Fast Convergence on Blind and Semi-Blind Channel Estimation for MIMO–OFDM Systems

Abstract

In this paper, blind and semi-blind subspace channel estimations with fast convergence rate are proposed for multiple-input multiple-output orthogonal frequency division multiplexing (OFDM) systems in the presence of virtual carriers. The subspace method has a main drawback of slow convergence rate when deriving the noise subspaces from the second-order statistics of received signals. This phenomenon is especially evident when the size of received signals is large. Inspired by this fact, we present a block matrix scheme (BMS) to generate a group of sub-vectors from each OFDM symbol when virtual carriers are used. The number of equivalent signals is increased and therefore, the convergence rate of channel estimation is enhanced. The semi-blind method is also investigated by incorporating subspace technology with least square scheme. The identifiability of the BMS-based channel estimation is analyzed to derive the applicable range of the BMS size. The computational complexity of the proposed channel estimation is calculated at the end. Computer simulations show that the proposed blind and semi-blind methods converge very well in channel estimation and equalization.

Appendix

Based on Theorem 2 and the Vandermonde matrix property [23], both H $_{G}$ and ${\bar{\mathbf {W}}}_0$ are full column rank matrices if $G \geqq D-L$. Then, ${\mathbf {H}}_G {\bar{\mathbf {W}}}_0$ has full column rank. Now we define some notations as follows:

$$\begin{aligned}&{\mathbf {B}}_1 ={\mathbf {H}}_G (1:(G+L-D)M_r ,1:(G+L-D)M_t ) \nonumber \\&{\mathbf {B}}_2 ={\mathbf {H}}_G (1:(G+L-D)M_r ,(G+L-D)M_t +1:(G+L)M_t ) \nonumber \\&{\mathbf {B}}_3 ={\mathbf {H}}_G ((G+L-D)M_r +1:(G+L-D)M_r , \nonumber \\&\quad 1:GM_t ,(G+L-D)M_t +1:(G+L)M_t ) \nonumber \\&{\bar{\mathbf {W}}}_1 ={\bar{\mathbf {W}}}_0 (1:(G+L-D)M_t ,1:DM_t ) \nonumber \\&{\bar{\mathbf {W}}}_2 ={\bar{\mathbf {W}}}_0 ((G+L-D)M_t +1:(G+L)M_t ,1:DM_t ) . \end{aligned}$$

(46)

Note that $\mathbf{B}_{3}$ is a generalized Sylvester matrix like $\mathbf{H}_{G}$. From Theorem 1 in [19], the sub-matrix $\mathbf{B}_{3}$ has a full column rank if the condition $L\le (D-1)/(M_{T}+1)$ is fulfilled. From (46), ${\mathbf {H}}_G {\bar{\mathbf {W}}}_0$ becomes

$$\begin{aligned} {\mathbf {H}}_G {\bar{\mathbf {W}}}_0 =\left( {{\begin{array}{l} {{\mathbf {B}}_1 {\bar{\mathbf {W}}}_1 +{\mathbf {B}}_2 {\bar{\mathbf {W}}}_2 } \\ {{\mathbf {B}}_3 {\bar{\mathbf {W}}}_2 } \\ \end{array} }}\right) . \end{aligned}$$

(47)

From (22), the eigenvalue-eigenvector pairs can be expressed by

$$\begin{aligned} {\mathbf {U}}=[{\mathbf {U}}_s \quad {\mathbf {U}}_n ]=\left( {{\begin{array}{lll} {{\mathbf {U}}_{s1} } &{} {{\mathbf {U}}_{n11} } &{} {{\mathbf {U}}_{n12} } \\ {{\mathbf {U}}_{s2} } &{} {{\mathbf {U}}_{n21} } &{} {{\mathbf {U}}_{n22} } \\ \end{array} }}\right) \!. \end{aligned}$$

(48)

$$\begin{aligned} \mathrm{{diag}}(\lambda _1 \ldots \lambda _{GM_r } )=\left( {{\begin{array}{ccc} {{\mathbf {\Lambda }}_s } &{} {\mathbf {0}} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\sigma _n^2 {\mathbf {I}}_{(D-L)M_r -DM_t } } &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\sigma _n^2 {\mathbf {I}}_{(G+L-D)M_r } } \\ \end{array} }}\right) \!. \end{aligned}$$

(49)

where $\mathbf{U}_{s1}$ and $\mathbf{U}_{s2}$ have ( $G+L-D$) $M_{r}$ and ($D-L$) $M_{r}$ rows, respectively. From (21) and (22), we have $\sigma _s {\mathbf {H}}_G {\bar{\mathbf {W}}}_0 ={\mathbf {U}}_s ({\mathbf {\Lambda }}_s -\sigma _n^2 {\mathbf {I}}_{DM_t } )^{1/2}{\mathbf {V}}^H.$ where V is an unitary matrix. Using (47)–(49), we obtain

$$\begin{aligned} \sigma _s {\mathbf {B}}_3 {\bar{\mathbf {W}}}_2 ={\mathbf {U}}_{s2} ({\mathbf {\Lambda }}_s -\sigma _n^2 {\mathbf {I}}_{DM_t } )^{1/2}{\mathbf {V}}^H. \end{aligned}$$

(50)

Since ${\bar{\mathbf {W}}}_2$ is a $DM_{t} \times DM_{t}$ sub-matrix from ${\bar{\mathbf {W}}}_0$, it is nonsingular and Range$({\mathbf {B}}_3 )=\mathrm{{Range}}({\mathbf {U}}_{s2} )$, which means that ${\mathbf {U}}_{s2}$ also has a full column rank. Let ${\mathbf {U}}_{s2}^\bot \in C^{(D-L)M_r \times [(D-L)M_r -DM_t ]}$ be the orthogonal complement of ${\mathbf {U}}_{s2}$. Then the noise subspace can be written as

$$\begin{aligned} {\mathbf {U}}_n =\left( {{\begin{array}{*{20}c} 0 &{} {{\tilde{\mathbf {U}}}_{n12} } \\ {{\mathbf {U}}_{s2}^\bot } &{} {{\tilde{\mathbf {U}}}_{n22} }\\ \end{array} }}\right) {\mathbf {\Pi }}. \end{aligned}$$

(51)

where ${\tilde{\mathbf {U}}}_{n12}$ and ${\tilde{\mathbf {U}}}_{n22}$ have the same dimensions as those of ${\mathbf {U}}_{n12}$ and ${\mathbf {U}}_{n22}$, and ${\mathbf {\Pi }}$ is a nonsingular matrix. If range$({\mathbf {H}}_G {\bar{\mathbf {W}}}_0 )= \mathrm{{range}}({\mathbf {{H}'}}_G {\bar{\mathbf {W}}}_0 )$, then ${\mathbf {U}}_n$ is also orthogonal to ${\mathbf {{H}'}}_G {\bar{\mathbf {W}}}_0$. Let ${\mathbf {{B}'}}_3$ be the sub-matrix of ${\mathbf {{H}'}}_G$ as with ${\mathbf {B}}_3$ in ${\mathbf {H}}_G$. Then we get

$$\begin{aligned} \quad {\mathbf {U}}_{s2}^\bot {\mathbf {{B}'}}_3 {\bar{\mathbf {W}}}_2 ={\mathbf {0}} \nonumber \\ \text{ or } {\mathbf {U}}_{s2}^\bot {\mathbf {{B}'}}_3 ={\mathbf {0}} . \end{aligned}$$

(52)

From (48), we know that Range$({\mathbf {B}}_3 )=\mathrm{{Range}}({\mathbf {{B}'}}_3 )$. By assuming that ${\mathbf {{h}'}}$ is of full column rank and using Theorem 2 in [19], we conclude that ${\mathbf {{h}'}}={\mathbf {h\Omega }}$ where ${\varvec{\Omega }}$ is a $M_{t}\times M_{t}$ nonsingular matrix.

In addition, if ${\mathbf {{h}'}}={\mathbf {h\Omega }}$, then ${\mathbf {{H}'}}_G {\bar{\mathbf {W}}}_0$ can be written as

$$\begin{aligned}&{\mathbf {{H}'}}_G {\bar{\mathbf {W}}}_0 ={\mathbf {H}}_G ({\mathbf {I}}_{(G+L)} \otimes {\varvec{\Omega }})({\mathbf {W}}_0 \otimes {\mathbf {I}}_{M_t } ) \nonumber \\&={\mathbf {H}}_G ({\mathbf {W}}_0 \otimes {\mathbf {I}}_{M_t } )({\mathbf {I}}_D \otimes {\varvec{\Omega }}) \nonumber \\&={\mathbf {H}}_G {\bar{\mathbf {W}}}_0 ({\mathbf {I}}_D \otimes {\varvec{\Omega }}) . \end{aligned}$$

(53)

Since $({\mathbf {I}}_D \otimes {\varvec{\Omega }})$ is nonsingular, we conclude that range$({\mathbf {H}}_G {\bar{\mathbf {W}}}_0 )= \mathrm{{range}}({\mathbf {{H}'}}_G {\bar{\mathbf {W}}}_0 )$.