American Institute of Mathematical Sciences

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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

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 and 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), Academic Press, New York, 1953, 273-287.

[2]

D. Blackwell, Comparison of experiments, in Proc. Second Berkeley Symposium on Math. Stat. Probab., 1950, University of California Press, Berkeley and Los Angeles, 1951, 93-102.

[3]

B. Damtie, M. Lehtinen, M. Orispää and J. Vierinen, Mismatched filtering of aperiodic quadriphase codes, IEEE Trans. Inform. Theory, 54 (2008), 1742-1749. doi: 10.1109/TIT.2008.917655.

[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), 797-806. doi: 10.5194/angeo-27-797-2009.

[5]

M. J. E. Golay, Complementary series, IRE Trans., IT-7 (1961), 82-87.

[6]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, $7^{th}$ edition, Elsevier/Academic Press, 2007.

[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), 453-463. doi: 10.1016/0021-9169(95)00048-8.

[8]

K. E. Iverson, A Programming Language, New York: Wiley, p. 11, 1962.

[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), 745-808. doi: 10.1002/j.1538-7305.1960.tb03942.x.

[10]

D. Knuth, Two notes on notation, Amer. Math. Monthly, 99 (1992), 403-422. doi: 10.2307/2325085.

[11]

L. Le Cam, Sufficiency and approximate sufficiency, Ann. Math. Statist., 35 (1964), 1419-1455. doi: 10.1214/aoms/1177700372.

[12]

L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer Series in Statistics, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4946-7.

[13]

M. S. Lehtinen, On optimization of incoherent scatter measurements, Adv. Space Res., 9 (1989), 133-141. doi: 10.1016/0273-1177(89)90351-7.

[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), 1623-1632. doi: 10.5194/angeo-22-1623-2004.

[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), 2291-2301. doi: 10.5194/angeo-26-2291-2008.

[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), 465-486. doi: 10.3934/ipi.2009.3.465.

[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), 281-286. doi: 10.1016/j.jastp.2012.10.010.

[18]

N. Levanon and E. Mozeson, Radar Signals, John Wiley & Sons, Inc. Hoboken, New Jersey, 2004. doi: 10.1002/0471663085.

[19]

P. Piiroinen, Statistical measurements, experiments and applications, Ann. Acad. Sci. Fenn. Math. Diss. No., 143 (2005), 89pp.

[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), 1375-1390. doi: 10.1175/JAM2288.1.

[21]

L. Roininen and M. S. Lehtinen, Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions, Inverse Problems and Imaging, 7 (2013), 649-661. doi: 10.3934/ipi.2013.7.649.

[22]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York, 1974.

[23]

C. E. Shannon, Communication in the presence of noise, Proc. I.R.E., 37 (1949), 10-21.

[24]

A. N. Shiryaev and V. G. Spokoiny, Statistical Experiments and Decisions, Advanced Series on Statistical Science & Applied Probability, 8, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/9789812779243.

[25]

M. I. Skolnik, Radar Handbook, $2^{nd}$ edition, McGraw-Hill Publishing Company, 1990.

[26]

E. Torgersen, Comparison of Statistical Experiments, Encyclopedia of Mathematics and its Applications, 36, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511666353.

[27]

H. L. van Trees, Detection, Estimation and Modulation theory, part III, John Wiley and Sons, 1971.

[28]

J. Vierinen, On Statistical Theory of Radar Measurements, Ph.D. Dissertation, Aalto University, 2012.

[29]

A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60637-3.

show all references

References:
[1]

R. H. Barker, Group synchronizing of binary digital systems, in Communication Theory (ed. W. Jackson), Academic Press, New York, 1953, 273-287.

[2]

D. Blackwell, Comparison of experiments, in Proc. Second Berkeley Symposium on Math. Stat. Probab., 1950, University of California Press, Berkeley and Los Angeles, 1951, 93-102.

[3]

B. Damtie, M. Lehtinen, M. Orispää and J. Vierinen, Mismatched filtering of aperiodic quadriphase codes, IEEE Trans. Inform. Theory, 54 (2008), 1742-1749. doi: 10.1109/TIT.2008.917655.

[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), 797-806. doi: 10.5194/angeo-27-797-2009.

[5]

M. J. E. Golay, Complementary series, IRE Trans., IT-7 (1961), 82-87.

[6]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, $7^{th}$ edition, Elsevier/Academic Press, 2007.

[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), 453-463. doi: 10.1016/0021-9169(95)00048-8.

[8]

K. E. Iverson, A Programming Language, New York: Wiley, p. 11, 1962.

[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), 745-808. doi: 10.1002/j.1538-7305.1960.tb03942.x.

[10]

D. Knuth, Two notes on notation, Amer. Math. Monthly, 99 (1992), 403-422. doi: 10.2307/2325085.

[11]

L. Le Cam, Sufficiency and approximate sufficiency, Ann. Math. Statist., 35 (1964), 1419-1455. doi: 10.1214/aoms/1177700372.

[12]

L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer Series in Statistics, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4946-7.

[13]

M. S. Lehtinen, On optimization of incoherent scatter measurements, Adv. Space Res., 9 (1989), 133-141. doi: 10.1016/0273-1177(89)90351-7.

[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), 1623-1632. doi: 10.5194/angeo-22-1623-2004.

[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), 2291-2301. doi: 10.5194/angeo-26-2291-2008.

[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), 465-486. doi: 10.3934/ipi.2009.3.465.

[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), 281-286. doi: 10.1016/j.jastp.2012.10.010.

[18]

N. Levanon and E. Mozeson, Radar Signals, John Wiley & Sons, Inc. Hoboken, New Jersey, 2004. doi: 10.1002/0471663085.

[19]

P. Piiroinen, Statistical measurements, experiments and applications, Ann. Acad. Sci. Fenn. Math. Diss. No., 143 (2005), 89pp.

[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), 1375-1390. doi: 10.1175/JAM2288.1.

[21]

L. Roininen and M. S. Lehtinen, Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions, Inverse Problems and Imaging, 7 (2013), 649-661. doi: 10.3934/ipi.2013.7.649.

[22]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York, 1974.

[23]

C. E. Shannon, Communication in the presence of noise, Proc. I.R.E., 37 (1949), 10-21.

[24]

A. N. Shiryaev and V. G. Spokoiny, Statistical Experiments and Decisions, Advanced Series on Statistical Science & Applied Probability, 8, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/9789812779243.

[25]

M. I. Skolnik, Radar Handbook, $2^{nd}$ edition, McGraw-Hill Publishing Company, 1990.

[26]

E. Torgersen, Comparison of Statistical Experiments, Encyclopedia of Mathematics and its Applications, 36, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511666353.

[27]

H. L. van Trees, Detection, Estimation and Modulation theory, part III, John Wiley and Sons, 1971.

[28]

J. Vierinen, On Statistical Theory of Radar Measurements, Ph.D. Dissertation, Aalto University, 2012.

[29]

A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60637-3.

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