Performance Analysis of the Blind Minimum Output Variance Estimator for Carrier Frequency Offset in OFDM Systems

Carrier frequency o ﬀ set (CFO) is a serious drawback in orthogonal frequency division multiplexing (OFDM) systems. It must be estimated and compensated before demodulation to guarantee the system performance. In this paper, we examine the performance of a blind minimum output variance (MOV) estimator. Based on the derived probability density function (PDF) of the output magnitude, its mean and variance are obtained and it is observed that the variance reaches the minimum when there is no frequency o ﬀ set. This observation motivates the development of the proposed MOV estimator. The theoretical mean-square error (MSE) of the MOV estimator over an AWGN channel is obtained. The analytical results are in good agreement with the simulation results. The performance evaluation of the MOV estimator is extended to a frequency-selective fading channel and the maximal-ratio combining (MRC) technique is applied to enhance the MOV estimator’s performance. Simulation results show that the MRC technique signiﬁcantly improves the accuracy of the MOV estimator.


INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) has been considered as a promising modulation scheme for the next generation wireless communication systems. OFDM signals are transmitted in parallel subchannels, which are frequency-nonselective and overlapped in spectra. Hence, OFDM systems are robust to frequency-selective fading and enjoy high bandwidth efficiency. As the symbol duration is extended, the OFDM scheme reduces the normalized delay spread and avoids intersymbol interference (ISI). Because of these excellent characteristics, OFDM has been suggested and standardized for high-speed communications in Europe for digital audio broadcasting (DAB) [1] and terrestrial digital video broadcasting (DVB) [2]. Furthermore, OFDM is standardized for broadband wireless local area networks, for example, ETSI-BRAN High-performance local area networks (Hiperlan/2) [3], IEEE 802.11a [4], and for broadband wireless access, for example, IEEE 802.16 [5]. One drawback of OFDM systems is that carrier frequency offset (CFO) between the transmitter and receiver may degrade system performance severely [6]. CFO causes a number of impairments, including the attenuation and phase rotation of each of the subcarriers and intercarrier interference (ICI). Many estimation techniques have been proposed to estimate and correct the CFO before demodulation. Moose proposed a scheme to estimate CFO by repeating a data symbol and comparing the phase of each of the subcarriers between successive symbols [7]. However, this scheme adds more overhead and reduces the bandwidth efficiency. Schmidl and Cox proposed an estimation algorithm for timing offset and carrier frequency offset estimation based on the training sequence [8]. To improve the bandwidth efficiency, many blind estimation techniques have been proposed. van de Beek et al. developed a maximum-likelihood (ML) estimator by exploiting the redundancy in the cyclic prefix (CP) [9]. This method, however, suffers from performance degradation when the delay spread is comparable to the length of CP. Luise et al. exploited the time-frequency domain exchange inherent to the modulation scheme and proposed a blind algorithm for CFO recovery [10]. Tureli et al. utilized the shift invariant feature of the signal structure and extended ESPRIT algorithm to estimate CFO [11]. Tureli et al. exploited the inherent orthogonality between information-bearing carriers and virtual carriers, and proposed an algorithm to estimate the CFO [12]. Because of the presence of the virtual carriers for the proposed algorithms in [11,12], the bandwidth efficiency is also reduced. In our 2 EURASIP Journal on Applied Signal Processing previous work [13], we proposed a new blind estimation algorithm which yields high bandwidth efficiency and high accuracy based on the minimum output variance (MOV). In [13], the performance of the MOV estimator is obtained by simulations. The main objective of this paper is to derive the theoretical mean and variance of the MOV estimator and examine the performance analytically. We also evaluate the performance of the MOV estimator over frequencyselective fading channels and apply the maximal-ratio combining (MRC) technique to improve its performance.
The rest of the paper is organized as follows. In Section 2, the OFDM system model is briefly introduced. In Section 3, the MOV estimator is described and its performance is analyzed over additive white Gaussian noise (AWGN) channels. Numerical results are presented in both AWGN and frequency-selective fading channels in Section 4. Finally, conclusions are drawn in Section 5.

OFDM SYSTEMS
OFDM is a form of mutlicarrier modulation and consists of a number of narrowband orthogonal subcarriers transmitted in a synchronous manner. The mth OFDM data block to be transmitted can be defined as s(m) = [s 0 (m), where N is the number of subcarriers. The OFDM signals can be formulated by inverse discrete Fourier transform (IDFT). Using the matrix representation, the mth block of the modulated signal is where W is the IDFT matrix given by with ω = 2π/N. After the IDFT modulation, a cyclic prefix (CP) is inserted and its length is assumed to be longer than the maximum delay spread of the channel to avoid intersymbol interference (ISI). The resultant baseband signal is upconverted to the radio frequency (RF) and transmitted over a multipath fading channel. At the receiver, the signal is downconverted and demodulated using discrete Fourier transform (DFT) to recover the desired signal. In the absence of CFO, the received signal is given by where represents the channel response in the frequency domain. After down-conversion, the carrier frequency offset Δ f is incurred because of the mismatch of carrier frequencies between the transmitter and receiver. The received signal, after the removal of the CP, is where Φ = diag 1, e jφ0 , . . . , e j(N−1)φ0 (6) denotes the CFO matrix. Note that φ 0 = 2πΔ f T s /N and T s is the OFDM block duration.

ESTIMATION OF CARRIER FREQUENCY OFFSET
Due to spectra overlapping, OFDM systems are sensitive to CFO. The CFO destroys the orthogonality among subcarriers, results in intercarrier interference (ICI) and causes system performance degradation. To improve the system performance, the CFO must be estimated and compensated before the DFT demodulation.

Minimum output variance (MOV) estimator
In the presence of CFO, the received signal, after DFT demodulation, is given by where W H represents the DFT demodulation matrix, T is the recovered signal vector and n (m) is the noise term after the DFT operation. Because of the presence of CFO, W H ΦW is no longer an identity matrix and the desired signal s(m) cannot be recovered properly.
For the sake of simplicity, the block number m will not be included in the subsequent derivation. The recovered signal on subcarrier k is given by where In (8), I 0 represents the attenuation and phase rotation of the desired signal and I l−k represents the ICI coefficient from other subcarriers. From (8) and (9), it can be seen that the coefficient I 0 of s k is independent of k. It implies that the desired signals on all subcarriers experience the same degree of attenuation and phase rotation. Note that the ICI term is the addition of many independent random variables and can be approximated as a complex Gaussian random variable with zero mean [15,16]. Thus, the output of each subcarrier can be considered as a dominant signal embedded in additive white Gaussian noise. To simplify the analysis, the BPSK modulation is employed in the OFDM scheme.
The ICI power is defined as Feng Yang et al.  and P ICI can be obtained as [17] where = Δ f T s ∈ (−0.5, 0.5) is the CFO normalized to the subcarrier spacing and sinc(x) = sin (πx)/(πx). The theoretical and the simulation results of the ICI power are presented in Figure 1. It is observed that the theoretical curve based on (11) matches the simulation results well.
Since the ICI can be approximated as a Gaussian random variable with zero mean, the variance of the ICI is equal to From (8) and (9), it is clear that the dominant signal on the kth subcarrier is I 0 s k . Hence, the magnitude mean of the dominant signal on the kth subcarrier is Based on the analysis above, the probability density function (PDF) of the output |d k | can be approximated as Gaussian mixture given by where u(x) is the unit step function. Let us consider the expectation and variance of the output |d k |. The expectation of where The second moment of |d k | is The variance of |d k | can be expressed as From (15), (17), and (18), it is observed that the expectation and variance of |d k | are functions of . Figures 2(a), 2(b), and 2(c) show the expectation, the variance, and the second moment of |d k | versus the normalized CFO, respectively. From these figures, it is observed that the simulation results match the theoretical results well. The results also show that the variance of |d k | monotonically increases and the expectation of |d k | monotonically decreases when the normalized CFO | | increases. In other words, the minimum variance and maximum mean are obtained when = 0. Intuitively, this property can be explained as follows. The CFO does not change the total signal power in one received symbol [14], but the CFO causes the uncertainty of the output. More frequency offset causes more uncertainties and thus, the variance of the output |d k | increases. This is the principal idea for the CFO estimator proposed in [13].
Based on this property, we have proposed a blind CFO estimator with minimum output variance (MOV) for OFDM systems in [13]. When N is sufficiently large, the expectation of |d k | is approximately equal to the average value of |d k | among N subcarriers. From the previous analysis, the average value of |d k | attains a maximum value if the CFO is properly compensated. Defining we search for φ e in the range of (−π/N, π/N) such that achieves its maximum value for Φ e = Φ, where W i is the ith column of the matrix W. If thermal noise is present, the accuracy of φ e may be improved by searching for the maximal value in (20) with more blocks. Hence, the proposed MOV estimator is to search for one φ e to maximize Λ(φ e ) [13], that is, where y(m) is defined in (5) and M is the number of blocks used by the MOV estimator. Based on (21), the DFT operation W H i in MOV estimator can be implemented efficiently by means of the fast Fourier transform (FFT) algorithm. Although the above analysis is based on BPSK modulation, it is also applicable to higher-order modulation schemes such as QPSK, 8-QAM, 16-QAM, and 64-QAM, and so forth. Simulations have been conducted to verify that the MOV estimator is applicable to higher-order modulation methods.

Performance analysis of the MOV estimator
For the MOV estimator, we need to obtain the expected value of |d k | for each trial value of φ e . The expectation of |d k | is approximated by the average value of |d k | among MN outputs, that is, When |d k | is embedded in AWGN, the accuracy of this approximation is affected and the difference between the expected value and the approximated average value is the major contribution to the error variance of the MOV estimator. It is known that μ φe follows the Gaussian distribution with mean μ φe and variance σ 2 w /MN [18], where σ 2 w represents the variance of AWGN noise. Thus, it has a probability that μ φe (for φ e = φ 0 ) is greater than the theoretical maximum mean value μ φ0 . When this happens, the estimation accuracy is affected. Equivalently, the analysis can be addressed as follows. Suppose that we have K trial values of φ e among the estimation range and we obtain the average output for each trial value independently. For the kth trial value, the average output is μ k = μ k +n k , where n k denotes the noise term with zero mean and variance σ 2 w /MN. The probability that μ k becomes the largest can be expressed as Pr μ k = max μ 1 , . . . , μ K = Pr μ k > μ 1 , μ k > μ 2 , . . . , μ k > μ K = Pr μ k + n k > μ 1 + n 1 , . . . , μ k + n k > μ K + n K . (23) Note that n 1 , n 2 , . . . , n K are independent noise. For a given n k , the conditional probability can be expressed as Pr μ k = max μ 1 , . . . , μ K | n k = Pr n 1 < μ k − μ 1 + n k , . . . , n K < μ k − μ K + n k = Pr n 1 < μ k − μ 1 + n k × · · · × Pr n K < μ k − μ K + n k Thus, the probability that μ k becomes the largest is obtained as where p nk (x) denotes the Gaussian PDF of n k given by Feng Yang et al.  When all the K probabilities in (25) are obtained, the mean and variance of the estimated offset can be obtained accordingly. Here, the variance represents the mean-square error (MSE) of the MOV estimator. The MSE results will be validated in the next section.

NUMERICAL RESULTS AND ANALYSIS
In this section, numerical results are presented to illustrate the performance of the proposed MOV estimator in OFDM systems. In particular, N = 64 subcarriers are used. The performance of the proposed MOV estimator is examined by the normalized MSE defined as where φ e is the estimate of φ 0 and P is the number of Monte Carlo runs. Simulations are conducted in both AWGN and frequency-selective fading channels. Figure 3 shows the normalized MSE of the proposed MOV estimator over an AWGN channel. From the simulation results, it is observed that the normalized MSE decreases as E b /N 0 increases. With more OFDM blocks, the accuracy of the proposed MOV estimator is in general improved. The approximated MSE values are the corresponding solid lines in Figure 3. The simulation results match the theoretical results well at high E b /N 0 , that is, E b /N 0 > 10 dB. The performance curve of the T&H estimator [12] is also included for comparison, where the number of subcarriers is N = 64, the number Normalized MSE of virtual carriers is L = 20, and the number of blocks for estimation is M = 4. For the value of MSE = 10 −3 , the proposed MOV algorithm outperforms the T&H algorithm by more than 6 dB in E b /N 0 . The reason for the significant improvement is that the proposed MOV estimator takes the average of the magnitude output and is robust to noise, but the orthogonality property utilized by the T&H estimator is more sensitive to noise. Also, the proposed MOV algorithm can be implemented by fast Fourier transform (FFT) with high efficiency. Thus, the computational complexity of MOV algorithm is generally lower than that of the T&H algorithm.

Performance over AWGN channels
In Figure 4, we present the MSE results of the proposed MOV algorithm with various numbers of OFDM blocks. From the simulation results, it is observed that the normalized MSE decreases with more OFDM blocks for estimation but using more blocks for estimation increases the complexity of the estimator. Also, the theoretical MSE results match the simulation MSE results well. Note that the normalized MSE is less than 10 −3 for moderately high E b /N 0 , which makes the degradation caused by the CFO negligible even when only one OFDM block is used for the estimation. Thus, the proposed MOV estimator is also suitable for burst transmission.

Performance over frequency-selective fading channels
Over frequency-selective fading channels, a 6-path multipath delay profile is assumed. The length of cyclic prefix (CP) is 6 and the maximal delay is not greater than the length of CP to avoid ISI. The MOV estimator takes the average of the output and searches for the maximum mean output. Thus, it is also applicable for frequency-selective fading channels even when  the channel information is not available. Figure 5 shows the normalized MSE results over frequency-selective fading channels. From the results, it is observed that the normalized MSE is less than 4×10 −3 when E b /N 0 ≥ 10 dB. The error floor is caused by the channel dispersion. It shows that the algorithm achieves high accuracy with only a few OFDM blocks under moderately high E b /N 0 . Thus, the proposed MOV algorithm is effective and reliable over frequency-selective fading channels.
In Figure 6, the normalized MSE of the proposed MOV algorithm is presented with different number of blocks over frequency-selective fading channels. It is observed that the accuracy is generally improved as the number of blocks increases but the improvement becomes insignificant for large M. Considering the system performance requirement and complexity, the number of estimation blocks M should be properly chosen.

Performance improvement with MRC technique
The MOV estimator takes the summation of the output over the N subcarriers. It is similar to the combining techniques in multicarrier CDMA (MC-CDMA) [19]. In MC-CDMA systems, many diversity-combining techniques have been applied to improve the system performance, for example, equal-gain combining (EGC) and maximal-ratio combining (MRC). With the MRC technique, the output SNR is maximized [16,21]. It is interesting to see whether the MRC technique can improve the performance of the proposed MOV estimator over frequency-selective fading channels.
In this subsection, we assume that the channel estimation has been achieved and the channel attenuation and the phase  shift on each subcarrier are known to the MOV estimator. Thus, the gain matrix for the MRC is given by [19,20] where H n is the channel gain on the nth subcarrier in (4) and the superscript (·) * denotes the complex conjugation. Multiplying with the gain matrix G, the strong signals on subcarriers carry larger weights than weak signals. Thus, the SNR of the output is maximized and the performance of MOV estimator is improved. According to the channel gain with MRC, the MOV estimator can be rewritten as where G represents the gain matrix for MRC. The normalized MSE results of the proposed MOV estimator versus E b /N 0 with MRC technique over frequencyselective fading channels are presented in Figure 7. From the simulation results, it is observed that the normalized MSE significantly decreases with MRC and the performance of the proposed MOV estimator with MRC is even comparable to that of the MOV estimator under the AWGN condition. It is also clear that the error floor is eliminated. This supports the argument that the error floor in Figure 6 is caused by channel dispersion. We also include the performance curve of the T&H estimator [12] for comparison. It is seen that the proposed MOV estimator still outperforms the T&H estimator over the same fading channel conditions. Feng Yang et al.

Compensation of the CFO
After the CFO estimation is accomplished, frequency offset can be properly compensated and corrected to allow for the data detection. Figure 8 shows the BER performance for three cases, namely, with 20% frequency offset, without frequency offset, and after offset correction over frequencyselective fading channels, respectively. It is observed that the BER has an error floor when there is a 20% frequency offset. The reason for the error floor is that the bit errors are mainly due to the ICI, not due to the noise for high E b /N 0 . The BER, after MOV estimation and correction, is very close to the BER without CFO, which validates that MOV estimator eliminates the effect of CFO effectively.

CONCLUSION
In this paper, we have examined the performance of the minimum output variance (MOV) estimator in OFDM systems. The variance and the expectation of the output magnitude have been derived for the MOV estimator. The theoretical MSE of the MOV estimator has been derived and the theoretical MSE results match the simulation results well. From the simulation results, it is observed that the MOV algorithm has high accuracy in both AWGN and frequency-selective fading channels. It has also been shown that the MOV algorithm outperforms the T&H algorithm. The MOV estimator can be implemented by FFT efficiently, which causes the computational complexity of the MOV estimator generally lower than that of the T&H estimator. Furthermore, the maximal-ratio combining (MRC) technique has been applied to the MOV estimator to improve the accuracy over frequency-selective fading channels. Simulation results verify that the MRC tech-  nique is effective and significantly improves the accuracy of the MOV estimator over frequency-selective fading channels.