Abstract
A novel algorithm for estimating the motion parameters of air maneuvering target by means of reconstructing time samples and signal is proposed in this paper. Firstly, the received data of multiple antennas are spliced together to reconstruct time samples of a single antenna by compensating a proper phase. This reconstruction is equivalent to increasing time samples within a single coherent processing interval for a single antenna. After that, an ideal signal whose time sample number is equal to the length of the reconstructed time samples is constructed. At last, the estimation results of initial velocity and acceleration of the air maneuvering target are obtained by applying the nonlinear least squares method to compare the similarity between the reconstructed time samples and signal. The proposed algorithm can achieve accurate parameter estimation with limited pulses. The effectiveness of this algorithm is verified and its performance is very close to the Cramer–Rao bound as shown by our simulation results.
Similar content being viewed by others
References
Barbarossa, S. (1995). Analysis of multicomponent lfm signals by a combined wigner-hough transform. IEEE Transactions on Signal Processing, 43(6), 1511–1515.
Boukamp, B. A. (1986). A nonlinear least squares fit procedure for analysis of immittance data of electrochemical systems. Solid State Ionics, 20(1), 31–44.
Chen, X.-L., Guan, J., Huang, Y., & He, Y. (2013). Application of fractional fourier transform in moving target detection and recognition: Development and prospect. Journal of Signal Processing, 29(1), 85–97.
Chirico, D., & Schirinzi, G. (2013). Multichannel interferometric sar phase unwrapping using extended Kalman Smoother. International Journal of Microwave and Wireless Technologies, 5(03), 429–436.
Clemente, C., & Soraghan, J. J. (2012). Range doppler and chirp scaling processing of synthetic aperture radar data using the fractional fourier transform. IET Signal Processing, 6(5), 503–510.
Dai, W., Hu, J., Xiao, S., Xu, D., & Wang, Z. (2013). Maneuvering weak target detection based on genetic algorithm for oth radar. Radar Science and Technology, 1, 013.
Fa, R., & De Lamare, R. C. (2011). Reduced-rank stap algorithms using joint iterative optimization of filters. IEEE Transactions on Aerospace and Electronic Systems, 47(3), 1668–1684.
Haykin, S., & Widrow, B. (2003). Least-mean-square adaptive filters (Vol. 31). Hoboken: Wiley.
Huang, L., Wu, S., Feng, D., & Zhang, L. (2006). Computationally efficient direction-of-arrival estimation based on partial a priori knowledge of signal sources. EURASIP Journal on Applied Signal Processing, 4–4, 2006.
Huang, L., Wu, S., & Mao, E. (2010). Recursion subspace-based method for bearing estimation: A comparative study. Chinese Journal of Electronics, 19(3), 477–480.
Jain, V., & Blair, W. D. (2009). Filter design for steady-state tracking of maneuvering targets with LFM waveforms. IEEE Transactions on Aerospace and Electronic Systems, 45(2), 765–773.
Klemm, R. (2002). Principles of space-time adaptive processing. UK: IET Publishers.
Le, B., Liu, Z., & Gu, T. (2010). Weak lfm signal dectection based on wavelet transform modulus maxima denoising and other techniques. International Journal of Wavelets, Multiresolution and Information Processing, 8(02), 313–326.
Levanon, N. (2009). Mitigating range ambiguity in high prf radar using inter-pulse binary coding. IEEE Transactions on Aerospace and Electronic Systems, 45(2), 687–697.
Liu, J. C. (2007). Study on accelerating target detection and tracking[D]. PhD Dissertation, National University of Defense Technology, Changsha, China.
Ma, N., & Wang, J. X. (2014). Wideband LFM signal parameter estimation based on compressed sensing theory. In Applied mechanics and materials (Vol. 548, pp. 1160–1165). Trans Tech Publications.
Oberlin, T., Meignen, S., & McLaughlin, S. (2013). Analysis of strongly modulated multicomponent signals with the short-time fourier transform. In ICASSP (pp. 5358–5362).
Osmanoglu, B., Dixon, T. H., & Wdowinski, S. (2014). Three-dimensional phase unwrapping for satellite radar interferometry, I: Dem generation. IEEE Transactions on Geoscience and Remote Sensing, 52(2), 1059–1075.
Ren-biao, W., Xiao-han, W., Hai, L., & Dong-me, W. (2012). Detection and parameter estimation of air maneuveringtargets via reconstructing time samples. Journal of Electronics and Information Technology, 34(4), 936–942.
Singh, K., Saxena, R., & Kumar, S. (2013). Caputo-based fractional derivative in fractional fourier transform domain. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 330–337.
Stoica, P., & Moses, R. L. (2005). Spectral analysis of signals. Upper Saddle River, NJ: Pearson/Prentice Hall.
Stoica, P., & Nehorai, A. (1989). MUSIC, maximum likelihood, and Cramer–Rao bound. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37(5), 720–741.
Wang, Y., & Peng, Y. (2000). Space–time adaptive processing (Vol. 159). Beijing: Tsinghua University Press.
Wu, S. Y. (2011). Study on weak moving targets detection and localization for spaced-based early warning radar. Xi’an, China: Xidian University.
Xiujuan, H., Jiahao, D., Wushan, C., Zhifeng, Z., & Huiping, S. (2009). An adaptive compensation of moving target doppler shift for airborne radar. In 2009 IEEE aerospace conference (pp. 1–6). IEEE.
Yang, X., Liu, Y., & Long, T. (2010). Performance analysis of optimal and reduced-dimension stap for airborne phased array radar. In 2010 9th international symposium on antennas propagation and EM theory (ISAPE) (pp. 1116–1119). IEEE
Acknowledgements
This work was supported by National Nature Science Foundation of China (NSFC) under Grant 61471365, U1633106, 61231017, National University’s Basic Research Foundation of China under Grant No. 3122017007. The work is also supported by the Foundation for Sky Young Scholars of Civil Aviation University of China.
Author information
Authors and Affiliations
Corresponding author
Appendix 1: The proof of cost function
Appendix 1: The proof of cost function
Without losing generality, assume that the amplitude of signal is 1. Considering an N elements system, the number of samples of each antenna is K. Without considering the space phase, the signal after reconstructing time samples is a \(NK \times 1\) vector, given by
where
\(v_0\) and \(a_0\) are the actual motion parameters. Thus the reconstructed signal is
where
v and a are the unknown parameters. Thus, the cost function can be expressed as
Equation (25) can be simplified as
Equation (26) reaches the minimum when the values of the cosine function family in it are all 1, given by
Thus, the results of the cosine function family can be obtain by the linear function family as follow:
where \(p=0\) is the unambiguous situation and \(p \ne 0\) is the ambiguous situation caused by the periodicity of trigonometric function. The explicit solutions to the Eq. (28) is
It is obvious that the solution of the linear function family is \(v=v_0\), \(a=a_0\), \(p=0\), as shown in Fig. 8a. When \(p=0\), each line represents the solution of the equations in Eq. (29). The whole linear function family converges to only one common solution, which is the true value of the estimation results. When \(p \ne 0\), the whole linear function family has no common solution as shown in Fig. 8b.
With the above discussion, the cost function reaches the minimum when and only when \(v=v_0\) and \(a=a_0\). Through the proof of cost function, we can draw a conclusion that the parameter estimation results can be acquired successfully if Eq. (15) is minimized. Meanwhile, the right term of Eq. (15) can be rewritten as
If \({{{\hat{\mathbf{a}}}}(\omega _t )}\) is a full column rank vector, \(\left[ {{{\hat{\mathbf{a}}}}^H (\omega _t ){{\hat{\mathbf{a}}}}(\omega _t )} \right] ^{ - 1}\) could be easily acquired. Equation (30) could be rewritten as
Then, the estimated values of v and a are also obtained by this cost function
Actually, Eqs. (15) and (32) are equivalent in estimating these parameters. The parameters are estimated by Eq. (32) for convenience in engineering.
1.1 Cramer–Rao bound of parameter estimation for air maneuvering target
It is assumed that the unknown parameters are initial velocity, acceleration, amplitude and phase. The covariance matrix of clutter and noise could be obtained from
The parameters v and a are decided by the space–time steering vector \(\tilde{A}\mathbf{{a}}(\omega _s ,\omega _t )\). Since the snapshot is assumed to obey a multivariate Gaussian distribution, the probability density function is
The unknown parameters are arranged to form a \(4 \times 1\) vector
where \(\rho \) and \(\varphi \) are the amplitude and phase, i.e., \(\tilde{A} = \rho e^{j\varphi } \). There are four parameters needed to be estimated, namely, the amplitude, phase, initial velocity and acceleration. The log-likelihood function is
The Cramer–Rao bound for the error covariance matrix of an unbiased estimator \({{\hat{\varvec{\theta }}}}\) is given by (Stoica and Moses 2005)
where \(\mathbf{{J}}\) is the Fisher information matrix. It is convenient to compute \(\mathbf{{J}}\) as
The Fisher information matrix \(\mathbf{{J}}\) is
Let us define the derivative vectors
Through deduction, the Fisher information matrix is shown to have the compact form
The vectors are defined as
The CRB of \({{\hat{\varvec{\theta }}}}\) is obtained from the diagonal elements of \(\mathbf{{J}}^{ - 1}\) , that is
\(\mathbf{{J}}^{ - 1}\) obtained with the inverse lemma of partitioned matrix. As initial velocity and acceleration errors are the results we need, the upper left block of \(\mathbf{{J}}^{ - 1}\) is focused. The inverse partitioned matrix is
Define
where \(\mathbf{{A}},\mathbf{{U}},\mathbf{{V}},\mathbf{{D}}\) are \(2 \times 2\) matrix.
where
According to Eq. (52), the upper left \(2 \times 2\) matrix is
Hence, the CRB of initial velocity is obtained as
and the CRB of acceleration is
Rights and permissions
About this article
Cite this article
Li, H., Zhou, M., Wu, R. et al. Parameter estimation of air maneuvering target for multi-antenna system via reconstructing time samples and signal. Multidim Syst Sign Process 29, 621–641 (2018). https://doi.org/10.1007/s11045-017-0495-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-017-0495-7