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Iterative Soft-Kalman Filter-based Data Detection and Channel Estimation for Turbo Coded MIMO–OFDM Systems

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MIMO channels are often assumed to be constant over a block or packet. This assumption of block stationarity is valid for many fixed wireless scenarios. However, for communications in a mobile environment, the stationarity assumption will result in considerable performance degradation. In this paper, we focus on a new channel estimation technique for Turbo coded MIMO systems using OFDM. In the proposed MIMO–OFDM system, pilots are placed on selected subcarriers and used by a pair of Kalman filter (KF) channel estimators at the receiver. The KF channel estimates are then utilized by a MIMO–OFDM soft data detector based on the computationally efficient QRD-M algorithm. The soft detector output is fed back to the Kalman filters to iteratively improve the channel estimates. The extrinsic information generated by the Turbo decoder is also used as a priori information for the soft data detector. The overall receiver thus combines MIMO data detection, KF-based channel estimation, and Turbo decoding in a joint iterative structure yielding computational efficiency and improved bit-error rate (BER) performance.

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Correspondence to Ronald A. Iltis.

Computation of Composite Noise Covariance

Computation of Composite Noise Covariance

Recall that

$$ \tilde{{\mathbf z}}^{q}(n)\triangleq \sum_{p=1}^{N_t}\tilde{\mathbf D}^{p}(n){{\mathbf C}}^{T}{\mathbf f}^{p,q}(n)+{\mathbf z}^{q}(n) \mbox{.} $$
(A.1)

Hence

$$ E\left\{ \tilde{\mathbf z}^{q}(n)\tilde{\mathbf z}^{q}(n)^{H} \right\}=\frac{2N_0}{T_s}{\mathbf I}+\sum_{p=1}^{N_t} E\left\{ \tilde{\mathbf D}^{p}(n){{\mathbf C}}^{T}{\mathbf f}^{p,q}(n) {\mathbf f}^{p,q}(n)^{H}{{\mathbf C}}^{*}\tilde{\mathbf D}^{p}(n)^{H} \right\}. $$
(A.2)

Note that only in the special case \({E\left\{{\mathbf f}^{p,q}(n) {\mathbf f}^{p,q}(n)^{H}\right\} = \sigma_f^2 {\mathbf I}}\) , the matrix \({E\left\{ {\mathbf C}^T{\mathbf f}^{p,q}(n) {\mathbf f}^{p,q}(n)^H {\mathbf C}^{\ast} \right\}}\) becomes Toeplitz. Now using the following relationship

$$ \begin{aligned} &\tilde{\mathbf D}^{p}(n){\mathbf C}^{T}{\mathbf f}^{p,q}(n){\mathbf f}^{p,q}(n)^{H}{\mathbf C}^{\ast}\tilde{\mathbf D}^{p}(n)^{H}\\ &=\sum_{l=1}^{K} \left[\begin{array}{c} {\mathbf 0}_{1\times l-1}\\ ({\mathbf C}^{T}{\mathbf f}^{p,q}(n){\mathbf f}^{p,q}(n)^{H} {\mathbf C}^{\ast})(l,:)\\ {\mathbf 0}_{1\times K-l} \end{array} \right]\tilde d^{p}_{l-1}(n)\tilde{\mathbf D}^{p}(n)^{H}, \end{aligned} $$
(A.3)

we have

$$ \begin{aligned} &E\left\{ \tilde{\mathbf D}^{p}(n){\mathbf C}^{T}{\mathbf f}^{p,q}(n){\mathbf f}^{p,q}(n)^{H} {\mathbf C}^{\ast}\tilde{\mathbf D}^{p}(n)^{H}\right\}\\ &= \sum_{l=1}^{K} \left[\begin{array}{c} {\mathbf 0}_{1\times l-1}\\ E\left[ \left({\mathbf C}^{T}{\mathbf f}^{p,q}(n){\mathbf f}^{p,q}(n)^{H} {\mathbf C}^{\ast} \right)(l,:) \right]\\ {\mathbf 0}_{1\times K-l} \end{array} \right] E\left[ \tilde d^{p}_{l-1}(n)\tilde{\mathbf D}^{p}(n)^{H} \right]. \end{aligned} $$
(A.4)

The data error correlation matrix is

$$ \begin{aligned} E \left\{ \tilde d^{p}_k(n)\tilde{\mathbf D}^{p}(n)^H \right\}&= \hbox{diag}\left\{ E\{\tilde d^{p}_k(n) \tilde d_0^{p}(n)^{\ast} \}, \ldots, E\{ \tilde d^{p}_k(n)\tilde d_{K-1}^{p}(n)^{\ast} \right\},\\ &=\left(E\{|d_k^{p}(n)|^{2}\} - |\bar{d}_k^{p}(n)|^{2}\right) {\mathbf e}_{k+1}{\mathbf e}_{k+1}^{T}. \end{aligned} $$
(A.5)

In the computation of (A.6), we assume uncorrelated symbol errors across the carriers thus

$$ E\left\{\tilde d_k^{p}(n)\tilde d_{k'}^{p}(n)^{\ast} \right\}= \left( E \left|d_k^{p}(n)\right|^2-\left| \bar{d}_k^{p}(n) \right|^2 \right)\delta_{k,k'}. $$

Denoting by

$$ {\mathbf S}_l\left({\mathbf f}^{p,q}(n) \right)\triangleq \left[\begin{array}{c} {\mathbf 0}_{1\times l-1}\\ \left({\mathbf C}^{T}E \left[{\mathbf f}^{p,q}(n){\mathbf f}^{p,q}(n)^{H} \right]{\mathbf C}^{\ast} \right) (l,:)\\ {\mathbf 0}_{1\times K-l}\\ \end{array}\right]$$

the l-th channel expectation in (A.4), we have

$$ \begin{aligned} &E \left\{ {\mathbf D}^{p}(n){\mathbf C}^{T} {\mathbf f}^{p,q}(n){\mathbf f}^{p,q}(n)^{H} {\mathbf C}^{\ast}{\mathbf D}^{p}(n)^{H} \right\}\\ &=\sum_{k=0}^{K-1} {\mathbf S}_{k+1}\left({\mathbf f}^{p,q}(n)\right) \left[E\{ |d_k^{p}(n)|^2 \} -|\bar{d}_k^{p}(n)|^{2} \right]{\mathbf e}_{k+1}{\mathbf e}_{k+1}^{T}.\\ \end{aligned} $$
(A.6)

Now substituting (A.6) into (A.2) yields (13), which completes the derivation.

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Kim, K.J., Iltis, R.A. Iterative Soft-Kalman Filter-based Data Detection and Channel Estimation for Turbo Coded MIMO–OFDM Systems. Int J Wireless Inf Networks 14, 175–189 (2007). https://doi.org/10.1007/s10776-007-0059-0

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