Low-complexity tensor-based blind receivers for MIMO systems

Abstract

In the tensor-based MIMO receivers, the multidimensional MIMO signals first are expressed as a third-order tensor model, wherein the factor matrices of tensor model are corresponding time/frequency, symbols, code/diversity of signals. A algorithm then is used for fitting this tensor mode, in which the symbols are estimated as a independent factor matrix. Although the performance of tensor-based receivers strongly depends on the initializations of the factor matrices. However, due to the absence of a priori on channels, these initializations are done randomly in alternating least squares (ALS), a basic algorithm for fitting the tensor models. In order to avoid these random initializations, this paper proposes two algorithms for fitting the tensor models. The first one, called delta bilinear ALS (DBALS) algorithm, where we exploit the increment values between two iterations of the factor matrices, refine these predictions by using the enhanced line search and use these refined values to initialize for two factor matrices. The second one, called orthogonal DBALS algorithm that takes into account the potential orthogonal in factor matrix for the DBALS algorithm, to provide the initialization for this factor matrix. By this way, we avoid random initializations for three factor matrices of tensor model. The performance of proposed receivers is illustrated by means of simulation results and a comparison is made with traditional ALS algorithm and other receivers. Beside a performance improving, our receivers give a lower complexity due to avoid random initializations.

Appendices

Appendix A: Derivation of $\left( {\rho _\mathbf{A} ,\rho _\mathbf{B} } \right)$

Let us define $\mathbf{T}_3 =\left( {{\varvec{\Pi }} _\mathbf{A} \odot {\varvec{\Pi }} _\mathbf{B} } \right) \mathbf{C}^{\mathrm{T}}$, $\mathbf{T}_2 =\left( {\mathbf{A}\odot {\varvec{\Pi }} _\mathbf{B} } \right) \mathbf{C}^{\mathrm{T}} \mathbf{T}_2 =\left( {\mathbf{A}\odot {\varvec{\Pi }} _\mathbf{B} } \right) \mathbf{C}^{\mathrm{T}} \quad \mathbf{T}_1 =\left( {{\varvec{\Pi }} _\mathbf{A} \odot \mathbf{B}} \right) \mathbf{C}^{\mathrm{T}}$, $\mathbf{T}_0 =\left( {\mathbf{A}\odot \mathbf{B}} \right) \mathbf{C}^{\mathrm{T}}-\mathbf{X}\in {\mathbb {C}}^{IJ\times K}$. (12) then can be rewritten as

$$\begin{aligned} \varepsilon _{{\mathrm{new}}}^{\left( \mathrm{int} \right) }= & {} \left\| {\rho _\mathbf{A} \rho _\mathbf{B} \mathbf{T}_3 +\rho _\mathbf{B} \mathbf{T}_2 +\rho _\mathbf{A} \mathbf{T}_1 +\mathbf{T}_0 } \right\| _\mathbf{F}^2 =\left\| {\mathbf{T}{\varvec{\Delta }} } \right\| _\mathbf{F}^2\nonumber \\ \quad= & {} {\varvec{\Delta }} ^{{*}}{} \mathbf{T}^{{*}}\mathbf{T}{\varvec{\Delta }} \end{aligned}$$

(27)

where $\mathbf{T}=\left[ {vec\left( {\mathbf{T}_3 } \right) ,vec\left( {\mathbf{T}_2 } \right) ,vec\left( {\mathbf{T}_1 } \right) ,vec\left( {\mathbf{T}_0 } \right) } \right] \in {\mathbb {C}}^{IJK\times 4}$ and ${\varvec{\Delta }} =\left[ {\rho _\mathbf{A} \rho _\mathbf{B} ,\rho _\mathbf{B} ,\rho _\mathbf{A} ,1} \right] ^{\mathrm{T}}$. We used $\mathbf{T}_{ij} =\left[ {vec\left( {\mathbf{T}_3 } \right) } \right] _{i+\left( {j-1} \right) I} ,\forall \mathbf{T}\in {\mathbb {C}}^{I\times J}$. The estimated of $\varepsilon _{\mathrm{new}}^{\left( \mathrm{int} \right) }$ in (27) is given as

$$\begin{aligned} \varepsilon _{{\mathrm{new}}}^{\left( \mathrm{int} \right) }= & {} a_{11} \rho _\mathbf{A}^2 \rho _\mathbf{B}^2 +2\rho _\mathbf{A} \rho _\mathbf{B} \left( {a_{13} \rho _\mathbf{A} +a_{12} \rho _\mathbf{B} +a_{23} +a_{14} } \right) \nonumber \\&+\,a_{33} \rho _\mathbf{H}^2 +a_{22} \rho _\mathbf{B}^2 +2\left( {a_{34} \rho _\mathbf{A} +a_{24} \rho _\mathbf{B} } \right) +a_{44}.\nonumber \\ \end{aligned}$$

(28)

Les us define a complex matrix $\mathbf{P}=\mathbf{T}^{{*}}{} \mathbf{T}\in {\mathbb {C}}^{4\times 4}$, where, $\mathbf{P}_{m\mathrm{n}} =\hbox {Re}\left( {\mathbf{P}_{mn} } \right) +j\hbox {Im}\left( {\mathbf{P}_{mn} } \right)$, $m=1,2,\ldots ,4$ and $n=1,2,\ldots ,4$. Set $\hbox {Re}\left( {\mathbf{P}_{mn} } \right) =u_{mn}$, $\hbox {Im}\left( {\mathbf{P}_{mn} } \right) =v_{mn}$. As $\mathbf{P}$ is a Hermite matrix, we have $u_{mn} =u_{nm}$, $v_{mn} =-v_{nm}$, $v_{mm} =0$. By setting derivations of $\rho _\mathbf{A}$ and $\rho _\mathbf{B}$ in (28) be equal zero, $\left( {\rho _\mathbf{A} ,\rho _\mathbf{B} } \right)$ is satisfied

$$\begin{aligned} \frac{\partial \varepsilon _{{\mathrm{new}}}^{\left( \mathrm{int} \right) } }{\partial \rho _\mathbf{B} }=e_1 \rho _\mathbf{B} +e_0 =0, \end{aligned}$$

(29)

and

$$\begin{aligned} \frac{\partial \varepsilon _{{\mathrm{new}}}^{\left( \mathrm{int} \right) } }{\partial \rho _\mathbf{A} }=g_1 \rho _\mathbf{A} +g_0 =0, \end{aligned}$$

(30)

where

$$\begin{aligned} \left\{ {\begin{array}{l} e_1 =2u_{11} \rho _\mathbf{A}^2 +4u_{12} \rho _\mathbf{A} +2u_{22} \\ e_0 =2u_{13} \rho _\mathbf{A}^2 +2u_{14} \rho _\mathbf{A} +2u_{23} \rho _\mathbf{A} +2u_{24} \\ \end{array}} \right. , \end{aligned}$$

(31)

and

$$\begin{aligned} \left\{ {\begin{array}{l} g_1 =2u_{11} \rho _\mathbf{B}^2 +4u_{13} \rho _\mathbf{B} +2u_{23} \\ g_0 =2u_{12} \rho _\mathbf{B}^2 +2u_{14} \rho _\mathbf{B} +2u_{23} \rho _\mathbf{B} +2u_{34} \\ \end{array}} \right. . \end{aligned}$$

(32)

Replace $\rho _\mathbf{B}$ by $-{e_0 }/{e_1 }$ in (31), we obtain

$$\begin{aligned} \frac{\partial \varepsilon _{{\mathrm{new}}}^{\left( \mathrm{int} \right) } }{\partial \rho _\mathbf{A} }=\sum _{p=1}^5 {r_p \rho _\mathbf{A}^p } =0, \end{aligned}$$

(33)

where the values of $r_p$ are given as

$$\begin{aligned} \left\{ {\begin{array}{l} r_5 =2u_{11}^2 u_{33} -2u_{13}^2 u_{11} \\ r_4 =2u_{11}^2 u_{34} -6u_{13}^2 u_{12} -2u_{11} u_{13} u_{14}\\ \quad -2u_{11} u_{13} u_{23} +8u_{11} u_{12} u_{33} \\ r_3 =8u_{12}^2 u_{33} -4u_{13}^2 u_{22} -8u_{12} u_{13} u_{14}\\ \quad -8u_{12} u_{13} u_{23} +8u_{11} u_{12} u_{34} +4a_{11} a_{22} a_{33} \\ r_2 =8u_{12}^2 u_{34} -2u_{23}^2 a_{12} -2u_{14}^2 u_{12} +2u_{11} u_{14} u_{24}\\ \quad -4u_{12} u_{13} u_{24} -4u_{12} u_{14} u_{23} \\ \quad -6u_{13} u_{22} u_{23} +4u_{11} u_{22} u_{34} +8u_{12} u_{22} u_{33} \\ r_1 =-2u_{14}^2 u_{22} -4u_{14} u_{22} u_{33} +2u_{22}^2 u_{33}\\ \quad -2u_{23}^2 u_{22} -4u_{13} u_{22} u_{24} +8u_{12} u_{34} u_{22} \\ \quad +2u_{24}^2 u_{11} \\ \end{array}} \right. . \end{aligned}$$

(34)

By solving (33), we have five values of $\rho _\mathbf{A}$, then five corresponding values of $\rho _\mathbf{B}$ are obtained. $\left( {\rho _\mathbf{A} ,\rho _\mathbf{B} } \right)$ that has been used to update data are obtained after minimization $\varepsilon _{{\mathrm{new}}}^{\left( \mathrm{int} \right) }$.

Appendix B: Derivation of $\mathbf{C}$

We use the Lagrange multiplier method to solve the constrained least squares problem in (16).

Let us define ${{\varvec{\lambda }} }$ is a constant symmetric matrix, (16) can be rewritten as

$$\begin{aligned} f\left( {\mathbf{C},{{\varvec{\uptheta }} }} \right)= & {} \left\| {\mathbf{X}-\left( {\mathbf{A}\odot \mathbf{B}} \right) \mathbf{C}^{\mathrm{T}}} \right\| _\mathbf{F}^2 +{{\varvec{\lambda }} }\left( {\mathbf{C}^{\mathrm{T}}{} \mathbf{C}-\mathbf{I}} \right) \nonumber \\ \quad= & {} tr\left[ {\left( {\mathbf{X}-\left( {\mathbf{A}\odot \mathbf{B}} \right) \mathbf{C}^{\mathrm{T}}} \right) ^{\mathrm{H}}\left( {\mathbf{X}-\left( {\mathbf{A}\odot \mathbf{B}} \right) \mathbf{C}^{\mathrm{T}}} \right) } \right] \nonumber \\&\quad +\,{{\varvec{\lambda }} }\left( {\mathbf{C}^{\mathrm{T}}{} \mathbf{C}-\mathbf{I}} \right) , \end{aligned}$$

(35)

By set the derivative of $\mathbf{C}$ in (35) be equal zero, we have

$$\begin{aligned} \frac{\partial f\left( {\mathbf{C},{{\varvec{\lambda }} }} \right) }{\partial \mathbf{C}}= & {} \mathbf{C}\hbox {Re}\left( {\left( {\mathbf{A}\odot \mathbf{B}} \right) ^{\mathrm{H}}\left( {\mathbf{A}\odot \mathbf{B}} \right) } \right) \nonumber \\&\quad -\,\hbox {Re}\left( {\mathbf{X}^{\mathrm{H}}\left( {\mathbf{A}\odot \mathbf{B}} \right) } \right) +{{\varvec{\lambda }} \mathbf{C}}=0 \end{aligned}$$

(36)

$$\begin{aligned} \mathbf{C}= & {} \hbox {Re}\left( {\mathbf{X}^{\mathrm{H}}\left( {\mathbf{A}\odot \mathbf{B}} \right) } \right) \nonumber \\&\quad \left( {{{\varvec{\lambda }} }+\hbox {Re}\left( {\left( {\mathbf{A}\odot \mathbf{B}} \right) ^{\mathrm{H}}\left( {\mathbf{A}\odot \mathbf{B}} \right) } \right) } \right) ^{-1}. \end{aligned}$$

(37)

Because of $\mathbf{C}^{\Gamma }{} \mathbf{C}=\mathbf{I}$, we have

$$\begin{aligned} \left( {{{\varvec{\lambda }} }+\hbox {Re}\left( {\left( {\mathbf{A}\odot \mathbf{B}} \right) ^{\mathrm{H}}\left( {\mathbf{A}\odot \mathbf{B}} \right) } \right) } \right) ^{-1}=\left( {\mathbf{U}^{\mathrm{T}}{} \mathbf{U}} \right) ^{-0.5}. \end{aligned}$$

(38)

The estimation of $\mathbf{C}$ is given as (17).

Appendix C: Derivation of CRB

The log likelihood function of $\mathbf{X}$ can be expressed as

$$\begin{aligned} f\left( {{\varvec{\uptheta }} } \right)= & {} -\frac{IJK}{\ln \left( {2\pi \sigma ^{2}} \right) }\nonumber \\&-\,\frac{1}{2\sigma ^{2}}\sum _{ijk} {\left( {x_{ijk} -\sum _{f=1}^F {a_{i,f} b_{j,f} c_{k,f} } } \right) ^{2}} \end{aligned}$$

(39)

where the unknown parameters are $a_{i,f}$, $b_{j,f}$, $c_{k,f}$, $f=1,2,\ldots F$. ${{\varvec{\uptheta }} }$ can be written in vector form as: ${{\varvec{\uptheta }} }=\left[ \mathbf{A}\left( {1,:} \right) ,\ldots ,\mathbf{A}\left( {I,:} \right) ,\mathbf{B}\left( {1,:} \right) ,\ldots ,\mathbf{B}\left( {J,:} \right) ,\mathbf{C}\left( {1,:} \right) ,\ldots ,{} \mathbf{C}\left( {K,:} \right) \right]$, where $\mathbf{A}\left( {i,f} \right) =a_{i,f}$, $\mathbf{B}\left( {j,f} \right) =b_{j,f}$, $\mathbf{C}\left( {k,f} \right) =c_{k,f}$. The FIM of ${{\varvec{\uptheta }} }$ is given as

$$\begin{aligned} {{\varvec{\Omega }} }\left( {{\varvec{\uptheta }} } \right)= & {} E\left\{ {\left( {\frac{\partial ^{2}f\left( {{\varvec{\uptheta }} } \right) }{\partial ^{2}{{\varvec{\uptheta }} }}} \right) } \right\} =E\left\{ {\left( {\frac{\partial f\left( { {\varvec{\uptheta }} } \right) }{\partial {{\varvec{\uptheta }} }}} \right) ^{\mathrm{T}}\left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial {{\varvec{\uptheta }} }}} \right) } \right\} \nonumber \\= & {} \left[ {{\begin{array}{ccc} {{{\varvec{\uppsi }} }_{aa} }&{} {{{\varvec{\uppsi }} }_{ab} }&{} {{{\varvec{\uppsi }} }_{ac} } \\ {{{\varvec{\uppsi }} }_{ab}^\mathrm{T} }&{} {{{\varvec{\uppsi }} }_{bb} }&{} {{{\varvec{\uppsi }} }_{bc} } \\ {{{\varvec{\uppsi }} }_{ac}^\mathrm{T} }&{} {{{\varvec{\uppsi }} }_{bc}^\mathrm{T} }&{} {{{\varvec{\uppsi }} }_{cc} } \\ \end{array} }} \right] \end{aligned}$$

(40)

Since the noise ${{\underline{\mathbf{E}}}}$ is not correlative, we get

$$\begin{aligned} E\left( {\underline{\mathbf{E}}\left( {i_1 j_1 k_1 } \right) {\underline{\mathbf{E}}}\left( {i_2 j_2 k_2 } \right) } \right) =\left\{ {\begin{array}{l} \sigma ^{2},\quad i_1 =i_2 ,j_1 =j_2 ,k_1 =k_2 \\ 0,\quad { otherwise} \\ \end{array}} \right. . \end{aligned}$$

(41)

The second-order derivative of $f\left( {{\varvec{\uptheta }} } \right)$ is estimated with logarithms $a_{i,f}$, $b_{j,f}$ and $c_{k,f}$, since $\mathbf{C}^{\Gamma }{} \mathbf{C}=\mathbf{I}$, $\sum _k {c_{k,f1} c_{k,f2} } ={{\varvec{\updelta }} }_{f1,f2}$, we have

$$\begin{aligned}&{\varvec{\uppsi }}_{aa} \left( {\left( {i_1 -1} \right) I+f_1 ,\left( {i_2 -1} \right) I+f_2 } \right) \nonumber \\&\quad =E\left\{ {\left( {\frac{\partial f\left( \theta \right) }{\partial a_{i_1 ,f_1 } }} \right) \left( {\frac{\partial f\left( \theta \right) }{\partial a_{i_2 ,f_2 } }} \right) } \right\} \nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _j {\left( {b_{j,f_1 } b_{j,f_2 } } \right) } \sum _k {\left( {c_{k,f_1 } c_{k,f_2 } } \right) } {\varvec{\updelta }}_{i_1 ,i_2 }\nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _j {\left( {b_{j,f_1 } b_{j,f_2 } } \right) } {\varvec{\updelta }}_{i_1 ,i_2 } {\varvec{\updelta }}_{f_1 ,f_2 } \end{aligned}$$

(42)

$$\begin{aligned}&\quad {\varvec{\uppsi }}_{bb} \left( {\left( {j_1 -1} \right) J+f_1 ,\left( {j_2 -1} \right) J+f_2 } \right) \nonumber \\&\quad =E\left\{ {\left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial b_{j_1 ,f_1 } }} \right) \left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial b_{j_2 ,f_2 } }} \right) } \right\} \nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _i {\left( {a_{i,f_1 } a_{i,f_2 } } \right) } \sum _k {\left( {c_{k,f_1 } c_{k,f_2 } } \right) } {\varvec{\updelta }}_{j_1 ,j_2 }\nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _i {\left( {a_{i,f_1 } a_{i,f_2 } } \right) } {\varvec{\updelta }}_{j_1 ,j_2 } {\varvec{\updelta }}_{f_1 ,f_2 } \end{aligned}$$

(43)

$$\begin{aligned}&\quad {\varvec{\uppsi }}_{cc} \left( {\left( {k_1 -1} \right) K+f_1 ,\left( {k_2 -1} \right) K+f_2 } \right) \nonumber \\&\quad =E\left\{ {\left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial c_{k_1 ,f_1 } }} \right) \left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial c_{k_2 ,f_2 } }} \right) } \right\} \nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _i {\left( {a_{i,f_1 } a_{i,f_2 } } \right) } \sum _k {\left( {b_{j,f_1 } b_{j,f_2 } } \right) } {\varvec{\updelta }}_{k_1 ,k_2 } . \end{aligned}$$

(44)

$$\begin{aligned}&\quad {\varvec{\uppsi }}_{ab} \left( {\left( {i-1} \right) I+f_1 ,\left( {j-1} \right) J+f_2 } \right) \nonumber \\&\quad =E\left\{ {\left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial a_{i,f_1 } }} \right) \left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial b_{j,f_2 } }} \right) } \right\} \nonumber \\&\quad =\frac{1}{\sigma ^{2}}a_{i,f_1 } b_{j,f_2 } {\varvec{\updelta }}_{f_1 ,f_2 } . \end{aligned}$$

(45)

$$\begin{aligned}&\quad {\varvec{\uppsi }}_{ac} \left( {\left( {i-1} \right) I+f_1 ,\left( {k-1} \right) K+f_2 } \right) \nonumber \\&\quad =E\left\{ {\left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial a_{i,f_1 } }} \right) \left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial c_{k,f_2 } }} \right) } \right\} \nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _j {\left( {b_{j,f_1 } b_{j,f_2 } } \right) a_{i,f_1 } c_{k,f_2 } } . \end{aligned}$$

(46)

$$\begin{aligned}&\quad {\varvec{\uppsi }}_{bc} \left( {\left( {j-1} \right) J+f_1 ,\left( {k-1} \right) K+f_2 } \right) \nonumber \\&\quad =E\left\{ {\left( {\frac{\partial f\left( {{\varvec{\theta }} } \right) }{\partial b_{j,f_1 } }} \right) \left( {\frac{\partial f\left( {{\varvec{\uptheta }} } \right) }{\partial c_{k,f_2 } }} \right) } \right\} \nonumber \\&\quad =\frac{1}{\sigma ^{2}}\sum _i {\left( {a_{i,f_1 } a_{i,f_2 } } \right) b_{j,f_1 } c_{k,f_2 } }. \end{aligned}$$

(47)

According to (40) and (41)–(47), we get

$$\begin{aligned} f_{{ CRB}} \left( {{\varvec{\uptheta }} } \right) ={{\varvec{\Omega }} }\left( {{\varvec{\uptheta }} } \right) ^{-1}=\left[ {{\begin{array}{ccc} {{{\varvec{\uppsi }} }_{aa} }&{} {{{\varvec{\uppsi }} }_{ab} }&{} {{{\varvec{\uppsi }} }_{ac} } \\ {{{\varvec{\uppsi }} }_{ab}^\mathrm{T} }&{} {{{\varvec{\uppsi }} }_{bb} }&{} {{{\varvec{\uppsi }} }_{bc} } \\ {{{\varvec{\uppsi }} }_{ac}^\mathrm{T} }&{} {{{\varvec{\uppsi }} }_{bc}^\mathrm{T} }&{} {{{\varvec{\uppsi }} }_{cc} } \\ \end{array} }} \right] . \end{aligned}$$

(48)