# American Institute of Mathematical Sciences

May  2013, 7(2): 649-661. doi: 10.3934/ipi.2013.7.649

## Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions

 1 University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland

Received  December 2011 Revised  March 2013 Published  May 2013

We propose a method to construct perfect pulse-compression codes with autoregressive moving average algorithms. We first show the relation between the study of coding and decoding techniques in radar engineering and the study of unimodular polynomials with constrained coefficients. Then we extend the study to unimodular Fourier series and unimodular rational functions. We use the Fourier series and rational functions as transfer functions in the autoregressive moving average algorithms. We show that by a suitable choice of the coefficients, the autoregressive moving average algorithms are realisable, stable and causal. We show examples of some almost perfect codes, i.e. numerically truncated perfect codes. We end by proposing perfect code design principles for practical radar engineering purposes.
Citation: Lassi Roininen, Markku S. Lehtinen. Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions. Inverse Problems and Imaging, 2013, 7 (2) : 649-661. doi: 10.3934/ipi.2013.7.649
##### References:
 [1] C. Chatfield, "The Analysis of Time Series: An Introduction, Sixth Edition," Chapman & Hall/CRC, 2003. [2] B. Damtie and M. S. Lehtinen, Comparison of the performance of different radar pulse compression techniques in an incoherent scatter radar measurement, Annales Geophysicae, 27 (2009), 797-806. doi: 10.5194/angeo-27-797-2009. [3] P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957), 291-300. doi: 10.1307/mmj/1028997963. [4] J. P. Kahane, Sur les polynômes à coefficients unimodulaires, Bull. London Math. Soc., 12 (1980), 321-342. doi: 10.1112/blms/12.5.321. [5] M. Lehtinen, B. Damtie, M. Orispää and P. Piiroinen, 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. [6] N. Levanon and E. Mozeson, "Radar Signals," Wiley-Blackwell, 2004. doi: 10.1002/0471663085. [7] R. Marsalek, P. Jardin and G. Baudoin, From post-distortion to pre-distortion for power amplifiers linearization, IEEE Communications Letters, 7 (2003), 308-310. doi: 10.1109/LCOMM.2003.814714. [8] R. Nikoukar, "Near-Optimal Inversion of Incoherent Scatter Radar Measurements: Coding Schemes, Processing Techniques, and Experiments," Ph.D. thesis, University of Illinois, 2010. [9] M. B. Priestley, "Spectral Analysis and Time Series. Vol. 1.," Academic Press, 1981. [10] J. X. Qiu, D. K. Abe, T. M. Antonsen, B. G. Danly and B. Levush, Linearizability of TWTAs using predistortion techniques, IEEE Transactions on Electron Devices, 52 (2005), 718-727. doi: 10.1109/TED.2005.845838.

show all references

##### References:
 [1] C. Chatfield, "The Analysis of Time Series: An Introduction, Sixth Edition," Chapman & Hall/CRC, 2003. [2] B. Damtie and M. S. Lehtinen, Comparison of the performance of different radar pulse compression techniques in an incoherent scatter radar measurement, Annales Geophysicae, 27 (2009), 797-806. doi: 10.5194/angeo-27-797-2009. [3] P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957), 291-300. doi: 10.1307/mmj/1028997963. [4] J. P. Kahane, Sur les polynômes à coefficients unimodulaires, Bull. London Math. Soc., 12 (1980), 321-342. doi: 10.1112/blms/12.5.321. [5] M. Lehtinen, B. Damtie, M. Orispää and P. Piiroinen, 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. [6] N. Levanon and E. Mozeson, "Radar Signals," Wiley-Blackwell, 2004. doi: 10.1002/0471663085. [7] R. Marsalek, P. Jardin and G. Baudoin, From post-distortion to pre-distortion for power amplifiers linearization, IEEE Communications Letters, 7 (2003), 308-310. doi: 10.1109/LCOMM.2003.814714. [8] R. Nikoukar, "Near-Optimal Inversion of Incoherent Scatter Radar Measurements: Coding Schemes, Processing Techniques, and Experiments," Ph.D. thesis, University of Illinois, 2010. [9] M. B. Priestley, "Spectral Analysis and Time Series. Vol. 1.," Academic Press, 1981. [10] J. X. Qiu, D. K. Abe, T. M. Antonsen, B. G. Danly and B. Levush, Linearizability of TWTAs using predistortion techniques, IEEE Transactions on Electron Devices, 52 (2005), 718-727. doi: 10.1109/TED.2005.845838.
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