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Blind estimation of carrier frequency offset, I/Q imbalance and DC offset for OFDM systems
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 105 (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
Consider an OFDM system with totally N subcarriers, among which K subcarriers occupied by data transmission are referred to as real subcarriers, and the other N  K unoccupied ones are referred to as virtual subcarriers. Let ${T}_{s}\triangleq \frac{T}{N}$denotes the sampling rate in digital signal processing (DSP) stage, where T is the duration of OFDM block without cyclic prefix (CP). After CP removal, the received samples that belong to the m th OFDM block can be expressed as
where k refers to the index of subcarrier, ${C}_{r}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left\{{k}_{0},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{k}_{K1}\right\}$is the indices set of all the K real subcarriers. S(k, m) is the modulated symbol mapped onto the k th subcarrier of m th OFDM block and H(k, m) is the corresponding frequency domain channel response. Both S(k, m) and H(k, m) are assumed to be zeromean independent random variables. ϕ(m) ≜ 2πε(m(N_{ CP } + N) + N_{ CP } )/N denotes a cumulative phase offset, where ε and N_{ CP } refer to the CFO normalized to subcarrier spacing and length of CP in samples, respectively. The impacts of I/Q imbalance on OFDM signal are characterized by two parameters α and β, which were widely employed in literatures [12, 13, 21]. Since only the ratio between α and β is needed for I/Q imbalance compensation, I/Q imbalance in this article is defined as $\gamma \triangleq \frac{\beta}{\alpha}$. DCO and zeromean AWGN are denoted by the last two terms d and w(n, m), respectively. For notation simplicity, (1) is rewritten as
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)
In our recent study [13], we proposed an eigendecomposition based blind estimator (EDE) for joint estimation of CFO, I/Q imbalance and DCO in OFDM systems. The basic idea of EDE can be summarized as (1) the impacts of I/Q imbalance and DCO on OFDM signal can be compensated by linear combination of the received signal, its complex conjugate, and an arbitrary dc signal; (2) the established MUSIClike CFO estimation algorithm [5] can then be applied to compensated signal. Based on the ideas, a key cost function was first constructed in EDE as
where V_{ N } is an N×(N  K) matrix, whose (n, l)th entry is ${e}^{j\frac{2\pi}{N}{v}_{l}n}$ where v_{ l } ∈D_{ r } with D_{ r } ≜ {k_{ K },... , k_{N1}} being the indices set of all the N  K virtual subcarriers. Note that the columns of U_{ N } are orthogonal to those of V_{ N } , i.e., ${\mathbf{V}}_{N}^{H}{\mathbf{U}}_{N}={0}_{N}.$${\mathbf{R}}_{3}\left(m\right)\triangleq \left[{\mathbf{r}}_{N}\left(m\right),\phantom{\rule{2.77695pt}{0ex}}{\mathbf{r}}_{N}^{*}\left(m\right),\phantom{\rule{2.77695pt}{0ex}}{\mathbf{1}}_{N}\right]$consists of the received signal, its complex conjugate, and a unit dc signal for the linear combination with a_{3} ≜ [a(0), a(1), a(2)] ^{T} being the corresponding weighting factors. The cost function in (3) is actually the same as that proposed in [5] except that the received signal r_{ N } (m) is replaced by the combination R_{3}(m)a_{3}. It had been illustrated in [13] that f_{EDE}(ν, a_{3}) achieves its minimum when ν = ± ε and a_{3} = g_{3} with g_{3} being the optimal weighting vector that can completely cancel the I/Q imbalance and DCO in received signal [13]. Therefore, CFO estimation can be achieved by minimization of f_{EDE}(ν, a_{3}) as
where${\mathbf{\Omega}}_{3}\left(v\right)\triangleq {\sum}_{m}{\mathbf{R}}_{3}^{H}\left(m\right){\mathbf{P}}_{N}\left(v\right){\mathbf{V}}_{N}{\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left(v\right){\mathbf{R}}_{3}\left(m\right)$. λ_{min}{.} denotes the operation of getting the smallest eigenvalue of matrix. To get rid of the sign ambiguity in ${\widehat{\epsilon}}_{a,\mathsf{\text{EDE}}}$, a metric T_{CFO,}_{EDE} was employed in EDE [13]. Finally, the CFO estimate without sign ambiguity was obtained by
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
The first improvement of TDAEDE comes from compensation of DCO before estimation of CFO and I/Q imbalance. Inspired by [11], we first make a coarse estimation for DCO by time domain average as
Then we cancel it from the received signal according to
By substituting (6) into (7), we have
where ${\mathbf{E}}_{N}\triangleq \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbf{I}}_{N}\frac{1}{N}{1}_{N}{1}_{N}^{T}$ with I_{ N } being the N×N identity matrix. Due to the fact that E_{ N }l_{ N } = 0, substituting (2) into (8) yields
4.2 Estimation of CFO by eigendecomposition
We can find in (9) that not only DCO is completely removed but also a key property of (2) that the desired signal component with CFO ε, i.e., the first term on the righthand side (RHS), and its mirror component with CFO ε introduced by I/Q imbalance, i.e., the second term on the RHS, are conjugates to each other up to a scaling factor is remained. As analyzed in [13], either of the two signal components can be removed by linear combination
where
The optimal weighting vector denoted by g_{2} ≜ [g(0), g(1)] ^{T} should satisfy either
to mitigate the component with CFO ε, or
to suppress the component with CFO ε. In absence of I/Q imbalance, we can directly apply NBE to the DCO compensated signal y_{ N } (m). The cost function should be constructed as [11]
where the columns of ${\stackrel{\u0303}{\mathbf{V}}}_{N}\left(v\right)$ are eigenvectors corresponding to the zero eigenvalues of ${\mathbf{Q}}_{N}\left(v\right)\triangleq {\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(v\right){\mathbf{U}}_{N}{\mathbf{U}}_{N}^{H}{\mathbf{P}}_{N}\left(v\right){\mathbf{E}}_{N}.$ As E_{ N } is of rank N  1, Q_{ N } (ν) has at least one zero eigenvalue. In order to involve the compensation of I/Q imbalance, a new cost function is constructed for TDAEDE as
where a_{2} ≜ [a(0), a(1)] ^{T} . By substituting (8) into (11), we have
where ${\mathbf{R}}_{2}\left(m\right)\triangleq \left[{\mathbf{r}}_{N}\left(m\right),\phantom{\rule{2.77695pt}{0ex}}{\mathbf{r}}_{N}^{*}\left(m\right)\right]$. Consequently, (15) can be rewritten as
Based on the constructed cost function, estimation of CFO and optimal weighting factor can be achieved by the following optimization approach as
where ${\mathbf{\Omega}}_{2}\left(\nu \right)\triangleq {\sum}_{m}{\mathbf{r}}_{2}^{H}\left(m\right){\mathbf{E}}_{N}{\stackrel{\u0303}{\mathbf{V}}}_{N}\left(v\right){\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{E}}_{N}{\mathbf{r}}_{2}\left(m\right)$. To avoid homogenous solution ${\mathbf{\u011d}}_{\mathsf{\text{2}}}={0}_{2}$, we impose a constraint ∥a_{2}∥^{2} = 1 to (17). The wellknown solution to this constrained minimization is [25]
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
It should be noted that the CFO estimate ${\widehat{\epsilon}}_{a,\mathsf{\text{TDA{EDE}}}$ obtained by (19) will involve sign ambiguity, i.e., ${\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}$ may approaches either ε or ε. As mentioned in Section 3, similar problem was also encountered by EDE [13]. From physical point of view, the ambiguity comes from the fact that the original received signal (2) contains component with CFO ε as well as component with CFO ε. From mathematical point of view, it can be deduced that ${\widehat{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}={\widehat{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}={0}_{N}$. Consequently, both Ω_{2}(ε) and Ω_{2}(ε) are rank deficient and therefore have eigenvalues of zero in absence of noise. To distinguish from sign ambiguity in CFO estimation, we take a similar approach as EDE with however a different metric ${T}_{\mathsf{\text{CFO}},\mathsf{\text{TDAEDE}}}\triangleq {\sum}_{m}\left\{{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{y}}_{N}\left(m\right)\u2225}^{2}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{y}}_{N}\left(m\right)\u2225}^{2}\right\}$. Substituting (9) into T_{CFO,TDAEDE} yields that
Recall that ${\stackrel{\u0303}{\mathbf{V}}}_{N}\left(v\right)$ consists of eigenvectors corresponding to the zero eigenvalues of Q_{ N } (ν), we have ${\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(v\right){\mathbf{U}}_{N}={0}_{N}$. In absence of noise, (20) can be simplified to
or
where $p\triangleq {\sum}_{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}{\mathbf{x}}^{*}\left(m\right)\u2225}^{2}={\sum}_{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{T}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}\mathbf{x}\left(m\right)\u2225}^{2}$ and $q\triangleq {\sum}_{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}\mathbf{x}\left(m\right)\u2225}^{2}.$ For most practical cases, p and q are comparable. If the real subcarriers are located symmetrically about dc, or mathematically if any column of U_{ N } is also a column of ${\mathbf{U}}_{N}^{*}$, we have ${\mathbf{\Omega}}_{2}^{*}\left(\epsilon \right)={\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}{\mathbf{U}}_{N}^{T}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{E}}_{N}={\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}{\mathbf{U}}_{N}^{H}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{E}}_{N}={\mathbf{\Omega}}_{2}\left(\epsilon \right)$ Recall that ${\stackrel{\u0303}{\mathbf{V}}}_{N}^{T}\left(\epsilon \right)$ and ${\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right)$ consist of the eigenvectors of ${\mathbf{\Omega}}_{2}^{*}\left(\epsilon \right)$ and Ω_{2}(ε), respectively, we have ${\stackrel{\u0303}{\mathbf{V}}}_{N}^{*}\left(\epsilon \right){\stackrel{\u0303}{\mathbf{V}}}_{N}^{T}\left(\epsilon \right)={\stackrel{\u0303}{\mathbf{V}}}_{N}\left(\epsilon \right){\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right)$ which leads to p = q. And because α^{2}≫β^{2}[14–16] in practice, the sign ambiguity can be distinguished by
The final CFO estimate is obtained as
4.4 Estimation and validation of I/Q imbalance
Integrated with the CFO estimation, the eigenvector corresponding to the smallest eigenvalue of ${\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right).$ is taken as ${\widehat{\mathbf{g}}}_{2}$, which is the estimate of the optimal weighting vector for the linear combination. Consequently, I/Q imbalance can be derived either from (12) as
for the cases of T_{CFO,TDAEDE}< 0, or otherwise from (13) as
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
Given the estimates of CFO and I/Q imbalance, their impacts can be compensated in the original received signal by
Suppose that ${\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=\gamma $ and then substitute (2) into (27), we have
where
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
Suppose that ${\widehat{\epsilon}}_{TDAEDE}=\epsilon $, it can be derived from (28) that
Through least square (LS) approach, estimate of c can be obtained by
where [.]^{+} denotes pseudoinversion and
Finally, fine estimate of DCO can be derived from (29) as
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].
Performance of CFO, I/Q imbalance and DCO estimation with different estimators are shown in Figures 1, 2, and 3, respectively. It can be observed that TDAMUE and NBE exhibit error floor for all the estimations due to the impacts of I/Q imbalance. On the contrary, performance of both EDE and TDAEDE achieve CRLB approximately with increasing SNR. It is also demonstrated by Figures 1, 2, and 3 that, given the reduction of computation efforts, TDAEDE achieves the same accuracy as EDE for estimation of CFO and I/Q imbalance and outperforms EDE for DCO estimation especially with lowtomedium SNR. This is because TDAEDE refines DCO estimation with validated estimates of I/Q imbalance. Comparison between the cases with and without validation also reveals the importance of I/Q imbalance validation.
Please note that TDAEDE and EDE achieve better performance than CRLB for I/Q imbalance estimation with relative low SNR when validation according to practical range of I/Q imbalance is performed. This is because that the validation makes additional error reduction. Recall that the validation will reset ${\widehat{\gamma}}_{TDAEDE}$ to 0 if it exceeds T_{IQI,TDAEDE}, which results in NMSE of γ^{2}. If on the contrary ${\widehat{\gamma}}_{TDAEDE}$ does not exceeds T_{IQI,TDAEDE}, the worst case as shown in Figure 4 will be ${\widehat{\gamma}}_{TDAEDE}={T}_{IQI,TDAEDE}{e}^{j\left[\text{arg}\left(\gamma \right)+\pi \right]}$, which results in NMSE of (γ + T_{IQI,TDAEDE})^{2}. As (γ + T_{IQI,TDAEDE})^{2} is greater than γ^{2}, (γ + T_{IQI,TDAEDE})^{2} is an upper error bound set by the validation for I/Q imbalance estimation. In low SNR region, CRLB exceeds this upper bound while NMSE of I/Q imbalance estimation keeps lower than it.
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.
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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.
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Liu, T., Li, H. Blind estimation of carrier frequency offset, I/Q imbalance and DC offset for OFDM systems. EURASIP J. Adv. Signal Process. 2012, 105 (2012). https://doi.org/10.1186/168761802012105
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Keywords
 Orthogonal Frequency Division Multiplex
 Orthogonal Frequency Division Multiplex System
 Carrier Frequency Offset
 Cyclic Prefix
 Orthogonal Frequency Division Multiplex Symbol