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 & Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831
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show all references

References:
[1]

in Communication Theory (ed. W. Jackson), Academic Press, New York, 1953, 273-287. Google Scholar

[2]

in Proc. Second Berkeley Symposium on Math. Stat. Probab., 1950, University of California Press, Berkeley and Los Angeles, 1951, 93-102.  Google Scholar

[3]

IEEE Trans. Inform. Theory, 54 (2008), 1742-1749. doi: 10.1109/TIT.2008.917655.  Google Scholar

[4]

Ann. Geophys., 27 (2009), 797-806. doi: 10.5194/angeo-27-797-2009.  Google Scholar

[5]

IRE Trans., IT-7 (1961), 82-87.  Google Scholar

[6]

$7^{th}$ edition, Elsevier/Academic Press, 2007.  Google Scholar

[7]

J. Atmos. Sol.-Terr. Phy., 58 (1996), 453-463. doi: 10.1016/0021-9169(95)00048-8.  Google Scholar

[8]

New York: Wiley, p. 11, 1962.  Google Scholar

[9]

The Bell System Technical Journal, 39 (1960), 745-808. doi: 10.1002/j.1538-7305.1960.tb03942.x.  Google Scholar

[10]

Amer. Math. Monthly, 99 (1992), 403-422. doi: 10.2307/2325085.  Google Scholar

[11]

Ann. Math. Statist., 35 (1964), 1419-1455. doi: 10.1214/aoms/1177700372.  Google Scholar

[12]

Springer Series in Statistics, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4946-7.  Google Scholar

[13]

Adv. Space Res., 9 (1989), 133-141. doi: 10.1016/0273-1177(89)90351-7.  Google Scholar

[14]

Ann. Geophys., 22 (2004), 1623-1632. doi: 10.5194/angeo-22-1623-2004.  Google Scholar

[15]

Ann. Geophys., 26 (2008), 2291-2301. doi: 10.5194/angeo-26-2291-2008.  Google Scholar

[16]

Inverse Problems and Imaging, 3 (2009), 465-486. doi: 10.3934/ipi.2009.3.465.  Google Scholar

[17]

J. Atmos. Sol.-Terr. Phy., 105-106 (2013), 281-286. doi: 10.1016/j.jastp.2012.10.010.  Google Scholar

[18]

John Wiley & Sons, Inc. Hoboken, New Jersey, 2004. doi: 10.1002/0471663085.  Google Scholar

[19]

Ann. Acad. Sci. Fenn. Math. Diss. No., 143 (2005), 89pp.  Google Scholar

[20]

J. Appl. Meteor., 44 (2005), 1375-1390. doi: 10.1175/JAM2288.1.  Google Scholar

[21]

Inverse Problems and Imaging, 7 (2013), 649-661. doi: 10.3934/ipi.2013.7.649.  Google Scholar

[22]

Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York, 1974.  Google Scholar

[23]

Proc. I.R.E., 37 (1949), 10-21.  Google Scholar

[24]

Advanced Series on Statistical Science & Applied Probability, 8, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/9789812779243.  Google Scholar

[25]

$2^{nd}$ edition, McGraw-Hill Publishing Company, 1990. Google Scholar

[26]

Encyclopedia of Mathematics and its Applications, 36, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511666353.  Google Scholar

[27]

John Wiley and Sons, 1971. Google Scholar

[28]

Ph.D. Dissertation, Aalto University, 2012. Google Scholar

[29]

Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60637-3.  Google Scholar

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