CFO Estimation Based on Auto-Correlation With Flexible Intervals for OFDM Systems With 1-bit ADCs

For orthogonal frequency division multiplexing (OFDM) systems, the carrier frequency offset (CFO) estimation is required for the frequency synchronization before the data reception. When 1-bit ADCs are employed in OFDM systems, the available information is limited to the sign of the received signal, and the acquisition of the CFO is not easy problem. To overcome this limitation, we utilize the extended preamble and construct the bank of the CFO estimators based on auto-correlation (AC) taking the samples from the preamble with flexible intervals, which produces the candidates for CFO estimate. We propose the CFO estimation algorithms using the normalized squared errors (NSEs) as the concept to check the quality of the candidates of CFO estimate. The simulation results show that the proposed algorithm works well in the multipath fading channel, and is superior to the conventional CFO estimation method based on AC with fixed interval.


I. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) as the waveform for the multicarrier transmission has been used in wireless communication systems [1]- [3]. In practical OFDM systems, the time and frequency synchronization procedures are important to mitigate intersymbol and intercarrier interferences, and performed by utilizing preambles or pilots as reference signal. In [4]- [6], the carrier frequency offset (CFO) estimation is dealt with, and based on auto-correlation (AC) using the repetition structure of the preambles.
Also, the OFDM is suitable for the wideband communication systems. However, the expanded bandwidth causes the growth of the sampling rate of the analog-to-digital converters (ADCs), which poses problems for high power consumption and hardware complexity. Thus, 1-bit ADCs as the extreme case of low resolution ADCs can be considered as the solutions for low power consumption and hardware complexity [7]. The OFDM is employed in massive MIMO (multiple input multiple output) [8], [9] and millimeter wave (mmWave) [10] with low resolution ADCs.
The associate editor coordinating the review of this manuscript and approving it for publication was Emrecan Demirors .
In [11]- [13], the channel estimation is considered for the OFDM systems with low resolution ADCs. In [12], the deep learning based methods are utilized to estimate channel, and the autoencoder for data symbol detection is also proposed. In [13], the Turbo-like channel estimation is proposed, and the prototyping system is presented for the reliable data transmission.
When 1-bit ADCs are deployed in the OFDM systems, the available information is only the sign of received signal. Thus, the phase differences from the received signal are distorted, so the CFO estimation is difficult in the OFDM receiver with 1-bit ADCs. In addition, there are several works dealing with the synchronization issues in the mmWave systems with 1-bit ADCs [14]- [16]. The directional frame synchronization method is developed in [14], and the message passing based algorithm is proposed to jointly estimate CFO and channel in [15]. In [16], the synchronization sequences is designed, and the CFO estimation methods based on the ratio metric exploiting those sequences proposed. However, previous works for the CFO estimation are not addressed well in OFDM systems with 1-bit ADCs.
In this paper, we focus on the CFO estimation based on AC in the OFDM systems with 1-bit ADCs. The contributions of this paper are summarized as follows. • We present the relation between CFOs and flexible intervals of repetition parts in the extended preamble used to overcome the lack of the available information. We suggest the feasibility of the perfect CFO estimation from the relation that makes the phase difference between repetition parts undistorted.
• We propose the CFO estimation algorithm to be able to perfectly estimate CFO corresponding to the relation by using the AC with the flexible intervals. By adopting the normalized squared error (NSE) evaluating the candidates of the CFO estimate, the proposed algorithm can determine the final CFO estimate for the fractional CFO range by comparing the NSEs. Notations: For a complex number x, x R and x I denote the real and imaginary parts of x, and x denotes the phase of x. For a real number x, the sign function is slightly modified and defined as S(x) = 1 For a real number x and a positive real number y, (x) y denotes the limited value x ∈ [−y, y]. For a set X , |X | denotes the cardinality of X . Z is the set of integer number. Z + is the set of positive integer number. E{·} is the expectation operation.

A. SIGNAL MODEL
The received signal distorted by CFO in OFDM system is written by is the l-th tap of multipath fading channel with the length L, w[n] is additive white Gaussian noise (AWGN) with zero mean and variance σ 2 w , N is OFDM symbol length, and N cp is cyclic prefix length. We define the signal-to-noise ratio (SNR) by SNR = 1/σ 2 w . ε is a CFO normalized by subcarrier spacing. For the description, y[n] is simply represented as where is the received signal without CFO and AWGN. The quantized received signal is given by Fig. 1 shows the block diagram of the OFDM system with 1-bit ADCs for CFO estimation, and y ADC [n] is the output of the 1-bit ADCs. We assume the perfect time synchronization.

B. LIMITATIONS OF CONVENTIONAL CFO ESTIMATION METHODS AND MOTIVATIONS
Among the conventional CFO estimation methods as mentioned above, we focus on the simple and efficient methods based on AC, which use preambles as reference signals [4]- [6]. Assuming one OFDM symbol containing B identical repetition parts as the preamble, the samples of y[n] with the fixed interval N B are taken to calculate AC, and then the CFO ε in (1) (4). However, if the phase differences between the samples of y ADC [n] are multiple of π 2 as undistorted values, it is possible to estimate ε by using (4). In other words, if we can take the samples of y ADC [n] with flexible intervals to make the phase difference caused by ε be multiple of π 2 as an undistorted value, ε can be estimated by CFO estimation method based on AC.
Let E be the set of CFOs producing the phase difference of multiple of π 2 as an undistorted value, and ε i be the i-th element of E. If the phase differences 2π|ε i |n i N = qπ 2 where q ∈ {1, 2}, 1 then This indicates the specific corresponding relation of CFO ε i and interval n i denoted by (ε i , n i ) for convenience. For ε i , the undistorted phase difference is obtained by taking the samples of y ADC [n] with the flexible interval n i from the relation of (ε i , n i ). Hence, ε i can be directly estimated by using the conventional CFO estimation method based on the AC with the flexible interval n i . In (4), B 2π is replaced by N 2πn i from (ε i , n i ) to estimate ε i . To deal with an unknown ε, we adopt the NSE defined as e i = |(ε i − ε i )/ε i | 2 whereε i is the estimate of ε i . Since the NSE e i can evaluate the similarity betweenε i and ε i , i.e., the quality ofε i , we consider the CFO estimation method for the unknown ε by utilizing the NSE e i as the additional information.
In the following examples, it is shown that there is the limitation of CFO estimation method based on the AC with fixed interval, and the effectiveness of the corresponding relation in (5). In addition, the meaning of the NSE for each CFO ε i is presented.

EXAMPLE
Assuming N = 4, B = 2, and ε = 1 6 , let y 0 = [y 0 [0], . . . , y 0 [11] 8 , e j 3π 8 ] as the identical repetition part, be the received signal vector without CFO ε. Let = [ 0 , . . . , 11 ] where n = e j2πεn N , be the phase rotation vector due to ε. When y = ⊗ y 0 where ⊗ is the element-wise multiplication, the quantized version of y is y ADC .   Fig. 2(a), the samples of y and y ADC are represented. For y, each average phase difference between the repetition parts with the fixed interval is π 6 . Calculating AC with the fixed interval in (4), the total average phase difference is also π 6 , and thenε = 1 6 is obtained. For y ADC , each average phase difference with the fixed interval N B is limited to 0 or π 4 , and thus total average phase difference 3π 20 . Then,ε = 3 20 , so there is the limitation in AC based CFO estimation with the fixed interval. Fig. 2(b) shows the effectiveness of the corresponding relation (ε i , n i ). In Fig. 2(b), we consider the average phase differences of the repetition parts with flexible interval. For , and (ε 5 , n 5 ) = ( 1 10 , 10). For y, the average phase differences are π 6 , π 3 , π 2 , 2π 3 , and 5π 6 , which are proportional to the flexible intervals in order of n 1 , n 2 , n 3 , n 4 , and n 5 , respectively. However, for y ADC , the average phase differences corresponding to the flexible intervals are π 4 , π 4 , π 2 , 3π 4 , and 3π 4 . π 6 , π 3 , 2π 3 , and 5π 6 as the average phase differences of y are distorted, and changed to π 4 and 3π 4 due to the quantization. π 2 for n 3 is undistorted and not changed, since ε equals to ε 3 and satisfies the corresponding relation (ε 3 , n 3 ) with n 3 . In addition, by considering the flexible intervals, the estimates of ε are obtained byε 1 = 1 4 , ε 2 = 1 8 ,ε 3 = 1 6 ,ε 4 = 3 16 , andε 5 = 3 20 , and then the NSEs are calculated as e 1 = 1 4 , e 2 = 1 4 , e 3 = 0, e 4 = 1 4 , and e 5 = 1 4 . As a result, the NSE of ε 3 is zero as the minimum value due to corresponding relation (ε 3 , n 3 ), and thusε 3 has the highest quality.

III. PROPOSED CFO ESTIMATION
In the OFDM systems with 1-bit ADCs, we employ the CFO estimation methods based on AC using a preamble. However, since the available information is limited to the quantized received signal, we use the extended preamble to take the samples with the flexible intervals. Then, E is specifically defined from the extended preamble by using the corresponding relation (ε i , n i ).
In order to estimate not only ε i ∈ E but also the unknown ε, we propose the CFO estimation algorithm adopting the NSE. In consideration of the inherent convexity of the NSE 2 and the periodicity of phase, e i are properly compared to estimate the unknown ε.

A. TRAINING SYMBOL
As shown in Fig. 3, the extended preamble of length N T contains multiple OFDM symbols of length N , which consist of the repetition parts. Each OFDM symbol includes B identical repetition parts, and each repetition part denoted by A is the basic unit of length N B . n i is the flexible interval between repetition parts for AC, and multiple of N B . W is the size of the window for AC. The maximum value of n i is denoted by n max . For example, in Fig. 3, the two grayed repetition parts of the length W are the interval as n i apart, and used for AC to obtain a phase difference. For AC, the windows of size W slide while maintaining the interval n i , and sequentially cover the all repetition parts. From the corresponding relation in (5), letting K = |E| 2 , we define E based on the structure of the extended preamble as below Specifically, ε i is constructed for the cases of q = 1 and q = 2 in (5) as follows.
. To construct the repetition part A of length N B , the complex Gaussian random vector of length N B is generated, and then normalized to make the 2-norm of the vector be N B . Also, to generate the OFDM symbols contained in the extended preamble, the pilots can be obtained from the output of FFT of one OFDM symbol with B identical repetition parts, and then nonzero pilots appear at the subcarrier positions of 0, B, 2B, . . . , N − B as the comb-type. The extended preamble is constructed by concatenating N T N OFDM symbols, and then the CP consisting of multiple of the repetition parts is attached to the extended preamble.

B. PROPOSED CFO ESTIMATION METHODS
According to E, the bank of the CFO estimators based on AC with the flexible intervals is represented bŷ where i = 1, . . . , |E|, and 2|ε i | π replaces B 2π in (4) from (ε i , n i ) for q = 1. The intervals n i of the repetition parts can be flexibly selected to obtain the phase differences needed to estimate ε i . When each term of AC y ADC [n + N B m + n i ]y * ADC [n+ N B m] is e jπ/2 for ε i > 0, the i-th estimator in (7) results inε i = ε i as the best performance. When y ADC [n + N B m + n i ]y * ADC [n+ N B m] is e −π/2 for ε i < 0, with the same way for ε i > 0, the ith CFO estimator in (7) producesε i = ε i and then achieves e i = 0, so is called the CFO estimator corresponding to (ε i , n i ). The conventional CFO estimator in (4) is equivalent to the CFO estimator corresponding to (ε 1 , n 1 ) or (ε 2K , n 2K ). From the result, y ADC [l + n i ]y * ADC [l] = e jπ/2 , so the CFO estimator corresponding to (ε i , n i ) producesε i = ε i , and then achieves e i = 0 at high SNR.
For ε i < 0, y ADC [l + n i ]y * ADC [l] = e −jπ/2 is achieved with the same way for ε i > 0, so the CFO estimator corresponding to (ε i , n i ) producesε i = ε i , and then achieves e i = 0 at high SNR.
Although Lemma 1 shows that ε i can be perfectly estimated, it should be able to estimate unknown ε ∈ E. To solve this problem, we consider methods adopting the NSE with convex property. The bank of the CFO estimators producesε i , and then e i are calculated for i = 1, . . . , |E|. Comparing e i in consideration of the convexity of the NSE,ε is obtained by determiningε =ε i 0 whereε i 0 minimizes e i , i.e., the estimate with the highest quality is selected.
However, for n k = (2z + 1)n i where z ∈ Z + , the CFO estimator corresponding to (ε k , n k ) can even achieve e k = 0 for ε i due to the periodicity of phase. For ε i > 0, the phase difference 2πε i n k N can become π 2 + zπ , and then the valid phase of 2πε i n k N is ± π 2 . This means that the CFO estimator corresponding to (ε k , n k ) can produceε k = ε k for ε i > 0, so e i = e k = 0 can be achieved. In addition, for ε i < 0, the phase difference 2πε i n k N for n k can become − π 2 −zπ . With the same way for ε i > 0, the valid phase of 2πε i n k N is ± π 2 . The CFO estimator corresponding to (ε k , n k ) can also producê ε k = ε k for ε i < 0. Hence, there can exist the ambiguity of the CFO estimation for unknown ε ∈ E by using the CFO estimation method comparing with the NSEs.
Lemma 2: For ε i ∈ E, if n k = (2z + 1)n i where z ∈ Z + , then the CFO estimator corresponding to (ε k , n k ) achieves e k = 0 at high SNR.
Theorem 1: For ε ∈ E, if |ε i | > |ε k | and e i = e k = 0, ε is estimated by determiningε =ε i , using the bank of the CFO estimators in (7) at high SNR.
From the Theorem 1, we can estimate the unknown ε ∈ E regardless of the ambiguity of the CFO estimation at high SNR. When σ w = 0, for ε i ∈ E, e i generally has a small enough value as a local minimum rather than zero. Thus, we search local minimum points of e i , and determineε according to the Theorem 1. In addition, for ε ∈ E, the CFO estimation based on the NSE is equivalent to select the most suitable CFO estimator from the bank of the CFO estimators.
Finally, although we can try to estimate the unknown ε / ∈ E, there is no a CFO estimator corresponding to ε. However, since the elements of E can cover the CFO range of [− B 2 , B 2 ] densely enough, it is feasible to estimate ε by utilizing the bank of the CFO estimators in (7). 3 Therefore, we propose the CFO estimation algorithm based on the NSE, which is depicted as the block diagram in Fig. 4 and summarized in Algorithm 1.
Considering the CFO range of interest, e.g., [−0.5, 0.5] as the fractional CFO, we define the threshold of e i as e th = |(ε i th − ε i th +1 )/ε i th | 2 where ε i th is the maximum value among the elements of E belonging to the CFO range of interest, and ε i th +1 is the second maximum value. If ε i is out of the CFO range of interest,ε i is not selected as theε. In step 1 and 2, the CFO estimator corresponding to (ε i , n i ) in (7) yieldsε i , and then e i are calculated for i = 1, . . . , |E|. In step 3, the local minimum points of e i are searched through comparison with three consecutive values, 4 and I = {i 1 , i 2 , . . .} is defined as the set of indexes corresponding to the local minimum points. In step 4, the indexes satisfying that e i k > e th for i k ∈ I are picked out, and then excluded from I. In the remaining steps,ε is determined according to |I|.
Additionally, the compensation of the phase rotation caused by the CFO ε is performed by applying e −j2πεn/N in the data path of the OFDM receiver in Fig. 1. We consider the repetition parts of length N B in Section III-A, and the phase difference between the repetition parts of the interval n i+k . The phase difference 2πε i n i+k N = 2πε i n i N n i+k n i for ε i > 0 becomes π 2 + π 2 k i−1 for 2 ≤ i + k ≤ K . π 2 is the portion of 2πε i n i N , and π 2 k i−1 corresponds to the additional phase, which is valid on its own in unquantization case. Since n i+k = n 2K +1−i−k , the phase difference 2πε i n 2K +1−i−k N is also π 2 + π 2 k i−1 due to the symmetry of n i . However, in quantization case, since each repetition part contains N B QPSK symbols, the additional phase is not π 2 k i−1 as nominal value. Thus, the additional phase is dependent on the sample pairs of the interval n i+k within repetition parts whose phase difference is not π 2 . The number of the sample pairs is denoted by c i,k , and determined by the inequality

Algorithm 1 CFO Estimation Algorithm Based on NSE
N means the effective phase difference of two adjacent elements of each repetition part. When m < | k i−1 | < m + 1 where m ∈ Z + , the average phase difference between repetition parts with the interval n i+k is π is limited to [−π, π] due to the periodicity of the phase, the average phase difference is represented by ( π 2 (1 + B N c i,k )) π to specify the limitation of the phase, and (1 + B N c i,k ) is also represented by (1 + B N c i,k ) 2 . Then, the estimate and the NSE of ε i+k are obtained aŝ ε i+k = ε i+k (1 + B N c i,k ) 2 and e i+k = |(1 + B N c i,k ) 2 − 1| 2 , respectively. In addition, (1 + B N c i,k ) is monotonic increasing with regard to k, and thus (1 + B N c i,k ) 2 is the sawtooth pattern with regard to k. Therefore, (1 + B N c i,k ) 2 − 1 is also the sawtooth pattern with regard to k, so there exist the local minima at cusps, which are corresponding to the points with the ambiguity of the CFO estimation in Lemma 2. Thus, e i+k has also the local minima at the same points. Therefore, e i+k is not only convex around the points of local minima, but also cyclic with regard to k.
For ε / ∈ E and ε > 0, assuming ε i is most similar with ε, . In the same manner as above, the average phase difference is ( ε Therefore, e i+k is also convex and cyclic with regard to k.

2) COMPUTATIONAL COMPLEXITY
Both the conventional and the proposed CFO estimation methods are based on AC to calculate the phase difference, and the proposed method contains the additional steps in Algorithm 1 to determine the estimate of CFOε.
First, to compare the computational complexity of the AC, we consider the number of complex multiplications. In the conventional method, the W complex multiplications are repeatedly performed B N (N T − W ) times. In the proposed method, the W complex multiplications are repeatedly performed B N (N T − n i ) times for the flexible interval n i , and are summed as  Table 1. As a result, for the AC, the conventional and the proposed methods require the computational complexity of O(N T ) and O(N 2 T ), respectively.  Moreover, the proposed method requires the additional computational complexity for the steps in Algorithm 1, which listed in Table 2. Each step requires simple operations, which performed repeatedly. For example, the calculation of NSE e i needs 1 real subtraction, 1 real division, and 1 real multiplication, which are repeated |E| times. The other steps also require very simple operations, which performed repeatedly on |E| scale at most. As a result, the steps except for the computation of AC in Algorithm 1 can be handled as the combinations of simple operations, and require the computational complexity of O(N T ).
Therefore, the conventional and the proposed methods require the computational complexity of O(N T ) and O(N 2 T )+ O(N T ), respectively.

IV. SIMULATION RESULTS
We verify the performances of the CFO estimation algorithm based on the NSE in the OFDM systems with 1-bit ADCs over the multipath fading channel.    5 shows the mean squared errors (MSEs) of Algorithm 1 and conventional method (denoted by Alg. and Conv.) according to the number of the OFDM symbols in the preamble, which is equivalent to the preamble length N T , at SNR = 30 dB. The MSEs are evaluated by E{|ε − ε| 2 }. As N T increasing, the MSEs tend to be enhanced, and then converge. This means that there is limitation in strictly measuring the phase difference for ε / ∈ E for even large N T due to 1-bit quantization. However, the MSEs of Algorithm 1 are superior to that of the conventional method, since Algorithm 1 can select the CFO estimator more suitable for ε from the bank of the CFO estimators. We observe that the MSEs are slightly improved for about N T ≥ 8N for all cases, and even for B = 16 of Algorithm 1 the decrement of MSE for the increment of OFDM symbol is less than 10%. Thus, the preamble length is selected as N T = 8N for the following simulations.
In Fig. 6, the MSEs are observed to be improved as SNR increasing, and then converge at high SNR. This means that there is limitation in strictly measuring the phase difference for ε / ∈ E at even high SNR due to 1-bit quantization. In the case of B = 16, the MSE of Algorithm 1 is 22.2 times smaller than that of conventional method at SNR = 30 dB. In the cases of conventional method and Algorithm 1 of B = 2 and B = 4, the MSEs at the SNR region around SNR = 8 dB are somewhat lower than the saturated MSEs at SNR = 30. This is because the AWGN noise can aid estimating CFO by varying slightly the patterns of the repetition parts favorably. In Fig. 7, the MSEs show the dependence on the CFO ε, and are symmetric for ε = 0. In Algorithm 1, the peaks of MSEs appear contiguously in zigzags. The downward peaks correspond to the elements of E, i.e. these peak points can ideally achieve e i = 0 for ε i by using the CFO estimator oriented to (ε i , n i ). The peaks become denser as B increasing, since the elements of E are defined more densely. In the case of B = 16 for Algorithm 1, the peak points correspond to [0.50, 0.44, 0.40, 0.36, . . .] as the elements of E. Likewise, the upward peaks occur due to the inherent differences between ε / ∈ E and the elements of E. Since the inherent differences becomes smaller as the CFO getting near zero, the value of the upward peaks become smaller. Unusually, the two centered upward peaks happen from the fact that it is hard to estimate the CFO near zero not included in E. Compared with the conventional method, the Algorithm 1 has the better MSE performances except near the two centered peaks.
In the conventional method, the downward peaks happen at ε = ±0.5 due to the CFO estimators corresponding to (0.5, N 2 ). The MSE is better near ε = ±0.25, which correspond to ± π 4 that can be expressed as the accurate phase difference relatively, even if distorted by quantization. Fig. 8 shows the bit error rates (BERs) of the coded OFDM systems. The Turbo code with code rate 1 3 [2] for 80 data bits and BPSK modulation are applied, and log-MAP algorithm is used for the decoding. Assuming the perfect channel, the zero-forcing equalization is applied. To evaluate the BER of the OFDM systems, we consider the frame consisting of the preamble part and the data part. The preamble of length N T = 8N and B = 16 is followed by the zero guard interval of length 16 and the OFDM symbols for the data in serial order. Compared with the BER of the OFDM systems with full precision ADCs (denoted by Full precision), the BER of the OFDM system with 1-bit ADCs (denoted by 1-bit w/o CFO) is degraded due to the distortion from 1-bit quantization. When the CFO is estimated and compensated, the BERs of the OFDM system is additionally deteriorated due the CFO estimation error. The BER of the Algorithm 1 (denoted by 1-bit w/ Alg.) is superior to that of the conventional method (1-bit w/ Conv.). In the case of 1-bit w/ Conv., the data are not recovered. It is remarkable that the BERs are degraded for the cases of 1bit w/o CFO and 1-bit w/ Alg. at high SNR region. This is because the SNR = 1/σ 2 w becomes more inaccurate as the distortion due to 1-bit quantization becomes more dominant at high SNR region.

V. CONCLUSIONS
To obtain the accurate CFO, we presented the CFO estimation algorithm in the OFDM systems with 1-bit ADCs over the multipath fading channel. We suggest the related pair for (ε i , n i ) based on the structure of the extended preamble, and the feasibility of the perfect CFO estimation. The bank containing the CFO estimators corresponding to (ε i , n i ) produces the candidates of CFO estimate through the AC with flexible intervals. By comparing the NSEs adopted to check the quality of the candidates of the CFO estimate, the proposed algorithm can estimate the CFO with sufficiently high accuracy. The simulation results show that the proposed algorithm is superior to the conventional method, and has the MSEs dependent on the CFO values.