A Blind Recognition Algorithm for Real Orthogonal STBC MC-CDMA Underdetermined Systems Based on LPCA and SCA

Abstract

This paper proposes a blind recognition algorithm for real orthogonal space–time block code multi-carrier code division multiple access (OSTBC MC-CDMA) underdetermined systems based on Laplacian potential clustering algorithm (LPCA) and sparse component analysis. In our work, the received signal first has been constructed to satisfy the instantaneous underdetermined model, where the mixing matrix (virtual channel matrix) includes the information of space–time block code. The virtual channel matrix is then can be separated by using the LPCA. We show that, for OSTBC, the correlation matrix of virtual channel matrix is a diagonal matrix, while with non-OSTBC (NOSTBC) signal, such correlation matrix of virtual channel matrix was not. According to this property, two characteristic parameters of correlation matrix of virtual channel matrix are extracted, such as sparsity and energy ratio of non-main and main diagonal elements. In recognition process, the energy ratio will be used in pre-decision step, thus it avoids the influence of noise and making sure that the correlation matrix of virtual channel matrix is a diagonal matrix for OSTBC. The last decision will be done through comparing sparsity parameter with the number of transmitted symbols, where the sparsity parameter of OSTBC will be equal the number of transmitted symbols and the such parameter of NOSTBC will not. Simulation results demonstrate the effectiveness of the proposed algorithm.

Appendices

Appendix 1

1.1 Correlation Comparison Algorithm

This section analyzes the effect of parameter of $\gamma$ and proposes a tool to acquire a good estimating. We have to note that that $\gamma$ can determine the location of peaks in the objective function $J\left( {\mathbf{z}} \right)$. A good estimation of $\gamma$ will induce a good clustering result. Thus, $\beta$ is not longer sensitive to the result.

As maximizing the total similarity measure $J\left( {\mathbf{z}} \right)$ is a way to find the peaks of the objective function $J\left( {\mathbf{z}} \right)$. The parameter $\gamma$ can determine the location of the peaks of $J\left( {\mathbf{z}} \right)$. Let $\tilde{J}\left( {{\mathbf{y}}_{j} } \right)$ be the total similarity of the data point ${\mathbf{y}}_{j}$ to all data points with

$$\tilde{J}\left( {{\mathbf{y}}_{j} } \right) = \sum\limits_{i = 1}^{m} {\left( {\exp - \frac{{\left| {{\mathbf{y}}_{i} - {\mathbf{y}}_{j} } \right|}}{\beta }} \right)^{\gamma } } \quad j = 1,2, \ldots ,m$$

(25)

This function can be seen closely related to the density shape of the data points in the neighborhood of ${\mathbf{y}}_{j}$. A large value for $\tilde{J}\left( {{\mathbf{y}}_{j} } \right)$ means that the data point ${\mathbf{y}}_{j}$ is close to some cluster centers and has many data points around. This function is equivalent to the mountain function proposed in [38]. According to (25), we here provide a tool for analyzing the effect of parameter $\gamma$ and give a method for a better choice of $\gamma$ based on $\tilde{J}\left( {{\mathbf{y}}_{j} } \right)$.

The selection of $\gamma$ can be solved using a correlation comparison procedure with $\left( {\gamma = 5,\gamma = 10} \right)$,$\left( {\gamma = 10,\gamma = 15} \right)$,…etc. We can easily find a good estimate of $\gamma$ via the density shape estimation concept and the underestimate (small $\gamma$) and overestimate (large $\gamma$) are undesirable. In order to execute the correlation comparison method as a computer program, (25) is rewritten as

$$\tilde{J}\left( {{\mathbf{y}}_{j} } \right)_{{\gamma_{\kappa } }} = \sum\limits_{i = 1}^{m} {\left( {\exp - \frac{{\left| {{\mathbf{y}}_{i} - {\mathbf{y}}_{j} } \right|}}{\beta }} \right)^{{\gamma_{\kappa } }} } \quad$$

(26)

where $\gamma_{\kappa } = 5\kappa \quad \kappa = 1,2,3 \ldots$. The correlation comparison method can be summarized as in the Table 5.

Appendix 2

2.1 Maximum Value of $J\left( {\mathbf{z}} \right)$ Estimation

For estimating the maximum value of $J\left( {\mathbf{z}} \right)$, first we can estimate the derivative ${{dJ\left( {\mathbf{z}} \right)} \mathord{\left/ {\vphantom {{dJ\left( {\mathbf{z}} \right)} {dz_{iq} }}} \right. \kern-0pt} {dz_{iq} }}$. As we have absolute value $\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|$, at $z_{iq} = \tilde{y}_{i} \left( t \right)$, $J\left( {\mathbf{z}} \right)$ is non differentiable, we can assume that

$$F\left( {z_{iq} } \right) = \left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right| = \left\{ \begin{aligned} \tilde{y}_{i} \left( t \right) - z_{iq} \quad \tilde{y}_{i} \left( t \right) \ge z_{iq} \hfill \\ z_{iq} - \tilde{y}_{i} \left( t \right)\quad \tilde{y}_{i} \left( t \right) < z_{iq} \hfill \\ \end{aligned} \right.$$

(27)

So we have

$${{dF} \mathord{\left/ {\vphantom {{dF} {dz_{iq} }}} \right. \kern-0pt} {dz_{iq} }} = \left\{ \begin{aligned} 1\quad \tilde{y}_{i} \left( t \right) > z_{iq} \hfill \\ - 1\quad \tilde{y}_{i} \left( t \right) < z_{iq} \hfill \\ \end{aligned} \right. = sign\left( {\left( {\tilde{y}_{i} \left( t \right) - z_{iq} } \right)} \right)$$

(28)

Let

$$\frac{{dJ\left( {\mathbf{z}} \right)}}{{dz_{iq} }} = \sum\limits_{t = 1}^{T} { - \frac{\gamma }{\beta }} \left( {\exp \left( { - {{\left( {\sum\limits_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } sign\left( {\tilde{y}_{i} \left( t \right) - z_{iq} } \right)$$

(29)

Set ${{dJ} \mathord{\left/ {\vphantom {{dJ} {dz_{iq} }}} \right. \kern-0pt} {dz_{iq} }}$ to zero. As the equation have symbol function, when $\tilde{y}_{i} \left( t \right) = z_{iq}$, the symbol function was not be determined. Therefore, it was so hard to obtain the expression of $z_{iq}$. However, the fixed point iterative method can solve this problem.

$$\begin{aligned} \frac{{dJ\left( {\mathbf{z}} \right)}}{{dz_{iq} }} & = \sum\limits_{t = 1}^{T} {\frac{\gamma }{\beta }\left( {\exp \left( { - {{\left( {\sum\limits_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } sign\left( {\tilde{y}_{i} \left( t \right) - z_{iq} } \right)} \\ & = \sum\limits_{t = 1}^{T} {\frac{\gamma }{\beta }\left( {\exp \left( { - {{\left( {\sum\limits_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } \frac{{\tilde{y}_{i} \left( t \right) - z_{iq} }}{{\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|}}} \\ & = 0 \\ \end{aligned}$$

(30)

We have

$$z_{iq} = \frac{{\sum_{t = 1}^{T} {\left( {\exp \left( { - {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } } \frac{{\tilde{y}_{i} \left( t \right)}}{{\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|}}}}{{{{\sum_{t = 1}^{T} {\left( {\exp \left( { - {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } } } \mathord{\left/ {\vphantom {{\sum_{t = 1}^{T} {\left( {\exp \left( { - {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } } } {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|}}} \right. \kern-0pt} {\left| {\tilde{y}_{i} \left( t \right) - z_{iq} } \right|}}}}$$

(31)

We can use the iterative algorithm to estimate $z_{iq}$, the iterative will be stop until ${\mathbf{z}}$ be unchanged. In order to prevented divide by zero, we can add small mount in the equation. The update of $z_{iq}$ is given as

$$z_{{_{iq} }}^{u + 1} \leftarrow \frac{{\sum\nolimits_{t = 1}^{T} {\left( {\exp \left( { - {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } \frac{{\tilde{y}_{i} \left( t \right)}}{{\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right| + 10^{ - 9} }}} }}{{{{\sum\nolimits_{t = 1}^{T} {\left( {\exp \left( { - {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } } } \mathord{\left/ {\vphantom {{\sum\nolimits_{t = 1}^{T} {\left( {\exp \left( { - {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right|} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum_{i = 1}^{m} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right|} } \right)} \beta }} \right. \kern-0pt} \beta }} \right)} \right)^{\gamma } } } {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right| + 10^{ - 9} }}} \right. \kern-0pt} {\left| {\tilde{y}_{i} \left( t \right) - z_{{_{iq} }}^{u} } \right| + 10^{ - 9} }}}}$$

(32)

where $u$ denotes the number of iteration.