CDKF approach for estimating a static parameter of carrier frequency offset based on nonlinear measurement equations in OFDM systems

Abstract

“Central difference Kalman filtering (CDKF)” is proposed as a new state of the art approach for carrier frequency offset estimation in orthogonal frequency division multiplexing systems. The parameter of interest to be estimated in this problem is a static value rather than a dynamically varying parameter. Therefore, classical approaches (e.g., maximum likelihood method or best linear unbiased estimator) might be more pertinent than Bayesian approaches if it is assumed to be a deterministic value. Nonetheless, it is shown and justified that a recently developed extended Kalman variant, i.e., CDKF, outperforms previously proposed methods in terms of mean squared error with efficient processing speed. Particularly, it is shown that CDKF outperforms recently proposed Gaussian particle filter for this one-dimensional static parameter estimation problem.

Appendix: Cramer–Rao bound (CRB)

The variance of the estimate, $\hat{\varepsilon }$ for any unbiased estimator is bounded by the inverse of the Fisher information as follows.

$$\begin{aligned} \mathsf Var (\hat{\varepsilon }) \ge J^{-1} = \frac{1}{-\mathsf E \left[\frac{\partial ^2 \ln p(y_{0:N-1};\varepsilon )}{\partial \varepsilon ^2} \right]} \end{aligned}$$

(9)

where $\mathsf E [\cdot ]$ denotes the expected value and $J$ is the Fisher information.

Since the likelihood function, (note that the noise is complex Gaussian)

$$\begin{aligned} p(y_{0:N-1};\varepsilon ) = \frac{1}{\left(\pi \sigma _w^2 \right)^N} \exp \left[-\frac{1}{\sigma _w^2} \sum _{k=0}^{N-1} |y_k - f_k|^2 \right]\end{aligned}$$

(10)

where $\sigma _w^2$ is the variance of the measurement noise, the bound can be derived as follows.

$$\begin{aligned} \mathsf Var (\hat{\varepsilon }) \ge \frac{\sigma _w^2}{2 \sum _{k=0}^{N-1} \left({f_{kr}^{\prime }}^2 + {f_{ki}^{\prime }}^2 \right)} \end{aligned}$$

(11)

where $f_{kr}, f_{ki}$ denote the real and imaginary parts of $f_k (\varepsilon )$, respectively, $f'_{kr} = \frac{\partial f_{kr}}{\partial \varepsilon }$, and $f'_{ki} = \frac{\partial f_{ki}}{\partial \varepsilon }$.