Abstract
A novel framework of sub-Nyquist sampling and reconstruction for linear frequency modulation (LFM) radar signals based on the theory of blind compressed sensing (BCS) is proposed. This structure takes LFM signals as a sparse linear combination under an unknown transform order \(p\) in fractional Fourier transform (FRFT) domain. First, making good use of energy concentration of LFM signal in the proper FRFT domain, we determine the optimal order \(p\) which meets the convergence conditions under subsampling condition. Second, discrete fractional Fourier transform (DFRFT) sparse basis is constructed according to the specific sparse FRFT domain dominated by \(p\). Third, based on the DFRFT basis dictionary, using the random demodulator and block reconstruction algorithm, a LFM signal subsampling and reconstruction system is designed in the framework of BCS theory. With this system, the unknown LFM signal in radar system can be sampled at about 1/8 of Nyquist rate without the knowledge of priori sparse basis, but still can be reconstructed with overwhelming probability. Finally, simulations are taken on verifying the feasibility and efficiency of the proposed method, the novel framework can bring a new way to subsample and reconstruct LFM signals under the environment of non-collaboration.
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References
E. Candes, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)
D. Donoho, M. Elad, Maximal sparsity representation via l1 minimization. Proc. Nat. Acad. Sci. 100(3), 2197–2202 (2003)
D. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)
M. Duarte, Y.C. Eldar, Structured compressed sensing: from theory to applications. IEEE Trans. Signal Process. 59(9), 4053–4085 (2011)
Y.C. Eldar, H. Bolcskei, Block sparsity: uncertainty relations and efficient recovery, in ICASSP, 2009, pp. 2885–2888
S. Gleichman, Y.C. Eldar, Multichannel blind compressed sensing, in Sensor Array and Multichannel Signal Processing Workshop (SAM), 2010, pp. 129–132
S. Gleichman, Y.C. Eldar, Blind compressed sensing. IEEE Trans. Inf. Theory 57(10), 6958–6975 (2011)
S. Kirolos, J. Laska, M. Wakin, et al., Analog-to-information conversion via random demodulation, in Proceedings of the IEEE Dallas Circuits and Systems Workshop (DCAS), 2006.
A. Majumdar, R.K. Ward, Fast group sparse classification. Can. J. Electri. Comput. Eng. 34(4), 136–144 (2009)
M. Mishali, Y.C. Eldar, Blind multi-band signal reconstruction: compressed sensing for analog signals. IEEE Trans. Signal Process. 57(3), 993–1009 (2009)
V. Namias, The fractional Fourier transform and its application in quantum mechanics. J. Inst. Math. 25, 241–265 (1980)
H.M. Ozaktas, O. Arikan et al., Digital computation of fractional Fourier transform. IEEE Trans. Signal Process. 44(9), 1737–1740 (1996)
F. Parvaresh, H. Vikalo, S. Misra, B. Hassibi, Recovering sparse signals using sparse measurement matrices in compressed DNA microarrays. IEEE J. Sel. Top. Sign. Process. 2(3), 275–285 (2008)
S.C. Pei, J.J. Ding, Closed-form discrete fractional and affine Fourier transform. IEEE Trans. Signal Process. 48(5), 1338–1353 (2000)
G.M. Shi, J. Lin, X.Y. Chen et al., UWB echo signal detection with ultra-low rate dampling based on compressed sensing. IEEE Trans. Circuits Syst. II Express Briefs 55(4), 379–383 (2008)
R. Tao, B. Deng, Y. Wang, Fractional Fourier transform and its evolution in signal processing. Sci. China Ser. E 36(2), 113–136 (2006)
R. Tao, B. Deng, Y. Wang, Fractional Fourier transform and its application (Tshua University Press, Beijing, 2009)
J.A. Tropp, J. Laska, M. Duarte et al., Beyond Nyquist: efficient sampling of sparse of sparse bandlimited signals. IEEE Trans. Inf. Theory 56(1), 520–544 (2010)
M.P. Vetterli, P. Marziliano, T. Blu, Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50(6), 1417–1428 (2002)
Y.J. Zhao, X.Y. Zhuang, H.J. Wang et al., Model-based multichannel compressive sampling with ultra-low sampling rate. Circuits Syst. Signal Process. 31(4), 1475–1486 (2012)
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Fang, B., Huang, G. & Gao, J. Sub-Nyquist Sampling and Reconstruction Model of LFM Signals Based on Blind Compressed Sensing in FRFT Domain. Circuits Syst Signal Process 34, 419–439 (2015). https://doi.org/10.1007/s00034-014-9859-5
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DOI: https://doi.org/10.1007/s00034-014-9859-5