Perfect radar pulse compression via unimodular fourier multipliers

Previous Article
Approximate marginalization of unknown scattering in quantitative photoacoustic tomography
IPI Home
This Issue
Next Article
Active arcs and contours

August 2014, 8(3): 831-844. doi: 10.3934/ipi.2014.8.831

Perfect radar pulse compression via unimodular fourier multipliers

Lassi Roininen ^1, , Markku S. Lehtinen ^1, , Petteri Piiroinen ^2, and Ilkka I. Virtanen ^3,

1.	University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland
2.	University of Helsinki, Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki
3.	University of Oulu, Department of Physics, P.O.Box 3000, FI-90014 University of Oulu, Finland

Received March 2014 Revised April 2014 Published August 2014

Abstract
Related Papers

We propose a novel framework for studying radar pulse compression with continuous waveforms. Our methodology is based on the recent developments of the mathematical theory of comparison of measurements. First we show that a radar measurement of a time-independent but spatially distributed radar target is rigorously more informative than another one if the modulus of the Fourier transform of the radar code is greater than or equal to the modulus of the Fourier transform of the second radar code. We then motivate the study by spreading a Gaussian pulse into a longer pulse with smaller peak power and re-compressing the spread pulse into its original form. We then review the basic concepts of the theory and pose the conditions for statistically equivalent radar experiments. We show that such experiments can be constructed by spreading the radar pulses via multiplication of their Fourier transforms by unimodular functions. Finally, we show by analytical and numerical methods some examples of the spreading and re-compression of certain simple pulses.

Keywords: radar experiments, comparison of measurements., Pulse compression.

Mathematics Subject Classification: Primary: 94A12, 62M99; Secondary: 86A2.

Citation: Lassi Roininen, Markku S. Lehtinen, Petteri Piiroinen, Ilkka I. Virtanen. Perfect radar pulse compression via unimodular fourier multipliers. Inverse Problems & Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831

References:

[1]	R. H. Barker, Group synchronizing of binary digital systems,, in Communication Theory* (ed. W. Jackson)*, (1953), 273. Google Scholar
[2]	D. Blackwell, Comparison of experiments,, in Proc. Second Berkeley Symposium on Math. Stat. Probab., (1950), 93. Google Scholar
[3]	B. Damtie, M. Lehtinen, M. Orispää and J. Vierinen, Mismatched filtering of aperiodic quadriphase codes,, IEEE Trans. Inform. Theory, 54 (2008), 1742. doi: 10.1109/TIT.2008.917655. Google Scholar
[4]	B. Damtie and M. S. Lehtinen, Comparison of the performance of different radar pulse compression techniques in an incoherent scatter radar measurement,, Ann. Geophys., 27 (2009), 797. doi: 10.5194/angeo-27-797-2009. Google Scholar
[5]	M. J. E. Golay, Complementary series,, IRE Trans., IT-7* (1961), 82. Google Scholar*
[6]	I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,, $7^{th}$ edition, (2007). Google Scholar
[7]	A. Huuskonen and M. S. Lehtinen, The accuracy of incoherent scatter measurements: Error estimates valid for high signal levels,, J. Atmos. Sol.-Terr. Phy., 58 (1996), 453. doi: 10.1016/0021-9169(95)00048-8. Google Scholar
[8]	K. E. Iverson, A Programming Language,, New York: Wiley, (1962). Google Scholar
[9]	J. R. Klauder, A. C. Price, S. Darlington and W. J. Albersheim, The theory and design of chirp radars,, The Bell System Technical Journal, 39 (1960), 745. doi: 10.1002/j.1538-7305.1960.tb03942.x. Google Scholar
[10]	D. Knuth, Two notes on notation,, Amer. Math. Monthly, 99 (1992), 403. doi: 10.2307/2325085. Google Scholar
[11]	L. Le Cam, Sufficiency and approximate sufficiency,, Ann. Math. Statist., 35 (1964), 1419. doi: 10.1214/aoms/1177700372. Google Scholar
[12]	L. Le Cam, Asymptotic Methods in Statistical Decision Theory,, Springer Series in Statistics, (1986). doi: 10.1007/978-1-4612-4946-7. Google Scholar
[13]	M. S. Lehtinen, On optimization of incoherent scatter measurements,, Adv. Space Res., 9 (1989), 133. doi: 10.1016/0273-1177(89)90351-7. Google Scholar
[14]	M. S. Lehtinen, B. Damtie and T. Nygrén, Optimal binary phase codes and sidelobe-free decoding filters with application to incoherent scatter radar,, Ann. Geophys., 22 (2004), 1623. doi: 10.5194/angeo-22-1623-2004. Google Scholar
[15]	M. S. Lehtinen, I. I. Virtanen and J. Vierinen, Fast comparison of IS radar code sequences for lag profile inversion,, Ann. Geophys., 26 (2008), 2291. doi: 10.5194/angeo-26-2291-2008. Google Scholar
[16]	M. Lehtinen, B. Damtie, P. Piiroinen and M. Orispää, Perfect and almost perfect pulse compression codes for range spread radar targets,, Inverse Problems and Imaging, 3 (2009), 465. doi: 10.3934/ipi.2009.3.465. Google Scholar
[17]	M. S. Lehtinen and B. Damtie, Radar baud length optimisation of spatially incoherent time-independent targets,, J. Atmos. Sol.-Terr. Phy., 105-106* (2013), 105. doi: 10.1016/j.jastp.2012.10.010. Google Scholar*
[18]	N. Levanon and E. Mozeson, Radar Signals,, John Wiley & Sons, (2004). doi: 10.1002/0471663085. Google Scholar
[19]	P. Piiroinen, Statistical measurements, experiments and applications,, Ann. Acad. Sci. Fenn. Math. Diss. No., 143* (2005). Google Scholar*
[20]	J. Pirttilä, M. S. Lehtinen, A. Huuskonen and M. Markkanen, A proposed solution to the range-doppler dilemma of weather radar measurements by using the SMPRF codes, practical results, and a comparison with operational measurements,, J. Appl. Meteor., 44 (2005), 1375. doi: 10.1175/JAM2288.1. Google Scholar
[21]	L. Roininen and M. S. Lehtinen, Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions,, Inverse Problems and Imaging, 7 (2013), 649. doi: 10.3934/ipi.2013.7.649. Google Scholar
[22]	H. H. Schaefer, Banach Lattices and Positive Operators,, Die Grundlehren der mathematischen Wissenschaften, (1974). Google Scholar
[23]	C. E. Shannon, Communication in the presence of noise,, Proc. I.R.E., 37 (1949), 10. Google Scholar
[24]	A. N. Shiryaev and V. G. Spokoiny, Statistical Experiments and Decisions,, Advanced Series on Statistical Science & Applied Probability, (2000). doi: 10.1142/9789812779243. Google Scholar
[25]	M. I. Skolnik, Radar Handbook,, $2^{nd}$ edition, (1990). Google Scholar
[26]	E. Torgersen, Comparison of Statistical Experiments,, Encyclopedia of Mathematics and its Applications, (1991). doi: 10.1017/CBO9780511666353. Google Scholar
[27]	H. L. van Trees, Detection, Estimation and Modulation theory, part III,, John Wiley and Sons, (1971). Google Scholar
[28]	J. Vierinen, On Statistical Theory of Radar Measurements,, Ph.D. Dissertation, (2012). Google Scholar
[29]	A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces,, Springer-Verlag, (1997). doi: 10.1007/978-3-642-60637-3. Google Scholar

show all references

References:

[1]	R. H. Barker, Group synchronizing of binary digital systems,, in Communication Theory* (ed. W. Jackson)*, (1953), 273. Google Scholar
[2]	D. Blackwell, Comparison of experiments,, in Proc. Second Berkeley Symposium on Math. Stat. Probab., (1950), 93. Google Scholar
[3]	B. Damtie, M. Lehtinen, M. Orispää and J. Vierinen, Mismatched filtering of aperiodic quadriphase codes,, IEEE Trans. Inform. Theory, 54 (2008), 1742. doi: 10.1109/TIT.2008.917655. Google Scholar
[4]	B. Damtie and M. S. Lehtinen, Comparison of the performance of different radar pulse compression techniques in an incoherent scatter radar measurement,, Ann. Geophys., 27 (2009), 797. doi: 10.5194/angeo-27-797-2009. Google Scholar
[5]	M. J. E. Golay, Complementary series,, IRE Trans., IT-7* (1961), 82. Google Scholar*
[6]	I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,, $7^{th}$ edition, (2007). Google Scholar
[7]	A. Huuskonen and M. S. Lehtinen, The accuracy of incoherent scatter measurements: Error estimates valid for high signal levels,, J. Atmos. Sol.-Terr. Phy., 58 (1996), 453. doi: 10.1016/0021-9169(95)00048-8. Google Scholar
[8]	K. E. Iverson, A Programming Language,, New York: Wiley, (1962). Google Scholar
[9]	J. R. Klauder, A. C. Price, S. Darlington and W. J. Albersheim, The theory and design of chirp radars,, The Bell System Technical Journal, 39 (1960), 745. doi: 10.1002/j.1538-7305.1960.tb03942.x. Google Scholar
[10]	D. Knuth, Two notes on notation,, Amer. Math. Monthly, 99 (1992), 403. doi: 10.2307/2325085. Google Scholar
[11]	L. Le Cam, Sufficiency and approximate sufficiency,, Ann. Math. Statist., 35 (1964), 1419. doi: 10.1214/aoms/1177700372. Google Scholar
[12]	L. Le Cam, Asymptotic Methods in Statistical Decision Theory,, Springer Series in Statistics, (1986). doi: 10.1007/978-1-4612-4946-7. Google Scholar
[13]	M. S. Lehtinen, On optimization of incoherent scatter measurements,, Adv. Space Res., 9 (1989), 133. doi: 10.1016/0273-1177(89)90351-7. Google Scholar
[14]	M. S. Lehtinen, B. Damtie and T. Nygrén, Optimal binary phase codes and sidelobe-free decoding filters with application to incoherent scatter radar,, Ann. Geophys., 22 (2004), 1623. doi: 10.5194/angeo-22-1623-2004. Google Scholar
[15]	M. S. Lehtinen, I. I. Virtanen and J. Vierinen, Fast comparison of IS radar code sequences for lag profile inversion,, Ann. Geophys., 26 (2008), 2291. doi: 10.5194/angeo-26-2291-2008. Google Scholar
[16]	M. Lehtinen, B. Damtie, P. Piiroinen and M. Orispää, Perfect and almost perfect pulse compression codes for range spread radar targets,, Inverse Problems and Imaging, 3 (2009), 465. doi: 10.3934/ipi.2009.3.465. Google Scholar
[17]	M. S. Lehtinen and B. Damtie, Radar baud length optimisation of spatially incoherent time-independent targets,, J. Atmos. Sol.-Terr. Phy., 105-106* (2013), 105. doi: 10.1016/j.jastp.2012.10.010. Google Scholar*
[18]	N. Levanon and E. Mozeson, Radar Signals,, John Wiley & Sons, (2004). doi: 10.1002/0471663085. Google Scholar
[19]	P. Piiroinen, Statistical measurements, experiments and applications,, Ann. Acad. Sci. Fenn. Math. Diss. No., 143* (2005). Google Scholar*
[20]	J. Pirttilä, M. S. Lehtinen, A. Huuskonen and M. Markkanen, A proposed solution to the range-doppler dilemma of weather radar measurements by using the SMPRF codes, practical results, and a comparison with operational measurements,, J. Appl. Meteor., 44 (2005), 1375. doi: 10.1175/JAM2288.1. Google Scholar
[21]	L. Roininen and M. S. Lehtinen, Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions,, Inverse Problems and Imaging, 7 (2013), 649. doi: 10.3934/ipi.2013.7.649. Google Scholar
[22]	H. H. Schaefer, Banach Lattices and Positive Operators,, Die Grundlehren der mathematischen Wissenschaften, (1974). Google Scholar
[23]	C. E. Shannon, Communication in the presence of noise,, Proc. I.R.E., 37 (1949), 10. Google Scholar
[24]	A. N. Shiryaev and V. G. Spokoiny, Statistical Experiments and Decisions,, Advanced Series on Statistical Science & Applied Probability, (2000). doi: 10.1142/9789812779243. Google Scholar
[25]	M. I. Skolnik, Radar Handbook,, $2^{nd}$ edition, (1990). Google Scholar
[26]	E. Torgersen, Comparison of Statistical Experiments,, Encyclopedia of Mathematics and its Applications, (1991). doi: 10.1017/CBO9780511666353. Google Scholar
[27]	H. L. van Trees, Detection, Estimation and Modulation theory, part III,, John Wiley and Sons, (1971). Google Scholar
[28]	J. Vierinen, On Statistical Theory of Radar Measurements,, Ph.D. Dissertation, (2012). Google Scholar
[29]	A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces,, Springer-Verlag, (1997). doi: 10.1007/978-3-642-60637-3. Google Scholar

[1]	Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems & Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[2]	Lassi Roininen, Markku S. Lehtinen. Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions. Inverse Problems & Imaging, 2013, 7 (2) : 649-661. doi: 10.3934/ipi.2013.7.649
[3]	Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[4]	Eric Falcon. Laboratory experiments on wave turbulence. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 819-840. doi: 10.3934/dcdsb.2010.13.819
[5]	Rafail Krichevskii and Vladimir Potapov. Compression and restoration of square integrable functions. Electronic Research Announcements, 1996, 2: 42-49.
[6]	Matthias Ngwa, Ephraim Agyingi. A mathematical model of the compression of a spinal disc. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1061-1083. doi: 10.3934/mbe.2011.8.1061
[7]	Philip N. J. Eagle, Steven D. Galbraith, John B. Ong. Point compression for Koblitz elliptic curves. Advances in Mathematics of Communications, 2011, 5 (1) : 1-10. doi: 10.3934/amc.2011.5.1
[8]	Joseph D. Fehribach. Using numerical experiments to discover theorems in differential equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 495-504. doi: 10.3934/dcdsb.2003.3.495
[9]	Amin Boumenir, Vu Kim Tuan. Recovery of the heat coefficient by two measurements. Inverse Problems & Imaging, 2011, 5 (4) : 775-791. doi: 10.3934/ipi.2011.5.775
[10]	Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems & Imaging, 2011, 5 (3) : 659-674. doi: 10.3934/ipi.2011.5.659
[11]	Gang Bao, Jun Lai. Radar cross section reduction of a cavity in the ground plane: TE polarization. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 419-434. doi: 10.3934/dcdss.2015.8.419
[12]	Alan J. Terry. Pulse vaccination strategies in a metapopulation SIR model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 455-477. doi: 10.3934/mbe.2010.7.455
[13]	Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations & Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036
[14]	G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783
[15]	Gabriella Bretti, Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 379-394. doi: 10.3934/dcdss.2014.7.379
[16]	Maide Bucolo, Federica Di Grazia, Luigi Fortuna, Mattia Frasca, Francesca Sapuppo. An environment for complex behaviour detection in bio-potential experiments. Mathematical Biosciences & Engineering, 2008, 5 (2) : 261-276. doi: 10.3934/mbe.2008.5.261
[17]	Iain Duff, Jonathan Hogg, Florent Lopez. Experiments with sparse Cholesky using a sequential task-flow implementation. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 237-260. doi: 10.3934/naco.2018014
[18]	Wen-Li Li, Jing-Jing Wang, Xiang-Kui Zhang, Peng Yi. A novel road dynamic simulation approach for vehicle driveline experiments. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1035-1052. doi: 10.3934/dcdss.2019071
[19]	Guillaume Bal, Ian Langmore, François Monard. Inverse transport with isotropic sources and angularly averaged measurements. Inverse Problems & Imaging, 2008, 2 (1) : 23-42. doi: 10.3934/ipi.2008.2.23
[20]	Amin Boumenir, Vu Kim Tuan, Nguyen Hoang. The recovery of a parabolic equation from measurements at a single point. Evolution Equations & Control Theory, 2018, 7 (2) : 197-216. doi: 10.3934/eect.2018010

2018 Impact Factor: 1.469

American Institute of Mathematical Sciences