Semi-Blind Multi-CFO Estimation and Equalization for Multiuser CoMP OFDM Systems

Abstract

In the previous chapter, a single-CFO estimation method is presented. In this chapter, we present a low-complexity semi-blind structure with multi-CFO estimation and ICA based equalization structure for multiuser CoMP OFDM systems. A short pilot is carefully designed for each user and has a two-fold advantage. On the one hand, using the pilot structure, a complex multi-dimensional search for multiple CFOs is divided into a number of low-complexity mono-dimensional searches. On the other hand, the cross-correlations between the transmitted and the received pilots are explored to allow simultaneous elimination of permutation ambiguity and quadrant ambiguity in the ICA equalized signals. Simulation results show that, with a low training overhead of 1.6 %, the presented semi-blind system not only outperforms the existing multi-CFO estimation schemes in terms of BER and MSE of multi-CFO estimation, but also achieves a BER performance close to the ideal case with perfect CSI and no CFO at the receiver.

Appendix

Proof of Theorem 1.

This appendix contains the proof that maximizing the cross-correlation $\rho _{\pi _{ k}(n),k}(n)$ in Eq. (5.16) is a solution to the problem of the permutation and quadrant ambiguities.

By using the correlation in Eq. (5.16), the cross-correlation between the π _k (n)-th ICA separated pilot $\check{s}_{\pi _{k}(n)}(n,i)$ and the k-th user’s pilot on subcarrier n over P blocks in the noiseless case can be written as

$$\displaystyle{ \frac{1} {P}\sum _{i=0}^{P-1}\check{s}_{\pi _{ k}(n)}(n,i)s_{k}^{{\ast}}(n,i) = \left \{\begin{array}{ll} e^{-j\theta }&\mbox{ $\pi _{ k}(n) = k$} \\ 0 &\mbox{ $\pi _{k}(n)\neq k$} \end{array} \right.. }$$

(5.18)

Substituting Eqs. (5.18) to (5.16) yields to

$$\displaystyle{ \rho _{\pi _{k}(n),k}(n) = \left \{\begin{array}{ll} e^{j(\tilde{\theta }_{\pi _{k}(n)}-\theta )} & \mbox{ $\pi _{ k}(n) = k$} \\ 0 &\mbox{ $\pi _{k}(n)\neq k$} \end{array} \right.. }$$

(5.19)

Since the phase rotation $\theta \in \{ 0, \frac{\pi } {2},\pi, \frac{3\pi } {2}\}$ and the trial rotation $\tilde{\theta }_{\pi _{k}(n)} \in \{ 0, \frac{\pi } {2},\pi, \frac{3\pi } {2}\}$, their substraction is equal to $(\tilde{\theta }_{\pi _{k}(n)}-\theta ) \in \{ 0,\pm \frac{\pi }{2},\pm \pi \}$. As a result, $e^{j(\tilde{\theta }_{\pi _{k}(n)}-\theta )} \in \{ 1,-1,j,-j\}$. Only when the phase rotation is found as $\tilde{\theta }_{\pi _{k}(n)} =\theta$, the cross-correlation $\rho _{\pi _{k}(n),k}(n)$ becomes one. This also provides a solution to the permutation ambiguity problem, as the phase rotation can only be found on the correct substream as π _k (n) = k. The real part of $\rho _{\pi _{ k}(n),k}(n)$ is maximized, while the imaginary part of $\rho _{\pi _{k}(n),k}(n)$ reduces to zero. Therefore, in the presence of noise, the real part of cross-correlation $\rho _{\pi _{ k}(n),k}(n)$ can be maximized to eliminate the permutation and phase ambiguities simultaneously.