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On the Blind Channel Identifiability of MIMO-STBC Systems Using Noncircular Complex FastICA Algorithm

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Abstract

This paper studies the channel identifiability for Multiple-Input Multiple-Output Space–Time Block Code (MIMO-STBC) systems using the noncircular complex Fast Independent Component Analysis (nc-FastICA) algorithm. In contrast to the previous blind MIMO-STBC channel estimation methods in literature, the method proposed in this paper is more suitable for non-cooperative scenario since it needs less prior knowledge and does not require any modification of the transmitter. The main contribution of the paper consists in the theoretical proof that, although the sources among different antennas of MIMO-STBC systems are not independent, they can be retrieved from the received data by directly using nc-FastICA algorithm in most cases. The conclusion is also demonstrated by simulation. This shows that the classical nc-FastICA algorithm can be applied to a wider range of situations rather than strictly independent sources.

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References

  1. S. Alamouti, A simple transmit diversity technique for wireless communication. IEEE J. Sel. Areas Commun. 16(8), 1451–1458 (1998)

    Article  Google Scholar 

  2. N. Ammar, Z. Ding, Channel identifiability under orthogonal space–time coded modulations without training. IEEE Trans. Wirel. Commun. 5(5), 1003–1013 (2006)

    Article  Google Scholar 

  3. N. Ammar, Z. Ding, Blind channel identifiability for generic linear space–time block codes. IEEE Trans. Signal Process. 55(1), 202–217 (2007)

    Article  MathSciNet  Google Scholar 

  4. S. Aouada, A. Zoubir, S. Cee, A comparative study on source number detection, in Proceedings of IEEE ISSPA (2003), pp. 173–176

    Google Scholar 

  5. T. Bell, An ICA page-paper, code, demos, link. http://www.cnl.salk.edu/~tony/ica.html. Accessed 10 Nov. 2005

  6. E. Bingham, A. Hyvärinen, A fast fixed-point algorithm for independent component analysis of complex valued signals. Int. J. Neural Syst. 10(1), 1–8 (2000)

    Article  Google Scholar 

  7. A. Boarui, D. Ionescu, A class of nonorthogonal rate-one space–time-block codes with controlled interference. IEEE Trans. Wirel. Commun. 2(2), 270–276 (2003)

    Article  Google Scholar 

  8. V. Choqueuse, A. Mansour, G. Burel, L. Collin, K. Yao, Blind channel estimation for STBC systems using higher-order statistics. IEEE Trans. Wirel. Commun. 10(2), 495–505 (2011)

    Article  Google Scholar 

  9. A. Dapena, H.J. Pérez-Iglesias, V. Zarzoso, Blind channel estimation based on maximizing the eigenvalue spread of cumulant matrices in 2×1 Alamouti’s coding schemes. Wirel. Commun. Mob. Comput. 12(6), 516–528 (2012)

    Article  Google Scholar 

  10. G. Ganesan, P. Stoica, Space-time block codes: a maximum SNR approach. IEEE Trans. Inf. Theory 47(4), 1650–1656 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Gao, X. Zhu, A.K. Nandi, Independent component analysis for multiple-input multiple-output wireless communication systems. Signal Process. 91(4), 607–623 (2011)

    Article  MATH  Google Scholar 

  12. A. Hyvärinen, Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 10(3), 626–634 (1999)

    Article  Google Scholar 

  13. A. Hyvarinen, J. Karunen, E. Oja, Independent Component Analysis (Wiley, New York, 2001)

    Book  Google Scholar 

  14. H. Jafarkhani, A quasi-orthogonal space–time block code. IEEE Trans. Commun. 49(1), 1–4 (2001)

    Article  MATH  Google Scholar 

  15. H. Jafarkhani, Space–Time Coding: Theory and Practice (Cambridge University Press, Cambridge, 2005)

    Book  Google Scholar 

  16. E. Larsson, P. Stoica, Space–Time Block Coding for Wireless Communication (Cambridge University Press, Cambridge, 2003)

    Book  Google Scholar 

  17. E. Larsson, P. Stoica, J. Li, On maximum-likelihood detection and decoding for space–time coding systems. IEEE Trans. Signal Process. 50(4), 937–944 (2002)

    Article  Google Scholar 

  18. E. Larsson, P. Stoica, J. Li, Orthogonal space–time block codes: maximum likelihood detection for unknown channels and unstructured interferences. IEEE Trans. Signal Process. 51(2), 362–372 (2003)

    Article  MathSciNet  Google Scholar 

  19. J. Liu, A. Iserte, M. Lagunas, Blind separation of OSTBC signals using ICA neural networks, in Proceedings of IEEE ISSPIT, Darmstadt, Germany (2003), pp. 502–505

    Google Scholar 

  20. B. Loesch, B. Yang, Cramér–Rao bound for circular and noncircular complex independent component analysis. IEEE Trans. Signal Process. 61(2), 365–379 (2013)

    Article  MathSciNet  Google Scholar 

  21. M. Luo, L. Li, G. Qian et al., Multidimensional blind separation algorithm suitable for STBC systems. Chin. J. Syst. Eng. Electron. (to be published)

  22. W. Ma, B. Vo, T. Davidson, P. Ching, Blind detection of orthogonal space–time block codes: efficient high-performance implementations. IEEE Trans. Signal Process. 54(2), 738–751 (2006)

    Article  Google Scholar 

  23. M. Novey, T. Adali, On extending the complex FastICA algorithm to noncircular sources. IEEE Trans. Signal Process. 56(5), 2148–2154 (2008)

    Article  MathSciNet  Google Scholar 

  24. S. Shahbazpanahi, A. Gershman, J. Manton, Closed form blind MIMO channel estimation for orthogonal space–time codes. IEEE Trans. Signal Process. 53(12), 4506–4517 (2005)

    Article  MathSciNet  Google Scholar 

  25. A. Swindlehurst, G. Leus, Blind and semi-blind equalization for generalized space–time block codes. IEEE Trans. Signal Process. 50(10), 2489–2498 (2002)

    Article  Google Scholar 

  26. V. Tarokh, H. Jafarkhani, A. Calderbank, Space–time block codes from orthogonal designs. IEEE Trans. Inf. Theory 45(5), 744–765 (1999)

    Article  MathSciNet  Google Scholar 

  27. A. Van Den Bos, Complex gradient and Hessian, in Proceedings of Inst. Elect. Eng. Image Signal Process (1994), pp. 380–382

    Google Scholar 

  28. J. Via, I. Santamaria, On the blind identifiability of orthogonal space–time block codes from second order statistics. IEEE Trans. Inf. Theory 54(2), 709–722 (2008)

    Article  MathSciNet  Google Scholar 

  29. H. Zhang, L. Li, W. Li, Independent component analysis based on fast proximal gradient. Circuits Syst. Signal Process. 10(1), 1–8 (2012)

    Google Scholar 

Download references

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Correspondence to Guobing Qian.

Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

Our proof of Theorem 1 is based on the fact that the optimal solutions for these non-independent sources of different antennas produce extrema of the nc-FastICA cost function for the algorithm to converge to.

As shown in Sect. 2, Alamouti code encodes a block of two independent symbols into a 2×2 matrix C(s v ) at vth block as

$$ \mathbf{C}(\mathbf{s}_{v}) = \left [ \begin{array}{c@{\quad}c} s_{1} & - s_{_{2}}^{*} \\ s_{2} & s_{_{1}}^{*} \end{array} \right ]. $$
(27)

As our main discussion is the channel identifiability, we tentatively omit the additive noise here and later we assess the performance of channel identifiability in environments with additive noise in simulation of Sect. 5. The MIMO-STBC communication model is expressed as

$$ \mathbf{Y}_{v} =\mathbf{ HX}_{v}\quad \mathrm{with}\ \mathbf{X}_{v} = \mathbf{C}(\mathbf{s}_{v}) = \left [ \begin{array}{c@{\quad}c} s_{1} & - s_{2}^{*} \\ s_{2} & s_{1}^{*} \end{array} \right ], $$
(28)

where v=1,…,N b , N b is the number of received blocks. The kth received samples, denoted as \(\mathbf{y}(k) = [\begin{array}{c@{\ }c@{\ }c@{\ }c} y_{1}(k) & y_{2}(k) & \cdots & y_{n_{r}}(k) \end{array}]^{T}\), can be expressed as

$$ \mathbf{y}(k) = \mathbf{Hx}(k), $$
(29)

where \(\mathbf{x}(k) = [\begin{array}{c@{\ }c} x_{1}(k) & x_{2}(k) \end{array}]^{T}\), n r is the number of receive antennas.

After whitening, the model is expressed as

$$ \mathbf{z}(k) =\mathbf{ VHx}(k) \equiv \mathbf{Bx}(k), $$
(30)

where V=Λ −1/2 U H and \(\boldsymbol{\Lambda} = \operatorname {diag}[ \lambda_{1}, \ldots, \lambda_{n_{t}} ]; n_{t}\) is the number of transmit antennas of MIMO-STBC systems (here it is 2 for Alamouti code), \(\lambda_{1}, \ldots, \lambda_{n_{t}}\) are the n t largest eigenvalues of the covariance matrix R=E[y(k)y H(k)], U is a matrix with the corresponding eigenvectors as its columns. Hereinafter, the subscript k is omitted for simplicity of expression.

We make the orthogonal change of coordinates q=B H w as in [23], resulting in the cost function J(q)=E{G 3(uu )}, where u=w H z=q H x. We assume an optimal solution for x 1 at \(\mathbf{q}_{1} = [\begin{array}{c@{\ }c} e^{j\theta_{1}} & 0 \end{array}]^{T}\). The cost J is not analytic in q but is analytic in q and q independently. The partial derivative can be found directly by differentiating with respect to q while treating q as a constant. Therefore,

$$ \frac{\partial J}{\partial q_{i}} = g\bigl(uu^{*}\bigr)ux_{i}^{*} $$
(31)

and the second derivatives can be found similarly:

$$\begin{aligned} a_{i,k} =& E \biggl( \frac{\partial^{2}J}{\partial q_{i}^{*}\partial q_{k}} \biggr) = E \bigl\{ x_{i}x_{k}^{*}\bigl[g' \bigl(uu^{*}\bigr)uu^{*} + g\bigl(uu^{*}\bigr)\bigr] \bigr\} , \end{aligned}$$
(32)
$$\begin{aligned} b_{i,k} =& E \biggl( \frac{\partial^{2}J}{\partial q_{i}^{*}\partial q_{k}^{*}} \biggr) = E \bigl\{ x_{i}x_{k}g'\bigl(uu^{*} \bigr)u^{*2} \bigr\} , \end{aligned}$$
(33)
$$\begin{aligned} c_{i,k} =& E \biggl( \frac{\partial^{2}J}{\partial q_{i}\partial q_{k}} \biggr) = b_{i,k}^{*}, \end{aligned}$$
(34)

where g is the derivative of G and g′ is the derivative of g. The gradient and Hessian can be written as

$$\begin{aligned} \tilde{\nabla}_{q}J =& E\left ( \begin{array}{c} \frac{\partial J}{\partial q_{1}} \\ \frac{\partial J}{\partial q_{1}^{*}} \\ \frac{\partial J}{\partial q_{2}} \\ \frac{\partial J}{\partial q_{2}^{*}} \end{array} \right ) = \left ( \begin{array}{c} E\{ g(uu^*)ux_{1}^{*}\} \\ E\{ g(uu^*)u^{*}x_{1}\} \\ E\{ g(uu^*)ux_{2}^{*}\} \\ E\{ g(uu^*)u^{*}x_{2}\} \end{array} \right ), \end{aligned}$$
(35)
$$\begin{aligned} \tilde{\mathrm{H}}_{q}J =& E \biggl( \frac{\partial^{2}J}{\partial \mathbf{q}^{*}\partial \mathbf{q}^{T}} \biggr) = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} a_{1,1} & b_{1,1} & a_{1,2} & b_{1,2} \\ b_{1,1}^{*} & a_{1,1} & b_{1,2}^{*} & a_{1,2} \\ a_{2,1} & b_{2,1} & a_{2,2} & b_{2,2} \\ b_{2,1}^{*} & a_{2,1} & b_{2,2}^{*} & a_{2,2} \end{array} \right ]. \end{aligned}$$
(36)

In this paper, we use G 3 in the cost function. Therefore,g(u)=u,g′(u)=1. Evaluating the gradient and Hessian at \(\mathbf{q}_{1} = [\begin{array}{c@{\ }c} e^{j\theta_{1}} & 0 \end{array}]^{T}\) and using the whiteness and the property of Alamouti code, we get

$$\begin{aligned} E\bigl\{ g\bigl(uu^*\bigr)ux_{2}^{*}\bigr\} =& E \biggl\{ \frac{1}{2} \times \bigl[ \vert s_{1} \vert ^{2}e^{ - j\theta_{1}}s_{1}s_{2}^{*} + \vert s_{2} \vert ^{2}e^{ - j\theta_{1}}\bigl( - s_{2}^{*}\bigr)s_{1} \bigr] \biggr\} \\ =& \frac{1}{2}E \bigl\{ \vert s_{1} \vert ^{2}e^{ - j\theta_{1}}s_{1} \bigr\} E\bigl(s_{2}^{*} \bigr) + \frac{1}{2}E \bigl\{ \vert s_{2} \vert ^{2}e^{ - j\theta_{1}}\bigl( - s_{2}^{*}\bigr) \bigr\} E(s_{1}) = 0. \end{aligned}$$
(37)

Similarly, E{g(uu )u x 2}=0;

$$\begin{aligned} &\begin{array}{@{}l} E\bigl\{ g\bigl(uu^*\bigr)ux_{1}^{*}\bigr\} = \displaystyle\frac{1}{2}E \bigl\{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \bigr\} e^{ - j\theta_{1}},\\ E\bigl\{ g\bigl(uu^*\bigr)u^{*}x_{1} \bigr\} = \displaystyle\frac{1}{2}E \bigl\{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \bigr\} e^{j\theta_{1}}, \end{array} \end{aligned}$$
(38)
$$\begin{aligned} &a_{1,2} = E \biggl( \frac{\partial^{2}J}{\partial q_{1}^{*}\partial q_{2}} \biggr) = E \bigl\{ x_{1}x_{2}^{*}\bigl[g' \bigl(uu^{*}\bigr)uu^{*} + g\bigl(uu^{*}\bigr)\bigr] \bigr\} \\ &\hphantom{a_{1,2}} = \frac{1}{2}E \bigl\{ s_{1}s_{2}^{*} \bigl[g'\bigl(\vert s_{1} \vert ^{2}\bigr) \vert s_{1} \vert ^{2} + g\bigl(\vert s_{1} \vert ^{2}\bigr)\bigr] + \bigl( - s_{2}^{*} \bigr)s_{1}\bigl[g'\bigl(\vert s_{2} \vert ^{2}\bigr)\vert s_{2} \vert ^{2} + g\bigl( \vert s_{2} \vert ^{2}\bigr)\bigr] \bigr\} \\ &\hphantom{a_{1,2}} = \frac{1}{2}E \bigl\{ s_{1}\bigl(2\vert s_{1} \vert ^{2}\bigr) \bigr\} E \bigl\{ s_{2}^{*} \bigr\} + \frac{1}{2}E \bigl\{ \bigl( - s_{2}^{*}\bigr) \bigl(2\vert s_{2} \vert ^{2}\bigr) \bigr\} E \{ s_{1} \} = 0. \end{aligned}$$
(39)

Similarly,

$$\begin{aligned} &a_{2,1} = 0;\qquad b_{2,1} = 0;\qquad b_{1,2} = 0, \end{aligned}$$
(40)
$$\begin{aligned} &\begin{array}{@{}rcl} a_{1,1} &=& E \bigl\{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \bigr\} ; \qquad a_{2,2} = 2E\bigl(\vert s_{1} \vert ^{2}\bigr)E\bigl(\vert s_{2} \vert ^{2}\bigr); \\ b_{1,1} &=& \frac{1}{2}E \bigl\{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \bigr\} e^{j2\theta_{1}}; \qquad b_{2,2} = \frac{1}{2}E \bigl\{ s_{1}^{*2}s_{2}^{2} + s_{2}^{*2}s_{1}^{2} \bigr\} e^{j2\theta_{1}}. \end{array} \end{aligned}$$
(41)

Therefore,

$$\begin{aligned} &\tilde{\nabla}_{q}J(\mathbf{q}_{1}) = \frac{1}{2} \left ( \begin{array}{c} E \{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \}e^{ - j\theta_{1}} \\ E \{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \}e^{j\theta_{1}} \\ 0 \\ 0 \end{array} \right ), \end{aligned}$$
(42)
$$\begin{aligned} & \tilde{\mathrm{H}}_{q}J(\mathbf{q}_{1}) \\ & \quad = \frac{1}{2} \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2E \{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \} & E \{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \}e^{j2\theta_{1}} & 0 & 0 \\ E \{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \}e^{ - j2\theta_{1}} & 2E \{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \} & 0 & 0 \\ 0 & 0 & 4E(\vert s_{1} \vert ^{2})E(\vert s_{2} \vert ^{2}) & e^{j2\theta} E \{ s_{1}^{*2}s_{2}^{2} + s_{2}^{*2}s_{1}^{2} \} \\ 0 & 0 & e^{ - j2\theta} E \{ s_{1}^{2}s_{2}^{*2} + s_{2}^{2}s_{1}^{*2} \} & 4E(\vert s_{1} \vert ^{2})E(\vert s_{2} \vert ^{2}) \end{array} \right ). \end{aligned}$$
(43)

We now cause a small perturbation ε=[ε 1,ε 2]T around the optimal solution \(\mathbf{q}_{1} = [\begin{array}{c@{\ }c} e^{j\theta_{1}} & 0 \end{array}]^{T}\) as in [23], using the complex Taylor series expansion derived from [27] as

$$\begin{aligned} J(\mathbf{q}_{1} + \boldsymbol{\varepsilon} ) =& J( \mathbf{q}_{1}) + \widetilde{\boldsymbol{\varepsilon}}^{T} \tilde{\nabla}_{q}J(\mathbf{q}_{1}) + \frac{1}{2} \widetilde{\boldsymbol{\varepsilon}}^{H}\tilde{\mathbf{H}}_{q}J( \mathbf{q}_{1})\widetilde{\boldsymbol{\varepsilon}} +\mathbf{ o}\bigl( \Vert \boldsymbol{\varepsilon} \Vert ^{2}\bigr) \\ =& J(\mathbf{q}_{1}) + \frac{1}{2}E \bigl\{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \bigr\} \bigl(\varepsilon_{1}e^{ - j\theta_{1}} + \varepsilon_{1}^{*}e^{j\theta_{1}}\bigr) + \vert \varepsilon_{1} \vert ^{2}a_{1,1} \\ &{} + \frac{1}{2}\bigl(\varepsilon_{1}^{2}b_{1,1}^{*} + \varepsilon_{1}^{*2}b_{1,1}\bigr) \\ &{}+ \vert \varepsilon_{2} \vert ^{2}a_{2,2} + \frac{1}{2}\bigl(\varepsilon_{2}^{2}b_{2,2}^{*} + \varepsilon_{2}^{*2}b_{2,2}\bigr) + o\bigl(\Vert \boldsymbol{\varepsilon} \Vert ^{2}\bigr). \end{aligned}$$
(44)

Noting that \(\Vert \mathbf{q}_{1} + \boldsymbol{\varepsilon} \Vert ^{2} = 1 + e^{ - j\theta_{1}}\varepsilon_{1} + e^{j\theta_{1}}\varepsilon_{1}^{*} + \vert \varepsilon_{2} \vert ^{2}\) and the constraint ∥q2=1, we obtain

$$ e^{ - j\theta_{1}}\varepsilon_{1} + e^{j\theta_{1}} \varepsilon_{1}^{*} = - \vert \varepsilon_{2} \vert ^{2}. $$
(45)

Substituting (45) into the Taylor series expansion, we get

$$\begin{aligned} J(\mathbf{q}_{1} + \boldsymbol{\varepsilon} ) =& J( \mathbf{q}_{1}) + a_{1,1}\vert \varepsilon_{1} \vert ^{2} + \biggl[a_{2,2} - \frac{1}{2}E\bigl(\vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4}\bigr)\biggr]\vert \varepsilon_{2} \vert ^{2} \\ &{}+ \frac{1}{2}\bigl(\varepsilon_{2}^{2}b_{2,2}^{*} + \varepsilon_{2}^{*2}b_{2,2}\bigr) + o\bigl( \Vert \boldsymbol{\varepsilon} \Vert ^{2}\bigr) \\ =& J(\mathbf{q}_{1}) + E \bigl\{ \vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4} \bigr\} \vert \varepsilon_{1} \vert ^{2} + \biggl\{ 2E\bigl(\vert s_{1} \vert ^{2}\bigr)E\bigl(\vert s_{2} \vert ^{2}\bigr) \\ &{} - \frac{1}{2}E\bigl(\vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4}\bigr) \biggr\} \vert \varepsilon_{2} \vert ^{2} \\ &{}+ \frac{1}{2} \biggl\{ \frac{1}{2}E \bigl[ s_{1}^{2}s_{2}^{*2} + s_{1}^{*2}s_{2}^{2} \bigr] \bigl( \varepsilon_{2}^{2}e^{j2\theta_{1}} + \varepsilon_{2}^{*2}e^{ - j2\theta_{1}} \bigr) \biggr\} + o\bigl(\Vert \boldsymbol{\varepsilon} \Vert ^{2} \bigr), \end{aligned}$$
(46)

where the term |ε 1|2is of the order o(∥ε2) according to (45) and can be neglected, thus resulting in

$$\begin{aligned} J(\mathbf{q}_{1} + \boldsymbol{\varepsilon} ) =& J( \mathbf{q}_{1}) + \biggl\{ 2E\bigl(\vert s_{1} \vert ^{2}\bigr)E\bigl(\vert s_{2} \vert ^{2}\bigr) - \frac{1}{2}E\bigl(\vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4}\bigr) \biggr\} \vert \varepsilon_{2} \vert ^{2} \\ &{}+ \frac{1}{2} \biggl\{ \frac{1}{2}E \bigl[ s_{1}^{2}s_{2}^{*2} + s_{1}^{*2}s_{2}^{2} \bigr] \bigl( \varepsilon_{2}^{2}e^{j2\theta_{1}} + \varepsilon_{2}^{*2}e^{ - j2\theta_{1}} \bigr) \biggr\} + o\bigl(\Vert \boldsymbol{\varepsilon} \Vert ^{2} \bigr) \\ =& J(\mathbf{q}_{1}) + \biggl\{ 2E\bigl(\vert s_{1} \vert ^{2}\bigr)E\bigl(\vert s_{2} \vert ^{2} \bigr) - \frac{1}{2}E\bigl(\vert s_{1} \vert ^{4} + \vert s_{2} \vert ^{4}\bigr) \\ &{} + \biggl\vert \frac{1}{2}E \bigl[ s_{1}^{2}s_{2}^{*2} + s_{1}^{*2}s_{2}^{2} \bigr] \biggr\vert \cos (\psi ) \biggr\} \vert \varepsilon_{2} \vert ^{2} + o \bigl(\Vert \boldsymbol{\varepsilon} \Vert ^{2}\bigr) \\ =& J(\mathbf{q}_{1}) + \bigl\{ 2E^{2}\bigl(\vert s \vert ^{2}\bigr) - E\bigl(\vert s \vert ^{4}\bigr) + \bigl\vert E\bigl(s^{2}\bigr) \bigr\vert \bigl\vert E \bigl(s^{*2}\bigr) \bigr\vert \cos (\psi ) \bigr\} \vert \varepsilon_{2} \vert ^{2} \\ &{} + o\bigl(\Vert \boldsymbol{ \varepsilon} \Vert ^{2}\bigr), \end{aligned}$$
(47)

where \(\psi = \arg ( \varepsilon_{2}^{2} ) + 2\theta_{1}\).

The last line of Eq. (47) is derived due to the assumption A3. When BPSK modulation is employed,

$$ 2E^{2}\bigl(\vert s \vert ^{2}\bigr) - E\bigl(\vert s \vert ^{4}\bigr) + \bigl\vert E\bigl(s^{2}\bigr) \bigr\vert \bigl\vert E\bigl(s^{*2}\bigr) \bigr\vert \cos (\psi ) = 1 + \cos (\psi ) > 0; $$
(48)

when other PSK modulations are employed,

$$ 2E^{2}\bigl(\vert s \vert ^{2}\bigr) - E\bigl(\vert s \vert ^{4}\bigr) + \bigl\vert E\bigl(s^{2}\bigr) \bigr\vert \bigl\vert E\bigl(s^{*2}\bigr) \bigr\vert \cos (\psi ) = 1 $$
(49)

due to the circular and constant modulus property of these PSK modulations and it is obvious that J(q 1+ε)>J(q 1), therefore, a local minimum is reached at the point \(\mathbf{q}_{1} = [\begin{array}{c@{\ }c} e^{j\theta_{1}} & 0 \end{array}]^{T}\); when 4QAM, 16QAM or 32QAM is employed,

$$ 2E^{2}\bigl(\vert s \vert ^{2}\bigr) - E\bigl(\vert s \vert ^{4}\bigr) + \bigl\vert E\bigl(s^{2}\bigr) \bigr\vert \bigl\vert E\bigl(s^{*2}\bigr) \bigr\vert \cos (\psi ) = 2E^{2}\bigl(\vert s \vert ^{2}\bigr) - E\bigl(\vert s \vert ^{4}\bigr) > 0 $$
(50)

due to the circular and sub-Gaussian property of these QAMs and it is obvious that J(q 1+ε)>J(q 1), therefore, a local minimum is reached at the point \(\mathbf{q}_{1} = [\begin{array}{c@{\ }c} e^{j\theta_{1}} & 0 \end{array}]^{T}\); when 8QAM is employed,

$$ E\bigl(\vert s \vert ^{2}\bigr) = 6,\qquad E\bigl(\vert s \vert ^{4}\bigr) = 52,\qquad E\bigl(s^{2}\bigr) = 4,\qquad E \bigl(s^{*2}\bigr) = 4, $$
(51)

therefore

$$ 2E^{2}\bigl(\vert s \vert ^{2}\bigr) - E\bigl(\vert s \vert ^{4}\bigr) + \bigl\vert E\bigl(s^{2}\bigr) \bigr\vert \bigl\vert E\bigl(s^{*2}\bigr) \bigr\vert \cos (\psi ) = 20 + 16\cos (\psi ) > 0, $$
(52)

it is obvious that J(q 1+ε)>J(q 1), and a local minimum is reached at the point \(\mathbf{q}_{1} = [\begin{array}{c@{\ }c} e^{j\theta_{1}} & 0 \end{array}]^{T}\).

In the same way we can prove that a local minimum is reached at the point \(\mathbf{q}_{2} = [\begin{array}{c@{\ }c} 0 & e^{j\theta_{2}} \end{array}]^{T}\) under PSK and QAM. Thus, the extrema of the nc-FastICA cost function coincide with the components of different antennas and Theorem 1 is proved.

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Qian, G., Li, L. & Luo, M. On the Blind Channel Identifiability of MIMO-STBC Systems Using Noncircular Complex FastICA Algorithm. Circuits Syst Signal Process 33, 1859–1881 (2014). https://doi.org/10.1007/s00034-013-9722-0

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