Adaptive Detection of a Subspace Signal in Structured Random Interference Plus Thermal Noise

Abstract

This paper studies adaptive radar detection of a subspace signal embedded in two disturbance sources. The former is thermal noise with known power. The latter is a Gaussian subspace interference with zero mean and unknown covariance matrix (CM). It is assumed that the signal and interference subspaces are known and partially related. As customary, several secondary data containing only interference and thermal noise are used to estimate this interference CM. This paper derives the generalized likelihood ratio test (GLRT), and theoretically deduces the probabilities of false alarm (PFA) and detection of the new detector. This PFA shows that the new detector has the constant false alarm rate (CFAR) property against the interference CM. Several numerical experiments are performed to evaluate the detection performance of the new detector. The results show that the performance of the new detector is better than the natural counterparts in some scenarios.

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References

  1. 1.

    T.W. Anderson, An introduction to multivariate statistical analysis (Wiley, New York, 2003)

    Google Scholar 

  2. 2.

    A. Aubry, A. De Maio, D. Orlando, M. Piezzo, Adaptive detection of point-like targets in the presence of homogeneous clutter and subspace interference. IEEE Signal Process. Lett. 21(7), 848–852 (2014)

    Google Scholar 

  3. 3.

    F. Bandiera, O. Besson, D. Orlando, G. Ricci, L.L. Scharf, GLRT-based direction detectors in homogeneous noise and subspace interference. IEEE Trans. Signal Process. 55(6), 2386–2394 (2007)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    F. Bandiera, O. Besson, G. Ricci, Direction detector for distributed targets in unknown noise and interference. Electron. Lett. 49(1), 68–69 (2013)

    Google Scholar 

  5. 5.

    F. Bandiera, A. De Maio, A.S. Greco, G. Ricci, Adaptive radar detection of distributed targets in Homogeneous and partially Homogeneous noise plus subspace interference. IEEE Trans. Signal Process. 55(4), 1223–1237 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    S. Bose, A.O. Steinhardt, Adaptive array detection of uncertain rank one waveforms. IEEE Trans. Signal Process. 44(11), 2801–2809 (1996)

    Google Scholar 

  7. 7.

    O. Besson, L.L. Scharf, S. Kraut, Adaptive detection of a signal known only to lie on a line in a known subspace, when primary and secondary data are partially homogeneous. IEEE Trans. Signal Process. 54(12), 4698–4705 (2006)

    MATH  Google Scholar 

  8. 8.

    D. Ciuonzo, A. De Maio, D. Orlando, A unifying framework for adaptive radar detection in homogeneous plus structured interference—part I: on the maximal invariant statistic. IEEE Trans. Signal Process. 64(11), 2894–2906 (2016)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    D. Ciuonzo, A. De Maio, D. Orlando, A unifying framework for adaptive radar detection in homogeneous plus structured interference—part II: detectors design. IEEE Trans. Signal Process. 64(11), 2907–2919 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    D. Ciuonzo, A. De Maio, D. Orlando, On the statistical invariance for adaptive radar detection in partially homogeneous disturbance plus structured interference. IEEE Trans. Signal Process. 65(5), 1222–1234 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    G.L. Cui, L.J. Kong, X.B. Yang, GLRT-based detection algorithm for polarimetric MIMO radar against SIRV clutter. Circuits Syst. Signal Process. 31(3), 1033–1048 (2012)

    MathSciNet  Google Scholar 

  12. 12.

    G.L. Cui, L.J. Kong, X.B. Yang, Performance analysis of colocated MIMO radars with randomly distributed arrays in compound-Gaussian clutter. Circuits Syst. Signal Process. 31(4), 1407–1422 (2012)

    MathSciNet  Google Scholar 

  13. 13.

    G.L. Cui, L.J. Kong, X.B. Yang, J.Y. Yang, The Rao and Wald tests designed for distributed targets with polarization MIMO radar in compound-Gaussian clutter. Circuits Syst. Signal Process. 31(1), 237–254 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    D. Ciuonzo, D. Orlando, L. Pallotta, On the maximal invariant statistic for adaptive radar detection in partially homogeneous disturbance with persymmetric covariance. IEEE Signal Process. Lett. 23(12), 1830–1834 (2016)

    Google Scholar 

  15. 15.

    A. De Miao, A new derivation of the adaptive matched filter. Signal Process. Lett. 11(10), 792–793 (2004)

    Google Scholar 

  16. 16.

    A. De Maio, Rao test for adaptive detection in Gaussian interference with unknown covariance matrix. IEEE Trans. Signal Process. 55(7), 3577–3584 (2007)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    A. De Maio, D. Orlando, Adaptive radar detection of a subspace signal embedded in subspace structured plus Gaussian interference via invariance. IEEE Trans. Signal Process. 64(8), 2156–2167 (2016)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    A. De Maio, D. Orlando, I. Soloveychik, A. Wiesel, Invariance theory for adaptive detection in interference with group symmetric covariance matrix. IEEE Trans. Signal Process. 64(23), 6299–6312 (2016)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    A. De Maio, G. Ricci, A polarimetric adaptive matched filter. Signal Process. 81, 2583–2589 (2001)

    MATH  Google Scholar 

  20. 20.

    Y. Fujikoshi, V.V. Ulyanov, R. Shimizu, Multivariate statistics: high-dimensional and large-sample approximations (Wiley, Hoboken, 2010)

    Google Scholar 

  21. 21.

    Y.C. Gao, H.B. Ji, W.J. Liu, Persymmetric adaptive subspace detectors for range-spread targets. Digit. Signal Process. 89, 116–123 (2019)

    MathSciNet  Google Scholar 

  22. 22.

    D.A. Harville, Matrix algebra from a statistician’s perspective (Springer, New York, 1997)

    Google Scholar 

  23. 23.

    M. Hurtado, A. Nehorai, Polarimetric detection of targets in heavy inhomogeneous clutter. IEEE Trans. Signal Process. 56(4), 1349–1361 (2008)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    C. Hao, J. Yang, X. Ma, C. Hou, D. Orlando, Adaptive detection of distributed targets with orthogonal rejection. IET Radar Sonar Navig. 6(6), 483–493 (2012)

    Google Scholar 

  25. 25.

    T. Jian, Y. He, F. Su, C. Qu, Adaptive range-spread target detection based on modified generalised likelihood ratio test in non-Gaussian clutter. IET Radar Sonar Navig. 5(9), 970–977 (2011)

    Google Scholar 

  26. 26.

    E.J. Kelly, An adaptive detection algorithm. IEEE Trans. Aerosp. Electron. Syst. 22(1), 115–127 (1986)

    Google Scholar 

  27. 27.

    S. Kraut, L.L. Scharf, R.W. Butler, The adaptive coherence estimator: a uniformly most powerful invariant adaptive detection statistic. IEEE Trans. Signal Process. 53(2), 427–438 (2005)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    N. Li, G.L. Cui, L.J. Kong, X.B. Yang, MIMO radar moving target detection against compound-Gaussian clutter. Circuits Syst. Signal Process. 33(6), 1819–1839 (2014)

    Google Scholar 

  29. 29.

    W.J. Liu, H. Han, J. Liu, H.L. Li, K. Li, Y.L. Wang, Multichannel radar adaptive signal detection in interference and structure nonhomogeneity. Sci. China Inf. Sci. 60(11), 112302 (2017)

    Google Scholar 

  30. 30.

    J. Liu, J.W. Han, W.J. Liu, S.W. Xu, Z.J. Zhang, Persymmetric Rao test for MIMO radar in Gaussian disturbance. Signal Process. 165, 30–36 (2019)

    Google Scholar 

  31. 31.

    H.B. Li, Y. Jiang, J. Fang, M. Rangaswamy, Adaptive subspace signal detection with uncertain partial prior knowledge. IEEE Trans. Signal Process. 65(16), 4394–4405 (2017)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    J. Liu, W.J. Liu, B. Chen, H.W. Liu, H.B. Li, C.P. Hao, Modified Rao test for multichannel adaptive signal detection. IEEE Trans. Signal Process. 64(3), 714–725 (2016)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    W.J. Liu, J. Liu, L. Huang, D. Zou, Y. Wang, Rao tests for distributed target detection in interference and noise. Signal Process. 117, 333–342 (2015)

    Google Scholar 

  34. 34.

    W.J. Liu, J. Liu, H. Li, Q.L. Du, Y.L. Wang, Multichannel signal detection based on Wald test in subspace interference and Gaussian noise. IEEE Trans. Aerosp. Electron. Syst. 55(3), 1370–1381 (2019)

    Google Scholar 

  35. 35.

    J. Liu, S.Y. Sun, W.J. Liu, One-step persymmetric GLRT for subspace signals. IEEE Trans. Signal Process. 67(14), 3639–3648 (2019)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    W.J. Li, H.B. Tong, K. Li, Y. Yang, W.J. Liu, Wald tests for direction detection in noise and interference. Multidimens. Syst. Sign. Process. 29(4), 1563–1577 (2018)

    Google Scholar 

  37. 37.

    H.L. Li, H. Wang, H. Han, H.Y. Dai, W.J. Liu, Performance analysis of adaptive detectors for a distributed target based on subspace model. Circuits Syst. Signal Process. 37(6), 2651–2664 (2018)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    W.J. Liu, W.C. Xie, J. Liu, Y.L. Wang, Adaptive double subspace signal detection in Gaussian background—part I: homogeneous environments. IEEE Trans. Signal Process. 62(9), 2345–2357 (2014)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    W.J. Liu, W.C. Xie, J. Liu, Y.L. Wang, Adaptive double subspace signal detection in Gaussian background—part II: partially homogeneous environments. IEEE Trans. Signal Process. 62(9), 2358–2369 (2014)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    W.J. Liu, W.C. Xie, J. Liu, D.J. Zou, H.L. Wang, Y.L. Wang, Detection of a distributed target with direction uncertainty. IET Radar Sonar Navig. 8(9), 1177–1183 (2014)

    Google Scholar 

  41. 41.

    N. Li, H.N. Yang, G.L. Cui, L.J. Kong, Q.H. Liu, Adaptive two-step Bayesian MIMO detectors in compound-Gaussian clutter. Signal Process. 161, 1–13 (2019)

    Google Scholar 

  42. 42.

    J. Liu, Z. Zhang, Y. Cao, M. Wang, Distributed target detection in subspace interference plus Gaussian noise. Signal Process. 95, 88–100 (2014)

    Google Scholar 

  43. 43.

    S.W. Lei, Z.Q. Zhao, Z.P. Nie, Analyses of the performance of adaptive subspace detector on fluctuating target detection in system-dependent clutter background. IET Radar Sonar Navig. 10(9), 1635–1642 (2017)

    Google Scholar 

  44. 44.

    S.W. Lei, Z.Q. Zhao, Z.P. Nie, Q.H. Liu, A CFAR adaptive subspace detector based on a single observation in system-dependent clutter background. IEEE Trans. Signal Process. 62(20), 5260–5269 (2014)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    J. Liu, Z.J. Zhang, Y. Yang, H. Liu, A CFAR adaptive subspace detector for first-order or second-order Gaussian signals based on a single observation. IEEE Trans. Signal Process. 59(11), 5126–5140 (2011)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    H.R. Park, J. Li, H. Wang, Polarization-space-time domain generalized likelihood ratio detection of radar targets. Signal Process. 41, 153–164 (1995)

    MATH  Google Scholar 

  47. 47.

    F.C. Robey, D.R. Fuhrmann, E.J. Kelly, R. Nitzberg, A CFAR adaptive matched filter detector. IEEE Trans. Aerosp. Electron. Syst. 28(1), 208–216 (1992)

    Google Scholar 

  48. 48.

    G.A.F. Seber, The non-central Chi squared and Beta distributions. Biometrika 50(3/4), 542–544 (1963)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    L.L. Scharf, Statistical signal processing: detection estimation and time series analysis (Addison, Boston, 1991)

    Google Scholar 

  50. 50.

    A.H. Sayed, Fundamentals of adaptive filtering (Wiley, New York, 2003)

    Google Scholar 

  51. 51.

    G.H. Sun, Z.S. He, Y.L. Zhang, Distributed airborne MIMO radar detection in compound-Gaussian clutter without training data. Circuits Syst. Signal Process. 37(10), 4617–4636 (2018)

    Google Scholar 

  52. 52.

    Z.R. Shang, X. Li, Y.X. Liu, Y.L. Wang, W.J. Liu, GLRT detector based on knowledge aided covariance estimation in compound Gaussian environment. Signal Process. 155, 377–383 (2019)

    Google Scholar 

  53. 53.

    L.L. Scharf, M. McCloud, Blind adaptation of zero forcing projections and oblique pseudo-inverses for subspace detection and estimation when interference dominates noise. IEEE Trans. Signal Process. 50(12), 2938–2946 (2002)

    Google Scholar 

  54. 54.

    I. Soloveychik, A. Wiesel, Group symmetric robust covariance estimation. IEEE Trans. Signal Process. 64(1), 244–257 (2016)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    M. Tang, Y. Rong, J. Zhou, X.R. Li, Invariant adaptive detection of range-spread targets under structured noise covariance. IEEE Trans. Signal Process. 65(12), 3048–3061 (2017)

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Z. Wang, M. Li, H. Chen, L. Zuo, P. Zhang, Y. Wu, Adaptive detection of a subspace signal in signal-dependent interference. IEEE Trans. Signal Process. 65(18), 4812–4820 (2017)

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Y.K. Wang, W. Xia, Z.S. He, H.B. Li, A.P. Petropulu, Polarimetric detection in compound Gaussian clutter with kronecker structured covariance matrix. IEEE Trans. Signal Process. 65(17), 4562–4576 (2017)

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Z.Z. Wang, Z.Q. Zhao, C.H. Ren, Z.P. Nie, CFAR subspace detectors with multiple observations in system-dependent clutter background. Signal Process. 153, 58–70 (2018)

    Google Scholar 

  59. 59.

    Z.Z. Wang, Z.Q. Zhao, C.H. Ren, Z.P. Nie, Adaptive GLR-, Rao- and Wald-based CFAR detectors for a subspace signal embedded in structured Gaussian interference. Digit. Signal Process. 92, 139–150 (2019)

    Google Scholar 

  60. 60.

    Z.Z. Wang, Z.Q. Zhao, C.H. Ren, Z.P. Nie, W. Yang, Adaptive detection of point-like targets based on a reduced-dimensional data model. Signal Process. 158, 36–47 (2019)

    Google Scholar 

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Appendices

Appendix A: Derivations of the GLRT

In this section, first we expand \( \ln f_{1} \left( {\mathbf{X}} \right) \) and derive the MLEs of \( {\mathbf{p}} \) and \( {\mathbf{R}} \). Then, we derive \( \mathop {\hbox{max} }\limits_{{{\mathbf{p}},{\mathbf{R}}}} \ln f_{1} \left( {\mathbf{X}} \right) \) with \( {\mathbf{p}} \) and \( {\mathbf{R}} \) being replaced by their MLEs. \( \mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{0} \left( {\mathbf{X}} \right) \) can be obtained in a similar approach.

Applying the Corollaries 18.1.2 in [22], we expand \( \ln \left| {\mathbf{C}} \right| \) as

$$ \ln \left| {\mathbf{C}} \right| = \left( {M - Q} \right)\ln \sigma^{2} + \ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \ln \left| {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right|. $$
(20)

Resorting to the Corollaries 18.2.9 in [22], we rewrite \( {\mathbf{C}}^{ - 1} \) as

$$ {\mathbf{C}}^{ - 1} = \frac{1}{{\sigma^{2} }}{\mathbf{P}}_{{\mathbf{J}}}^{ \bot } + {\mathbf{U}}^{H} \left[ {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right]^{ - 1} {\mathbf{U}}, $$
(21)

where \( {\mathbf{U}} = \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} {\mathbf{J}}^{H} \).

Substituting (20) and (21) into (4) leads to the expended \( \ln f_{1} \left( {\mathbf{X}} \right) \), which is

$$ \begin{aligned} \ln f_{1} \left( {\mathbf{X}} \right) & = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| - \left( {K + 1} \right)\ln \left| {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right| \\ & - \frac{1}{{\sigma^{2} }}tr\left( {{\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{T}}_{1} } \right) - tr\left\{ {\left[ {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right]^{ - 1} {\mathbf{UT}}_{1} {\mathbf{U}}^{H} } \right\}. \\ \end{aligned} $$
(22)

According to the Lemma 3.2.2 in [1], the MLE of \( {\mathbf{R}} \) under \( H_{1} \), noted as \( {\hat{\mathbf{R}}}_{1} \), is derived as

$$ {\hat{\mathbf{R}}}_{1} = \frac{1}{K + 1}{\mathbf{UT}}_{1} {\mathbf{U}}^{H} - \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} . $$
(23)

Let \( {\mathbf{R}} = {\hat{\mathbf{R}}}_{1} \), the maximum of \( \ln f_{1} \left( {\mathbf{X}} \right) \) w.r.t. \( {\mathbf{R}} \) is obtained as

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{1} \left( {\mathbf{X}} \right) & = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \left( {K + 1} \right)Q\ln \left( {K + 1} \right) - \left( {K + 1} \right)Q \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{UT}}_{1} {\mathbf{U}}^{H} } \right| - \frac{1}{{\sigma^{2} }}tr\left( {{\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{T}}_{1} } \right). \\ \end{aligned} $$
(24)

Nulling the partial derivative of (24) w.r.t. \( {\mathbf{x}} \) leads to the MLE of \( {\mathbf{p}} \) under \( H_{1} \) (denoted by \( {\hat{\mathbf{p}}}_{1} \))

$$ {{\partial \left[ {\mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{1} \left( {\mathbf{X}} \right)} \right]} \mathord{\left/ {\vphantom {{\partial \left[ {\mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{1} \left( {\mathbf{X}} \right)} \right]} {\partial {\mathbf{p}}}}} \right. \kern-0pt} {\partial {\mathbf{p}}}} = {\mathbf{0}}_{P} \Rightarrow {\mathbf{x}} - {\mathbf{Hp}} = {\mathbf{0}}_{M} \Rightarrow {\hat{\mathbf{p}}}_{1} = \left( {{\mathbf{H}}^{H} {\mathbf{H}}} \right)^{ - 1} {\mathbf{H}}^{H} {\mathbf{x}}. $$
(25)

Let \( {\mathbf{p}} = {\hat{\mathbf{p}}}_{1} \), the maximum of (24) w.r.t. \( {\mathbf{p}} \) is expressed as

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{{\mathbf{p}},{\mathbf{R}}}} \ln f_{1} \left( {\mathbf{X}} \right) = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \left( {K + 1} \right)Q\ln \left( {K + 1} \right) \\ - \left( {K + 1} \right)Q - \left( {K + 1} \right)\ln \left| {{\mathbf{USU}}^{H} } \right| \\ - \left( {K + 1} \right)\varPhi \left( {\mathbf{x}} \right), \\ \end{aligned} $$
(26)

where

$$ \varPhi \left( {\mathbf{x}} \right) = \frac{{{\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{x}}}}{{\left( {K + 1} \right)\sigma^{2} }} + \ln \left[ {1 + {\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{x}}} \right]. $$
(27)

As \( {\mathbf{Hp}} \) does not exist under \( H_{0} \), \( \mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{0} \left( {\mathbf{X}} \right) \) can be obtained from (26) by setting \( {\mathbf{H}} = {\mathbf{0}}_{M \times P} \), which is

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{0} \left( {\mathbf{X}} \right) & = \left. {\mathop {\hbox{max} }\limits_{{{\mathbf{p}},{\mathbf{R}}}} \ln f_{1} \left( {\mathbf{X}} \right)} \right|_{{{\mathbf{H}} = {\mathbf{0}}_{M \times P} }} \\ & = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \left( {K + 1} \right)Q\ln \left( {K + 1} \right) \\ & - \left( {K + 1} \right)Q - \left( {K + 1} \right)\ln \left| {{\mathbf{USU}}^{H} } \right| \\ & - \left( {K + 1} \right)\left( {\lambda_{1} + \lambda_{2} } \right), \\ \end{aligned} $$
(28)

where

$$ \left\{ \begin{aligned} \lambda_{1} = \frac{{{\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{x}}}}{{\left( {K + 1} \right)\sigma^{2} }} \\ \lambda_{2} = \ln \left[ {1 + {\mathbf{x}}^{H} {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{x}}} \right] \\ \end{aligned} \right.. $$
(29)

Appendix B: PDFs of \( \lambda_{1} \), \( \lambda_{2} \) and \( \lambda = \lambda_{1} + \lambda_{2} \)

B.1 PDFs of \( \lambda_{1} \) in (29)

According to the descriptions on page 65 of [49], the PDFs of \( \lambda_{1} \) under \( H_{0} \) and \( H_{1} \) can be derived as

$$ \left\{ {\begin{array}{*{20}c} {2\left( {K + 1} \right)\lambda_{1} \sim\chi_{{2\left( {M - Q} \right)}}^{2} \;\;\;\;\;\;} & {{\text{under}}\;H_{0} } \\ {2\left( {K + 1} \right)\lambda_{1} \sim\chi_{{2\left( {M - Q} \right)}}^{2} \left( {\mu_{1} } \right)} & {{\text{under}}\;H_{1} } \\ \end{array} } \right., $$
(30)

where \( \mu_{1} \) is a non-centrality parameter

$$ \mu_{1} = \frac{{2{\mathbf{p}}^{H} {\mathbf{H}}^{H} {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{Hp}}}}{{\sigma^{2} }}. $$
(31)

Therefore, the PDFs of \( \lambda_{1} \) under \( H_{0} \) and \( H_{1} \) can be written as

$$ h_{1} \left( {\left. {\lambda_{1} } \right|H_{0} } \right) = \frac{K + 1}{{\varGamma \left( {\delta_{1} + 1} \right)}}\gamma_{1}^{{\delta_{1} }} e^{{ - \gamma_{1} }} , $$
(32)
$$ h_{1} \left( {\left. {\lambda_{1} } \right|H_{1} } \right) = \frac{K + 1}{{e^{{{{\left( {2\gamma_{1} + \mu_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {2\gamma_{1} + \mu_{1} } \right)} 2}} \right. \kern-0pt} 2}}} }}\left( {{{2\gamma_{1} } \mathord{\left/ {\vphantom {{2\gamma_{1} } {\mu_{1} }}} \right. \kern-0pt} {\mu_{1} }}} \right)^{{{{\delta_{1} } \mathord{\left/ {\vphantom {{\delta_{1} } 2}} \right. \kern-0pt} 2}}} {\mathcal{I}}_{{\delta_{1} }} \left( {\sqrt {2\gamma_{1} \mu_{1} } } \right), $$
(33)

respectively, where \( \delta_{1} = M - Q - 1 \) and \( \gamma_{1} = \left( {K + 1} \right)\lambda_{1} \).

B.2 PDFs of \( \lambda_{2} \) in (29)

According to the Theorem 5.2.2 in [1], we have

$$ \left\{ {\begin{array}{*{20}c} {e^{{\lambda_{2} }} - 1\sim\frac{{\chi_{2Q}^{2} }}{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} }}} & {{\text{under}}\;H_{0} } \\ {e^{{\lambda_{2} }} - 1\sim\frac{{\chi_{2Q}^{2} \left( {\mu_{2} } \right)}}{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} }}} & {{\text{under}}\;H_{1} } \\ \end{array} ,} \right. $$
(34)

where \( \mu_{2} \) is a non-centrality parameter

$$ \mu_{2} = 2{\mathbf{p}}^{H} {\mathbf{H}}^{H} {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{CJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{Hp}}, $$
(35)

where \( {\mathbf{C}} \) is defined in (5).

According to (34), we have

$$ \left\{ {\begin{array}{*{20}c} {e^{{ - \lambda_{2} }} \sim\frac{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} }}{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} + \chi_{2Q}^{2} }}} & {{\text{under}}\;H_{0} } \\ {1 - e^{{ - \lambda_{2} }} \sim\frac{{\chi_{2Q}^{2} \left( {\mu_{2} } \right)}}{{\chi_{2Q}^{2} \left( {\mu_{2} } \right) + \chi_{{2\left( {K - Q + 1} \right)}}^{2} }}} & {{\text{under}}\;H_{1} } \\ \end{array} } \right.. $$
(36)

Referring to [48], we know that \( e^{{ - \lambda_{2} }} \) follows a Beta distribution with two shape parameters \( \left\{ {K - Q + 1,Q} \right\} \) under \( H_{0} \). \( 1 - e^{{ - \lambda_{2} }} \) follows a Beta distribution with two shape parameters \( \left\{ {Q,K - Q + 1} \right\} \) and a non-centrality parameter \( \mu_{2} \) under \( H_{1} \). Namely,

$$ \left\{ {\begin{array}{*{20}l} {e^{{ - \lambda_{2} }} \sim{\mathcal{B}\mathcal{E}\mathcal{T}\mathcal{A}}\left( {K - Q + 1,Q} \right)} \hfill & {{\text{under}}\;H_{0} } \hfill \\ {1 - e^{{ - \lambda_{2} }} \sim{\mathcal{B}\mathcal{E}\mathcal{T}\mathcal{A}}\left( {Q,K - Q + 1,\mu_{2} } \right)} \hfill & {{\text{under}}\;H_{1} .} \hfill \\ \end{array} } \right. $$
(37)

Based on (37), we derive the PDFs of \( \lambda_{2} \) under \( H_{0} \) and \( H_{1} \) as

$$ h_{2} \left( {\left. {\lambda_{2} } \right|H_{0} } \right) = \frac{{\left( {1 - e^{{ - \lambda_{2} }} } \right)^{Q - 1} }}{{{\mathcal{B}}\left( {\delta_{2} ,Q} \right)e^{{\gamma_{2} }} }}, $$
(38)
$$ h_{2} \left( {\left. {\lambda_{2} } \right|H_{1} } \right) = \sum\limits_{j = 0}^{ + \infty } {\frac{{\left( {{{\mu_{2} } \mathord{\left/ {\vphantom {{\mu_{2} } 2}} \right. \kern-0pt} 2}} \right)^{j} }}{{e^{{{{\left( {2\gamma_{2} + \mu_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {2\gamma_{2} + \mu_{2} } \right)} 2}} \right. \kern-0pt} 2}}} \cdot j!}} \cdot \frac{{\left( {1 - e^{{ - \lambda_{2} }} } \right)^{Q + j - 1} }}{{{\mathcal{B}}\left( {Q + j,\delta_{2} } \right)}}} , $$
(39)

respectively, where \( \delta_{2} = K - Q + 1 \) and \( \gamma_{2} = \delta_{2} \lambda_{2} \).

B.3 PDFs of \( \lambda = \lambda_{1} + \lambda_{2} \) in (13)

As given in (29), \( \lambda_{1} \propto {\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{x}} \) and \( \lambda_{2} \propto {\mathbf{x}}^{H} {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{x}} \). Observe that \( {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } \times {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} = {\mathbf{0}}_{M \times Q} \), by applying the Theorem 2.4.2 (2) in [20], we know that \( \lambda_{1} \) and \( \lambda_{2} \) are independent. Thus, the PDF of \( \lambda = \lambda_{1} + \lambda_{2} \) under \( H_{i} \), denoted by \( g\left( {\left. \lambda \right|H_{i} } \right) \), is the convolution between \( h_{1} \left( {\left. {\lambda_{1} } \right|H_{i} } \right) \) and \( h_{2} \left( {\left. {\lambda_{2} } \right|H_{i} } \right) \), \( i = 0,1 \). Namely,

$$ \begin{aligned} g\left( {\left. \lambda \right|H_{i} } \right) & = h_{1} \left( {\left. {\lambda_{1} } \right|H_{i} } \right) \otimes h_{2} \left( {\left. {\lambda_{2} } \right|H_{i} } \right) \\ & = \int_{0}^{\lambda } {h_{1} \left( {\left. t \right|H_{i} } \right)h_{2} \left( {\left. {\lambda - t} \right|H_{i} } \right){\text{d}}t} ,\quad i = 0,1. \\ & = \int_{0}^{\lambda } {h_{1} \left( {\left. {\lambda - t} \right|H_{i} } \right)h_{2} \left( {\left. t \right|H_{i} } \right){\text{d}}t} \\ \end{aligned} $$
(40)

Substituting (32) and (38) into (40) leads to the PDF of \( \lambda \) under \( H_{0} \), as shown in (14). Substituting (33) and (39) into (40) results in the PDF of \( \lambda \) under \( H_{1} \), as given in (15).

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Wang, Z. Adaptive Detection of a Subspace Signal in Structured Random Interference Plus Thermal Noise. Circuits Syst Signal Process 39, 4047–4066 (2020). https://doi.org/10.1007/s00034-020-01352-7

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Keywords

  • Adaptive radar detection
  • Subspace signal
  • Subspace interference
  • Thermal noise
  • Generalized likelihood ratio test
  • Constant false alarm rate