Parameter Based Channel Estimation for OFDM Systems Over Time-Varying Channels

Abstract

This paper investigates the parameter-based (PB) channel estimation for orthogonal frequency division multiplexing over time-varying channels. The time-varying channel impulse response (CIR) can be represented using limited number of channel parameters. Instead of estimating the CIR directly, the PB approach estimates these parameters, from which, the CIR can be regenerated. To estimate these parameters, a differential scheme, utilizing the repetitive structure of the training signal, is proposed in this paper. When expressing the received signal in time–frequency-representation form, the repetitive structure can cause a phase difference between consecutive signal branches. This provides the chance for parameter estimation. Space domain sampling using multiple receive antennas are also employed to gain unique solution for parameter estimation. The effectiveness of the proposed algorithm is demonstrated by computer simulations.

Appendix

From the structure of $\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} }$, to prove matrix $\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} }$ is a column full rank matrix, we first assume that the estimation of Doppler frequency is ideal.

To prove C is a column full rank matrix, we assume there are a set of coefficients α _l,q that can make

$$ \sum\limits_{l = 0}^{L - 1} {\sum\limits_{q = 0}^{{\tfrac{Q - 1}{2}}} {\alpha_{l,q} {\varvec{\Omega}}\left( {v_{q} } \right)\varvec{s}_{l} \lambda_{qL + 1}^{r} = 0} } $$

(41)

where λ ^r _qL+1 is the (qL + 1)th element of ${\varvec{\Lambda}}^{\left( r \right)}$, $\varvec{s}_{l} = \varvec{F}^{H} t_{l}$ and t _l is the lth column of product matrix $\varvec{TF}_{L}$. Actually, we have $\tfrac{Q + 1}{2}$ such equations with r changing from 0 to $\tfrac{Q - 1}{2}$. Multiplying $\varvec{s}_{{l_{0} }}^{H}$ on both sided of (41) with where 0 ≤ l ₀ ≤ L − 1, we have

$$ \sum\limits_{l = 0}^{L - 1} {\sum\limits_{q = 0}^{{\tfrac{Q - 1}{2}}} {\alpha_{l,q} \varvec{s}_{{l_{0} }}^{H} {\varvec{\Omega}}\left( {v_{q} } \right)\varvec{s}_{l} = 0} } $$

(42)

Note that

$$ {\varvec{\Omega}}\left( {v_{q} } \right)\varvec{s}_{l} = {\varvec{\Omega}}\left( {v_{q} } \right)\varvec{F}^{H} \varvec{t}_{l} = \varvec{F}^{H} \left[ {\varvec{t}_{l} \odot\varvec{\omega}_{F} \left( q \right)} \right] = \varvec{F}^{H} \left[ {t_{l}^{\left( 0 \right)} ,t_{l}^{\left( 1 \right)} , \ldots ,t_{l}^{{\left( {N - 1} \right)}} } \right]\varvec{\omega}_{F} \left( q \right) $$

(43)

where $\odot$ denotes the circular convolution, $\varvec{\omega}_{F} \left( q \right) = \varvec{F}^{H}\varvec{\omega}\left( q \right)$ and $\varvec{\omega}\left( q \right)$ is the vector representation of elements on the diagonal line of ${\varvec{\Omega}}\left( {v_{q} } \right)$, that is $\varvec{\omega}\left( q \right) = {\varvec{\Omega}}\left( {v_{q} } \right){\mathbf{I}}_{N}$ with ${\mathbf{I}}_{N}$ being an N × 1 all ones vector. The third equation in (43) is due to the fact that the circular convolution can be represented as the form of matrix product, and $\varvec{t}_{l}^{\left( s \right)}$ denotes the circular shift of $\varvec{t}_{l}^{{}}$. By multiplying $\varvec{s}_{{l_{0} }}^{H}$ on both sides of (43), we can obtain

$$ \varvec{s}_{{l_{0} }}^{H} {\varvec{\Omega}}\left( {v_{q} } \right)\varvec{s}_{l} = \varvec{t}_{{l_{0} }}^{H} \left[ {t_{l}^{\left( 0 \right)} ,t_{l}^{\left( 1 \right)} , \ldots ,t_{l}^{{\left( {N - 1} \right)}} } \right]\varvec{\omega}_{F} \left( q \right) $$

(44)

Recall the fact that the frequency domain training sequence is obtained by interleaving the m-sequence with null subcarriers. Hence, we have $\varvec{t}_{{l_{0} }}^{H} \varvec{t}_{l}^{(s)} = 0$ when s is odd. When s is even, by noting that $\varvec{t}_{l}^{{}}$ is the m-sequence modulated by a carrier $e^{{ - j\tfrac{2\pi ln}{N}}}$, from the derivation in [8], it is easy to find that $\varvec{t}_{{l_{0} }}^{H} \varvec{t}_{l}^{(s)} = N\sigma_{t}^{2}$ only when l = l ₀, s = 0, and $\varvec{t}_{{l_{0} }}^{H} \varvec{t}_{l}^{(s)} = 0$ for other l’s and s’s. In summary, we can derive the following the relation

$$ \varvec{t}_{{l_{0} }}^{H} \varvec{t}_{l}^{(s)} = \left\{ {\begin{array}{*{20}l} {N\sigma_{t}^{2} } \hfill & {l = l_{0} ,s = 0} \hfill \\ 0 \hfill & {others} \hfill \\ \end{array} } \right. $$

(45)

With the relation (45), (44) can be simplified as

$$ \varvec{s}_{{l_{0} }}^{H} {\varvec{\Omega}}\left( {v_{q} } \right)\varvec{s}_{l} = \left\{ {\begin{array}{*{20}l} {\left[ {\varvec{\omega}_{F} (q)} \right]_{(0)} } \hfill & {l = l_{0} } \hfill \\ 0 \hfill & {l \ne l_{0} } \hfill \\ \end{array} } \right. $$

(46)

where $\left[ {\varvec{\omega}_{F} \left( q \right)} \right]_{(0)}$ denotes the 0th element of vector $\varvec{\omega}_{F} \left( q \right)$, that is, the d.c. component of $\varvec{\omega}_{F} \left( q \right)$. For large enough N, it is easy to derive an approximation of $\left[ {\varvec{\omega}_{F} \left( q \right)} \right]_{(0)}$ as

$$ \left[ {\varvec{\omega}_{F} \left( q \right)} \right]_{(0)} \approx {\text{sinc}}(\pi v_{q} )e^{{j\pi v_{q} }} $$

(47)

If denoting $\left[ {\varvec{\omega}_{F} \left( q \right)} \right]_{(0)} \lambda_{qL + l}^{r} = m(r,q)$, (42) can be simplified as

$$ \sum\limits_{q = 0}^{{\tfrac{Q - 1}{2}}} {\alpha_{{l_{0} ,q}} } m(r,q) = 0 $$

(48)

When r changes from 0 to $\tfrac{Q - 1}{2}$, we can obtain $\tfrac{Q + 1}{2}$ such equations, and the matrix form of those equations can be given as

$$ \varvec{M\alpha }_{{l_{0} }} = 0 $$

(49)

where $\left[ \varvec{M} \right]_{r,q} = m(r,q)$ and $\varvec{\alpha}_{{l_{0} }} = \left( {\alpha_{{l_{0} ,0}} ,\alpha_{{l_{0} ,1}} , \ldots ,\alpha_{{l_{0} ,\tfrac{Q - 1}{2}}} } \right)^{T}$. For given system parameters, e.g. d ₀ = 0.35 and Q = 9, M becomes a deterministic matrix, which depends only on maximum Doppler frequency V. It is easy to verify that matrix M is invertable when 0 ≤ V < 1. Therefore, there is only zero vector as the solution for (49), that is, $\alpha_{{l_{0} ,q}} = 0$ for $q = 0,1, \ldots ,\tfrac{Q - 1}{2}$. Similar conclusion can be obtained for other l ₀ ∊ [0, L − 1]. This means that only all zeros coefficients can satisfy the relation in (41). Hence, $\varvec{C}$ is a column full rank matrix.

In above, we assumed the Doppler frequency estimation is ideal. In the presence of Doppler frequency estimation error, the noise causes some perturbation $\Delta \varvec{M}$ to (49) and thus we obtain

$$ \left( {\varvec{M + }\Delta \varvec{M}} \right)\varvec{\alpha}_{{l_{0} }} = 0 $$

(50)

However, as shown in [20], small perturbation cannot reduce the matrix rank, that is

$$ r\left( {\varvec{M + }\Delta \varvec{M}} \right) \ge r\left( \varvec{M} \right) $$

(51)

where $r( \cdot )$ denotes the matrix rank. Since $\varvec{M}$ is a full rank matrix, $\varvec{M + }\Delta \varvec{M}$ is also a full rank matrix and thus invertable. This makes $\varvec{\alpha}_{{l_{0} }} = {\mathbf{0}}$ become the unique solution of (50). Correspondingly, matrix $\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} }$ is column full rank matrix as well.