# April 2006, Volume 15, Number 1

**P. Ziska, M. Laipert**
[references] [full-text]

Analog Group Delay Equalizers Design Based on Evolutionary Algorithm

This paper deals with a design method of the analog all-pass filter designated for equalization of the group delay frequency response of the analog filter. This method is based on usage of evolutionary algorithm, the Differential Evolution algorithm in particular. We are able to design such equalizers to be obtained equal-ripple group delay frequency response in the pass-band of the low-pass filter. The procedure works automatically without an input estimation. The method is presented on solving practical examples.

- DAVIDEK, V., LAIPERT, M., VLCEK, M. Analog and Digital Filters (Analogove a cislicove filtry in Czech). CVUT Praha, 2000.
- MARTINEK, P., VONDRAS, J. New approach to filters and group delay equaliser transfer function design. In The 8th IEEE Int. Conf. on Electronics, Circuits and Systems ICECS 2001, vol. 1, p. 70.
- MARTINEK, P., VONDRAS, J. Multi-criterion filter design via Differential Evolution method for function minimization. In 1st IEEE Int. Conf. on Circuits and Systems for Communications ICCSC'02 Proceedings. St. Petersburg (Russia): Saint-Petersburg State Technical University, 2002, vol. 1, p. 106-109.
- STORN, R. Differential Evolution Design of an IIR-Filter with Requirements for Magnitude and Group Delay. Technical Report TR-95-026, ICSI, May 1995.
- MICHALEWICZ, Z. Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, Berlin Heidelberg, .1996.
- HOSPODKA, J., BICAK, J. Syntfil - Synthesis of electric filters in Maple. In MSW 2004 [CD-ROM]. Waterloo, 2004, vol. 1. http://syntfil.feld.cvut.cz/syntfil.html
- GREGORIAN, R., TEMES, G. Design techniques for digital and analog all-pass circuits. IEEE Trans. on Circuits and Systems, 1978, vol. 25, no. 12, p. 981-988.
- HENK, T. The generation of arbitrary-phase polynomials by recurrence formulae. J. Circuit Theory Appl., 1981, vol. 9, p. 461-478.
- ZAPLATILEK, K., HAJEK, K., DENK, M. Optimal all-pass network design using numerical optimization loop. In Proc. of the 11th Electr. Devices Systems Conf. EDS 2004, Brno, 2004, p. 98-103.

Keywords: Analog all-pass filters, group delay frequency response, Differential Evolution algorithm

**J. Petrzela, Z. Kolka, S. Hanus**
[references] [full-text]

Simple Chaotic Oscillator: From Mathematical Model to Practical Experiment

This paper shows the circuitry implementation and practical verification of the autonomous nonlinear oscillator. Since it is described by a single third-order differential equation, its state variables can be considered as the position, velocity and acceleration and thus have direct connection to a real physical system. Moreover, for some specific configurations of internal system parameters, it can exhibit a period doubling bifurcation leading to chaos. Two different structures of the nonlinear element were verified by a comparison of numerically integrated trajectory with the oscilloscope screenshots .

- CHUA, L. O., KOMURO, M., MATSUMOTO, T. The double scroll family. IEEE Trans. on CAS I, 1986, vol. 33, no. 11, p. 1073 - 1117.
- POSPISIL, J., KOLKA, Z., HORSKA, J., BRZOBOHATY, J. New reference state model of the third-order piecewise-linear dynamical system. Radioengineering, 2000, vol. 9, no. 3, p. 1 - 4.
- SPROTT, J. C., LINZ, S. J. Algebraically simple chaotic flows. International Journal of Chaos Theory and Applications, 2000, vol. 5, no. 2, p. 1 - 20.
- POSPISIL, J., KOLKA, Z., HORSKA, J., BRZOBOHATY, J. Simplest ODE equivalents of Chua's equations. International Journal of Bifurcation and Chaos, 2000, vol. 10, no. 1, p. 1 - 23.
- CHUA, L. O., LIN, G. N. Canonical realization of Chua's circuit family. IEEE Trans. on CAS I, 1990, vol. 37, no. 7, p. 885 - 902.
- SILVA, CH. P., CHUA, L. O. The overdamped double-scroll family. International Journal of Circuit Theory and Applications, 1988, vol. 16, p. 233 - 302.
- SILVA, CH. P. ShilnikovÂ´s theorem - A tutorial. IEEE Trans. on CAS I, 1993, vol. 40, no. 10, p. 675 - 682.

Keywords: Nonlinear oscillator, chaos, Lyapunov exponents, circuit realization, measurement

**P. Vagner, P. Kutin**
[references] [full-text]

X-Band PLL Synthesizer

This paper deals with design and realization of a PLL synthesizer for the microwave X−band. The synthesizer is intended for use as a local oscillator in a K−band downconverter. The design goal was to achieve very low phase noise and spurious free signal with a sufficient power level. For that purpose a low phase noise MMIC VCO was used in phase locked loop. The PLL works at half the output frequency, therefore there is a frequency doubler at the output of the PLL. The output signal from the frequency doubler is filtered by a band-pass filter and finally amplified by a single stage amplifier.

- ROHDE, U. L. Microwave and Wireless Synthesizers: Theory and Design. New York: John Wiley & Sons, 1997. 638 p.
- GRAVEL, J-F., WIGHT, J. S. On the conception and analysis of a 12-GHz push-push phase-locked DRO. IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 1, January 2006.
- ROHDE, U. L. Synthesizer Design for Microwave Applications. . Synergy Microwave Corporation - Technical Articles, 1999. 51 p.
- VAGNER, P. Kmitoctovy syntezator v pasmu X (X-Band PLL Synthesizer). Diploma thesis. Brno: VUT. 2005 (in Czech).
- HEWLETT-PACKARD COMPANY HP E7404A Specifications and Characteristics. Reference Guide. 1999.

Keywords: Microwave PLL synthesizer, phase locked loop, low phase noise, local oscillator, X-band

**V. Schejbal, P. Bezousek, D. Cermak, Z. Nemec, O. Fiser, M. Hajek**
[references] [full-text]

UWB Propagation through Walls

The propagation of ultra wide band (UWB) signals through walls is analyzed. For this propagation studies, it is necessary to consider not only propagation at a single frequency but in the whole band. The UWB radar output signal is formed by both transmitter and antenna. The effects of antenna receiving and transmitting responses for various antenna types (such as small and aperture antennas) are studied in the frequency as well as time domain. Moreover, UWB radar output signals can be substantially affected due to electromagnetic wave propagation through walls and multipath effects.

- HEYMAN, E., MANDELBAUM, B., SHILOH, J. Ultra-Wideband Short-Pulse Electromagnetics 4. New York: Plenum Press, 1999.
- TAYLOR, J. D. Ultra-Wideband Radar Technology. N. York: CRC, 2001.
- ANDREWS, J. R. UWB Signal sources, antennas & propagation. Application Note AN -14a. Picosecond Pulse Labs, Boulder, USA. August, 2003.
- BEZOUSEK, P., SCHEJBAL, V. Radar technology in the Czech Republic. IEEE Aerospace & Electronic Systems Magazine, 2004, vol. 19, no. 8, p. 27 - 34.
- JOHNK, R.T et al. Time-domain measurements of radiated a conducted UWB Emissions. IEEE Aerospace & Electronic Systems Magazine, 2004, vol. 19, no. 8, p. 18 - 26.
- SCHANTZ, H. The Art and Science of Ultrawideband Antennas. Boston, Artech House, 2005.
- MILLER, E. K. Time-Domain Measurements in Electromagnetics. New York: Van Nostrand Reinhold, 1986.
- COLLIN, R. E. Antennas and Radiowave Propagation. N. York: McGraw-Hill, 1985.
- SCHEJBAL, V. Directivity of planar antennas. IEEE Antennas and Propagation Magazine, 1999, vol. 41, no. 2, p. 60 - 62.
- SCHEJBAL, V. et al. UWB radar signal propagation through walls and multipath effects. IEEE Aerospace & Electronic Systems Magazine. Submitted to publication.
- RAMO, S., WHINNERY, J. R., VAN DUZER, T. Fields and Waves in Communication Electronics. N. York: J. Wiley & Sons, 1994.
- BHATTACHARYYA, A. K. High-Frequency Electromagnetic Techniques: Recent Advances and Applications. N. York: J. Wiley & Sons, 1995.
- PENA, D., FEICK, R., HRISTOV, H. D., GROTE, W. Measurement and modeling of propagation losses in brick and concrete walls for the 900-MHz band. IEEE Transactions on Antennas and Propagation, 2003, vol. 51, no. 1, p. 31 - 39.
- SOUTSOS, M. N., BUNGEY, J. H., MILLARD, S. G., SHAW, M. R., PATTERSON, A. Dielectric properties of concrete and their influence on radar testing. NDT&E International, 2001, vol. 34, p. 419 - 425.
- GIBSON, T. B., JENN, D. C. Prediction and measurement of wall insertion loss. IEEE Transactions on Antennas and Propagation, 1999, vol. 47, no. 1, p. 55 - 57.

Keywords: Ultra wide band, UWB antennas, UWB propagation, multipath effects

**V. I. Djigan**
[references] [full-text]

Lattice RLS for Nonstationary Signal Processing

The paper presents the modification of lattice RLS adaptive filtering algorithms for the case of nonstationary signal processing. The modification includes the using of sliding window and dynamic regularization in the adaptive filter correlation matrix estimation. The algorithms can be implemented by means of sequential or parallel computations. Based on sequential computations, 30 regularized prewindowed, sliding widow and regularized sliding window lattice RLS algorithms were developed. A family of the same algorithms with parallel computations includes 21 units. Lattice algorithms are developed for the application in single-channel adaptive filters with complex-valued weights. Some computation procedures of the algorithms are listed. The results of the algorithm simulation are also presented.

- DJIGAN, V. I. Multichannel RLS and fast RLS adaptive filtering algorithms. Successes of Modern Radioelectronics, 2004, no. 11, p. 48 - 77 (in Russian).
- WIDROW, B. Thinking about thinking: the discovery of the LMS algorithm - DSP history. IEEE Signal Processing Magazine, 2005, vol. 22, no. 1, p. 100 - 106.
- DJIGAN, V. I. Efficiency of adaptive signal processing algorithms implementation on basis of MULTICORE SoC. In Proceedings of the All-Russian Scientific and Technical Conference on Problems of Prospective Microelectronic Systems Development. Moscow (Russia), 2005, p. 453 - 460 (in Russian).
- GAY, S. L. Dynamically regularized fast RLS with application to echo cancellation. In Proceedings of the International Conference on Acoustic Speech and Signal Processing. Atlanta (USA), 1996, p. 957 - 960.
- DJIGAN V. I. Unified approach to the fast time recursive least square adaptive filtering algorithms development. In Proceedings of the 3-rd International Conference on Antennas, Radiocommunication Systems & Means (ICARSM-97). Voronezh (Russia), 1997, vol. 3, p. 33 - 42.
- DJIGAN, V. I. RLS adaptive filtering algorithms based on parallel computations. Radioengineering: Proceedings of Czech and Slovak Technical Universities and URSI Committees, 2005, vol. 14, no. 3, p. 28 - 36.
- ZELNIKER, G., TAYLOR, F. J. Advanced Digital Signal Process-ing: Theory and Applications. New York: Marcel Dekker, Inc., 1994.
- CARAYANNIS, G., MANOLAKIS, D., KALOUPTSIDIS, N. A unified view of parametric processing algorithms for prewindowed signals. Signal Processing, 1986, vol. 10, p. 335 - 368.
- HAYKIN, S. Adaptive Filter Theory. 4-th edition. Prentice Hall, 2001.
- SAYED, A. H. Fundamentals of Adaptive Filtering. Hoboken, NJ: John Wiley and Sons, Inc., 2003.
- DJIGAN, V. I. Lattice RLS adaptive filtering algorithms diversity. Digital Signal Processing, 2005, no. 3, p. 2 -12 (in Russian).
- ZHAO, K., LING, F., LEV-ARI, H., PROAKIS, J. G. Sliding window order-recursive least-squares algorithms. IEEE Trans. Signal Processing, 1994, vol. 42, no. 8, p. 1961 - 1972.
- BAYKAL, B., CONSTANTINIDES, A. G. Sliding window adaptive fast QR and QR-lattice algorithms. IEEE Trans. Signal Processing, 1998, vol. 46, no. 11, p. 2864 - 2876.
- DJIGAN, V. I. Sliding window in lattice RLS algorithms of adaptive filtering. In Proceedings of the 13-th International Conference on Information Systems and Technologies. Moscow (Russia), 2005, vol. 1, p. 151 - 154 (in Russian).
- DJIGAN, V. I. Regularization of lattice RLS algorithms. In Proceedings of the 5-th International Scientific-Technical Conference on Electronics and Informatics 2005. Moscow (Russia), 2005, vol. 2, p. 131 - 132 (in Russian).
- DJIGAN, V. I. A family of lattice adaptive filtering algorithms with parallel computations. In Proceedings of the 14-th International Scientific and Technical Conference on Problem of Information Transmitting and Processing in Telecommunication Networks and Systems. Ryazan (Russia), 2005, p. 203 - 204 (in Russian).
- DJIGAN, V.I. Library of adaptive filtering algorithms. In Proceedings of the 6-th International Conference on Digital Signal Processing and its Applications (DSPA-2004). Moscow (Russia), 2004, vol. 1, pp. 88 - 94.

Keywords: Adaptive filtering, RLS, fast RLS, lattice RLS

**S. Jurko, G. Rozinaj**
[references] [full-text]

High Resolution of the ECG Signal by Polynomial Approximation

Averaging techniques as temporal averaging and space averaging have been successfully used in many applications for attenuating interference [6], [7], [8], [9], [10]. In this paper we introduce interference removing of the ECG signal by polynomial approximation, with smoothing discrete dependencies, to make up for averaging methods. The method is suitable for low-level signals of the electrical activity of the heart often less than 10 m V. Most low-level signals arising from PR, ST and TP segments which can be detected eventually and their physiologic meaning can be appreciated. Of special importance for the diagnostic of the electrical activity of the heart is the activity bundle of His between P and R waveforms. We have established an artificial sine wave to ECG signal between P and R wave. The aim focus is to verify the smoothing method by polynomial approximation if the SNR (signal-to-noise ratio) is negative (i.e. a signal is lower than noise).

- BARTKOVJAK, J., KAROVICOVA, M. The smoothing and derivation with smoothing of discrete dependencies. Metrologia a skusobnictvo, 4/2000; p. 9 - 10 (in Slovak).
- JURKO, S., ROZINAJ, G. Interference removing from ECG signal by polynomial approximation. In 17-th Biennial International EURASIP Conference Biosignal, 2004, p. 158 - 161.
- RANGAYYAN, R. M. Biomedical Signal Analysis. IEEE Press, Wiley-Interscience, 2002.
- MALMIVUO, J., PLONSEY, R.. Bioelectromagnetism, Principles and Application of Bioelectric and Biomagnetic. New York: Oxford University Press, 1995.
- KILPATRICK, D., JOHNSTON, P. R. Origin of the electrocardiogram. IEEE Engineering in Medicine and Biology, 8/9 1994, p. 479 - 486.
- BERBARI, E. J., SCHERLAG, B., J. EL-SHERIF, N., BEFELER, B., ARANDA, J. M.,. LAZARA, R. His-Purkinje electrocardiogram in man: An initial assessment of its uses and limitations. Circulation, Aug. 1976, vol. 54, p. 219 - 224.
- BERBARI, E. J., SCHERLAG, B., LAZARA, R. A computerized technique to record new components of the electrocardiogram. Proc. IEEE, 1977, vol. 65, p. 799 - 802.
- LEXA, J., BYTESNIK, J., STUPKA, J. Technique of non-invasive recording of very low electrical activieties including the His bundle electrical activity. Lekar a technika, 1985, vol.. 16, no. 2, p. 28 - 31 (in Czech).
- LEXA, J., BYTESNIK, J., Non-invasive (beat-by-beat) micropotential detection. Lekar a technika, 1989, vol. 20, no. 1, p.8 - 14 (in Czech).
- SPERANZA, G., BONATO, P., ANTOLINI, R. Analyzing late ventricular potentials: Using an improced alignment technique for signal averaging to increase detection reliability. IEEE Engineering in Medicine and Biology, May/June 1996, p. 88 - 94.

Keywords: Polynomial approximation, electrocardiogram ECG, high resolution of the ECG, low-level signals, interference

**J. Kolouch**
[references] [full-text]

FPGA Based Test Module for Error Bit Evaluation in Serial Links

A test module for serial links is described. In the link transmitter, one module generates pseudorandom pulse signal that is transmitted by the link. Second module located in the link receiver generates the same signal and compares it to the received signal. Errors caused by the signal transmission can be then detected and results sent to a master computer for further processing like statistical evaluation. The module can be used for long-term error monitoring without need for human operator presence.

- ABRAMOVICI, M., BREUER, M. A., FRIEDMAN, A. D. Digital Systems Testing and Testable Design. J. Wiley and Sons, Inc., 1995.
- Recommendation of ITU-T O.151, 1992.
- SAWYER, N. Data Recovery. XAPP224, Xilinx, 2005.
- WILFERT, O., KOLKA, Z., BIOLKOVA, V. Free-space optical communication. In Proc. of Optical Communications 2004. Prague, 2004, p. 207 - 216.
- KOLOUCH, J., KOLKA, Z. Embedded data tester for FSO links. In Proc. of EDS 2005. Electronic Devices and Systems International Conference, Brno, 2005, p. 39 - 43.

Keywords: Serial link, bit error rate, LFSR counter, free-space optical link, FPGA

**B. Enyedi, L. Konyha, K. Fazekas**
[references] [full-text]

Motion Compensated Video Compression with 3D Wavelet Transform and SPIHT

The following paper introduces a low bitrate video coding method on the basis of 3D motion compensated wavelet transform and SPIHT algorithm. In contrast to the conventional algorithms applying motion compensation and differential coding, here wavelet transform is used to exploit the opportunities of time redundancy. For coefficient collection, the 3D version of SPIHT algorithm was selected from the various procedures developed for wavelet transform. Motion vectors are compressed, too (also by wavelet transform), therefore the time and spatial redundancy of coding is exploited here as well. These procedures result in effective video compressing and can easily be aligned to the MPEG4 standard.

- ENYEDI, B., KONYHA, L., FAZEKAS, K. Fast video compression based on 3D wavelet transform and SPIHT. In 7th COST 276 Workshop on Information and Knowledge Management for Integrated Media Communication. Ankara (Turkey), 4-5 November 2004.
- TURAN, J. Fast Translation Invariant Transform and Their Applications. Kosice, Slovakia: elfa Publ. H. ISBN-80-88964-19-9, pp.156, 1999.
- TURAN, J., FAZEKAS, K., GAMEC, J., KOVESI, L. Railway station crowd motion estimation using invertible rapid transform. Image Processing & Communications, International Journal, 1997, vol.3, no.1-2, pp.12-23.
- MALLAT, S. G. A Theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Machine Intell., 1989, vol. 11, pp. 674-693.
- AMIR SAID, PEARLMAN; W. A. A New fast and efficient image codec based on set Partitioning in Hierarchical Trees. IEEE Transaction on Circuit and Systems for Video Technology, vol.5, June 1996, pp 243-250.
- BOTTREAU, V., BENETICRE, M., FELTS, B., PESQUET-POPESCU, B. A fully scalable 3D subband video codec. Image Processing, vol.2, 2001, pp. 1017-1020.
- DAUBECHIES, I. Ten lectures on wavelets. CBMS-NSF Lecture Notes nr. 61, SIAM, 1992.
- http://www.bearcave.com/misl/misl_tech/wavelets/haar.html
- AMIR SAID, PEARLMAN, W. A. A new fast and efficient image codec based on set partitioning in hierarchical trees. IEEE Transactions on Circuits and Systems for Video Technology, vol.6, 1996, pp. 243-250.

Keywords: Video coding, 3D wavelet transform, SPIHT, motion compensation, MPEG-4, motion vector compression

**D. Djebouri, A. Djebbari, M. Djebbouri**
[references] [full-text]

Robust GPS Satellite Signal Acquisition Using Lifting Wavelet Transform

A novel GPS satellite signal acquisition scheme that utilizes lifting wavelet to improve acquisition performance is proposed. Acquisition in GPS system is used to calculate the code phase (or shift) and find the pseudo-range, which is used to calculate the position. The performance of a GPS receiver is assessed by its ability to precisely measure the pseudo-range, which depends on noise linked to the signals in the receiver’s tracking loops. The level of GPS receiving equipment system noise determines in part how precisely pseudo-range can be measured.

Our objective, in this paper, is to achieve robust real-time positioning with maximum of accuracy in the presence of noise. Robust positioning describes a positioning system\'s ability to maintain position data continuity and accuracy through most or all anticipated operational conditions. In order to carry out a robust less complex GPS signals acquisition system and to facilitate its implementation, a substitute algorithm for calculating the convolution by using lifting wavelet decomposition is proposed.

Simulation is used for verifying the performance which shows that the proposed scheme based lifting wavelet transform outperforms both FFT search and signal decimation schemes in the presence of a hostile environment.

- KAPLAN, E. D. Understanding GPS: Principles and Applications. Norwood: Artech House, 1996.
- PARKINSON, B. W., SPILKER, J. J. JR. Global Positioning System: Theory and Application. American Inst. of Astronautics: 1996.
- JAMES, B-Y., TSUI. Fundamentals of Global Positioning System Receivers: A Software Approach. John Wiley & Sons, 2000.
- BRAASCH, M., VAN DIERENDONCK, A. GPS receiver architectures and measurements. Proc. IEEE. 1999, vol. 87, no. 1, p. 48-64.
- PSIAKI, M.-L. Block acquisition of weak GPS signals in a software receiver. In 14th International Technical Meeting of the Satellite Division of the Inst. of Navigation (ION GPS). Salt Lake City, 2001.
- KRUMVIEDA, K., MADHANI, P., THOMAS, J., AXELRAD, P., KOBER, W., HEDDINGS, C., HOWE, P., LEONARD, J. A complete IF software GPS receiver: A tutorial about the details. In 14th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS). Salt Lake City, 2001.
- DJEBBOURI, M., DJEBOURI, D. Fast GPS satellite signal acquisition. ElectronicsLetters.com Journal. 2003, p. 1-12.
- HANNAH, BRUCE, M. Modelling and Simulation of GPS Multipath Propagation. Ph.D. Dissertation. Queensland Univ. of Techn., 2001.
- DJEBOURI, D. New robust GPS signal acquisition technique using lifting wavelet transform. In Proceedings of the IEEE 11th Symposium on Communications and Vehicular Technology (SCVT 2004). Ghent University, Ghent (Belgium), 2004.
- FOLKERS, A., JANNASCH, S., HOFMANN, U.-G. Lifting scheme - an alternative to wavelet transforms for real time applications. GlobalDPS Magazine. 2003, vol. 2, no. 12.
- DAUBECHIES I., SWELDENS, W. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl., 98, vol. 4, no. 3, p. 247-269.
- SWELDENS, W. The lifting scheme: A new philosophy in biorthogonal wavelet constructions. In Wavelet Applications in Signal and Image Processing III (SPIE). 1995.
- CHAYPOOK, R. L. JR., BARANIUK, R.-G. Flexible wavelet transforms using lifting. In Proceedings of the 68th SEG Meeting. New Orleans (Louisiana ,USA), 1998.
- MALLAT, S. A wavelet tour of signal processing. 2 ed. San Diego: Academic Press, 1998.
- HARRIS, F. Multirate Signal Processing for Communication Systems. 1st ed. Prentice Hall PTR, 2004.
- SUTER, B., CHUI, C. Multirate and Wavelet Signal Processing: Wavelet Analysis and Its Applications. Academic Press, Vol. 8, 1997.
- FERNANDEZ, G., PERIASWAMY, S., SWELDENS, W. LIFTPACK: A software package for wavelet transforms using lifting. Wavelet Applications in Signal and Image Processing IV, Proc. SPIE 2825. 1996, http://www.cse.sc.edu/~fernande/liftpack.
- HERLEY, C., VETTERLI, M. Orthogonal time-varying filter banks and wavelets. In Proc. IEEE Int. Symp. Circuits Systems. 1993, vol. 1, p. 391 - 394.
- HERLEY, C. Boundary filters for finite-length signals and time-varying filter banks. IEEE Trans. on Circuits and Systems-II: Analog and Digital Signal Processing. 1995, vol. 42, no. 2, p. 102 - 114.
- SWELDENS, W., SCHRODER, P. Building your own wavelets at home. Tech. Rep. Industrial Mathematics Initiative, Mathematics Department, University of South Carina, 1995, no. 5.
- ALAQEELI, ABDULQADIR, A. Global Positioning System Signal Acquisition and Tracking using Field Programmable Gate Arrays. Ph.D. Dissertation. Ohio University, 2002.
- JOHANSSON, F., MOLLAEI, R., THOR, J., UUSTITALO J. GPS satellite Signal Acquisition and Tracking. Lulea University of Technology, Division of Signal Processing Undergraduate Projects. Sweden, 1998.
- DONOHO, D. De-noising by soft thresholding. IEEE Transactions on Information Theory. 1995, vol. 41, p. 613-627.

Keywords: GPS, acquisition, convolution, FFT, decimation, lifting wavelet