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SCIENCE CHINA Information Sciences, Volume 62, Issue 2: 022302(2019) https://doi.org/10.1007/s11432-017-9322-2

Quantitative SNR analysis of QFM signals in the LPFT domain with Gaussian windows

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  • ReceivedApr 7, 2017
  • AcceptedJan 8, 2018
  • PublishedOct 15, 2018

Abstract

The purpose of this paper is to present a quantitative SNR analysis of quadratic frequency modulated (QFM) signals. This analysis is located in the continuous-time local polynomial Fourier transform (LPFT) domain using a Gaussian window function based on the definition of 3 dB signal-to-noise ratio (SNR). First, the maximum value of the local polynomial periodogram (LPP), and the 3 dB bandwidth in the LPFT domain for a QFM signal is derived, respectively. Then, based on these results, the 3 dB SNR of a QFM signal with Gaussian window function is given in the LPFT domain with one novel idea highlighted: the relationship among standard SNR, parameters of QFM signals and Gaussian window function is clear, and the potential application is demonstrated in the parameter estimation of a QFM signal using the LPFT. Moreover, the 3 dB SNR in the LPFT domain is compared with that in the linear canonical transform (LCT) domain. The validity of theoretical derivations is confirmed via simulation results. It is shown that, in terms of SNR, QFM signals in the LPFT domain can achieve a significantly better performance than those in the LCT domain.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61671063), and also by Foundation for Innovative Research Groups of National Natural Science Foundation of China (Grant No. 61421001).


References

[1] Wang J Z, Su S Y, Chen Z P. Parameter estimation of chirp signal under low SNR. Sci China Inf Sci, 2015, 58: 020307. Google Scholar

[2] Cohen L. Time-frequency distributions-a review. Proc IEEE, 1989, 77: 941-981 CrossRef Google Scholar

[3] Whitelonis N, Ling H. Radar Signature Analysis Using a Joint Time-Frequency Distribution Based on Compressed Sensing. IEEE Trans Antennas Propagat, 2014, 62: 755-763 CrossRef ADS Google Scholar

[4] Chen V C, Hao Ling V C. Joint time-frequency analysis for radar signal and image processing. IEEE Signal Process Mag, 1999, 16: 81-93 CrossRef ADS Google Scholar

[5] Pitton J W, Kuansan Wang J W, Biing-Hwang Juang J W. Time-frequency analysis and auditory modeling for automatic recognition of speech. Proc IEEE, 1996, 84: 1199-1215 CrossRef Google Scholar

[6] Amin M G. Interference mitigation in spread spectrum communication systems using time-frequency distributions. IEEE Trans Signal Process, 1997, 45: 90-101 CrossRef ADS Google Scholar

[7] Xiang-Gen Xia . A quantitative analysis of SNR in the short-time Fourier transform domain for multicomponent signals. IEEE Trans Signal Process, 1998, 46: 200-203 CrossRef ADS Google Scholar

[8] Bai G, Tao R, Zhao J. Fast FOCUSS method based on bi-conjugate gradient and its application to space-time clutter spectrum estimation. Sci China Inf Sci, 2017, 60: 082302 CrossRef Google Scholar

[9] Mu W F, Amin M G. SNR analysis of time-frequency distributions. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Turkey, 2000. II645--II648. Google Scholar

[10] Li X, Bi G, Ju Y. Quantitative SNR Analysis for ISAR Imaging using LPFT. IEEE Trans Aerosp Electron Syst, 2009, 45: 1241-1248 CrossRef ADS Google Scholar

[11] Xiang-Gen Xia , Genyuan Wang , Chen V C. Quantitative SNR analysis for ISAR imaging using joint time-frequency analysis-Short time Fourier transform. IEEE Trans Aerosp Electron Syst, 2002, 38: 649-659 CrossRef ADS Google Scholar

[12] Stankovic L, Ivanovic V, Petrovic Z. Unified approach to noise analysis in the Wigner distribution and spectrogram. Ann Telecommun, 1996, 11: 585--594. Google Scholar

[13] Stankovic L, Stankovic S. Wigner distribution of noisy signals. IEEE Trans Signal Process, 1993, 41: 956-960 CrossRef ADS Google Scholar

[14] Ouyang X, Amin M G. Short-time Fourier transform receiver for nonstationary interference excision in direct sequence spread spectrum communications. IEEE Trans Signal Process, 2001, 49: 851-863 CrossRef ADS Google Scholar

[15] Song J, Niu Z Y, Zhang J Y. OFD-LFM signal design and performance analysis for distributed aperture fully coherent radar. Sci China Inf Sci, 2015, 45: 968--984. Google Scholar

[16] Xiang-Gen Xia , Chen V C. A quantitative SNR analysis for the pseudo Wigner-Ville distribution. IEEE Trans Signal Process, 1999, 47: 2891-2894 CrossRef ADS Google Scholar

[17] Li X, Bi G, Stankovic S. Local polynomial Fourier transform: A review on recent developments and applications. Signal Processing, 2011, 91: 1370-1393 CrossRef Google Scholar

[18] Wu Y, Li B Z, Cheng Q Y. A quantitative SNR analysis of LFM signals in the linear canonical transform domain with Gaussian windows. In: Proceedings of IEEE International Conference on Mechatronic Sciences, Electric Engineering and Computer (MEC), Shengyang, 2013. 1426--1430. Google Scholar

[19] Li Y, Liu K, Tao R. Adaptive Viterbi-Based Range-Instantaneous Doppler Algorithm for ISAR Imaging of Ship Target at Sea. IEEE J Ocean Eng, 2015, 40: 417-425 CrossRef ADS Google Scholar

[20] O'Shea P. A Fast Algorithm for Estimating the Parameters of a Quadratic FM Signal. IEEE Trans Signal Process, 2004, 52: 385-393 CrossRef ADS Google Scholar

[21] Bai X, Tao R, Wang Z. ISAR Imaging of a Ship Target Based on Parameter Estimation of Multicomponent Quadratic Frequency-Modulated Signals. IEEE Trans Geosci Remote Sens, 2014, 52: 1418-1429 CrossRef ADS Google Scholar

[22] Wang Y, Zhao B. Inverse Synthetic Aperture Radar Imaging of Nonuniformly Rotating Target Based on the Parameters Estimation of Multicomponent Quadratic Frequency-Modulated Signals. IEEE Senss J, 2015, 15: 4053-4061 CrossRef ADS Google Scholar

[23] Katkovnik V. Discrete-time local polynomial approximation of the instantaneous frequency. IEEE Trans Signal Process, 1998, 46: 2626-2637 CrossRef ADS Google Scholar

[24] Djurovi? I, Thayaparan T, Stankovi? L. Adaptive Local Polynomial Fourier Transform in ISAR. EURASIP J Adv Signal Process, 2006, 2006: 36093 CrossRef ADS Google Scholar

[25] Katkovnik V, Gershman A B. A local polynomial approximation based beamforming for source localization and tracking in nonstationary environments. IEEE Signal Process Lett, 2000, 7: 3-5 CrossRef ADS Google Scholar

[26] Guo Y, Li B Z. Blind image watermarking method based on linear canonical wavelet transform and QR decomposition. IET Image Process, 2016, 10: 773-786 CrossRef Google Scholar

[27] Xu T Z, Li B Z. Linear Canonical Transform and Its Application. Beijing: Science Press, 2013. Google Scholar

[28] Bu H X, Bai X, Tao R. Compressed sensing SAR imaging based on sparse representation in fractional Fourier domain. Sci China Inf Sci, 2012, 55: 1789-1800 CrossRef Google Scholar

[29] Liu F, Xu H F, Tao R. Research on resolution between multi-component LFM signals in the fractional Fourier domain. Sci China Inf Sci, 2012, 55: 1301-1312 CrossRef Google Scholar

[30] Wei D, Li Y M. Generalized Sampling Expansions with Multiple Sampling Rates for Lowpass and Bandpass Signals in the Fractional Fourier Transform Domain. IEEE Trans Signal Process, 2016, 64: 4861-4874 CrossRef ADS Google Scholar

[31] Feng Q, Li B Z. Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications. IET Signal Process, 2016, 10: 125-132 CrossRef Google Scholar

[32] Shi J, Liu X P, He L, et al. Sampling and reconstruction in arbitrary measurement and approximation spaces associated with linear canonical transform. IEEE Trans Signal Process, 2016, 64: 6379--6391. Google Scholar

[33] Zhang Z C. Tighter uncertainty principles for linear canonical transform in terms of matrix decomposition. Digital Signal Processing, 2017, 69: 70-85 CrossRef Google Scholar

[34] Wei D, Li Y M. The dual extensions of sampling and series expansion theorems for the linear canonical transform. Optik - Int J Light Electron Opt, 2015, 126: 5163-5167 CrossRef ADS Google Scholar

[35] Kou K I, Xu R H. Windowed linear canonical transform and its applications. Signal Processing, 2012, 92: 179-188 CrossRef Google Scholar

[36] Kou K I, Zhang R H, Zhang Y H. Paley-Wiener theorems and uncertainty principles for the windowed linear canonical transform. Math Method Appl Sci, 2012, 35: 2212--2132. Google Scholar

[37] Tao R, Li Y L, Wang Y. Short-Time Fractional Fourier Transform and Its Applications. IEEE Trans Signal Process, 2010, 58: 2568-2580 CrossRef ADS Google Scholar

[38] Yin Q, Shen L, Lu M. Selection of optimal window length using STFT for quantitative SNR analysis of LFM signal. J Syst Eng Electron, 2013, 24: 26-35 CrossRef Google Scholar

[39] Varadarajan V S. Some problems involving Airy functions. Commun Stoch Anal, 2012, 1: 65--68. Google Scholar

[40] Popescu S A. Mathematical analysis II integral calculus. http://civile.utcb.ro/cmat/cursrt/ma2e.pdf. Google Scholar

  • Figure 1

    Geometry of the problem. Points $P_1$ and $P_2$ determine the limits of the rectangular area enclosing the baseline of the 3 dB surface and $P_0$ is the position of the maximum.

  • Figure 3

    (Color online) (a) The relation between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and LPFT parameter $w_1$; (b) the relation between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and Gaussian window parameter $\alpha$; (c) the relation between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and $w_2-a_2$.

  • Figure 4

    (Color online) The ratio between ${\rm~SNR}_{\rm~LPFT}^{3~{\rm~dB}}$ and ${\rm~SNR}_{\rm~LCT}^{3~{\rm~dB}}$ versus $a/b$

  • Figure 5

    (Color online) LPP of a QFM signal in the noise environment $y(n)$ in (a) $w-w_1$ plane under optimal $w_2=18$, (b) $w-w_2$ plane under optimal $w_1=45$, and (c) $w_1-w_2$ plane under optimal $w=12$.

  • Figure 6

    (Color online) LCT spectrum of a QFM signal in the noise environment $y(n)$ in $u-k$ plane.

  • Figure 7

    (Color online) LCT of a QFM signal in the noise environment $y(n)$ projected in (a) LCT-frequency $u$ axis, (b) parameter-ratio $k$ axis.

  • Figure 8

    (Color online) MSEs of the parameter estimations via LPFT-based method. (a) Constant frequency $a_0$; (b) first chirp rate $a_1$; (c) second chirp rate $a_2$.

  • Table 1   Numerical results for ${\rm~LPP}_x^3(t,\bar~w)\mid_{w_2=a_2}^{3~{\rm~dB}}$ examples in Figure
    Example $t$ $A$ $a_0$ $a_1$ $a_2$$\alpha$ $P_0$ $P_1$ $P_2$ Average Avg/$P_0$
    I$0$$1$ $0.5$$-2$$-1$ $1$ $3.54$ $1.77$ $1.77$ $2.34$ $0.66$
    II$2$$2$ $1$ $4$$-2$ $2$ $10.03$ $5.01$ $5.01$ $6.63$ $0.66$
    III$10$$4$ $3.5$$1$ $-1.5$ $1.5$ $46.31$ $23.16$ $23.16$ $30.62$ $0.66$
    IV$-100$$8$ $-1$$0.5$ $2$ $3$ $130.99$ $65.49$ $65.49$ $86.60$ $0.66$

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