Pilot-based joint ML estimation of timing offset, CFO and channel for OFDMA uplink using minimum residue decomposition technique

Abstract

This paper addresses the challenging issue of joint maximum-likelihood estimation of carrier frequency offset, timing offset and channel response for orthogonal frequency division multiple access (OFDMA) uplink transmissions. A pilot preamble based approach is proposed that makes use of the concept of signal decomposition. Earlier works have demonstrated that it is possible to decompose the multidimensional optimization involved in this joint estimation problem into a series of one-dimensional searches. However, the computational complexity involved with such methods is also very high. In this work, we propose a new signal decomposition method based on a minimum residue criterion. The method with appropriate signal structures provides a computationally-efficient solution for the joint estimation of all required parameters of all users at the base station in an OFDMA link, leading to energy-efficient system design. Another advantage of the proposed technique is that it offers the flexibility of application to OFDMA systems with any subcarrier assignment scheme. Performance advantages of the proposed estimation method are substantiated through extensive computer simulation studies and are compared with that of some important recently-developed algorithms for the problem. Proof of convergence of the new estimation algorithm is also illustrated.

Appendix: Proof of convergence of MRD algorithm

In this Appendix, we prove that the value of the cost function for the mth user decreases in successive iterates. From Eqs. (28) and (34), this implies that the separation of the mth user’s cost function becomes more distortionless through consecutive iterations. Thus the cost function in the $(i+1)$th iteration will give a better estimate than the cost function in the ith iteration. The cost function in the ith iteration for the mth user is,

$$\begin{aligned} \mathbf J _{(m)}^{(i)} = \mathbf r ^{H}{} \mathbf W _{(m)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m)}^{(i-1)}\right] \mathbf W _{(m)}^{(i)}{} \mathbf r \end{aligned}$$

(51)

Substituting the expression for $\mathbf W _{m}^{(i)}$ in Eq. (36), we get

$$\begin{aligned}&\mathbf W _{(m)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m)}^{(i-1)}\right] \mathbf W _{(m)}^{(i)} \nonumber \\&\quad = \mathbf W _{(m-1)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m-1))}^{(i)}\right] \left[ \mathbf I - \mathbf P _{(m)}^{(i-1)}\right] \nonumber \\&\quad \left[ \mathbf I - \mathbf P _{(m-1)}^{(i)}\right] \mathbf W _{(m-1)}^{(i)} \nonumber \\&\quad = \mathbf W _{(m-1)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m-1)}^{(i)}\right] \mathbf W _{(m-1)}^{(i)} \nonumber \\&\quad - \mathbf W _{(m-1)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m-1))}^{(i)}\right] \mathbf P _{(m)}^{(i-1)}\nonumber \\&\quad \left[ \mathbf I - \mathbf P _{(m-1)}^{(i)}\right] \mathbf W _{(m-1)}^{(i)} \end{aligned}$$

(52)

The $\mathbf P$ and $[\mathbf I - \mathbf P ]$ matrices satisfy the properties of idempotent projection matrices. Hence $\mathbf W _{(m)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m)}^{(i-1)}\right] \mathbf W _{(m)}^{(i)}$ is less positive than the positive definite matrix, $\mathbf W _{(m-1)}^{(i)H}\left[ \mathbf I - \mathbf P _{(m-1)}^{(i)}\right] \mathbf W _{(m-1)}^{(i)}$. Due to the iterative update procedure adopted in the algorithm, this means that

$$\begin{aligned} \mathbf J _{(m)}^{(i)}< \mathbf J _{(m-1)}^{(i)}<\cdots< \mathbf J _{(m+1)}^{(i-1)} < \mathbf J _{(m)}^{(i-1)}. \end{aligned}$$

(53)

A graphical representation of the efficacy of the separation operation and convexity of the separated objective functions for the case of four users is shown in Fig. 9. The cost functions are plotted with high resolution values of the CFOs in the x-axis, to reveal the finer variations. Successive CFO values have a difference of 0.0001. Figure 9a depicts the behavior of the separated cost functions for all four users after two iterations and Fig. 9b after four iterations. It can be seen that the separated cost functions are smooth and convex in nature, whose optimum values can be easily determined using any suitable one-dimensional optimization procedure. The plotted minimum values move closer towards the actual CFO values to be estimated with more number of iterations. The figure also corroborates Eq. (53). This is so because the minimum of each separated cost function which corresponds to the best estimate possible from it decreases with more number of iterations. Also, such a descent in the value of the separated cost functions gives better estimates for all users.