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Carrier frequency synchronization for WLAN systems based on MIMOOFDMIM
EURASIP Journal on Wireless Communications and Networking volume 2024, Article number: 77 (2024)
Abstract
Multipleinput multipleoutput (MIMO) orthogonal frequency division multiplexing (OFDM) with index modulation (IM) MIMOOFDMIM is a modulation technique that has garnered significant interest in recent times owing to its attraction of smooth transition to greener communications, enhancing efficiency in terms of energy, spectrum and compatibility to previous existing standards without the need for drastic changes in the physical layer. It is also proved that even though OFDMIM provides better immunity to carrier frequency offset compared to traditional OFDM, the sensitivity to frequency error is still a major issue and it has to be resolved on priority basis to fully exploit the potential offered by the IM technique. In this work, we propose a novel nondataaided algorithm that efficiently estimates and eliminates the offset present in the received signal. The algorithm has been analyzed for the MIMOOFDMIMbased wireless local area network system using the high throughput task group (TGn) MIMO channel models in the presence of AWGN. In addition, it is compared with the other popular blind algorithms based on both pilots and inactive data tones. The simulation plots depict a clear improvement of the new estimator over the existing methods not only in the low signaltonoise ratio (SNR) regions but also in the high SNR region.
1 Introduction
Wireless local area network (WLAN) has made giant leaps and bounds ever since its inception in 1997 with WiFi 1 to the most recent standard of WiFi 7 [1–5].With the evolution of newer versions, WLAN has not only significantly improved the user experience in terms of increased data rate, coverage and reduced latency but also inadvertently supported various applications. Every new release was built on the strengths of the previous standards, maintaining backward compatibility while adding flexibility and scalability adapting to everchanging market dynamics [6]. Historically also, it is observed that every generation of cellular has offloaded much of its traffic onto the WLAN systems because of its easy economics and superior speeds [7]. The future wireless networks are predicted that the nextgen standards of WLAN to not only support the expansion and densification of enterprise networks but also support new and emerging advanced applications such as Internet of things (IoT), Industrial IoT (IIOT), 4K video, wireless office and ultra HD [8]. The latest standard under 802.11, WiFi 7 [9] is designed for utmost compatibility and coexisting cordially with all previous 802.11a/n/ac/ax devices.
To facilitate these features, WiFi 7 has brought in many physical layer changes such as the usage of higher modulation schemes up to 4096 QAM, orthogonal frequency division multiple access (OFDMA) technology, increasing the channel bandwidth, symbol duration and reducing the subcarrier spacing [10]. MIMOOFDM has been an undisputed successful physical layer modulation method employed in many of the cellular, wireless and mobile networks achieving higher spectral efficiency, effective against interference and facilitating simple channel equalization [11]. Classical MIMOOFDM even though supports spectral efficiency but lacks in the energy efficiency aspect because of the scaled power consumption on parallel radio frequency (RF) chains. Also, large peaktoaverage power ratio (PAPR) is another key concern [12] which requires costly and complex power amplifiers. These alarms necessitate the requirement of new advanced modulation techniques which can address the concerns effectively [13, 14]. In addition, there is a vital need for smooth transit to greener communication techniques along with all the features to move towards a safer and better world [15–17] .
To address the new necessities [18–20], several novel techniques have been put forth in the physical layer. One of them is IM which has garnered significant research interests in both academia and industry [21–25]. Traditionally all the digital modulation techniques have relied on modulation of the amplitude, phase or frequency or in combination with the sinusoidal carrier signal for the transmissions. Whereas the IM technique has the unique ability to select a subset of certain communication resources like time slots, frequency subcarriers, antenna, virtual parallel channels, space–time matrix, signal constellation and activation pattern of antenna and so on; and adds a new dimension to carry the data [26]. The indices of these building blocks too are utilized to carry additional data effectively.
In a frequency domain IM technique, the actual data bits are divided into index bits and symbol bits. The index bits dictate the active region of the communication resources such as subcarriers, and the symbol bits are utilized for systematic constellation mapping which are transmitted through only the active resources. The unselected subcarriers are termed as inactive resources [27, 28]. The data bits are transmitted through the active RF subcarriers and also carried by the indices of those chosen active OFDM subcarriers. Thus the IM techniques have the energy efficient way of carrying the data by deactivating some of the subcarriers but using the indices to carry additional information [29, 30]. The resultant index modulated method significantly increases the spectral efficiency as a new dimension is added to carry the additional information and no new hardware complexities are required. Thus IM enhances both spectral and energy efficiency while also improving on the aspects of high PAPR and easing the intercarrierinterference (ICI) issues because of the unused subcarriers. Thus MIMOOFDMIM stands out as an endearing candidate for meeting the challenging needs of nextgeneration networks [31–37].
MIMOOFDM/MIMOOFDMIM both being multicarrier techniques [38] are bound to lose the orthogonality between the subcarriers owing to either the Doppler spread or due to the mismatch in the clocking circuitry of the transmitter/receiver which causes the transmission/reception to offset in frequency from the desired location leading to a condition termed as carrier frequency offset (CFO) [39]. The CFO causes the ICI which results in degraded performance by increasing the bit error rate [40]. This problem of CFO is more severe in nextgeneration mobile and wireless networks as the subcarrier spacing is further reduced and in addition higher QAM is utilized to map more bits onto a subcarrier which reduces the spacing between the constellation points. Thus, the performance suffers as the ratio of maximum frequency offset to subcarrier spacing increases. Hence, tracking the frequency error is particularly given priority in the latest wireless standards to completely harness all the features bragged by 5G and 6G. Generally, the WLAN standards specify the tolerable limits for the carrier frequency error in terms of parts per million (ppm) as how much frequency may be deviated from the center frequency. Where ppm refers to 1 out of 10^{6} parts. And, in the current standard of IEEE 802.11, the tolerable frequency error is set at ± 20 ppm [41] which is a very stringent limit.
2 Related works and motivation
Effective carrier frequency offset estimation and compensation technique is critical to any multicarrier system and is a wellestablished fact, and it is extensively studied leading to an abundance of literature. The CFO estimation is carried out using dataaided techniques like utilizing the training sequences which are preappended to the data sequence before transmission or blind procedures [42–44]. Even though plenty of research work has been carried out and published in the areas of OFDMIM substantiating its advantages in comparison with classical OFDM, giving out a considerate amount of potential applications and its usefulness in future wireless networks, but most of the prevailing IM techniques are only evaluated under ideal conditions [45  47] and further very few literature work talks on the impact of carrier frequency offset on OFDMIM.
The authors of [48] have reiterated the usefulness of index modulation in communication systems and have proved the sensitivity of the index modulated OFDM, similar to classical OFDMbased systems toward the carrier frequency offset which reduces the system performance. They have come up with the closedform expression of the BER to show the performance analysis but do not suggest any correction algorithm. In article [49], authors proposed a leveraging extreme learning machine (ELM) scheme to attain synchronization. Exactly, they exploit the training signals which are affected by the synchronization offsets. Two schemes of ELMs are combined with the classical MIMOOFDM system to assess the residual symbol timing offset (RSTO) and the residual carrier frequency offset (RCFO).
Similarly, authors in [50] have addressed the issue of CFO calculation for MIMOOFDM systems using blind technique. They have been aided by the banded structure of the circulant channel matrix from every transmitting stream to each of the receiver streams. Their algorithm works mainly for constant modulus signals with the assumption that the channel is constant for the duration of at least two successive OFDM symbols. Authors of [51] have come up with an MLbased CFO calculation algorithm for MIMO systems and implemented the hardware design. Their proposed method can precisely evaluate the CFO, particularly at a high SNR ratio. This is its major limitation.
In the paper [52], authors proposed a method to assess the frequency offset in OFDM systems with generalized index modulation (OFDMGIM). In this technique, initially the receiver computes and corrects the offset error with the help of the preallocated pilot subcarriers. The energy detection methodology is utilized to identify the unused data carriers. Lastly, both the identified unused data carriers and the pilot tones are utilized to recalculate the frequency error. The main disadvantage of the proposed method lies in the identification of the inactivated null subcarriers which always differ in count and location of each subblock. If any of the used data tone is diagnosed as unused null tone, then it causes further inaccuracies.
Authors in [53] have demonstrated bit error rate versus signaltonoise ratio for two of the IM schemes popularly used in frequency domain, GIM and SNM in the presence of radio frequency impairments including both frequency error and IQ imbalance. They substantiate through their results that although the IM scheme enhances spectral efficiency, they are not immune to the said RF impairments. And, they do not suggest any method to overcome the effects of CFO and IQ imbalance. In article [54], the authors perform the analysis of ESIMOFDM with wellestablished Schmidl and Cox algorithm synchronizer using LabVIEW Simulator. A BER analysis in the presence of AWGN and Rayleigh fading channel is carried out using a training based detector. The major drawback of this system is spectral inefficiency and, it is required to change the threshold value of the synchronizer in proportion to the changing SNR.
Authors of article [55] proposed a theoretical method and simulations to calculate the BER accurateness of IM schemes such as OFDM interleaved subcarrier index modulation (OFDM ISIM) and OFDM adjacent subcarrier index modulation (OFDM ASIM) in contrast to traditional OFDM in the presence of frequency offset and Rayleigh multipath fading channels. Their results confirm that OFDMIM performance is outperformed as compared to the regular OFDM when carrier frequency offset is added but they do not implement/suggest any correction algorithm for the CFO.
Similarly, authors in paper [56] proposed a hybrid OFDMIM, wherein depending on the channel conditions, the transmission mode can be alternately switched between the OFDM and OFDMIM. The BER performance is examined in both the Rayleigh and Rician fading channels without considering the effect of CFO. They presumed that the complete channel’s impulse response state is available at the receiver and also the SNR threshold value needs to be set manually for different configurations of hybrid OFDMIM. In the article [57], authors have given a closedform expression for the OFDMIM system using modulation schemes such as Mary quadrature amplitude modulation and Mary phase shift keying under the influence of Nakagamim fading channel. Their simulation outcomes demonstrate that OFDMIM systems provide better performance in highdatarate transmission as compared to normal OFDM when it is under the effect of various factors of m, in a Nakagami channel environment. But they do not consider the CFO effect.
3 Methods
With the above backdrop of study, we are proposing a new twolevel efficient computation and correction method that effectively eliminates the frequency error in MIMOWLAN system impacted by the various TGn MIMO channel models from A to F including the AWGN. The proposed algorithm is based on the blind approach as it is more suitable for the WLAN scenario because of its quasistatic channel conditions and packetbased transmission. Also, as MIMO configuration increases the synchronization structure also needs to be enhanced in dataaided techniques which reduce the system bandwidth efficiency; therefore, blind approaches are better for WLAN systems.
In our proposed research work, we show through the MATLAB simulations that the performance of the algorithm completely satisfies the standard prescribed limit. And there is no compromise on the spectral efficiency as in the case of the dataaided techniques, and also, since our algorithm makes use of the remodulated cyclic prefix (CP), the complexity is less and it is independent of type of modulation schemes used unlike other algorithms.
Our proposed algorithm is capable of efficient CFO estimation for both the MIMOOFDM WLAN and MIMOOFDMIM WLAN systems, although the IMbased WLAN system gives a better performance as compared to normal MIMOOFDM WLAN but it is very minimal. Our proposed method is also compared to that of other unaided algorithms based on pilot and unused null tones, and there is a clear improvement of the new estimator over the existing methods not only in the low SNR regions but also in the high SNR region.
The proposed algorithm in general can be applied to SISO/MIMOOFDM/OFDMIMbased WLAN systems as can be observed through the performance results.
Thus, we summarize our contributions as:

Developed algorithm using blind data approach for computing the frequency offset. The algorithm is based on the remodulated cyclic prefix; hence, the complexity is less and it is independent of type of modulation schemes used.

Proposed a new twolevel efficient computation and correction technique that effectively eliminates the frequency error in WLAN system influenced by the various TGn MIMO channel models from A to F including the AWGN.
The remaining segments of the current article are arranged as mentioned. Segment III briefly describes the MIMOOFDMIM system model with a schematic. Segment IV describes the proposed detection algorithm. The simulated plots are discussed in Section V. The article is overall concluded in segment VI.
Notation: The symbol notations and their mathematical representation are given in Table 1.
4 MIMOOFDMIM system model
A MIMOOFDMIM system with N_{t} transmit antennas is shown in the block schematic of Fig. 1. A data set of dN_{t} bits is split among N_{t} streams, each with d bits. The d data bits in each stream are fragmented into G groups, each of size g bits; the g bits are further divided into g1 index bits and g2 symbol bits. The symbol bits are mapped onto the selected subcarriers within the group chosen by the index bits, and the unselected subcarriers in the group are called the inactive or null subcarriers as depicted in Table 2. All the G groups are clubbed together and forwarded to the OFDM block creator, and IFFT is applied to it to create a time domain vector of N samples. To this signal vector, a guard period called the cyclic prefix which spans longer than the largest delay spread in the transmitted channel is preappended, and thus a cyclic structure is created. This process is repeated in all the N_{t} streams.
The nth MIMOOFDMIM symbol transmitted at the mth transmit antenna may be represented as in Eq. (1) where m = {1, 2, …., N_{t}}
where
The nth received signal at the pth antenna, impaired by the channel H, AWGN noise w and normalized frequency offset ϕ can be represented as in Eq. (4)
where
ϕ = NΔ_{f}/F_{s,} with the frequency offset Δ_{f} and the sampling frequency F_{s} in Hz. p = {1,2,3…N_{r}}.
The channel matrix \({{\varvec{H}}}^{p,m}\) of (N + L) × (N + 2L) dimension is arranged as a Toeplitz matrix whose first row is
[\({h}^{p,m }\left(L\right),\dots \dots ..,{h}^{p,m}\left(0\right),0\dots 0]\) and the first column is [\({h}^{p,m }\left(L\right),\dots \dots ..,{h}^{p,m}\left(0\right),0\dots 0]\)^{T}
where \({{\varvec{h}}}^{p,m}=[ {h}^{p,m}\left(0\right),{h}^{p,m}\left(1\right),\dots \dots \dots ,{h}^{p,m}(L)]\) is the channel coefficients at the pth receiver transmitted from mth transmitter. Here, it is assumed that the channel’s maximum delay spread does not exceed the guard interval period of L.
The nth AWGN signal with \({\sigma }_{n}^{2}\) variance on the pth receiving stream can be represented as
5 Proposed algorithm
The cyclic prefix or the guard interval is a prefix of length L, included at the beginning of each of the transmitted OFDM symbols which creates a continuous or periodic structure. Any frequency offset introduced can be estimated very accurately using the repetitive structure of the CP under flat fading channel conditions, but in any other fading or multipath condition the periodicity is lost due to the intersymbolinterference (ISI), and estimation using the guard interval does not provide accurate results. Therefore, the remodulation concept [58] is utilized, which retains the periodicity of the signal vector even in deep fade channel conditions, to accurately estimate the frequency error.
To construct a remodulation signal, two consecutive received symbols are required. Let the 2 consecutive symbols be \({{\varvec{y}}}_{{\varvec{n}}1}^{{\varvec{p}}}\) and \({{\varvec{y}}}_{{\varvec{n}}}^{{\varvec{p}}}\) at the pth receiver antenna, then the remodulation signal \({\widetilde{{\varvec{y}}}}_{{\varvec{n}}}^{{\varvec{p}}}\) is a concatenation of the N concluding samples of the (n − 1)th symbol and the L initial samples of the nth symbol thereby retaining the overall length of the remodulation vector at (N + L).
The nth constructed remodulation vector on the pth receiver antenna \({{\varvec{y}}}_{n}^{p}\) is represented as
where the noise component \({\widetilde{{\varvec{w}}}}_{n}^{p}\) is:
Considering the difference vector of nth received vector and the remodulation vector at the pth antenna
where the variable θ is used to estimate the offset introduced, substituting Eqs. (4) and (9) in Eq. (11) results into:
Computing the autocorrelation of Eq. (11), R_{rr}(θ) can be mathematically represented as in (13), considering the signal and noise to be uncorrelated.
Equation (13) can be alternately represented as
where \({\sigma }_{x}^{2}\) is the mean power of the signal at the pth antenna and ϖ is a (N + 2L) × (N + 2L) matrix:
with \({\alpha }_{1}=\text{cos}\left(2\uppi \left(\theta \phi \right)\right)\) and \({\alpha }_{2}=1{e}^{j2\uppi \left(\theta \phi \right)}\), \({{\varvec{R}}}_{\mathcalligra{n}}^{p}\left(\theta \right)\) is the matrix with dimension of (N + L) × (N + L):
From Eq. (14), it can be concluded that the diagonal values of \({{\varvec{R}}}_{rr}^{p}\left(\theta \right)\) are
where the sequence S = {0, 1, …, L − 1, N, N + 1 …N + L − 1}
and,
From Eqs. (18) and (19), it can be observed that both are independent of the parameter ϕ and θ. And, only the initial and concluding L diagonal data depends on θ. Therefore, the cost function can be summed as these 2L entries across all the receiver antennas can be represented as.
Expanding Eq. (20) using Eq. (13):
The cost function will be minimum and unique when θ = ϕ. Thus the first level of estimation is obtained using Eq. (23). Minimizing the cost function by taking the derivative of the cost function with respect to θ and equating it to zero.
Once the L1 estimate is computed using Eq. (23), the mean square error (MSE) for \({\widehat{\phi }}_{L1}\) exhibits error flooring even as the signaltonoise ratio is increased. To overcome this error flooring issue, the CFO is again computed after correcting the received signal from the L1 estimate and calculating only for those diagonal entries which have minimum value and restricting the set s to \({s}_{\text{min}}\) whose length is fixed to L and the L2 computation is represented as:
Equation (25) computes the MSE of the estimated parameter \(\widehat{\phi }\) and the introduced frequency error ϕ, the count for which the algorithm is simulated for the given parameter of frequency offset is given by R and the estimated error is averaged out across all of the received antennas.
6 Results and discussion
The performance of the method proposed for the estimation and correction of the frequency error caused during the signal transmission is evaluated using the MATLAB simulations for three different but equal antenna configurations of 2, 3 and 4 in both transmitter and receiver. The OFDM block dimension of 50 is considered, and the NFFT length is 64 with a CP period of 25% is preappended to the OFDM symbol. The IEEE TGn fading channels, including all the models from A to F which correspond to various indoor setups, are introduced along with the AWGN. A carrier frequency offset of ± 20 ppm is included uniformly along all the transmission routes. The channel impulse response and the frequency error are both assumed not to change over the entire duration of the estimation.
The simulation results in Figs. 2, 3, 4, 5, 6, 7 and 8 are obtained considering the parameters mentioned in Table 3: Configurations. Figure 2 is a plot of the mean square error comparison between the MIMOOFDM and MIMOOFDMIM systems for a 2 × 2 antenna configuration under the influence of various TGn channel models varying from TGn A to TGn F with respect to increasing SNR.
The best MSE performance can be observed for TGn A in both the cases of MIMOOFDM and MIMOOFDMIM. And it is also observed that the MSE varies linearly with the SNR, and there is no error flooring as seen in other channel models. Since model A is designed for flat fading or nonfrequency selective fading, whereas models B to F are frequency selective fading models with increasing delay spread, channel model A gives the best performance, and the MSE increases thereafter from B to F, with F having the least MSE performance compared to all the other models. This behavior can be commonly seen with all the antenna configurations.
We also observe that there is a marginal improvement in the case of MIMOOFDMIM compared to MIMOOFDM for the channel models between A and D, but the difference diminishes as observed in the case of channel model F, where they both overlap.
For TGn A, the MSE of the frequency offset varies linearly with the increase in the SNR and reaches an MSE of close to 10^{–9} in classical MIMOOFDM and crosses 10^{–9} with IM system. But for other channels from B to F, we see that MSE decreases initially with the increase in SNR and after that, there is an error flooring even with high SNR, and MSE of close to 10^{–8} is obtained for model B. In the case of TGn F, the MSE crosses 10^{–5} in both MIMOOFDM and MIMOOFDMIM, and their performance overlaps.
Figure 3 is a comparison plot of the MSE of the frequency error estimated with respect to varying SNR for MIMOOFDMIM and MIMOOFDM with a 3 × 3 antenna configuration. The characteristics of Fig. 3 follow those of Fig. 2, with a slight improvement over the 2 × 2 antenna configuration. The improvement can be seen in the lower SNR regions where the MIMOOFDM and MIMOOFDMIM MSE overlap. Still in high SNR regions, IM has marginally better performance.
Figure 4 is a comparison plot of MIMOOFDM and MIMOOFDMIM with a 4 × 4 antenna configuration. In this plot, it is clearly seen that the MSE of both the MIMOOFDM and MIMOOFDMIM not only overlap at the lower SNR regions but also at the higher SNR regions can be seen, and the performance also reaches close to a MSE of 10^{–10} for TGn A and close to 10^{–6} for TGn F.
From Figs. 2, 3 and 4, it is evident that at low SNR, the IM performs better compared to nonIM system and as the MIMO configuration increases, this difference diminishes, and for Model E and F, they completely overlap.
Figure 5 is a performance comparison plot of the proposed algorithm and the algorithm using the inactive and pilot subcarriers in the IM system. We observe that the proposed algorithm works well; although a performance improvement can be seen in the high SNR region, it is marginal as compared to our proposed algorithm.
Figure 6 is a comparison plot of OFDM and OFDMIM with SISO configuration, the MSE varies between 10^{–9} and 10^{–5} for A to F models, and the pattern continues similar to MIMO configuration. Reassuring that the proposed algorithm perfectly works for OFDM/OFDMIM/MIMOOFDM/MIMOOFDMIM modulation methods.
Figure 7 is the plot of MIMOOFDM and MIMOOFDMIM with frequency error further increased to ± 25 ppm for the 3 × 3 configuration, and our proposed algorithm equally works well, and the result pattern continues here too. When the frequency error is further increased to ± 30 ppm, a plot of Fig. 8 is obtained wherein the MSE slightly deteriorates and lies between 10^{–8} and 10^{–5}; also at the low SNR region, there is a performance dip till an SNR of 5 dB, and thereafter, it picks up, indicating a better signal quality requirement at such high frequency errors.
7 Conclusion
In this work, we have proposed a novel blind estimation algorithm that efficiently estimates and corrects the carrier frequency error introduced during transmission. The proposed algorithm is mainly targeted for highefficiency WLANs that are based on the MIMOOFDMIM modulation technique. The performance of the proposed method completely satisfies the expected tolerable frequency deviation limit of ± 20 ppm from the IEEE 802.11 standard. We have also compared its performance with the other algorithm based on the pilot and inactive data carriers, wherein the proposed algorithm excels at both the low SNR and high SNR regions. The proposed algorithm is not only limited to the MIMOOFDMIM technique but can be used with the same efficiency for MIMOOFDM/OFDM/OFDMIM modulation technique based WLAN, as shown in the results. As a part of future work, the proposed algorithm can also be implemented in cellular and broadband systems. It can be tested with their respective channel models for accuracy.
Availability of data and materials
The data sets were generated from the codes developed in the MATLAB software as per the IEEE 802.11n guidelines.
Abbreviations
 CFO:

Carrier frequency offset
 CP:

Cyclic prefix
 HD:

High definition
 IIOT:

Industrial IoT
 IM:

Index modulation
 IoT:

Internet of things
 MIMO:

Multipleinput multipleoutput
 OFDM:

Orthogonal frequency division multiplexing
 OFDMA:

Orthogonal frequency division multiple access
 PAPR:

Peaktoaverage power ratio
 PPM:

Parts per million
 QAM:

Quadrature amplitude modulation
 RF:

Radio frequency
 SNR:

Signaltonoise ratio
 TGn:

High throughput task group
 WiFi:

Wireless fidelity
 WLAN:

Wireless local area network
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Lalitha, H., Kumar, N. Carrier frequency synchronization for WLAN systems based on MIMOOFDMIM. J Wireless Com Network 2024, 77 (2024). https://doi.org/10.1186/s1363802402406z
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DOI: https://doi.org/10.1186/s1363802402406z