Blind estimation of carrier frequency offset, I/Q imbalance and DC offset for OFDM systems
 Tao Liu^{1}Email author and
 Hanzhang Li^{2}
https://doi.org/10.1186/168761802012105
© Liu and Li; licensee Springer. 2012
Received: 20 January 2012
Accepted: 9 May 2012
Published: 9 May 2012
Abstract
Sensitivity to carrier frequency offset (CFO) is one of the biggest drawbacks of orthogonal frequency division multiplexing (OFDM) system. A lot of CFO estimation algorithms had been studied for compensation of CFO in OFDM system. However, with the adoption of directconversion architecture (DCA), which introduces additional impairments such as dc offset (DCO) and inphase/quadrature (I/Q) imbalance in OFDM system, the established CFO estimation algorithms suffer from performance degradation. In our previous study, we developed a blind CFO, I/Q imbalance and DCO estimation algorithm for OFDM systems with DCA. In this article, we propose an alternative algorithm with reduced computation complexity and better accuracy. Performance of the proposed algorithm is demonstrated by simulations.
1 Introduction
As one of the most popular technologies for wireless communications, orthogonal frequency division multiplexing (OFDM) [1], on the one hand, has attractive advantages such as high spectrum efficiency, robustness to multipath fading and efficient implementation based on fast Fourier transform, which make it suitable for wideband wireless communications. However, on the other hand, OFDM suffers from performance degradation caused by carrier frequency offset (CFO) which damages the orthogonality among subcarriers and introduces intercarrier interference [2]. Since compensation of CFO is straightforward as long as estimate of CFO can be obtained, a lot of efforts have been taken on algorithms for CFO estimation in OFDM systems [3–10]. The established algorithms can be divided into two categories, dataaided and blind (or nondataaided). The dataaided algorithms, e.g., [3], can achieve better performance than the blind ones [4–10] at costs of data rate reduction due to transmission of pilots or training sequences. What we are concerned with in this article are blind estimation algorithms that on the contrary do not depend on pilots or training sequences and therefore retain data rate of system. Among the blind algorithms, the MUSIClike algorithm [5] was widely adopted in literatures [6, 11–13] as baseline due to the facts that first it was proved to be the maximum likelihood (ML) algorithm for CFO estimation in OFDM system under fading channel [7]; second it can be applied for both constant modulus modulation, e.g., phase shift keying (PSK), and inconstant modulus modulation, e.g., quadrature amplitude modulation (QAM), on subcarriers; third it is available for estimation of both integer CFO and fractional CFO; and finally it achieves reasonable compromise among performance and complexity within single OFDM symbol duration.
With the evolution of wireless transceiver, directconversion architecture (DCA) had been widely adopted in wireless terminals especially mobile terminals to replace the superheterodyne architecture [14–16]. DCA on the one hand simplifies the design of RF frontend and reduces costs, but on the other hand introduces disturbances such as inphase/quadrature (I/Q) imbalance and dc offset (DCO), etc., [14–17]. I/Q imbalance and DCO may not only cause interference by themselves but also violate established CFO estimation algorithms. A number of researchers had studied different approaches for joint estimation of CFO along with I/Q imbalance and/or DCO [11–13, 18–24]. Among these approaches, the blind algorithms [11–13] are of our great interests. Lin et al. [11] developed the MLequivalent nullspace based estimator (NBE) for joint CFO and DCO estimation and proved its equivalence to ML estimator. In our previous study, we studied the joint estimation of CFO and I/Q imbalance by exploring the similarity between the ideal OFDM signal and its mirror signal generated by I/Q imbalance [12]. Recently, we investigated the coexistence of CFO, I/Q imbalance and DCO in OFDM system and presented an eigendecomposition based estimator (EDE) [13] for all the three parameters.
In this article, we further study joint estimation of CFO, I/Q imbalance and DCO in OFDM systems and propose an alternative blind algorithm, EDE with time domain average (TDAEDE). When compared with EDE, TDAEDE achieves improvements in three aspects. First, it reduces computation efforts of the estimation to around 1/3 of that of EDE. Second, it improves accuracy of I/Q imbalance estimation within the whole signal to noise ratio (SNR) range of interest. Third, it significantly improves accuracy of DCO estimation with lowtomedium SNR. These improvements are achieved by compensation of DCO before estimation of CFO and I/Q imbalance, and validation of estimated I/Q imbalance according to its practical range.
The rest of this article is organized as follows. First, model of OFDM system with CFO, I/Q imbalance and DCO is established in Section 2. In Section 3, the EDE algorithm is briefly revisited and the proposed TDAEDE algorithm is then developed based on EDE and NBE in Section 4. In Section 5, simulation results and corresponding analysis are provided to demonstrate the performance of TDAEDE. Finally, conclusions are drawn in the last section.
2 Model of OFDM system with CFO, I/Q imbalance and DCO
where r_{ N }(m) ≜ [r(0, m),... , r(N  1, m)]^{ T }, x_{ K }(m) ≜ [H(k_{0}, m)S(k_{0}, m),..., H(k_{K1}, m) S(k_{K1}, m)]^{ T }e^{jϕ(m)}and w_{ N }(m) ≜ [w(0, m),... , w(N 1, m)]^{ T }. l_{ N }denotes an all ones column vector of length N. The inverse discrete Fourier transform (IDFT) on real subcarriers are denoted by an N×K matrix U _{ N }, whose (n, l)th entry is ${e}^{j\frac{2\pi}{N}{u}_{l}n}$ with u_{ l } ∈C_{ r } . The diagonal matrix ${\mathbf{P}}_{N}\left(\epsilon \right)\triangleq diag\left\{1,\phantom{\rule{2.77695pt}{0ex}}{e}^{j\frac{2\pi}{N}\epsilon},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{e}^{j\frac{2\pi \left(N1\right)}{N}\epsilon}\right\}$ represents the phase offset caused by CFO.
3 Eigendecomposition based estimator (EDE)
The eigenvector corresponding to the smallest eigenvalue of ${\mathbf{\Omega}}_{3}\left({\widehat{\epsilon}}_{\mathsf{\text{EDE}}}\right)$is taken as estimate of g_{3}, from which I/Q imbalance and DCO can be derived [13].
4 EDE with time domain average (TDAEDE)
When compared with EDE, TDAEDE achieves improvements in three aspects. First, it reduces computation efforts of the estimation to around 1/3 of that of EDE. Second, it improves the accuracy of I/Q imbalance estimation within the whole SNR range of interest. Third, it significantly improves the accuracy of DCO estimation with lowtomedium SNR.
4.1 Coarse estimation and compensation of DCO by time domain average
4.2 Estimation of CFO by eigendecomposition
and ${\widehat{\mathbf{g}}}_{2}$ equals to the eigenvector corresponding to the smallest eigenvalue of ${\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right).$ In case that ${\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right).$ has multiple minimum eigenvalues, ${\widehat{\mathbf{g}}}_{2}$ can be randomly selected from the eigenvectors corresponding to the minimum eigenvalue. Actually, since ${\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right).$ constitutes random data, it is of little possibility to have multiple minimum eigenvalues. Consequently, ${\widehat{\mathbf{g}}}_{2}$ obtained under such case has little influence of average performance.
4.3 Sign ambiguity of CFO estimation and its solution
4.4 Estimation and validation of I/Q imbalance
The second and third improvements of TDAEDE are achieved mainly by validation of estimated I/Q imbalance according to its practical range. Recall the fact that α^{2}≫β^{2}[14–16] in practice, which equivalently gives the range that γ^{2} ≪ 1. A too large ${\widehat{\gamma}}_{TDAEDE}$usually indicates failure of I/Q imbalance estimation. In TDAEDE, we validate the estimated I/Q imbalance according to a controllable threshold T_{IQI,TDAEDE} ∈ (0, 1). If the estimated result exceeds this threshold, i.e., ${\widehat{\gamma}}_{TDAEDE}>{T}_{\mathsf{\text{IQI,TDAEDE}}}$, it will be reset to${\widehat{\gamma}}_{TDAEDE}=0$, which means we would prefer to ignore I/Q imbalance for such cases rather than compensate it based on the unreasonable estimation results.
4.5 Compensation of CFO and I/Q imbalance
and ${\stackrel{\u0303}{w}}_{N}\left(m\right)\triangleq {w}_{N}\left(m\right){w}_{N}^{*}\left(m\right){\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}^{*}.$ It can be observed from the RHS of (28) that only the desired signal component with CFO ε is retained.
4.6 Fine estimation of DCO
4.7 Summary and discussion
The steps for joint estimation of CFO, DCO, and I/Q imbalance by TDAEDE can be summarized as:

After CP removal, group received samples that belong to the same OFDM block to construct the vector r_{ N }(m) ≜ [r(0, m),..., r(N  1, m)]^{ T };

Cancel the DCO in r_{ N }(m) by time domain average according to (7) to obtain y_{ N }(m);

Search for the trail ν that minimizes the smallest eigenvalue of Ω_{2}(ν) according to (19);

Set ${\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}$ to the ν found during the search and record the eigenvector corresponding to the smallest eigenvalue of ${\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right)$ in g_{2};

Calculate T_{CFO,TDAEDE} according to its definition, and detect sign ambiguity with (23) and correct it with (24);

Get estimate of I/Q imbalance according to (25) or (26) depending on the sign of T_{CFO,TDAEDE} and then reset it to 0 if it exceeds T_{IQI,TDAEDE} in the validation;

Compensate CFO and I/Q imbalance according to (27) to obtain z_{ N }(m);

Get fine estimation of DCO according to (31) and (33).
Computation efforts of TDAEDE are mainly determined by the eigendecomposition of Ω_{2}(ν) in the 1D search operation for CFO estimation, as it will execute in each searching step while other operations execute only once. Although computation of ${\u1e7c}_{N}^{H}\left(v\right)$ in TDAEDE is much more timeconsuming than that of ${\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left(v\right)$ in EDE, both of them can be calculated and stored in advance [11] and is therefore not necessary to be computed on the run. What has to be done online for TDAEDE is the eigendecomposition of Ω_{2}(ν). If the wellknown Power Method [25] is employed, eigendecomposition of the 2 × 2 matrix Ω_{2}(ν) is of complexity O(2^{3}) [26, 27], while eigendecomposition of the 3 × 3 matrix Ω_{3}(ν) in EDE is of complexity O(3^{3}) [26, 27]. Therefore, TDAEDE reduces about 1/3 of the computation efforts compared with EDE.
5 Simulation results
As has been illustrated, TDAEDE originates from NBE and EDE, which in turn come from MUE [5]. In order to demonstrate the performance of TDAEDE, it is compared by simulations with NBE [11], EDE [13] and TDAMUE, which compensates DCO with time domain average approach before estimation of CFO with MUE. Moreover, to illustrate the benefits of I/Q imbalance validation, EDE and TDAEDE with and without validation are evaluated. The threshold for the validation is set to 0.5. The OFDM system in the simulations is a WLAN [28] system with N = 64, K = 48, 64QAM modulation. The frequency selective Rayleigh fading channel is set to have power delay profile e^{p/ 5}, p = 0,..., 9. The estimation performance is evaluated by normalized mean square error (NMSE) defined as $E\left\{\widehat{\epsilon}\epsilon {}^{2}\right\},E\left\{\widehat{\gamma}\gamma {}^{2}\right\}$, and $E\left\{\widehat{d}d{}^{2}\right\}$, where E{.} denotes expectation operation. To track the variation of CFO, I/Q imbalance and DCO as quickly as possible, only one OFDM symbol is used in the estimation. In addition to the aforementioned estimators, we also provide the corresponding CramérRao lower bound (CRLB), whose derivation can be found in [13].
6 Conclusions
In this article, a novel blind estimator TDAEDE for joint estimation of CFO, I/Q imbalance and DCO in DCA OFDM systems is presented. Compared with our previous study EDE, TDAEDE reduces computation efforts to around 1/3 of that of EDE by compensation of DCO with time domain average approach before estimation of CFO, and improves accuracy of I/Q imbalance estimation and DCO estimation by validation of estimated I/Q imbalance according to its practical range. Performance of TDAEDE is demonstrated with established algorithms and CRLB by simulations.
Declarations
Acknowledgements
This study was supported by the National Natural Science Foundation of China under Grant 60903004, the Beijing Natural Science Foundation under Grant 4102042 and the Fundamental Research Funds for the Central Universities under Grant FRFTP12097A.
Authors’ Affiliations
References
 Prasad R: OFDM for Wireless Communications Systems. Artech House, London; 2004.Google Scholar
 Steendam H, Moeneclaey M: Sensitivity of orthogonal frequencydivision multiplexed systems to carrier and clock synchronization errors. Signal Process 2000, 80(7):12171229. 10.1016/S01651684(00)000311View ArticleMATHGoogle Scholar
 Schmidl T, Cox D: Robust frequency and timing synchronization for OFDM. IEEE Trans Commun 1997, 45(12):16131621. 10.1109/26.650240View ArticleGoogle Scholar
 Van de Beek J, Sandell M, Borjesson P: ML estimation of time and frequency offset in OFDM systems. IEEE Trans Signal Process 1997, 45(7):18001805. 10.1109/78.599949View ArticleMATHGoogle Scholar
 Liu H, Tureli U: A highefficiency carrier estimator for OFDM communications. IEEE Commun Lett 1998, 2(4):104106.View ArticleGoogle Scholar
 Tureli U, Liu H, Zoltowski M: OFDM blind carrier offset estimation: ESPRIT. IEEE Trans Commun 2000, 48(9):14591461. 10.1109/26.870011View ArticleGoogle Scholar
 Chen B: Maximum likelihood estimation of OFDM carrier frequency offset. IEEE Signal Process Lett 2002, 9(4):123126.View ArticleGoogle Scholar
 Yao Y, Giannakis G: Blind carrier frequency offset estimation in SISO, MIMO, and multiuser OFDM systems. IEEE Trans Commun 2005, 53: 173183. 10.1109/TCOMM.2004.840623MathSciNetView ArticleGoogle Scholar
 Jeon H, Kim K, Serpedin E: An efficient blind deterministic frequency offset estimator for OFDM systems. IEEE Trans Commun 2011, 59(4):11331141.View ArticleGoogle Scholar
 Oh J, Kim J, Lim J: Blind carrier frequency offset estimation for OFDM systems with constant modulus constellations. IEEE Commun Lett 2011, 15(9):971973.View ArticleGoogle Scholar
 Lin H, Senevirathna HM, Yamashita K: Blind estimation of carrier frequency offset and DC offset for OFDM systems. IEEE Trans Commun 2008, 56(5):704707.View ArticleGoogle Scholar
 Liu T, Li H: Blind carrier frequency offset estimation in OFDM systems with I/Q imbalance. Signal Process 2009, 89(11):22862290. 10.1016/j.sigpro.2009.04.026View ArticleMATHGoogle Scholar
 Liu T, Li H: Joint estimation of carrier frequency offset, dc offset and I/Q imbalance for OFDM systems. Signal Process 2011, 91(5):13291333. 10.1016/j.sigpro.2010.12.002View ArticleMATHGoogle Scholar
 Razavi B: Design considerations for directconversion receivers. IEEE Trans Circ Syst II Analog Digital Signal Process 1997, 44(6):428435. 10.1109/82.592569View ArticleGoogle Scholar
 Mak PI, U SP, Martins RP: Transceiver architecture selection: review, stateoftheart survey and case study. IEEE Circ Syst Mag 2007, 7(2):625.View ArticleGoogle Scholar
 Chastellain F, Botteron C, Farine P: Looking inside modern receivers. IEEE Microwave Mag 2011, 12(2):8798.View ArticleGoogle Scholar
 Deepaknath T, Marc M: Efficient compensation of transmitter and receiver IQ imbalance in OFDM systems. EURASIP J Adv Signal Process 2010, 2010: 114.Google Scholar
 Tubbax J, Fort A, Van der Perre L, Donnay S, Engels M, Moonen M, Man H De: Joint compensation of IQ imbalance and frequency offset in OFDM systems. In IEEE Global Telecommunications Conference (GLOBECOM). Volume 4. San Francisco, USA; 2003:23652369.Google Scholar
 Xing G, Shen M, Liu H: Frequency offset and I/Q imbalance compensation for directconversion receivers. IEEE Trans Wirel Commun 2005, 4(2):673680.View ArticleGoogle Scholar
 Lin H, Wang X, Yamashita K: A lowcomplexity carrier frequency offset estimator independent of DC offset. IEEE Commun Lett 2008, 12(7):520522.View ArticleGoogle Scholar
 Inamori M, Bostamam A, Sanada Y, Minami H: IQ imbalance compensation scheme in the presence of frequency offset and dynamic DC offset for a direct conversion receiver. IEEE Trans Wirel Commun 2009, 8: 22142220.View ArticleGoogle Scholar
 Lin H, Zhu X, Yamashita K: Lowcomplexity pilotaided compensation for carrier frequency offset and I/Q imbalance. IEEE Trans Commun 2010, 58(2):448452.View ArticleGoogle Scholar
 Chung Y, Phoong S: Joint estimation of I/Q imbalance, CFO and channel response for MIMO OFDM systems. IEEE Trans Commun 2010, 58(5):14851492.View ArticleGoogle Scholar
 Yan F, Zhu WP, Ahmad MO: Carrier frequency offset estimation and I/Q imbalance compensation for OFDM systems. EURASIP J Adv Signal Process 2007, 2007: 111.MATHGoogle Scholar
 Golub GH: CFV Loan. In Matrix Computations. JHU Press, Baltimore; 1996.Google Scholar
 Smale S: Complexity theory and numerical analysis. Acta Numerica 1997, 6: 523551.MathSciNetView ArticleMATHGoogle Scholar
 Demmel J, Dumitriu I, Holtz O: Fast linear algebra is stable. Numerische Mathematik 2007, 108: 5991. 10.1007/s002110070114xMathSciNetView ArticleMATHGoogle Scholar
 Part 11: Wireless LAN medium access control and physical layer (PHY) specifications: highspeed physical layer in the 5 GHz band. IEEE 802 LAN/MAN Standards 1999.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.