Semiblind Channel Estimation for Multiuser MIMO-CDMA Systems with Orthogonal Space-Time Block Codes

Abstract

In this paper, we concern the channel estimation for a wireless communication system in which the techniques of multiple-input multiple-output, code division multiple access (CDMA) and orthogonal space-time block codes (OSTBCs) are integrated together for the purpose of achieving high data rate. We show that a composite channel information (CCI) vector can be formed, which contains the effects of channel state information, spreading coding and OSTBCs. From the standpoint of the MUltiple SIgnal Classification method, such CCI vector must lie in the signal subspace spanned by the dominant eigenvectors of the received data covariance matrix. Also, this CCI vector is located in another subspace which is associated with the CDMA and OSTBC codes and can be computed off-line. Using the vector space projections method, this CCI vector can be viewed as the intersection of these two subspaces and thus can be computed by alternative projections. In order to reduce the computation complexity, we propose an equivalent but computationally effective single-step solution in which the channel estimation amounts to searching for the principal eigenvector of a certain matrix with moderate size. Additionally, only one training block is required to overcome the problem of sign ambiguity. Numerical results demonstrate that, in addition to improving the bandwidth efficiency, the proposed method offers better performance in terms of channel estimation accuracy and bit-error-rate as compared with the standard nonblind least-squares channel estimation approach.

Appendices

Appendix 1: Proof of (24)

In this “Appendix”, we prove (24). First, we can rewrite the matrix ${\varvec{\Upsilon }}_p$ as

$$\begin{aligned} {\varvec{\Upsilon }}_p=(\mathbf {I}_{4KN_r}\otimes \mathbf {C}_p){\varvec{\Phi }}=(\mathbf {I}_{4KTN_r}\otimes \mathbf {c}_p) \begin{bmatrix} {\varvec{\Phi }}_1\\ {\varvec{\Phi }}_2\\ \vdots \\ {\varvec{\Phi }}_{2K} \end{bmatrix} \end{aligned}$$

(37)

where $\mathbf {C}_p$ and ${\varvec{\Phi }}$ are defined in (1) and (12) respectively. Consequently, the matrix inner product ${\varvec{\Upsilon }}_p^T{\varvec{\Upsilon }}_p$ can be expressed as

$$\begin{aligned} {\varvec{\Upsilon }}_p^T{\varvec{\Upsilon }}_p&= \begin{bmatrix} {\varvec{\Phi }}_1^T&{\varvec{\Phi }}_2^T&\cdots&{\varvec{\Phi }}_{2K}^T \end{bmatrix} (\mathbf {I}_{4KTN_r}\otimes \mathbf {c}_p)^T (\mathbf {I}_{4KTN_r}\otimes \mathbf {c}_p) \begin{bmatrix} {\varvec{\Phi }}_1\\ {\varvec{\Phi }}_2\\ \vdots \\ {\varvec{\Phi }}_{2K} \end{bmatrix}\nonumber \\&= \begin{bmatrix} {\varvec{\Phi }}_1^T&{\varvec{\Phi }}_2^T&\cdots&{\varvec{\Phi }}_{2K}^T \end{bmatrix} (\mathbf {c}_p^T\mathbf {c}_p\mathbf {I}_{4KTN_r}) \begin{bmatrix} {\varvec{\Phi }}_1\\ {\varvec{\Phi }}_2\\ \vdots \\ {\varvec{\Phi }}_{2K} \end{bmatrix}\nonumber \\&= L_c\begin{bmatrix} {\varvec{\Phi }}_1^T&{\varvec{\Phi }}_2^T&\cdots&{\varvec{\Phi }}_{2K}^T \end{bmatrix} \begin{bmatrix} {\varvec{\Phi }}_1\\ {\varvec{\Phi }}_2\\ \vdots \\ {\varvec{\Phi }}_{2K} \end{bmatrix}\nonumber \\&= L_c\sum _{k=1}^{2K}{\varvec{\Phi }}_{k}^T{\varvec{\Phi }}_{k} \nonumber \\&= 2KL_c\mathbf {I}_{N_{T}N_r} \end{aligned}$$

(38)

where the properties $\mathbf {c}_p^T\mathbf {c}_p=L_c$ and ${\varvec{\Phi }}_{k}^T{\varvec{\Phi }}_{k}=\mathbf {I}_{N_{T}N_r}$ are used in the above. Now the projector matrix $\mathbf {P}_{cp}$ can be represented as follows

$$\begin{aligned} \mathbf {P}_{cp}&= {\varvec{\Upsilon }}_p({\varvec{\Upsilon }}_p^T{\varvec{\Upsilon }}_p)^{-1}{\varvec{\Upsilon }}_p^T \nonumber \\&= \frac{1}{2KL_c}{\varvec{\Upsilon }}_p{\varvec{\Upsilon }}_p^T\nonumber \\&= \frac{1}{2KL_c}(\mathbf {I}_{4KTN_r}\otimes \mathbf {c}_p){\varvec{\Phi }}{\varvec{\Phi }}^T(\mathbf {I}_{4KTN_r}\otimes \mathbf {c}_p)^T. \end{aligned}$$

(39)

Appendix 2: Training-Based Least-Squares Channel Estimation

Following the work of [8], we can formulate the training-based least-squares (LS) channel estimator readily. First we can rewrite (13) as

$$\begin{aligned} \underline{\mathbf {Y}}=\sum _{p=1}^{P}(\mathbf {I}_{2N_r}\otimes \mathbf {C}_p) \widetilde{\mathbf {A}}(\underline{\mathbf {s}_p})\mathbf {h}_p+\underline{\mathbf {V}} \end{aligned}$$

(40)

where $\widetilde{\mathbf {A}}(\underline{\mathbf {s}_p})$ is a $2N_rT \times 2N_rN_t$ matrix whose $k$th column is given by

$$\begin{aligned}{}[\widetilde{\mathbf {A}}(\underline{\mathbf {s}_p})]_k=\mathbf {A}(\mathbf {e}_k)\underline{\mathbf {s}_p} \end{aligned}$$

where $\mathbf {e}_k$, as before, is the $2N_rN_t$ vector having one in its $k$th position and zeros elsewhere. Then each user is assumed to transmits $J_t$ training blocks, and thus we have

$$\begin{aligned} \mathbf {y}(n)&= \underline{\mathbf {Y}(n)}\nonumber \\&= \sum _{p=1}^{P}(\mathbf {I}_{2N_r}\otimes \mathbf {C}_p) \widetilde{\mathbf {A}}(\underline{\mathbf {s}_p(n)})\mathbf {h}_p+\underline{\mathbf {V}(n)},~~n=1,2,\ldots ,J_t \end{aligned}$$

(41)

where $\mathbf {s}_p(n)$ represents the $n$th known training symbol transmitted by the $p$th user, and $\mathbf {Y}(n)$ and $\mathbf {V}(n)$ are, respectively, the received signal matrix and the noise matrix for the $n$th training blocks. By defining

$$\begin{aligned} \mathbf {g}&\overset{\vartriangle }{=}&[\mathbf {h}_1^T,\mathbf {h}_2^T,\ldots ,\mathbf {h}_p^T]^{T} \\ \overline{\mathbf {A}}(n)&\overset{\vartriangle }{=} [(\mathbf {I}_{2N_r}\otimes \mathbf {C}_p) \widetilde{\mathbf {A}}(\underline{\mathbf {s}_1(n)}),\ldots , (\mathbf {I}_{2N_r}\otimes \mathbf {C}_p) \widetilde{\mathbf {A}}(\underline{\mathbf {s}_P(n)})],\nonumber \end{aligned}$$

(42)

we can rewrite (41) as

$$\begin{aligned} \mathbf {y}(n)=\overline{\mathbf {A}}(n)\mathbf {g}+\underline{\mathbf {V}(n)},~~n=1,2,\cdots ,J_t. \end{aligned}$$

(43)

Stacking the vectors $\mathbf {y}(n) (n=1,\ldots ,J_t)$ in a longer $2TN_rL_cJ_t \times 1$ vector

$$\begin{aligned} \mathbf {z}=[\mathbf {y}^T(1),\mathbf {y}^T(2),\ldots ,\mathbf {y}^T(J_t)]^T \end{aligned}$$

(44)

and defining the $2TN_rL_cJ_t \times 2N_tN_r$ matrix

$$\begin{aligned} \mathcal {A}=[\overline{\mathbf {A}}^T(1),\overline{\mathbf {A}}^T(2),\ldots ,\overline{\mathbf {A}}^T(J_t)]^T \end{aligned}$$

(45)

and the $2TN_rL_cJ_t \times 1$ vector

$$\begin{aligned} \mathbf {n}=[\underline{\mathbf {V}^T(1)},\underline{\mathbf {V}^T(2)},\ldots ,\underline{\mathbf {V}^T(J_t)}]^T \end{aligned}$$

(46)

we can rewrite (43) in a more compact form as

$$\begin{aligned} \mathbf {z}=\mathcal {A}\mathbf {g}+\mathbf {n}. \end{aligned}$$

(47)

Based on (47), the LS estimate of the vector $\mathbf {g}$ can be expressed as

$$\begin{aligned} \widehat{\mathbf {g}}=(\mathcal {A}^H\mathcal {A})^{-1}\mathcal {A}^H\mathbf {z}. \end{aligned}$$

(48)