Coding to reduce the interference to carrier ratio of OFDM signals
 Besma Smida^{1}Email author
https://doi.org/10.1186/s1363801708093
© The Author(s) 2017
Received: 22 March 2016
Accepted: 7 January 2017
Published: 28 January 2017
Abstract
In this paper, we propose a simple approach to reduce the sensitivity of OFDM system to the carrier frequency offset (CFO). The main idea is to choose a wellknown channel code and to rotate each coordinate by a fixed phase shift such that the maximum intercarrier interference (ICI) taken over all subcarriers is minimized. This approach is based on a geometric interpretation of the peak interference to carrier ratio (PICR) of OFDM signals. Simulation results show that a reduction in PICR of 7 dB can be easily achieved. Furthermore, we addressed the fundamental limit of the proposed technique by providing both an exact and an approximate lower bound of PICR of phaseshifted binary codes. The bounds are applied to some codes: nonredundant binary code, BCH codes, and ReedMuller codes. Simulation results demonstrated that phaseshift designs approach lower bounds and are resilient to CFO change.
Keywords
1 Introduction
1.1 Motivations
Orthogonal frequencydivision multiplexing (OFDM) is a multicarrier modulation technique that has been implemented for many highspeed wireless and wireline applications. Indeed, the orthogonal properties among subcarriers of OFDM lead to high spectral efficiency and excellent ability to cope with multipath fading environment. Since the spectra of the subcarriers are overlapping, an accurate frequencysynchronization technique is needed. Due to oscillator inaccuracies and nonideal transmitter/receiver synchronization, the orthogonal properties are easily broken down and result in errors due to frequency offsets. It is wellknown that the carrier frequency offset (CFO) reduces the useful signal power and creates intercarrier interference (ICI) [1, 2].
1.2 Related works

Selfcancellation: A simple and effective method—known as the ICI selfcancelation scheme—was proposed by Zhao and Haggman in [18] and later generalized in [15, 19], where polynomial coding in the frequency domain is used to mitigate the effect of frequency offset. In this method, copies of the same data symbol are modulated on r adjacent subcarriers using specific weights. This method can reduce the ICI at the price of lowering the transmission rate by a factor r. To further improve the ICI selfcancellation performance, many researchers proposed more efficient mapping (optimized weighting coefficients) for the redundant data symbol modulated over adjacent or nonadjacent subcarriers [20–27]. The weights are designed such that, at the receiver, the ICI at each subcarrier is reduced when a frequency offset is present. Note that ICI selfcancellation techniques can be regarded as a coding scheme, where only the codewords with low ICI are used, this key idea was first proposed in [28, 29].Table 1
Coding techniques to reduce OFDM systems sensitivity to frequency shifts

Windowing: Seyedi and Saulnier [30] showed that the polynomial coding in the frequency domain—used by ICI selfcancellation schemes—is equivalent to using windowing in time domain. They hence used windowing at both the transmitter and the receiver to generalize and improve selfcancellation schemes. Recently, Real and Almenar [31] used windowing at the transmitter only allowing a reduction of the receiver complexity without any performance degradation. The windowing methods reduce ICI sensitivity significantly but they lower the transmission rate and increase the complexity of the transceiver.

Twopath transmission: The twopath transmission method halves the spectral efficiency similar to ICI selfcancellation techniques in [18]. However, unlike ICI selfcancellation method, twopath transmission method transmits the data copies in two concatenated OFDM blocks. The first path represents the standard OFDM signal and the second one is formed by a conjugate of the first path [32, 33]. This approach was generalized in [34–36] by introducing phase rotation and spacetime coding.

Correlativecoding: Contrary to all previous coding techniques, that reduce the sensitivity to CFO at the price of reducing the transmission rate, the correlative coding method do not reduce the data rate [37, 38]. This method is based on frequencydomain Partialresponse Coding (PRC). PRC were originally proposed in the time domain to reduce the sensitivity of singlecarrier systems to time offset.
The PRC does not sacrifice spectral efficiency but needs a maximumlikelihood sequence estimator (MLSE) thus increasing the receiver complexity. An interesting combination of correlativecoding and selfcancellation was proposed in [39].
1.3 Our contributions
The intend of this paper is to reduce the sensitivity of OFDM systems to CFO without reducing the transmission rate. Our technique is inspired by the following key observation: all current wireless OFDM systems include forwarderrorcorrection or channel codes to reduce the probability of error. Indeed, the forwarderrorcorrection or channel codes have been studied and used for controlling errors in data transmission for the last half century (since Shannon 1948 landmark paper). Sathananthan and Tellambura related ICI reduction to channel coding and showed the capability of any forwarderror correction code to reduce the errors caused by the ICI in [14]. We propose here to extend the role of those codes in OFDM systems. The main idea is to rotate each coordinate of the channel code by a fixed phaseshift such that the maximum intercarrier interference taken over all subcarriers is minimized. Clearly, the phaseshifted version of the code has the same error correcting capability, rate, decoding complexity as the original code. But it has lower maximum ICI and this reduction comes with no reduction of the spectral efficiency of the system. We proposed a simple technique—phaseshift—to reduce the sensitivity of OFDM system to CFO in [40]. This paper is an extension of this work—it provides an analytical evaluation of the proposed technique.
The key departure of our research from prior works on CFO effect reduction is that (a) we provide a geometric interpretation of the peak interference to carrier ratio (PICR) of OFDM signals, (b) we propose a simple phaseshift approach to reduce the sensitivity of OFDM to fine frequency offset, (c) we analyze the fundamental limit of the proposed technique and obtain a lower bound of the PICR of phaseshifted binary codes which we evaluate for nonredundant binary code, BCH codes, and ReedMuller codes, and (d) we numerically evaluate phaseshift designs and prove their resilience to CFO change.
2 Statement of the problem
2.1 Notations and preliminaries
 1.
Without frequency error (ε=0), b _{ k } reduces to the unit impulse sequence b _{ k }=δ[ k].
 2.
The coefficient b _{0} depends on ε but is independent of k. In other words, all subcarriers experience the same degree of attenuation and rotation of the wanted component.
 3.
b _{ k }=b _{ N+k }, so if we consider an N modulo numbering −k maps to N−k. This property is easy to prove and is used in [20] to propose a repetition code that reduces the ICI.
 4.
All the coefficients b _{ k } for k≠0 slowly vary with k, b _{ k+1}≃b _{ k }, the slow varying nature of b _{ k } is the key motivation for self ICI cancelation [19]. This property will be used here to derive a more practical (approximate) lower bound of PICR of phaseshifted binary codes.
 5.Under the assumption of small normalized frequency offset (ε≪1), the coefficients b _{ k } became$$b_{k} \simeq \frac{\sin \pi \varepsilon }{N \sin \frac{\pi k}{N}} \exp\left[j \pi\left(1\frac{1}{N}\right)\varepsilonj \pi \frac{k}{N} \right]. $$
 6.The average power of the intercarrier interference is derived as [1]$$ E\left[I_{k}^{2}\right]=\sum_{l=1}^{N1} b_{l}^{2}=1b_{0}^{2}, $$(4)
where E [.] denotes the expected value over the distribution of data.
3 Peak interference to carrier ratio evaluation
In this section, we provide a geometric interpretation of the maximum ICI taken over all subcarriers that explicitly incorporates the distance between the codewords c and some specific points. Based on this interpretation, we will propose a simple phaseshifted approach to reduce the sensitivity of OFDM systems to CFO.
3.1 PICR definition
Note that the expression of PICR is similar to the peaktomean envelope power ratio (PMEPR) that is considered as measure of the fluctuation of the OFDM signal^{2} \(\text {PMEPR}(\mathbf {c})= \max _{0\leq t \leq 1} \frac {S_{\mathbf {c}}(t)^{2}}{\parallel \mathbf {c} \parallel ^{2}}.\)
3.2 PICR geometric representation
We refer to \(\mathcal {J}(k,\varepsilon)\) as the maximum interferencetosignal ratio of code \(\mathcal {C}\) on subcarrier k and focus on its computation. The next theorem provides a geometric interpretation of the PICR per subcarrier [40].
Theorem 3.1
The value of \(\mathcal {J}(0,\varepsilon)\)is determined by the distance between the codewords \(\mathbf {c} \in \mathcal {C}\) and the two points w _{0} and j w _{ 0 }.
Proof
where c=[ c _{0},…,c _{ N−1}]. Without loss of generality, we assume that the elements of Q have unit amplitude (c _{ n }^{2}=1, 0≤n≤N−1) then c^{2}=N.^{3}
The proof of the second part of Theorem 3.1 is based on the definition of cyclic code. For every codeword c=[ c _{0},…,c _{ N−1}]∈ cyclic code \(\mathcal {C}\), the word [ c _{−k },…,c _{ N−k−1}] obtained by a cyclic shift of k components is again a codeword. In addition, based on property 3 of the coefficients b _{ n }, w _{ k } is obtained by k left cyclic shift of the element of w _{0}. If we assume that the \(\text {PICR}(\mathcal {C},\varepsilon)\) is reached for the subcarrier k≠0 and code c ^{ k+}, then we have just to shift the codeword c ^{ k+} by k to find a codeword c ^{0+} and it is easy to prove that the interference generated for the subcarrier 0 and code c ^{0+} is equals to \(\text {PICR}(\mathcal {C},\varepsilon)\). Hence, the \(\text {PICR}(\mathcal {C},\varepsilon)\) is also reached for the subcarrier k=0. □
The geometric interpretation of Theorem 3.1 is simple but important because it explicitly states that the value of \(\text {PICR}(\mathcal {C},\varepsilon)\) is determined by the distance between the codewords \(\mathbf {c} \in \mathcal {C} \) and the points w _{ k } and j w _{ k }, k=0,…,N−1. Furthermore for cyclic code, the value of PICR is only function of the distance between the codewords and the two points w _{0} and j w _{ 0 }. This representation leads to an understanding of the value of \(\text {PICR}(\mathcal {C},\varepsilon)\) and how to reduce it through phaseshift approach which is described in the following Section 4.
4 PICR reduction
4.1 Phaseshift technique: illustration
The reduction of interference for OFDM can now be tackled as follows: For a given code \(\mathcal {C}\), the reducing phaseshift problem is to find offset \(\mathbf {v}=[\!v_{0}, \ldots, v_{N1}] \in \mathbb {Z}_{2^{h}}^{N}\) such that \(\text {PICR}(\mathcal {C}_{v}, \epsilon)\) is minimized, where \( \mathcal {C}_{v}=\{ e^{j \pi \mathbf {v}/2^{h}} \mathbf {c} \  \mathbf {c} \in \mathcal {C} \} \), h is an integer that set the resolution of the offset. Notice that, when h tends to infinity, we can consider \(\text {PICR}(\mathcal {C}_{v}, \epsilon)\) as a continuous multivariable function of ϕ _{0}=2π v _{0}/2^{ h },…,ϕ _{ N−1}=2π v _{ N−1}/2^{ h }. Because of the continuity, the global minimum of this function over [ 0,2π]×[ 0,2π]×⋯×[ 0,2π] exists. But this function may have various local minima and saddle points over this compact set as well. There are various methods for minimization of \(\text {PICR}(\mathcal {C}_{v}, \epsilon)\) in the literature, but they can only guarantee convergence to a local minima. As representative of these methods, we have used the gradient method in our simulations. It is important to note that this approach can be applied to binary and nonbinary codes. Then, we used the lower bound, derived in the following section, to evaluate the optimality of our design.
4.2 Phaseshift technique: range
This section is dedicated to prove that phaseshift designs are robust to CFO change. The optimum offset v minimizes the PICR for a range of frequency offset (not for a specific frequency offset value). To do so we need the following proposition.
Proposition 4.1
Let \(\mathcal {C}\) be a code of length N defined over an equal energy constellation Q. Let ε _{1} and ε _{2} be small normalized frequency offsets (ε _{1}≪1 and ε _{2}≪1). If the offset \(\mathbf {v} \in \mathbb {Z}_{2^{h}}^{N}\) minimizes \(\text {PICR}(\mathcal {C}_{v}, \varepsilon _{1})\) that it minimizes \(\text {PICR}(\mathcal {C}_{v}, \epsilon _{2})\) too.
Proof
□
Hence, the offset \(\mathbf {v} \in \mathbb {Z}_{2^{h}}^{N}\), that minimizes \(\text {PICR}(\mathcal {C}_{v}, \varepsilon _{1})\), minimizes \(\text {PICR}(\mathcal {C}_{v}, \epsilon _{2})\) too. Therefore, there is no need to search for an offset v specific to each carrier frequency offset. The simulation results will show in Section 6 that the optimum set of shifts, derived for ε=0.1, reduce the intercarrier interference for a wide range of CFO (0<ε<0.5).
5 Minimum PICR for phaseshifted binary code
We provide a lower bound and an approximated lower bound for this limit. The bounds are used to evaluate the optimality of our design.
5.1 Main results
Our lower bound in the spirit of Schmidt’s [45] will be expressed in terms of the covering radius of the binary code \(\mathcal {B}\) defined below.
Definition
Theorem 5.1
The proof of Theorem 5.1 is provided in Appendix. The preceding lower bound is a decreased function of h. The corollary below states the asymptotic lower bound for h→∞.
Corollary 5.2
Proof
Similarly to [45], we used \( {\lim }_{h \rightarrow \infty } 2^{2h2} \sin ^{2} \left (\frac {\pi }{2^{h}} \right) = \frac {\pi ^{2}}{4}. \) □
5.2 Examples
5.2.1 Nonredundant BPSK signaling
The code {1,−1}^{ N } is a binary code without redundancy, ρ({1,−1}^{ N })=0, Theorem 5.1 and Corollary 5.2 imply the following.
Corollary 5.3
For N=16 and N=64 the PICR approximated lower bounds are 0.10 and 0.22, respectively. Since the PICR for nonredundant binary code are 0.19 (N=16) and 0.50 (N=64), the maximum PICR reduction of the proposed phaseshift designs is 3 dB for nonredundant code.
5.2.2 ReedMuller codes
In the following we inspect codes that are equivalent to first order ReedMuller codes of length 2^{ m }, RM(1,m).
Corollary 5.4
Proof
5.2.3 Binary primitive BCH codes
Next we present the lower bounds on the PICR of codes that equivalent to binary primitive terrorcorrecting BCH codes, \(\mathcal {B}(t,m)\).
Corollary 5.5
Proof
The authors of [47] proved that there exists an m _{ o } depending on t such that \(\rho (\mathcal {B}(t,m))=2t1\) for all m≥m _{ o }. □
6 Numerical results
The numerical results presented in this section aim to evaluate the ICI reduction performance of the proposed phaseshift approach. We considered a normalized frequency offset of ε=0.1 and two wellknown codes extended BoseChaudhuriHocqenghem (BCH) and Reed Muller (RM). We note that Pyndiah [48] has proposed a simple softinput softoutput (approximated MAP) decoder for BCH codes, based on Chase algorithm, that gives excellent BER performance. RM codes achieve capacity on erasure channels and reduce PMEPR [49, 50].
Optimum phase shifts for BCH(64,10) in radians
ϕ _{0}=2.4683  ϕ _{1}= 1.7502  ϕ _{2}= 4.4962  ϕ _{3}= 4.0882  ϕ _{4}= 0.1253  ϕ _{5}=2.9330  ϕ _{6}=5.3401  ϕ _{7}=2.2695 
ϕ _{8}=4.2003  ϕ _{9}=4.0068  ϕ _{10}=6.2288  ϕ _{11}=5.6747  ϕ _{12}=5.3153  ϕ _{13}=1.0958  ϕ _{14}=4.5622  ϕ _{15}=6.0023 
ϕ _{16}=3.2120  ϕ _{17}=1.5937  ϕ _{18}=4.9119  ϕ _{19}=3.7611  ϕ _{20}=4.1274  ϕ _{21}= 5.8449  ϕ _{22}=5.5075  ϕ _{23}=5.4723 
ϕ _{24}=4.0787  ϕ _{25}=4.3756  ϕ _{26}=4.4991  ϕ _{27}=3.1733  ϕ _{28}=3.6295  ϕ _{29}= 2.6383  ϕ _{30}=5.2078  ϕ _{31}=5.7397 
ϕ _{32}=3.4149  ϕ _{33}=0.9170  ϕ _{34}=0.8856  ϕ _{35}=5.4017  ϕ _{36}=5.2185  ϕ _{37}=2.2959  ϕ _{38}=0.6368  ϕ _{39}=1.1671 
ϕ _{40}=3.1718  ϕ _{41}=2.9789  ϕ _{42}=5.0463  ϕ _{43}=5.1208  ϕ _{44}=0.9384  ϕ _{45}=3.8918  ϕ _{46}=2.3587  ϕ _{47}=2.3106 
ϕ _{48}=2.7175  ϕ _{49}=3.2082  ϕ _{50}=0.9428  ϕ _{51}=1.0826  ϕ _{52}=5.4491  ϕ _{53}=5.7840  ϕ _{54}=4.6229  ϕ _{55}=1.2202 
ϕ _{56}=3.5798  ϕ _{57}=0.1929  ϕ _{58}=2.7888  ϕ _{59}= 5.5207  ϕ _{60}= 1.8680  ϕ _{61}=4.5057  ϕ _{62}=0.4838  ϕ _{63}=3.2663 
Optimum phase shifts for RM(1,6) in radians
ϕ _{0}=1.5620  ϕ _{1}=4.5390  ϕ _{2}=3.5491  ϕ _{3}=0.3789  ϕ _{4}=2.9522  ϕ _{5}=4.9135  ϕ _{6}=1.9776  ϕ _{7}=3.7244 
ϕ _{8}=3.7291  ϕ _{9}=5.2304  ϕ _{10}=3.8296  ϕ _{11}=2.8186  ϕ _{12}=5.5612  ϕ _{13}=4.4629  ϕ _{14}=3.4743  ϕ _{15}=6.1346 
ϕ _{16}=4.7435  ϕ _{17}=1.3264  ϕ _{18}=3.1416  ϕ _{19}=0.1649  ϕ _{20}=4.9018  ϕ _{21}=2.6437  ϕ _{22}=5.8063  ϕ _{23}=0.9283 
ϕ _{24}=4.0979  ϕ _{25}=3.9276  ϕ _{26}=5.6677  ϕ _{27}=5.7450  ϕ _{28}=3.8378  ϕ _{29}=1.3179  ϕ _{30}=4.9345  ϕ _{31}=5.4689 
ϕ _{32}=0.0817  ϕ _{33}=3.2805  ϕ _{34}=2.8142  ϕ _{35}=5.0262  ϕ _{36}=5.0639  ϕ _{37}=1.0568  ϕ _{38}=0.8457  ϕ _{39}=2.9594 
ϕ _{40}=5.7340  ϕ _{41}=5.5270  ϕ _{42}=3.2478  ϕ _{43}=3.5318  ϕ _{44}=1.1401  ϕ _{45}=1.7986  ϕ _{46}=2.3697  ϕ _{47}=0.3248 
ϕ _{48}=4.5588  ϕ _{49}=1.6792  ϕ _{50}=0.0914  ϕ _{51}=6.2392  ϕ _{52}=1.8721  ϕ _{53}=1.1146  ϕ _{54}=3.4850  ϕ _{55}=6.2436 
ϕ _{56}=1.3990  ϕ _{57}=5.3068  ϕ _{58}=0.0079  ϕ _{59}=6.0174  ϕ _{60}=4.3187  ϕ _{61}=4.3175  ϕ _{62}=6.1484  ϕ _{63}=5.8044 
7 Conclusions
In this paper, we proposed an interference reduction scheme to mitigate the effects of CFO in OFDM system. We provided a geometrical interpretation of \(\text {PICR}(\mathcal {C},\varepsilon)\), the PICR of \(\mathcal {C}\). To reduce PICR, we focused on a simple phaseshift approach. The idea is to take a code with good BER performance and to rotate each coordinate of the code by a fixed phaseshift such that the maximum PICR taken over all codewords is minimized. This shift leaves unchanged the errorcorrecting properties of the code. In addition, we can use the standard decoding algorithm of the original code after back rotation of the received signals. Simulation results show that a reduction of 7 dB can freely be obtained.
We also study the fundamental limit of the proposed technique by providing an approximated lower bound of the PICR. The bound is applied to some codes: nonredundant binary code, BCH codes, and ReedMuller codes. It is demonstrated that our phaseshift designs approach our bound.
8 Endnotes
^{1} One more approach uses signal processing, such as time and frequency equalization, to reduce the sensitivity of OFDM systems to frequency offset. Although efficient, signal processing techniques usually increase the complexity of the transceiver (see [16] and references therein).
^{2} The authors of [29] have noted that despite the similarities between PICR and PMEPR, the PICR problem differs from PMEPR issue in several ways.
^{3} In the case of nonuniform energy constellation, like QAM, \(\mathcal {J}(k,\varepsilon)\) will be determined for each energy level and then minimized over all the levels.
9 Appendix
10 Proof of Theorem 5.1
where x and ∠ x are the amplitude and the angle of x, respectively. Let \(\mathbb {Z}_{2^{h}}\) denote the ring of integers modulo 2^{ h }. We define \(\overline {c}_{i}\) and \(\overline {b}_{i}\) \(\in \mathbb {Z}_{2^{h}}\) as: \(\frac {2\pi \overline {c}_{i}}{2^{h}} \simeq \angle c_{i} \) and \(\frac {2\pi \overline {b}_{i} }{2^{h}} \simeq \angle b_{i}\). Note that when h tends to the infinity the phases can be approximated with arbitrary high probability.
where \(\rho (\mathcal {B})\) is covering radius of the binary code \(\mathcal {B} \subseteq \{1,1\}^{N}\). We replace Eq. (20) in Eqs. (17) and (19) to complete the proof.
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 T Pollet, MV Bladel, M Moeneclaey, Ber sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise. IEEE Trans. Commun.43(2/3/4), 191–193 (1995). doi:10.1109/26.380034.View ArticleGoogle Scholar
 K Sathananthan, C Tellambura, Probability of error calculation of OFDM systems with frequency offset. IEEE Trans. Commun.49(11), 1884–1888 (2001). doi:10.1109/26.966051.View ArticleMATHGoogle Scholar
 JJ van de Beek, M Sandell, PO Borjesson, Ml estimation of time and frequency offset in OFDM systems. IEEE Trans. Signal Process.45(7), 1800–1805 (1997). doi:10.1109/78.599949.View ArticleMATHGoogle Scholar
 U Tureli, D Kivanc, H Liu, Experimental and analytical studies on a highresolution OFDM carrier frequency offset estimator. IEEE Trans. Veh. Technol.50(2), 629–643 (2001). doi:10.1109/25.923074.View ArticleGoogle Scholar
 MJFG Garcia, O Edfors, JM PaezBorrallo, Frequency offset correction for coherent OFDM in wireless systems. IEEE Trans. Consum. Electron.47(1), 187–193 (2001). doi:10.1109/30.920438.View ArticleGoogle Scholar
 M Luise, M Marselli, R Reggiannini, Lowcomplexity blind carrier frequency recovery for OFDM signals over frequencyselective radio channels. IEEE Trans. Commun.50(7), 1182–1188 (2002). doi:10.1109/TCOMM.2002.800819.View ArticleGoogle Scholar
 PH Moose, A technique for orthogonal frequency division multiplexing frequency offset correction. IEEE Trans. Commun.42(10), 2908–2914 (1994). doi:10.1109/26.328961.View ArticleGoogle Scholar
 TM Schmidl, DC Cox, Robust frequency and timing synchronization for OFDM. IEEE Trans. Commun.45(12), 1613–1621 (1997). doi:10.1109/26.650240.View ArticleGoogle Scholar
 M Morelli, U Mengali, An improved frequency offset estimator for OFDM applications. IEEE Commun. Lett.3(3), 75–77 (1999). doi:10.1109/4234.752907.View ArticleGoogle Scholar
 M Morelli, U Mengali, Carrierfrequency estimation for transmissions over selective channels. IEEE Trans. Commun.48(9), 1580–1589 (2000). doi:10.1109/26.870025.View ArticleGoogle Scholar
 J Lei, TS Ng, A consistent ofdm carrier frequency offset estimator based on distinctively spaced pilot tones. IEEE Trans. Wirel. Commun.3(2), 588–599 (2004). doi:10.1109/TWC.2004.825350.View ArticleGoogle Scholar
 Y Li, H Minn, N AlDhahir, AR Calderbank, Pilot designs for consistent frequencyoffset estimation in OFDM systems. IEEE Trans. Commun.55(5), 864–877 (2007). doi:10.1109/TCOMM.2007.896105.View ArticleGoogle Scholar
 U Tureli, H Liu, MD Zoltowski, OFDM blind carrier offset estimation: Esprit. IEEE Trans. Commun.48(9), 1459–1461 (2000). doi:10.1109/26.870011.View ArticleGoogle Scholar
 K Sathananathan, C Tellambura, in ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No. 01CH37196), 4. Forward error correction codes to reduce intercarrier interference in OFDM (Sydney, 2001), pp. 566–569.Google Scholar
 J Armstrong, Analysis of new and existing methods of reducing intercarrier interference due to carrier frequency offset in OFDM. IEEE Trans. Commun.47(3), 365–369 (1999). doi:10.1109/26.752816.View ArticleGoogle Scholar
 AF Molisch, M Toeltsch, S Vermani, Iterative methods for cancellation of intercarrier interference in OFDM systems. IEEE Trans. Veh. Technol.56(4), 2158–2167 (2007). doi:10.1109/TVT.2007.897628.View ArticleGoogle Scholar
 P Singh, OP Sahu, in 2015 IEEE International Conference on Computational Intelligence & Communication Technology. An overview of ICI self cancellation techniques in OFDM systems (Ghaziabad, 2015). pp. 299–302.Google Scholar
 Y Zhao, SG Haggman, in Proceedings of Vehicular Technology Conference  VTC, 3. Sensitivity to Doppler shift and carrier frequency errors in OFDM systemsthe consequences and solutions (Atlanta, 1996), pp. 1564–1568.Google Scholar
 Y Zhao, SG Haggman, Intercarrier interference selfcancellation scheme for OFDM mobile communication systems. IEEE Trans. Commun.49(7), 1185–1191 (2001). doi:10.1109/26.935159.View ArticleMATHGoogle Scholar
 K Sathananthan, RMAP Rajatheva, SB Slimane, Cancellation technique to reduce intercarrier interference in OFDM. Electron. Lett.36(25), 2078–2079 (2000). doi:10.1049/el:20001456.View ArticleGoogle Scholar
 YH Peng, YC Kuo, GR Lee, JH Wen, Performance analysis of a new ICIselfcancellationscheme in OFDM systems. IEEE Trans. Consum. Electron.53(4), 1333–1338 (2007). doi:10.1109/TCE.2007.4429221.View ArticleGoogle Scholar
 A Singh, A Sharma, in 2015 International Conference on Advances in Computer Engineering and Applications. ICI cancellation in OFDM by phase rotated data transmission (Ghaziabad, 2015), pp. 952–956.Google Scholar
 K Sathananthan, CRN Athaudage, B Qiu, in Proceedings. ISCC 2004. Ninth International Symposium on Computers And Communications (IEEE Cat. No. 04TH8769), 2. A novel ICI cancellation scheme to reduce both frequency offset and IQ imbalance effects in OFDM (Alexandria, 2004), pp. 708–713.Google Scholar
 S Tang, K Gong, J Song, C Pan, Z Yang, Intercarrier interference cancellation with frequency diversity for OFDM systems. IEEE Trans. Broadcast.53(1), 132–137 (2007). doi:10.1109/TBC.2006.886835.View ArticleGoogle Scholar
 T ShuMing, H YuShun, A novel ICI selfcancellation scheme for OFDM systems. Int. J. Commun. Syst.24(11), 1496–1505 (2011).View ArticleGoogle Scholar
 MW Wen, X Cheng, X Wei, B Ai, BL Jiao, in 2011 IEEE Global Telecommunications Conference  GLOBECOM. A Novel Effective ICI SelfCancellation Method (Houston, 2011), pp. 1–5.Google Scholar
 TL Liu, WH Chung, SY Yuan, SY Kuo, ICI selfcancellation with cosine windowing in OFDM transmitters over fast timevarying channels. IEEE Trans. Wirel. Commun.14(7), 3559–3570 (2015). doi:10.1109/TWC.2015.2408325.View ArticleGoogle Scholar
 K Sathananthan, C Tellambura, in IEEE 54th Vehicular Technology Conference. VTC Fall 2001. Proceedings (Cat. No. 01CH37211), 2. New ICI reduction schemes for OFDM system (Atlantic City, 2001), pp. 834–838.Google Scholar
 K Sathananthan, C Tellambura, Partial transmit sequence arid selected mapping schemes to reduce ICI in OFDM systems. IEEE Commun. Lett.6(8), 313–315 (2002). doi:10.1109/LCOMM.2002.802067.View ArticleGoogle Scholar
 A Seyedi, GJ Saulnier, General ICI selfcancellation scheme for OFDM systems. IEEE Trans. Veh. Technol.54(1), 198–210 (2005). doi:10.1109/TVT.2004.838849.View ArticleGoogle Scholar
 O Real, V Almenar, in 2008 IEEE International Conference on Wireless and Mobile Computing, Networking and Communications. OFDM ICI SelfCancellation Scheme Based on Five Weights (Avignon, 2008), pp. 360–364.Google Scholar
 HG Yeh, CC Wang, in IEEE 60th Vehicular Technology Conference, 2004. VTC2004Fall, 3. New parallel algorithm for mitigating the frequency offset of OFDM systems (Los Angeles, 2004), pp. 2087–2091.Google Scholar
 HG Yeh, YK Chang, B Hassibi, A scheme for cancelling intercarrier interference using conjugate transmission in multicarrier communication systems. IEEE Trans. Wirel. Commun.6(1), 3–7 (2007). doi:10.1109/TWC.2007.04541.View ArticleGoogle Scholar
 CL Wang, YC Huang, Intercarrier interference cancellation using general phase rotated conjugate transmission for OFDM systems. IEEE Trans. Commun.58(3), 812–819 (2010). doi:10.1109/TCOMM.2010.03.080288.View ArticleGoogle Scholar
 HG Yeh, K Yao, in 2012 IEEE Global Communications Conference (GLOBECOM). A parallel ICI cancellation technique for OFDM systems (Anaheim, 2012), pp. 3679–3684. doi:10.1109/GLOCOM.2012.6503688.
 M Wen, X Cheng, L Yang, B Jiao, in 2013 IEEE Global Communications Conference (GLOBECOM). Twopath transmission framework for ICI reduction in OFDM systems (Atlanta, 2013), pp. 3716–3721.Google Scholar
 Y Zhao, JD Leclercq, SG Haggman, Intercarrier interference compression in OFDM communication systems by using correlative coding. IEEE Commun. Lett.2(8), 214–216 (1998). doi:10.1109/4234.709435.View ArticleGoogle Scholar
 H Zhang, Y Li, Optimum frequencydomain partial response encoding in OFDM system. IEEE Trans. Commun.51(7), 1064–1068 (2003). doi:10.1109/TCOMM.2003.814203.View ArticleGoogle Scholar
 JY Yun, SY Chung, YH Lee, Design of ICI canceling codes for OFDM systems based on capacity maximization. IEEE Signal Process. Lett.14(3), 169–172 (2007). doi:10.1109/LSP.2006.884035.View ArticleGoogle Scholar
 B Smida, in 2011 IEEE Global Telecommunications Conference  GLOBECOM. Reduction of the Peak Interference to Carrier Ratio of OFDM Signals (Houston, 2011), pp. 1–5. doi:10.1109/GLOCOM.2011.6133563.
 AE Jones, TA Wilkinson, SK Barton, Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission schemes. Electron. Lett.30(25), 2098–2099 (1994). doi:10.1049/el:19941423.View ArticleGoogle Scholar
 AE Jones, TA Wilkinson, in Vehicular Technology Conference, 1996. Mobile Technology for the Human Race., IEEE 46th, 2. Combined coding for error control and increased robustness to system nonlinearities in OFDM, (1996), pp. 904–9082. doi:10.1109/VETEC.1996.501442.
 V Tarokh, H Jafarkhani, On the computation and reduction of the peaktoaverage power ratio in multicarrier communications. IEEE Trans. Commun.48(1), 37–44 (2000). doi:10.1109/26.818871.View ArticleGoogle Scholar
 KG Paterson, V Tarokh, On the existence and construction of good codes with low peaktoaverage power ratios. IEEE Trans. Inf. Theory. 46(6), 1974–1987 (2000). doi:10.1109/18.868473.MathSciNetView ArticleMATHGoogle Scholar
 KU Schmidt, On the peaktomean envelope power ratio of phaseshifted binary codes. IEEE Trans. Commun.56(11), 1816–1823 (2008). doi:10.1109/TCOMM.2008.060652.View ArticleGoogle Scholar
 T Helleseth, T Klove, J Mykkeltveit, On the covering radius of binary codes (corresp.)IEEE Trans. Inf. Theory. 24(5), 627–628 (1978). doi:10.1109/TIT.1978.1055928.MathSciNetView ArticleMATHGoogle Scholar
 SG Vladuts, AN Skorobogatov, Covering radius for long BCH codes. Probl. Inf. Transm.25(1), 28–34 (1989).MathSciNetMATHGoogle Scholar
 RM Pyndiah, Nearoptimum decoding of product codes: block turbo codes. IEEE Trans. Commun.46(8), 1003–1010 (1998). doi:10.1109/26.705396.View ArticleMATHGoogle Scholar
 M Sabbaghian, Y Kwak, B Smida, V Tarokh, Near shannon limit and low peak to average power ratio turbo block coded OFDM. IEEE Trans. Commun.59(8), 2042–2045 (2011). doi:10.1109/TCOMM.2011.080111.090356.View ArticleGoogle Scholar
 E Abbe, A Shpilka, A Wigderson, Reed x2013;muller codes for random erasures and errors. IEEE Trans. Inf. Theory. 61(10), 5229–5252 (2015). doi:10.1109/TIT.2015.2462817.MathSciNetView ArticleMATHGoogle Scholar