlington, USA Guan, Shan, Eastman Kodak, USA Guillet, Bruno, University of Caen, France Guo, Zhen, New Jersey Institute of Technology, USA Gupta, Narendra Kumar, Napier University, UK Hadjiloucas, Sillas, The University of Reading, UK Haider, Mohammad R., Sonoma State University, USA Hashsham, Syed, Michigan State University, USA Hasni, Abdelhafid, Bechar University, Algeria Hernandez, Alvaro, University of Alcala, Spain Hernandez, Wilmar, Universidad Politecnica de Madrid, Spain Homentcovschi, Dorel, SUNY Binghamton, USA Horstman, Tom, U.S. Automation Group, LLC, USA Hsiai, Tzung (John), University of Southern California, USA Huang, Jeng-Sheng, Chung Yuan Christian University, Taiwan Huang, Star, National Tsing Hua University, Taiwan Huang, Wei, PSG Design Center, USA Hui, David, University of New Orleans, USA Jaffrezic-Renault, Nicole, Ecole Centrale de Lyon, France Jaime Calvo-Galleg, Jaime, Universidad de Salamanca, Spain James, Daniel, Griffith University, Australia Janting, Jakob, DELTA Danish Electronics, Denmark Jiang, Liudi, University of Southampton, UK Jiang, Wei, University of Virginia, USA Jiao, Zheng, Shanghai University, China John, Joachim, IMEC, Belgium Kalach, Andrew, Voronezh Institute of Ministry of Interior, Russia Kang, Moonho, Sunmoon University, Korea South Kaniusas, Eugenijus, Vienna University of Technology, Austria Katake, Anup, Texas A&M University, USA Kausel, Wilfried, University of Music, Vienna, Austria Kavasoglu, Nese, Mugla University, Turkey Ke, Cathy, Tyndall National Institute, Ireland Khan, Asif, Aligarh Muslim University, Aligarh, India Sapozhnikova, Ksenia, D.I.Mendeleyev Institute for Metrology, Russia

Kim, Min Young, Kyungpook National University, Korea South Ko, Sang Choon, Electronics. and Telecom. Research Inst., Korea South Kockar, Hakan, Balikesir University, Turkey Kotulska, Malgorzata, Wroclaw University of Technology, Poland Kratz, Henrik, Uppsala University, Sweden Kumar, Arun, University of South Florida, USA Kumar, Subodh, National Physical Laboratory, India Kung, Chih-Hsien, Chang-Jung Christian University, Taiwan Lacnjevac, Caslav, University of Belgrade, Serbia Lay-Ekuakille, Aime, University of Lecce, Italy Lee, Jang Myung, Pusan National University, Korea South Lee, Jun Su, Amkor Technology, Inc. South Korea Lei, Hua, National Starch and Chemical Company, USA Li, Genxi, Nanjing University, China Li, Hui, Shanghai Jiaotong University, China Li, Xian-Fang, Central South University, China Liang, Yuanchang, University of Washington, USA Liawruangrath, Saisunee, Chiang Mai University, Thailand Liew, Kim Meow, City University of Hong Kong, Hong Kong Lin, Hermann, National Kaohsiung University, Taiwan Lin, Paul, Cleveland State University, USA Linderholm, Pontus, EPFL - Microsystems Laboratory, Switzerland Liu, Aihua, University of Oklahoma, USA Liu Changgeng, Louisiana State University, USA Liu, Cheng-Hsien, National Tsing Hua University, Taiwan Liu, Songqin, Southeast University, China Lodeiro, Carlos, University of Vigo, Spain Lorenzo, Maria Encarnacio, Universidad Autonoma de Madrid, Spain Lukaszewicz, Jerzy Pawel, Nicholas Copernicus University, Poland Ma, Zhanfang, Northeast Normal University, China Majstorovic, Vidosav, University of Belgrade, Serbia Marquez, Alfredo, Centro de Investigacion en Materiales Avanzados, Mexico Matay, Ladislav, Slovak Academy of Sciences, Slovakia Mathur, Prafull, National Physical Laboratory, India Maurya, D.K., Institute of Materials Research and Engineering, Singapore Mekid, Samir, University of Manchester, UK Melnyk, Ivan, Photon Control Inc., Canada Mendes, Paulo, University of Minho, Portugal Mennell, Julie, Northumbria University, UK Mi, Bin, Boston Scientific Corporation, USA Minas, Graca, University of Minho, Portugal Moghavvemi, Mahmoud, University of Malaya, Malaysia Mohammadi, Mohammad-Reza, University of Cambridge, UK Molina Flores, Esteban, Benemérita Universidad Autónoma de Puebla, Mexico Moradi, Majid, University of Kerman, Iran Morello, Rosario, University "Mediterranea" of Reggio Calabria, Italy Mounir, Ben Ali, University of Sousse, Tunisia Mulla, Imtiaz Sirajuddin, National Chemical Laboratory, Pune, India Neelamegam, Periasamy, Sastra Deemed University, India Neshkova, Milka, Bulgarian Academy of Sciences, Bulgaria Oberhammer, Joachim, Royal Institute of Technology, Sweden Ould Lahoucine, Cherif, University of Guelma, Algeria Pamidighanta, Sayanu, Bharat Electronics Limited (BEL), India Pan, Jisheng, Institute of Materials Research & Engineering, Singapore Park, Joon-Shik, Korea Electronics Technology Institute, Korea South Penza, Michele, ENEA C.R., Italy Pereira, Jose Miguel, Instituto Politecnico de Setebal, Portugal Petsev, Dimiter, University of New Mexico, USA Pogacnik, Lea, University of Ljubljana, Slovenia Post, Michael, National Research Council, Canada Prance, Robert, University of Sussex, UK Prasad, Ambika, Gulbarga University, India Prateepasen, Asa, Kingmoungut's University of Technology, Thailand Pullini, Daniele, Centro Ricerche FIAT, Italy Pumera, Martin, National Institute for Materials Science, Japan Radhakrishnan, S. National Chemical Laboratory, Pune, India Rajanna, K., Indian Institute of Science, India Ramadan, Qasem, Institute of Microelectronics, Singapore Rao, Basuthkar, Tata Inst. of Fundamental Research, India Raoof, Kosai, Joseph Fourier University of Grenoble, France Reig, Candid, University of Valencia, Spain Restivo, Maria Teresa, University of Porto, Portugal Robert, Michel, University Henri Poincare, France Rezazadeh, Ghader, Urmia University, Iran Royo, Santiago, Universitat Politecnica de Catalunya, Spain Rodriguez, Angel, Universidad Politecnica de Cataluna, Spain Rothberg, Steve, Loughborough University, UK Sadana, Ajit, University of Mississippi, USA Sadeghian Marnani, Hamed, TU Delft, The Netherlands Sandacci, Serghei, Sensor Technology Ltd., UK

Saxena, Vibha, Bhbha Atomic Research Centre, Mumbai, India Schneider, John K., Ultra-Scan Corporation, USA Seif, Selemani, Alabama A & M University, USA Seifter, Achim, Los Alamos National Laboratory, USA Sengupta, Deepak, Advance Bio-Photonics, India Shearwood, Christopher, Nanyang Technological University, Singapore Shin, Kyuho, Samsung Advanced Institute of Technology, Korea Shmaliy, Yuriy, Kharkiv National Univ. of Radio Electronics, Ukraine Silva Girao, Pedro, Technical University of Lisbon, Portugal Singh, V. R., National Physical Laboratory, India Slomovitz, Daniel, UTE, Uruguay Smith, Martin, Open University, UK Soleymanpour, Ahmad, Damghan Basic Science University, Iran Somani, Prakash R., Centre for Materials for Electronics Technol., India Srinivas, Talabattula, Indian Institute of Science, Bangalore, India Srivastava, Arvind K., Northwestern University, USA Stefan-van Staden, Raluca-Ioana, University of Pretoria, South Africa Sumriddetchka, Sarun, National Electronics and Computer Technology Center, Thailand Sun, Chengliang, Polytechnic University, Hong-Kong Sun, Dongming, Jilin University, China Sun, Junhua, Beijing University of Aeronautics and Astronautics, China Sun, Zhiqiang, Central South University, China Suri, C. Raman, Institute of Microbial Technology, India Sysoev, Victor, Saratov State Technical University, Russia Szewczyk, Roman, Industrial Research Inst. for Automation and Measurement, Poland Tan, Ooi Kiang, Nanyang Technological University, Singapore, Tang, Dianping, Southwest University, China Tang, Jaw-Luen, National Chung Cheng University, Taiwan Teker, Kasif, Frostburg State University, USA Thumbavanam Pad, Kartik, Carnegie Mellon University, USA Tian, Gui Yun, University of Newcastle, UK Tsiantos, Vassilios, Technological Educational Institute of Kaval, Greece Tsigara, Anna, National Hellenic Research Foundation, Greece Twomey, Karen, University College Cork, Ireland Valente, Antonio, University, Vila Real, - U.T.A.D., Portugal Vaseashta, Ashok, Marshall University, USA Vazquez, Carmen, Carlos III University in Madrid, Spain Vieira, Manuela, Instituto Superior de Engenharia de Lisboa, Portugal Vigna, Benedetto, STMicroelectronics, Italy Vrba, Radimir, Brno University of Technology, Czech Republic Wandelt, Barbara, Technical University of Lodz, Poland Wang, Jiangping, Xi'an Shiyou University, China Wang, Kedong, Beihang University, China Wang, Liang, Advanced Micro Devices, USA Wang, Mi, University of Leeds, UK Wang, Shinn-Fwu, Ching Yun University, Taiwan Wang, Wei-Chih, University of Washington, USA Wang, Wensheng, University of Pennsylvania, USA Watson, Steven, Center for NanoSpace Technologies Inc., USA Weiping, Yan, Dalian University of Technology, China Wells, Stephen, Southern Company Services, USA Wolkenberg, Andrzej, Institute of Electron Technology, Poland Woods, R. Clive, Louisiana State University, USA Wu, DerHo, National Pingtung Univ. of Science and Technology, Taiwan Wu, Zhaoyang, Hunan University, China Xiu Tao, Ge, Chuzhou University, China Xu, Lisheng, The Chinese University of Hong Kong, Hong Kong Xu, Tao, University of California, Irvine, USA Yang, Dongfang, National Research Council, Canada Yang, Wuqiang, The University of Manchester, UK Yang, Xiaoling, University of Georgia, Athens, GA, USA Yaping Dan, Harvard University, USA Ymeti, Aurel, University of Twente, Netherland Yong Zhao, Northeastern University, China Yu, Haihu, Wuhan University of Technology, China Yuan, Yong, Massey University, New Zealand Yufera Garcia, Alberto, Seville University, Spain Zakaria, Zulkarnay, University Malaysia Perlis, Malaysia Zagnoni, Michele, University of Southampton, UK Zamani, Cyrus, Universitat de Barcelona, Spain Zeni, Luigi, Second University of Naples, Italy Zhang, Minglong, Shanghai University, China Zhang, Qintao, University of California at Berkeley, USA Zhang, Weiping, Shanghai Jiao Tong University, China Zhang, Wenming, Shanghai Jiao Tong University, China Zhang, Xueji, World Precision Instruments, Inc., USA Zhong, Haoxiang, Henan Normal University, China Zhu, Qing, Fujifilm Dimatix, Inc., USA Zorzano, Luis, Universidad de La Rioja, Spain Zourob, Mohammed, University of Cambridge, UK

Sensors & Transducers Journal (ISSN 1726-5479) is a peer review international journal published monthly online by International Frequency Sensor Association (IFSA). Available in electronic and on CD. Copyright © 2009 by International Frequency Sensor Association. All rights reserved.

Sensors & Transducers Journal

Contents Volume 112 Issue 1 January 2010

www.sensorsportal.com

ISSN 1726-5479

Research Articles Design of an Acoustic Displacement Transducer Tariq Al Mograbi, Mohammad A. K. Alia, Mohammad Abuzalata ......................................................

1

Vibration Analysis Based on Hammer Impact for Fouling Detection using Microphone and Accelerometer as Sensors Jaidilson Silva, Antonio Marcus Lima, Helmut Neff and José Sérgio Rocha Neto ............................

10

Simulation of the Two-Phase Liquid – Gas Flow through Ultrasonic Transceivers Application in Ultrasonic Tomography Zulkarnay Zakaria, Mohd Hafiz Fazalul Rahiman, Ruzairi Abdul Rahim ...........................................

24

Image Reconstructions of a Portable Optical CT-Scan Using an NIR Light Source Margi Sasono and Hariyadi Soetedjo.................................................................................................

39

Statistical Feature Extraction and Recognition of Beverages Using Electronic Tongue P. C. Panchariya and A. H. Kiranmayee. ...........................................................................................

47

Modeling and Verification of Heat Fields by Virtual Instrumentation Libor Hargaš, Dušan Koniar, Miroslav Hrianka, Anna Príkopová .......................................................

64

PC Based Instrument for the Measurement of Dielectric Constant of Liquids V. V. Ramana C. H. and Malakondaiah K. .........................................................................................

73

Development of Laser LEDs Based a Programmable Optical Sensor for Detection of Environmental Pollutants Amit K. Sharma and R. K. Tiwari........................................................................................................

80

Performance Evaluation and Robustness Testing of Advanced Oscilloscope Triggering Schemes Shakeb A. Khan, Alka Nigam, A. K. Agarwala, Mini S.Thomas .........................................................

95

Design and Development of an Embedded System for Testing the Potentiometer Linearity Raghavendra Rao Kanchi, Nagamani Gosala ...................................................................................

107

Development of an FPGA Based Embedded System for High Speed Object Tracking Chandrashekar Matham, Nagabhushan Raju Konduru .....................................................................

118

A New Algorithm of Compensation of the Time Interval Error GPS-Based Measurements Jonny Paul Zavala de Paz, Yuriy S. Shmaliy .....................................................................................

124

Colour Determination and Change of Sensory Properties of Mayonnaise with Different Contents of Oil depending on Length of Storage Višnja M. Sikimić, Jovanka V. Popov-Raljić, Branislav P. Zlatković, Nada Lakić ..............................

138

Dynamically Functioning Structure and Problem of Measurements of Rapidly Time-Varying Processes: Dream or Reality George Abramchuk, Kristina Abramchuk.. .........................................................................................

166

Authors are encouraged to submit article in MS Word (doc) and Acrobat (pdf) formats by e-mail: [email protected] Please visit journal’s webpage with preparation instructions: http://www.sensorsportal.com/HTML/DIGEST/Submition.htm International Frequency Sensor Association (IFSA).

Sensors & Transducers Journal, Vol. 112, Issue 1, January 2010, pp. 124-137

Sensors & Transducers ISSN 1726-5479 © 2010 by IFSA http://www.sensorsportal.com

A New Algorithm of Compensation of the Time Interval Error GPS-Based Measurements 1,2

Jonny Paul ZAVALA DE PAZ, 2Yuriy S. SHMALIY

1

Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Instituto Politécnico Nacional, Cerro Blanco 141, Col. Colinas del Cimatario, C.P. 76090, Querétaro, México Tel.: (442) 229 0804, ext. 81002 2 FIMEE, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago Km 3.5+1.8 Km Comunidad de Palo Blanco, C.P. 36885 Salamanca, Gto., México E-mail: [email protected]

Received: 29 October 2010 /Accepted: 22 January 2010 /Published: 29 January 2010

Abstract: In this paper we present a new algorithm of compensation of the time interval error (TIE) applying an unbiased p-step predictive finite impulse response (FIR) filter at the signal of the receiver Global Positioning System (GPS)-based measurements. The practical use of the system GPS involves various inherent problems of the signal. Two of the most important problems are the TIE and the instantaneous loss of the signal of the GPS by a small interval of time, called "holdover". The error holdover is a problem that at present does not possess solution and the systems that present this type of error produce lines of erroneous synchronization in the signal of the GPS. Basic holdover algorithms are discussed along with their most critical properties. Efficiency of the predictive filter in holdover is demonstrated in applications to GPS-based measurements of the TIE. Copyright © 2010 IFSA. Keywords: GPS-based measurements, Time interval error, Holdover, Unbiased predictive FIR filter

1. Introduction The problem of accuracy of wirelessly disciplined local clocks and oscillators arises whenever a synchronizing signal is temporary not available. The relevant operation mode is termed “holdover” and technical solutions for holdover are often crucial for both ground and space applications. Holdover can also be implied for available, but “bad” synchronizing signals. In different applications, it is required to obtain normal functioning in holdover during all time. In deep 124

Sensors & Transducers Journal, Vol. 112, Issue 1, January 2010, pp. 124-137

space, oscillators and clocks typically operate without synchronizing signals meaning that normal functioning is required for years and decades of years [1]. The holdover time differs for different clocks being specified via the allowed fractional frequency departure that, for instance, in digital communication network nodes should not be greater than 1  10 11 [2]. In a short term (days and months), the holdover problem may be solved by predicting current errors via the nearest past history. On a long time base (months and years), internal resources may be exploited if additional resonances of the oscillator resonant system are observable. In applications to the oven controlled crystal oscillators (OCXOs), the long-term holdover approach was developed in [3] as the modulational method. The method employs a high aging correlation of the fundamental and harmonic frequencies of crystal resonators [4] to form a synchronizing signal. Its application to an OCXO with the AT-cut quartz crystal resonator has demonstrated an extremely small long-term frequency drift of about  3  10 10 /year [5]. Based upon prediction, the clock or oscillator holdover technology prolongs continued functioning time after loss of a synchronizing signal. For the clock to operate in holdover, prediction is provided of the time error at a current discrete time n by processing the nearest available past measurements. A control loop is then organized such that the synchronizing signal is produced by prediction. Based upon, different approaches can be used. One may employ extrapolation methods or prolong an interpolating function (Lagrange and Newton methods, splines, etc.) [6] and regression curve from the past to n that have been applied to clocks by many authors [7-13]. Setting aside these methods, the 1step toward predictive FIR filters were derived by Heinonen and Neuvo in [14] employing the Lagrange multipliers. It was also stated in [7] that linear predictors are optimal or close to optimal for the prediction of clock instabilities. Representation of the clock model in state space allows solving the problem from the standpoint of system theory using the Kalman filter. Suitable for holdover, the predictive Kalman filtering algorithms are used in many works as [15], [16] owing to a strong advantage: the current clock state is predicted via its past state. A disadvantage is that the Kalman filter needs the covariance of commonly unknown for users the clock noise. Moreover, Kalman claims the model to be known and all noises white Gaussian. Otherwise, under the model temporary uncertainty and non Gaussian noises, the Kalman filter may produce biased and noisy estimates [17]. The errors can be reduced using finite impulse response (FIR) filters having inherent bounded input/bounded output (BIBO) stability and better robustness against temporary model uncertainties and round-off errors. By FIR filters, measurement on a horizon of N past points is processed to estimate the state at n. Extensive investigations of optimal FIR filtering of state space models have been provided by Kwon et. al. [18-20] resulting in the theory of receding horizon control [21]. Of importance is that gains of the unbiased FIR filters do not depend on noise properties [14] and past system states [19, 22] that makes such filters simple for applications. An unbiased predictive FIR filter was proposed by [24] of a class of discrete time state space models associated with holdover problems in remote wireless clock synchronization, such as the Global Positioning System (GPS)-based. In this paper, we present a new algorithm of compensation of the time interval error (TIE) applying the unbiased p-step predictive finite impulse response (FIR) filter at the signal of the receiver Global Positioning System (GPS)-based measurements.

2. Clock Time Model and Problem Formulation The discrete time-invariant model of a local clock can be represented in state space at a current time point n with the state and observation equations, respectively, as follows 125

Sensors & Transducers Journal, Vol. 112, Issue 1, January 2010, pp. 124-137

n  An 1  wn ,

(1)

 n  Cn  vn ,

(2)

where K  1 is the vector of the clock states x kn , k  1, K  , is

n  x1n x2 n  x Kn T ,

(3)

in which x1n is the time interval error (TIE), x 2 n is the fractional frequency offset, x3n is the fractional frequency drift rate, etc. The K  K transition matrix

1  0 A  0   0 



2 2

1 

 

0

1 





0

0 

   ( K  2 )!   K 3  ( K 3)!   1   K 1

( K 1)! K 2

(4)

projects a nearest past clock state  n 1 to the current state  n , where  is a sampling time identical to 1s in GPS-based clock synchronization using the one pulse per second (1PPS) signal. For the measurement  n of the clock first state x1n used in synchronization, the 1  K measurement matrix is

C  1 0  0 .

(5)

wn  w1n w2 n  wKn 

(6)

The K  1 clock noise vector T

contains zero-mean components with known covariance. In GPS-based synchronization, via 1PPS signals, the TIE excursions are compensated on a horizon of N points in the presence of the zeromean measurement noise vn that commonly dominates. Let us now suppose that the 1PPS-based measurement  n (2) has been provided from some far past point up to n-1. At n, measurement is not available or wrong (inapplicable). The problem formulates as follows. We would like to derive an unbiased p-step predictive FIR filter of the clock first state x1 at n by processing the data taken from the time interval n  N  1  p, n  p , p  0 , of the past history (see Fig. 1). In such a formulation, the problem seems to be most general for holdover. We also would like to test the filter by simulation and GPS-based sawtooth and sawtooth-less measurements of the TIE of a crystal clock.

3. An Unbiased p-step Predictive FIR Filter Below, following the diagram shown in Fig. 1, we derive a real time unbiased FIR estimator intended for p-step predictive filtering of the clock state x1n .

126

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Fig. 1. The p-step toward predictive FIR filtering problem for holdover.

3.1. General Relations To find the estimate, we start at n  N  1  p and represent (1) and (2), neglecting wn , as

n  A N 1 p n  N 1 p ,

(7)

 n  Cn  vn ,

(8)

where 1  0 i A  0    0

i

 2i 2 2



1

i



0

1





0

0



   .    1 

(i ) K 1 ( K 1)! (i ) K  2 ( K  2 )! (i ) K  3 ( K 3)!

(9)

Now, the noiseless state model (7) projects ahead from n  N  1  p to n with the degree Taylor polynomial such that the first state (TIE) is represented with K 1

x1n   x1 q q 0

 q ( N  1  p) q q!

 x1  x 2 ( N  1  p )  x3

 2 ( N  1  p) 2 2

   xK

 ( N  1  p)K 1 ,

(10)

( K  1)!

where xk xk ( n N 1 p ) , k  1, K  , is the clock k-state at n  N  1  p . To provide the p-step toward predictive estimate x1n x1n n p; N of x1n via (7) and (8) utilizing N points starting at n  N  1  p , the discrete-time convolution operator can be applied to (8), ~ x1n 

N 1 p

h  i p

li

n i

 Wl T  ,

(11)

where l  0 is the degree of thee FIR filter polynomial gain hli . The l-degree N  1 gain matrix and 127

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N  1 matrix of measurements are given with, respectively,



Wl  hlp hl ( p 1)  hl ( p  N 1)



T

,

(12)

   n p  n  p 1  n p  N 1 .

(13)





T

It is known that unbiasedness in the FIR estimate is guaranteed if hli , existing from p to N  1  p , obeys several fundamental properties. Modified for (11), these properties, namely N 1 p

h

( N , p)  1 ,

(14)

( N , p)i u  0 , u  1, l  ,

(15)

i p

N 1 p

h i p

li

li

assure that the prediction error is zero in the absence of noise when K  l  1.

3.2. Generic Coefficients for Polynomial Gains To specify Wl , it needs examining the mean square error (MSE) of the predicted estimate (11),





(16a)



(16b)

J  E  x1n  ~ x1n 



 E x1n  Wl T 



2

2

 E x1n  Wl T   Wl T v

 , 2

(16c)

where





(17)

v  v1( n p ) v1( n p 1) v1( n p  N 1) .

(18)

  x1( n  p ) x1( n  p 1)  x1( n  p  N 1) , T





T

By the commutativity property Wl T v v T Wl and zero-mean noise, Ev n   0 , we have





2

J  x1n  Wl T   Wl T RWl ,

(19)

NN where the covariance matrix R has a generic component Rij  E vi v j , i, j  n  p , n  p  N  1. as can be seen, the first term in (19) represents the square bias and the second one the variance, both in the p-step toward prediction of the clock 1-state. It also follows that, in order for the prediction to be unbiased at any point, the following equality must be satisfied, namely x1n  Wl T  .

(20) 128

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To solve (20) for Wl , the well known from the Kalman-Bucy filter theory condition may be applied, namely: the order of the optimal filter is the same as that of the system. That means that for x1n represented with (10) and the ( K  1) -degree polynomial, of the same degree l  K  1 a polynomial must be used to describe the gain l

hli   a jl i j ,

(21)

j 0

where a jl are still unknown coefficients. Now, following [22] substituting (10) with K  l  1, (12) with (21), and (18) to (20) yields the generic coefficient a jl  ( 1) j

M ( j 1)1 D

,

(22)

calculated via the determinant D and minor M ( j 1)1 of the (l  1)  (l  1) quadratic symmetric matrix d 0 d D 1   dl

d1 d2

 d l 1

dl   d l 1  ,     d 2l 



(23)

which component d r , r  0,2l , is defined with the Bernoulli polynomials Bn (x) [23] as d r ( N , p) 

N 1 p

i i p

r



1 Br 1 ( N  p)  Br 1 ( p). r 1

(24)

Determined hli , the unbiased p-step predictive FIR filtering of the clock 1-state is provided at n (Fig. 1) by (11).

3.3. Noise Power Gain The estimate variance  n2 is evaluated by the term Wl T RWl in (20). With large N, the noise induced by the GPS timing receiver becomes delta-correlated [22]. If also its variance  vn2 is supposed to be constant over the horizon, then R becomes diagonal with equal components  vn2 and we have

 n2  g l  vn2 , where g l ( N , p )  Wl T Wl

(25)

is the noise power gain (NG) [14]. For (15) and (21), the NG is defined to be gk 

N 1 p

h i p

2 li

(26a) 129

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l

  a jl

N 1 p

h i

j 0

i p

li

j

 a 0l ,

(26b)

meaning that the coefficient a 0l in (21) is responsible for noise in the unbiased estimate. As it will be shown below, a 0l becomes zero when N tends toward infinity.

4. A p-step Predictive Filter with a Ramp Gain Function Because of a lesser divergency with large p, linear predictors are often optimal or close to optimal in the sense of the minimum mean square error (MSE) in the predicted value. For linear prediction, l  1 , the filter gains is ramp, hl (i, N , p)  a01 ( N , p)  a11 ( N , p)i ,

(27)

having the coefficients produced by (21)-(24) to be a 01 

2( 2 N  1)( N  1)  12 p ( N  1  p ) , N ( N 2  1)

a11  

6( N  1  2 p ) . N ( N 2  1)

(28)

(29)

For small p  0,3 , (27) yields h1i ( N ,0) 

2(2 N  1)  6i , N ( N  1)

(30)

h1i ( N ,1) 

2(2 N  1)  6i , N ( N  1)

(31)

2 N 2  9 N  13  3( N  3)i , N ( N 2  1)

(32)

2 N 2  15 N  37  3( N  5)i . N ( N 2  1)

(33)

h1i ( N ,2)  2

h1i ( N ,3)  2

If p  0 , (27) degenerates to the gain (not predictive) earlier derived in [25] using linear regression and thereafter in [22] via the state space model. For (26b) and (28), the NG is defined to be g1 ( N , p ) 

2( 2 N  1)( N  1)  12 p ( N  1  p ) , N ( N 2  1)

(34)

and its particular values are provided by (30)-(33) if to set i  0 . Fig. 2 illustrates (34), manifesting that prediction is achieved at increase of noise. 130

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Fig. 2. NG of the 1-degree (ramp) unbiased p-step predictive FIR filter. The case of p  0 corresponds to filtering without prediction. An asymptotic line (dashed) corresponds to N  1 .

Indeed, when 2  N  N b , where N b is determined by solving g1 ( N b , p)  1 , the filter becomes inefficient, because noise is gained when NG exceeds unity. Clock synchronization, however, is commonly provided with large horizons. In this case, when N  N b  2 , the NG does not depend on p and fits the asymptotic function g1

N 1



4 N

(35)

shown as a dashed line in Fig. 2. Moreover, when N  1 and (35), an unbiased predictive FIR filtering estimate becomes optimal in the sense of the zero bias and zero noise.

5. A p-step predictive Filter with a Quadratic Gain Function For l  2 and arbitrary step p, (21)-(24) yield a quadratic gain function h2 (i, N , p )  a 02 ( N , p )  a12 ( N , p )i  a 22 ( N , p )i 2 ,

(36)

having the coefficients 3[3 N 4  12 N 3  17 N 2  12 N  4  12( N  1)(2 N 2  5 N  2) p  12(7 N 2  15 N  7) p 2 a 02 

 120( N  1) p 3  60 p 4 ] N ( N 4  5 N 2  4)

(37) , 131

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 18[2 N 3  7 N 2  7 N  2  2(7 N 2  15 N  7) p  30( N  1) p 2  20 p 3 ] a12  N ( N 4  5 N 2  4) a 22

(38) ,

N 2  3N  2  6( N  1) p  6 p 2  30 . N ( N 4  5 N 2  4)

(39)

Fig. 3 sketches g 2 ( N , p)  a02 ( N , p) for different p and instantly indicates that (37), first, becomes inefficient with much smaller p than in (28) and, second, a singularity occurs at N  2 causing instability with N  3 .

Fig. 3. NG of the 2-degree (quadratic) unbiased p-step predictive FIR filter. The case of p  0 corresponds to filtering without prediction. An asymptotic line (dashed) corresponds to N  1 . The range of instability is shadowed.

6. Holdover Algorithms Based upon the proposed filter, the holdover algorithm can be designed following one of the diagrams shown in Fig. 4. Here the measurement (input) is depicted with “  ”, predicted estimate (output) with “  ”, and unavailable or excluded “bad” measurement by “  ”. It is implied that every measured point (from 1 to 9) represents the last point of the averaging horizon of N points shown in Fig. 1. From each of these points the filter is able to produce the p-step toward predictive estimate. For example, the point 1 (see Fig. 4a) representing a horizon of N neighboring past points, produces the estimate at 2a associated with the measurement at 2.

132

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Fig. 4. Holdover algorithms with predictive filtering: (a) fixed-step ( p  1 ), (b) fixed-step ( p  2 ), and (c) variable-step.

The algorithm can be organized either with the fixed step (see Fig. 4a and 4b) or variable step (see Fig. 4c). In the fixed-step case of p  1 (see Fig. 4a), the algorithm operates in “Hold-in” (from 1 to 3) producing the estimates (form 2a to 4a). At 4, the measurement is not available. Therefore, a predicted value (4a) is used instead and the filter produces the next estimate at 5a. Such a “Holdover” procedure is applied from 4 to 6. The filter then returns to “Hold-in”. In a like manner, the algorithm is organized for the case shown in Fig. 4b and one can easily figure out how to design the fixed-step algorithm for arbitrary p  0 . In the variable-step case illustrated in Fig. 4c, the point 3 covers all the holdover range, producing individual estimates for the points 4a-7a. Here, predicted values are not involved in holdover and used in the subsequent hold-in range.

7. Design of the New Algorithm of Compensation of the TIE In the Fig. 5, is the signal of the GPS that present holdover, loss of the signal in an interval of time. This problem occurs always in a time from 20 to 30 minutes in the morning. To apply our filter is necessary to have an indicator (flag), which is going to say us in which moment itself active and defuses the holdover. These indicators are obtained of the Geometry DOP (Dilution of Presicion) of the signal of the TIE. The problem holdover depends of the Geometry DOP of the satellites in the moment of the calculation of the position (see Fig. 5). This means that is not the same thing that 4 satellites be much separated (better precision) to that be more next (smaller precision). With this algorithm we applied the unbiased predictive FIR filter at the signal of the TIE. When the indicator detects holdover, itself active the filter with prediction ( p  1 ) and predicts the future signal, 133

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until the indicator to stops detecting presence of holdover then is defused the filter without prediction ( p  0 ) and continuous the filter.

(a)

(b) Fig. 5. (a) Show the signal of the TIE where is present the holdover during an interval of time, (b) represent the holdover detection how an indicator or flag, this is obtained of the Geometry DOP.

8. Results The fixed-step, ( p  1) , holdover algorithm has been applied to GPS-based measurements of the TIE. The fixed-step algorithm is applied with l  1 and N  2500 (see Fig. 6). In the Fig. 7 is presented the FIR filtering with l  2 and N  7000 . We use how reference the measurement of the atomic clock Rubidium (Rb). 134

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The Fig. 7 assures of high efficiency of the solution. In fact, in the initial hold-in range, the values produced with p  0 and p  1 are indistinct. In the holdover range, the predicted estimate ( p  1) still does not get out of the actual linear behavior and keeps tracing it in the subsequent hold-in range. In turn, the filter ( p  0) exhibits inherent transients both within and beyond the holdover range.

x1 of the TIE with Fig. 6. Applications of the unbiased (l  1) , 1-step ( p  1) predictive FIR filtering estimate ~ x1 ( p  0) is provided with the 1-degree unbiased different measurement noises, by N  2500 . The estimate ~ FIR filter. To demonstrate holdover, “bad” points are excluded from the data.

x1 of the TIE with Fig. 7. Applications of the unbiased (l  2) , 1-step ( p  1) predictive FIR filtering estimate ~ x1 ( p  0) is provided with the 2-degree unbiased different measurement noises, by N  7000 . The estimate ~ FIR filter. To demonstrate holdover, “bad” points are excluded from the data. 135

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9. Conclusion In this paper we proposed a new algorithm of compensation of the TIE GPS-based measurement applying an unbiased predictive FIR filter. Unique coefficients were derived for a two-parameter family of the l-degree polynomial filter gain functions. The cases of linear and quadratic predictive filtering, they have been investigated in detail as near optimal, GPS-based measurements of the TIE. Based upon, the fixed-step ( p  1) holdover algorithms has been worked out and examined. It was shown that the filter of degree l  2 is a lot better than the filter of degree l  1 , due to that the estimation of the unbiased predictive FIR filter comes be optimum in the sense that the noise and the bias are almost zero. The filter of degree l  2 was more than sufficient, for which was not necessary to utilize a filter of greater degree to 2. The new algorithm depends of the geometry of the satellites in the moment of the calculated of the position, due that the geometry relates with the DOP. With this we find an indicator for the algorithm, when the DOP is 1, we detect holdover and the predictive filter it active, and when is 0 does not occur anything, the predictive filter is defused. With the new algorithm we reach the accuracy of the atomic clock of Rubidium and the possibility to reach that of an atomic clock of Cesium. We can rival an atomic clock using a receiver GPS, doing that the companies of communication have savings and they can apply it to the synchronization of their teams.

Acknowledgements The work presented in this paper was supported by CONACYT scholarship number 206481.

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