Carrier Frequency Offset Estimation for OFDM with

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2869772, IEEE Communications Letters IEEE COMMUNICATIONS LETTERS, VOL XX, NO. XX, MONTH, 2018

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Carrier Frequency Offset Estimation for OFDM with Generalized Index Modulation Systems Using Inactive Data Tones Zhibin Yang, Fangjiong Chen, Beixiong Zheng, Miaowen Wen, Wulong Yu

Abstract—As carrier frequency offset (CFO) causes the loss of orthogonality among subcarriers and further degrades the performance of orthogonal frequency division multiplexing (OFDM) systems, it is of vital importance to estimate and compensate for the CFO. In this letter, we develop a novel scheme to estimate the CFO for OFDM with generalized index modulation (OFDMGIM), in which the variable inactive data tones are exploited to improve the accuracy of CFO estimation. Specifically, in the proposed scheme, the receiver first estimates and compensates for the CFO based on the pre-assigned null subcarriers. Since only partial subcarriers are activated in OFDM-GIM systems, the energy detection approach is applied to detect the inactive data tones. Finally, both the detected inactive data tones and the pre-assigned null subcarriers are exploited to re-estimate the CFO. We also derive the asymptotic analytical results, which are validated by simulations and shown to achieve higher accuracy of the CFO estimation. Index Terms—Carrier frequency offset (CFO), orthogonal frequency division multiplexing (OFDM), index modulation, null subcarriers.

I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) has been widely adopted in various communication systems due to its effectiveness in combating multipath effects. However, carrier frequency offset (CFO) introduced by Doppler effect or the mismatch between transceiver oscillators impairs the orthogonality among subcarriers, which causes inter-carrier interference (ICI) and further degrades the performance of OFDM systems significantly. Therefore, it is paramount to perform CFO estimation and compensation before demodulation. In the existing work, the estimation of CFO relies on either the training sequence (including the cyclic-prefix) or the null subcarriers. In this letter, we focus on the CFO estimation with the leverage of null subcarriers. The basic idea for CFO estimation based on null subcarriers is quite straightforward [1]: in case of perfect compensation of CFO and noiseless, energy on null subcarriers should be zero, i.e., no ICI power leakage [2]. The maximum-likelihood (ML) estimator has been derived in [3], [4]. In [5], Wu et al. investigated the deployment of null subcarriers and proposed an optimal placement strategy. Zhibin Yang, Fangjiong Chen, Beixiong Zheng and Miaowen Wen are with the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China. Wulong Yu is with the Beijing Institute of technology, Zhuhai, China. This work was supported by the National Natural Science Foundation of China under Grant 61671211, Grant U1701265 and Grant 61871190, in part by the Natural Science Foundation of Guangdong Province under Grant 2016A030311024 and Grant 2016A030308006. (Corresponding author: Fangjiong Chen, [email protected]).

Sun et al. extended the null subcarrier based CFO estimation to underwater acoustic channels [6]. OFDM with index modulation (OFDM-IM) has emerged as a competitive alternative transmission technique for the future wireless communications [7]. In OFDM-IM, a subset of subcarriers are activated to transmit modulated symbols while the remaining subcarriers are set to be inactive. One generalized scheme of OFDM-IM, referred to as OFDM with generalized index modulation (OFDM-GIM), has been proposed in [8]. In OFDM-GIM, the number of active data tones in an OFDM subblock is changing according to the incoming bits to carry additional information. The performance of OFDM-IM and OFDM-GIM is proven to be better than that of classical OFDM under the same spectral efficiency in the medium to high signal-to-noise ratio (SNR) region [8], [9], [10]. The presence of inactive data tones in OFDM-GIM systems makes it possible to improve the performance of CFO estimation. However, as the receiver has no idea of the inactive data tones before demodulation, whose indices are selected by the transmitter and varying from block to block, the problem is how to identify which subcarrier is inactive at the receiver. In the proposed scheme, we assume some null subcarriers are known to the receiver beforehand. A maximum-likelihood (ML) estimator is first applied to estimate and compensate for the CFO based on the pre-assigned null subcarriers. After that, the energy detection approach is employed to determine those inactive data tones. The detected inactive data tones, together with the pre-assigned null subcarriers, are then exploited to re-estimate the CFO. Theoretical results are derived for the proposed scheme, which is validated by simulations and demonstrated to achieve higher accuracy of CFO estimation. II. S IGNAL M ODEL We consider an OFDM block consisting of N = NA + NZ subcarriers, in which there are NA data tones and NZ pre-assigned null subcarriers for initial CFO estimation. Let A = {n1 , n2 , · · · , nNA } denote the indices of data tones and Z = {m1 , m2 , · · · , mNZ } denote the indices of pre-assigned null subcarriers. The data tones indexed by A are generated using OFDMGIM scheme [8], then coupled with the null subcarriers indexed by Z to form a frequency-domain block {X(n)}N n=1 . The N -point inverse fast Fourier transform (IFFT) is applied to {X(n)}N n=1 to generate the time-domain signal, which is then appended with a cyclic prefix (CP) longer than the maximum delay spread of the channel.

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After removing the CP at the receiver, the signal received in the time domain can be written as ∑ 1 y(k) = √ ej2πkϵ0 /N α(n)ej2πkn/N + v(k) (1) N n∈A for k = 0, 1, . . . , N − 1, where ϵ0 is the CFO normalized to ∆f , α(n) = H(n)X(n) is the signal received in the frequency domain in the absence of CFO and noise, H(n) denotes the channel frequency response at the n-th tone, and v(k) is complex zero-mean additive Gaussian noise with variance σv2 = E{|v(k)|2 }. The expression in (1) can be rewritten in vector form as y = D(ϵ0 )ΦA α + v

(2)

where

Algorithm 1 Inactive Data Tones Aided ML Estimator Input: y and ΦA Output: ϵˆ0PROP 1) Generate a coarse estimate of CFO using (3); 2) Apply ϵˆ0ML to compensate for the signal received in timedomain y′ = DH (ˆ ϵ0ML )y; 3) Apply N -point fast Fourier transform (FFT) to generate the signal compensated in frequency-domain Y′ = FFT[y′ ]; 4) Detect active data tones with a pre-defined threshold γ; 2 A˜ = {k| |Y ′ (k)| > γ, k ∈ A} 5) Refine the CFO estimation as ϵˆ0PROP = arg max g2 (ϵ) (5) ϵ

where g2 (ϵ) = yH D(ϵ)ΨA˜DH (ϵ)y and ΨA˜ = ΦA˜ΦH ˜. A

T

y = [y(0), y(1), . . . , y(N − 1)] [ ] D(ϵ0 ) = diag 1, ej2πϵ0 /N , . . . , ej2π(N −1)ϵ0 /N ] 1 [ ΦA = √ φn1 φn2 . . . φnNA NA [ ]T φni = 1, ej2πni /N , . . . , ej2πni (N −1)/N , ni ∈ A [ ]T α = α(n1 ), α(n2 ), . . . , α(nNA ) T

v = [v(0), v(1), . . . , v(N − 1)]

for i = 1, 2, . . . , NA . A deterministic ML estimator was proposed in [3], which can be described as a one-dimensional optimization problem: ϵˆ0ML = arg max g1 (ϵ)

(3)

ϵ

IV. P ERFORMANCE A NALYSIS

where H

this probability of miss alarm event, inactive data tones are detected in best effort. As a result, there is little chance for an ˆ which is the set of inactive data tone indice to appear in A, detected active data tone indices. Remark 2: The threshold γ is an important controlling factor. If an inactive data tone is detected as an active one, it is equivalent to zero gain at the corresponding channel frequency response. This will not effect the identifiability of the CFO estimation. However, extra error occurs if an active data tone is detected as inactive data tone, which should be avoided. More discussion will be provided in the following section.

H

g1 (ϵ) = y D(ϵ)ΨA D (ϵ)y

(4)

ϵˆ0ML is the ML estimate of ϵ0 , and ΨA = ΦA ΦH A. III. T HE P ROPOSED E STIMATOR According to the generalized index modulation principle, NA data tones are divided into G subblocks, each of which has Ns = NA /G data tones. In each subblock, only Kr < Ns data tones are activated to carry modulated symbols and the remaining Ns − Kr data tones are set to be inactive. Notably, those inactive data tones in OFDM-GIM systems can be regarded as additional null subcarriers, which can be exploited to enhance the accuracy of CFO estimation. However, in OFDM-GIM, as the indices of inactive data tones are unknown to the receiver, we need to detect the indices of the inactive data tones before employing those inactive data tones to refine the estimate of CFO. Building on the above motivation, we propose a novel deterministic ML estimator with the aid of inactive data tones, which are summarized in Algorithm 1. We have the following remarks for the proposed estimator. Remark 1: In the OFDM-GIM scheme, the number of active data tones in each subblock is varying. Since they are unknown to the receiver, the number of detected inactive data tones is not limited. Instead, a thereshold γ is pre-defined to keep the probability of miss alarm event, which implies an active data tone is detected as an inactive one, with a small value. With

In this section, we investigate the analytical bias and meansquare error (MSE) of the proposed scheme. Note that the proposed scheme is based on the indices of detected active data tones and pre-assigned null subcarriers. We shall calculate the ˜ then weighted by the probability bias and MSE for a given A, of each A˜ to evaluate the performance of the proposed scheme. The estimator in (5) can be applied to deterministic channel model. The MSE of the proposed estimator can be calculated according to the nature of target function. Note that g(ϵ) is generally a continuous function with multiple local optima. We assume that after the phase compensation with the ML estimator, the residual CFO is small enough and close to the exact CFO ϵ0 . In the high SNR region, g(ϵ) with ϵ close to ϵˆ0 can be considered as a unimodal function. The expectation and MSE of the estimator in (5) can be approximated as [11], [12] E{g(ϵ ˙ 0 )} E{¨ g (ϵ0 )} E{[g(ϵ ˙ 0 )]2 } M SE{ˆ ϵ0 } ≈ [E{¨ g (ϵ0 )}]2 E{ˆ ϵ0 } ≈ ϵ0 −

(6) (7)

where g(ϵ ˙ 0 ) and g¨(ϵ0 ) denote the first and second order derivatives of g(ϵ) at ϵ = ϵ0 , which are given by g(ϵ ˙ 0 ) = j2πN −1 yH D(ϵ0 )B1 DH (ϵ0 )y 2

g¨(ϵ0 ) = 4π N

−2 H

H

y D(ϵ0 )B2 D (ϵ0 )y

(8) (9)



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in which B1 = MΨA˜ − ΨA˜M, B2 = 2MΨA˜M − ΨA˜M2 − M2 ΨA˜, M = diag[0, 1, . . . , N − 1]. Note that B1 and B2 are conjugate antisymmetry matrices with zero diagonals. Expand (8) and we have

′

(10)

where v = D (ϵ0 )v. The first term α , c is deterministic and the other terms, which contain noise v, are stochastic. The second and third terms of (10) can be regarded as linear combinations of elements in v′ so they have zero mean. The expectation of fourth term can be expressed as H

H

E{v′H B1 v′ } =

N −1 N −1 ∑ ∑

ΦH A B1 ΦA α

[B1 ]k,n E{v ′∗ (k)v ′ (n)}

(11)

k=0 n=0

and zero diagonal elements of B1 eliminate terms [B1 ]k,k E{v ′∗ (k)v ′ (k)}. Since E{v ′∗ (k)v ′ (n)} = 0 when k ̸= n, we can infer that it has zero mean. From (10) and (11), we get E{g(ϵ ˙ 0 )} = j2πN −1 c. (12) It can be deduced from (12) that c is an imaginary number for E{g(ϵ ˙ 0 )} is real. Furthermore, c can be zero in many cases. When the subset A˜ contains only correctly detected active data tones, it can be observed that √ NA˜ ΦA α = Φ ˜ α′ (13) NA A ]T [ where α′ = α(n1 ), α(n2 ), . . . , α(nNA˜ ) . When A˜ = A, i.e., no inactive data tone is detected and ML estimator only based on null subcarriers is adopted, (13) still holds. Substituting (13) into c results in c = αH ΦH A B1 ΦA α NA˜ ′H H H H ′ = α [ΦA˜ MΦH ˜ΦA ˜ − ΦA ˜MΦA ˜]α ˜ ΦA ˜ ΦA ˜ ΦA A NA =0 (14) where ΦH ˜ = I. Similarly, we have ˜ ΦA A ′ ′H [g(ϵ ˙ 0 )] = − 4π 2 N −2 {c2 + 2αH ΦH A B1 v v B1 ΦA α 2

′ ′H ′H ′ + 2c[αH ΦH A B1 v + v B1 ΦA α + v B1 v ]

+ [v′H B1 v′ ]2 }.

(15)

Recalling E{v′ v′H } = σv 2 I and (11), we can obtain E{[g(ϵ ˙ 0 )] } = − 4π 2 N −2 {c2 + 2σv 2 αH ΦH A B1 IB1 ΦA α 2

+ 2c ∗ 0 + [v′H B1 v′ ]2 }

≈ − 4π 2 N −2 {c2 + 2σv 2 αH ΦH A B1 B1 ΦA α} (16) where [v′H B1 v′ ]2 is negligible at high SNR. Let’s move on to g¨(ϵ0 ) in (7), which can be calculated as H H ′ g¨(ϵ0 ) = 4π 2 N −2 [αH ΦH A B2 ΦA α + α ΦA B2 v

+ v′H B2 ΦA α + v′H B2 v′ ] 2

E{¨ g (ϵ0 )} = 4π N

−2

H

α

ΦH A B2 ΦA α

2 [E{¨ g (ϵ0 )}]2 = 16π 4 N −4 [αH ΦH A B2 ΦA α]

(17) (18) (19)

jcN

E{ˆ ϵ0 } ≈ ϵ0 + M SE{ˆ ϵ0 } ≈

H H ′ g(ϵ ˙ 0 ) = j2πN −1 [αH ΦH A B1 ΦA α + α ΦA B1 v

+ v′H B1 ΦA α + v′H B1 v′ ]

Substituting (12) (16) (18) (19) into (6) and (7) yields 2παH ΦH A B2 ΦA α N 2 c2 + 2σv 2 αH ΦH A B1 B1 ΦA α − 2 H H 4π [α ΦA B2 ΦA α]2

(20) (21)

From the above calculations, the general analytical form of asymptotic expectation and MSE with a given A˜ are derived. The expressions in (20) and (21) reveal that expectation and MSE of an estimator depend on transmitted symbols, ˜ Interestingly, the value channel response, subsets A and A. of ϵ0 doesn’t affect MSE. When there are incorrectly detected inactive data tones and c ̸= 0, the proposed CFO estimator becomes biased. Then we go further to the MSE by considering the probability of each A˜ event. Rewrite (21) in form of conditional expectation as ˜ = M SE{ˆ M SE{ˆ ϵ0 |A} ϵ0 }.

(22)

The mean MSE of the proposed CFO estimator can be expressed as ∑ ˜ {A}. ˜ M SE{ˆ ϵ0 }PROP = M SE{ˆ ϵ0 |A}P (23) ˜ ∀A

Denoting I˜ as the set of detected inactive data tones, we have A˜ ∪ I˜ = A and ∏ ∏ ˜ = P {A} P0 (i) [1 − P0 (j)] (24) ˜ i∈I

˜ j∈A

where P0 (k) = P {|Y ′ (k)|2 < γ} denotes the probability of the k-th subcarrier to be decided as inactive, i.e., the miss alarm probability. Assuming ML estimator is precise enough, the received signal after compensation can be approximated as Y ′ (k) ≈ α(k) + v(k), whose real and imaginary parts are approximately Gaussian variables with mean values of Re{α(k)} and Im{α(k)}, respectively. Thus |Y ′ (k)|2 is noncentral chisquare random variable whose degrees of freedom is 2. After normalization, we get ( ) 2|Y ′ (k)|2 2|α(k)|2 ′2 ∼ χ 2, (25) k σv2 σv2 Based on (25), we can calculate P0 (k) for all k, with which we can work out all that are required for calculating the MSE of the proposed CFO estimator. Finally, we discuss the selection of a proper threshold γ. The detection of inactive subcarriers can be regarded as a hypothesis testing problem. For a general Rayleigh channel model, we assume each α(k) follows a Gaussian distribution with identical variance, then we can apply the NP detector for the Gaussian signal corrupted by additive white Gaussian noise, which is also referred to as energy detector. When the noise level is available at the transmitter, the miss alarm probability and the corresponding MSE can be calculated for a given value of γ. In other words, the optimal value of γ depends on the SNR. However, it is too complicated to derive the analytic expression for an optimal γ and we leave it to our future work.



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V. S IMULATIONS We consider an OFDM-GIM system with 32 subcarriers, where part of which are pre-assigned null subcarriers. The remaining subcarriers are data tones, which are segmented into multiple subblocks. Each subblock has 4 subcarriers. In each subblock, the set of all allowed numbers of active data tones are set to be {1, 2, 3} and BPSK modulation is adopted for active data tones. In all simulations, the normalized CFO ϵ0 is set to be 0.2 and SNR is defined as Es /N0 = E{|α(k)|2 }/E{|v(k)|2 }. We first evaluate the proposed CFO estimator with various numbers of γ and NZ in Rayleigh channels. The average of 100 independent trials are plotted in Fig. 1, where the maximum delay spread is equal to 4 sampling periods. Fig. 1(a) shows that the optimal value of γ depends on different SNRs, and it can be empirically obtained via log γ = −0.06 ∗ SNR(dB) − 0.3. In practice, one can decide a proper γ with a coarse estimate of SNR. In Fig. 1(b), we provide two comparisons. In the first one, the sum of NZ and NI (the number of inactive data tones) is fixed to 20 while NZ is adjusted to 8, 12, and 16 (corresponding to cases 1, 2, and 3). In the second one, the average number of inactive data tones per subblock is set to 2. NZ is adjusted to 8, 12, and 4 (corresponding to cases 1, 4, and 5). It can be observed that a larger NZ improves the estimator outperforms at low to medium SNR. At high SNR, the advantage of a larger NZ is marginal since a smaller NZ can also ensure enough precision of estimate in first stage. The second comparison also indicates that smaller NZ leads to poorer performance, especially at low SNR. (a)

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Fig. 1: (a) MSE vs. threshold γ under different SNRs; (b) MSE vs. SNR under different NZ . In Fig. 2, we compare the proposed CFO estimator with the existing ML estimator in a deterministic channel. The channel coefficients are given as follows: h = [−0.3091 − 0.3103i, −0.0397 − 0.2770i, 0.2535 + 0.2856i, −0.5314 − 0.5492i]T . There are 8 pre-assigned null subcarriers indexed by Z = [1, 2, 8, 14, 19, 25, 31, 32]. The optimal CFO estimator with all inactive data tones correctly detected is also presented as a performance benchmark. At high SNR, our proposed CFO estimator achieves better performance than the existing ML estimator. Moreover, the theoretical results of both estimators

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Fig. 2: MSE vs. SNR under different estimators. are shown in the figure, which fits the simulation results well as the SNR increases. In particular, our proposed CFO estimator approaches the optimal one at high SNR. As the SNR increases, there are more correctly detected inactive data tones to exploit and further improve the estimate performance. VI. C ONCLUSION In this letter, we have proposed a novel CFO estimator for the OFDM-GIM system with the aid of inactive data tones. The performance of proposed CFO estimator is validated via simulation and theoretical results. By exploiting inactive data tones, the proposed CFO estimator has improved the accuracy of CFO estimation without sacrificing spectral efficiency. R EFERENCES [1] X. Ma, C. Tepedelenlioglu, G. B. Giannakis, and S. Barbarossa, “Nondata-aided carrier offset estimators for OFDM with null subcarriers: Identifiability, algorithms, and performance,” IEEE J. Sel. Areas Commun., vol. 19, no. 12, pp. 2504–2515, Dec. 2001. [2] S. Barbarossa, M. Pompili, and G. B. Giannakis, “Channel-independent synchronization of orthogonal frequency division multiple access systems,” IEEE J. Sel. Areas Comm., vol. 20, no. 2, pp. 474–486, Feb. 2002. [3] M. Ghogho, A. Swami, and G. B. Giannakis, “Optimized null-subcarrier selection for CFO estimation in OFDM over frequency-selective fading channels,” in Proc. IEEE GLOBECOM, Nov. 2001, pp. 202–206. [4] B. Chen, “Maximum likelihood estimation of OFDM carrier frequency offset,” IEEE Sig. Proc. Lett., vol. 9, no. 4, pp. 123–126, Apr. 2002. [5] Y. Wu, S. Attallah, and J. W. M. Bergmans, “On the optimality of the null subcarrier placement for blind carrier offset estimation in OFDM systems,” IEEE Trans. Veh. Tech., vol. 58, no. 4, pp. 2109–2115, May 2009. [6] H. Sun, W. Shen, Z. Wang, and et al, “Joint carrier frequency offset and impulse noise estimation for underwater acoustic OFDM with null subcarriers,” in Proc. IEEE 2012 Oceans, 2012. [7] E. Basar, “Index modulation techniques for 5G wireless networks,” IEEE Comm. Mag., vol. 54, no. 7, pp. 168–175, Jul. 2016. [8] R. Fan, Y. J. Yu, and Y. L. Guan, “Generalization of orthogonal frequency division multiplexing with index modulation,” IEEE Trans. Wireless Commun., vol. 14, no. 10, pp. 5350–5359, May 2015. [9] E. Basar, U. Aygolu, E. Panayirci, and H. V. Poor, “Orthogonal frequency division multiplexing with index modulation,” IEEE Trans. Signal Process., vol. 61, no. 22, pp. 5536–5549, Nov. 2013. [10] M. Wen, X. Cheng, M. Ma, B. Jiao, and H. V. Poor, “On the achievable rate of ofdm with index modulation,” IEEE Trans. Signal Process., vol. 64, no. 8, pp. 1919–19 329, Apr 2016. [11] M. Morelli and U. Mengali, “Carrier-frequency estimation for transmissions over selective channels,” IEEE Trans. Comm., vol. 48, no. 9, pp. 1580–1589, Sept. 2000. [12] C. H., Chen, Yuan, Chan, T. Y., Ho, and C. K., “Simple formulas for bias and mean square error computation,” IEEE Signal Process. Mag., vol. 30, no. 4, pp. 161–164, Jul. 2013.
