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

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

Lassi Roininen 1, and  Markku S. Lehtinen 1,

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.
Keywords: inverse filter., unimodular polynomials, Perfect codes, autoregressive moving average.
Mathematics Subject Classification: Primary: 68P30; Secondary: 94A12, 93E11, 93E0.
Citation: 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
References:
[1]

C. Chatfield, "The Analysis of Time Series: An Introduction, Sixth Edition,", Chapman & Hall/CRC, (2003).   Google Scholar

[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.  doi: 10.5194/angeo-27-797-2009.  Google Scholar

[3]

P. Erdős, Some unsolved problems,, Michigan Math. J., 4 (1957), 291.  doi: 10.1307/mmj/1028997963.  Google Scholar

[4]

J. P. Kahane, Sur les polynômes à coefficients unimodulaires,, Bull. London Math. Soc., 12 (1980), 321.  doi: 10.1112/blms/12.5.321.  Google Scholar

[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.  doi: 10.3934/ipi.2009.3.465.  Google Scholar

[6]

N. Levanon and E. Mozeson, "Radar Signals,", Wiley-Blackwell, (2004).  doi: 10.1002/0471663085.  Google Scholar

[7]

R. Marsalek, P. Jardin and G. Baudoin, From post-distortion to pre-distortion for power amplifiers linearization,, IEEE Communications Letters, 7 (2003), 308.  doi: 10.1109/LCOMM.2003.814714.  Google Scholar

[8]

R. Nikoukar, "Near-Optimal Inversion of Incoherent Scatter Radar Measurements: Coding Schemes, Processing Techniques, and Experiments,", Ph.D. thesis, (2010).   Google Scholar

[9]

M. B. Priestley, "Spectral Analysis and Time Series. Vol. 1.,", Academic Press, (1981).   Google Scholar

[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.  doi: 10.1109/TED.2005.845838.  Google Scholar

show all references

References:
[1]

C. Chatfield, "The Analysis of Time Series: An Introduction, Sixth Edition,", Chapman & Hall/CRC, (2003).   Google Scholar

[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.  doi: 10.5194/angeo-27-797-2009.  Google Scholar

[3]

P. Erdős, Some unsolved problems,, Michigan Math. J., 4 (1957), 291.  doi: 10.1307/mmj/1028997963.  Google Scholar

[4]

J. P. Kahane, Sur les polynômes à coefficients unimodulaires,, Bull. London Math. Soc., 12 (1980), 321.  doi: 10.1112/blms/12.5.321.  Google Scholar

[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.  doi: 10.3934/ipi.2009.3.465.  Google Scholar

[6]

N. Levanon and E. Mozeson, "Radar Signals,", Wiley-Blackwell, (2004).  doi: 10.1002/0471663085.  Google Scholar

[7]

R. Marsalek, P. Jardin and G. Baudoin, From post-distortion to pre-distortion for power amplifiers linearization,, IEEE Communications Letters, 7 (2003), 308.  doi: 10.1109/LCOMM.2003.814714.  Google Scholar

[8]

R. Nikoukar, "Near-Optimal Inversion of Incoherent Scatter Radar Measurements: Coding Schemes, Processing Techniques, and Experiments,", Ph.D. thesis, (2010).   Google Scholar

[9]

M. B. Priestley, "Spectral Analysis and Time Series. Vol. 1.,", Academic Press, (1981).   Google Scholar

[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.  doi: 10.1109/TED.2005.845838.  Google Scholar

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