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Biquaternion noncircular MUSIC

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Abstract

A biquaternion-based direction-finding algorithm for noncircular sources is presented. The covariance and conjugate covariance matrices of the array output are utilized symmetrically within a frame of biquaternions. The direction-of-arrivals are found where the biquaternion steering vectors are orthogonal to the noise subspace in the biquaternion domain. Simulations show the improved performance of the proposed method compared to its complex counterparts.

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Notes

  1. In the case of coherent or highly-correlated sources, the smoothing preprocessing is required, which can be achieved by the approach proposed in Xu and Liu (2009).

  2. In the case of cyclostationary and noncircular signals, \({\varvec{R}}_{xx}\) and \({\varvec{R}}_{xx^*}\) are replaced with \({\varvec{R}}_{xx}^\alpha (T)\) and \({\varvec{R}}_{xx^*}^\alpha (T)\), respectively, where \(\alpha \) is the cyclostationary frequency and \(T\) is the time lag (Chargé and Wang 2005).

  3. The bicomplex algebra is a reduced biquaternion algebra with real part and \({\mathtt{i}}\), \({\mathtt{I}}\), \({\mathtt{iI}}\) imaginary parts left. The bicomplex numbers are multiplicatively commutative and are also known as complexified complex numbers.

  4. In the simulations, we set the uncertainties to \(2\,\%\) in magnitude and \(2^\circ \) in phase for the 6 sensors. In every simulation run, the measurement vector is generated by \({\varvec{x}}(t)={\varvec{GWAs}}(t)+{\varvec{n}}(t)\), \({\varvec{G}}={\varvec{I}}+diag\{0,\pm 0.02,\pm 0.02,\pm 0.02,\pm 0.02,\pm 0.02\}\), \({\varvec{W}}=diag\{1,e^{\pm \frac{{\mathtt{i}}\pi }{90}},e^{\pm \frac{{\mathtt{i}}\pi }{90}}, e^{\pm \frac{{\mathtt{i}}\pi }{90}},e^{\pm \frac{{\mathtt{i}}\pi }{90}}, e^{\pm \frac{{\mathtt{i}}\pi }{90}}\}\), respectively, where \({\varvec{I}}\) is an identity matrix.

    Fig. 4
    figure 4

    RMSEs of BN-MUSIC and NC-MUSIC versus SNR for \(K=100\) in the presence of channel mismatch

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    Fig. 5
    figure 5

    RMSEs of BN-MUSIC and NC-MUSIC versus \(K\) for SNR = 7 dB in the presence of channel mismatch

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Authors and Affiliations

  1. School of Information and Electronics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Xiaoming Gou, Zhiwen Liu & Yougen Xu

Authors
  1. Xiaoming Gou
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  2. Zhiwen Liu
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  3. Yougen Xu
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Corresponding author

Correspondence to Yougen Xu.

Additional information

This work was supported by the National Natural Science Foundation of China (61072098, 61072099) and Program for Changjiang Scholars and Innovative Research Team in University (IRT1005).

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Gou, X., Liu, Z. & Xu, Y. Biquaternion noncircular MUSIC. Multidim Syst Sign Process 26, 95–111 (2015). https://doi.org/10.1007/s11045-013-0238-3

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  • DOI: https://doi.org/10.1007/s11045-013-0238-3

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