A New Fast CFO Tracking Algorithm for OFDM Systems

In this paper, a new improved CFO tracking algorithm for OFDM systems is proposed. It is based on the fact that energy on virtual subcarriers could be quadratic or trigonometric with CFO estimation errors when it's small, so the CFO can be estimated by coarse search and interpolation. Simulation results suggest that performance of the proposed method is equivalent with enumeration method while the computation burden of the proposed algorithm is only ten percent of the latter's.


INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is a popular modulation technique for wireless communication. At receiver end, carrier frequency offset (CFO) should be estimated to improve system performance. Generally, the CFO estimation is divided into two phases, first training sequence is transmitted for coarse CFO estimation, and then fine CFO tracking is performed based on pilots in OFDM symbols [1].
Several methods have been proposed for CFO coarse estimation and fine estimation. Subspace method is a popular method for blind CFO estimation and it requires virtual subcarries [2][3][4][5][6][7]. And the optimal virtual subcarrier allocation scheme is uniform distribution [8]. In order to reduce the computation burden of subspace method, polynomial form cost function is suggested for CFO estimation in [2,4,6] and estimation of signal parameters via rotation invariance technique (ESPRIT) algorithm is suggested in [7]. In [9] the second order statistics of received OFDM signal is used for CFO tracking. A blind CFO estimation method based on constant envelop algorithm is proposed in [10] and it requires constant envelop modulation in frequency domain. Joint estimation of direct current offset and CFO for direct conversion architecture OFDM system is discussed in [11], and Cramer bound for frequency estimation error in the presence of direct current offset is analyzed. A low complexity blind CFO estimation method based on kurtosis is suggested in [12]. And a fast maximal likelihood CFO estimation method is proposed based on polynomial root formula in [13].
A new improved CFO tracking algorithm for OFDM systems is proposed in this paper. It is based on the fact that energy on virtual subcarriers could be quadratic or trigonometric when CFO estimation errors when it's small.
where ε is the normalized carrier frequency offset, F is FFT matrix , The set of positions of virtual subcarriers in OFDM symbol is denoted as V and the cost function is defined as where k f is kth row of matrix F . The problem with formula (2) is that a big computation burden is required for high resolution.

III. THE NEW CFO ESTIMATION METHOD
From (1), the data on kth subcarrier is , and the subscript m is omitted for convenience. If k V ∈ , then 0 In (6), when 1 k reaches its maximal π ε , and the energy of the latter is 1/100 of the former's when 0.1 ε = which validates the approximation of (6). Then we have , k θ ε are amplitude and phase of ( ) s σ denotes the variance of noise introduced by approximation.
From above discussion, we have a new CFO estimation method: We firstly use rough frequency search in (2), e.g., the frequency interval is set as 0.1 and then we use (8) for fine frequency estimation. Suppose that 0 ε is the coarse frequency search result, and 1 ε , 2 ε ,ε are its two neighboring frequencies, the real frequency offset, respectively, then we define where 0 F , 1 F and 2 F are calculated during phase one coarse frequency search. And the CFO is estimated according to . Equation (12) is a high efficient CFO estimator. If we use sine function instead of quadric function for approximation of (8), then we get sin π sin π A ε ε = + which is another CFO estimator, however, with a higher computation complexity than (12). Figure 1 illustrates the estimation results of the proposed algorithms, where method #1 and method #2 refer to (12) and (16) , respectively. During coarse frequency search phase, the frequency interval is set as 0.1 and frequency search range is (-0.5, 0.5), which indicates 11 frequency points should be calculated. It can be found that estimate results of (12) and (16) are similar and we will discuss method #1 only later in this paper for it has a lower computation complexity. The estimation error becomes larger at two ends of frequency range for that the data for interpolation in (12) and (16) all deviate from real CFO at one direction and the method for conquering this problem is finer frequency search at frequency ends.  Figure 2 illustrates the estimate variance of proposed algorithm, where Classen&Meyr method and V-method refer to the algorithms in [14] and [4], respectively. In simulation, the frequency interval is set as 0.01 for algorithms in [14] and [4] while 0.1 for algorithm in this paper which indicates its superiority.

V. CONCLUSIONS
A new CFO estimation algorithm is proposed in this paper to reduce computation complexity while keeping a comparative performance with existing algorithms. The new algorithm is supposed that the OFDM system is already time and coarse frequency synchronized. Simulation results suggest that performance of proposed method is equivalent with enumeration method while the computation burden of the proposed algorithm is only ten percent of the latter's.