Frequency Offset Tracking for OFDM Systems via Scattered Pilots and Virtual Carriers Tao Cui Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125, USA Email: taocui@caltech.edu Feifei Gao and A. Nallanathan Department of Electrical and Computer Engineering National University of Singapore Singapore 119260 Email: {feifeigao, elena}@nus.edu.sg Abstract—In this paper, we propose a new carrier frequency offset (CFO) tracking algorithm for orthogonal frequency divi- sion multiplexing (OFDM) systems. Assuming that the channel remains constant during two consecutive OFDM blocks, a CFO estimation algorithm is proposed based on the limited number of pilots in each OFDM block. Identifiability of this pilot based algorithm is fully discussed under the noise free environment. A weighted algorithm is then developed by considering both pilot carriers and virtual carriers. The asymptotic mean square error (MSE) of the proposed algorithm is provided, and simulation re- sults clearly show the performance improvement of the proposed algorithm over the existing methods. I. I NTRODUCTION Orthogonal frequency-division multiplexing (OFDM) [1], is a promising candidate for next generation high-speed wireless communication systems. In OFDM systems, it is well known that a CFO, caused by oscillators’ mismatch or Doppler effects, destroys the subcarriers orthogonality and results in a substantial bit error rate (BER) degradation [2]. Frequency synchronization for OFDM systems usually con- tains the acquisition stage and the tracking stage. Several CFO acquisition methods have been proposed in [3]- [7] by using the periodic nature of the time domain signal. However, using the periodic nature greatly reduces the CFO estimation region. Furthermore, all these methods, except [3]- [4], are only applicable in CFO acquisition stage because consecutive training blocks are required. In the tracking stage, the CP based method [3], [4], can still be used for residue CFO estimation. This method will be referred as Beek’s method in this paper. However, the performance of Beek’s method depends critically on the difference between the length of the CP and the channel length. Therefore, an alternative way is to implement CFO tracking based on the scattered pilots available in existing OFDM standards. However, all the above mentioned CFO estimation methods [3]- [7] fail to work with the limited number of pilots. To the best of the authors’ knowledge, only an algorithm in [8] considers the scattered pilot symbols. This method is referred as Classen&Meyr’s method in this paper. There are two drawbacks of Classen&Meyr’s method: 1) It is only applicable to a sufficiently small CFO; 2) An error floor appears at high SNR. In this paper, we first develop a new CFO tracking algorithm by using the scattered pilot carriers. Identifiability of this pilot aided algorithm is studied for the noise free case. We then consider further utilizing the virtual carriers existing in practical OFDM standards. A weighted algorithm, called pv- algorithm is then proposed by exploiting both scattered pilots and virtual carriers. We show that in the pv-algorithm, the p-algorithm part increases the estimation accuracy, while the v-algorithm part reduces the ambiguity effect. Moreover, we derive the asymptotic mean square error (MSE) of our pro- posed algorithm, and the optimal weight in the pv-algorithm is given in a closed-form. II. PROBLEM FORMULATION Let K denote the number of subcarriers in one OFDM block. The index sets for pilot carriers and virtual carriers are denoted as P and V , respectively.The transmitted symbol on the kth subcarrier in the mth OFDM block is s k (m)=    p k (m) ∈C p , k ∈P , 0, k ∈V , d k (m), ∈C d otherwise, (1) where d k (m) is the information symbol from the signal constellation C d , and p k (m) is the pilot symbol from the signal constellation C p . The power of the pilot symbols is normalized, i.e., |p k (m)| =1. Suppose the channel length is upper bounded by LT s . The equivalent discrete channel vector can then be represented as h =[h 0 , ..., h L ] T , with normalized power ‖h‖ 2 =1. For DFT-based OFDM, a CP with length P is added in the front of each transmitted block. If P ≥ L, the mth received block after the removal of the CP is given by y(m)= e j2πφ((m-1)Ks+P ) Ω(φ)FHs(m)+ n(m), (2) where K s represents K + P , for simplicity in notations; s(m) = [s 0 (m), ..., s K-1 (m)] T is the mth OFDM block transmitted on all subcarriers; F is the K × K normalized IDFT matrix with the (a, b)th entry given by 1 √ K e j2π(a-1)(b-1) K ; Ω(φ) = diag{1,e j2πφ , ..., e j2πφ(K-1) }; φ is the normalized CFO by 1/T s ; H is the diagonal matrix with its (k,k)th element given by H(k,k)= H k-1  ∑ L l=0 h l e - j2π(k-1)l K , and each elements of n(m) is the sample of zero-mean white Gaussian noise with the variance σ 2 . 1-4244-0353-7/07/$25.00 ©2007 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.