Carrier frequency synchronization for WLAN systems based on MIMO-OFDM-IM

Abstract

Multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) with index modulation (IM) MIMO-OFDM-IM 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 OFDM-IM 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 non-data-aided algorithm that efficiently estimates and eliminates the offset present in the received signal. The algorithm has been analyzed for the MIMO-OFDM-IMbased 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 signal-to-noise 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 Wi-Fi 1 to the most recent standard of Wi-Fi 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 ever-changing 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 next-gen 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, Wi-Fi 7 [9] is designed for utmost compatibility and coexisting cordially with all previous 802.11a/n/ac/ax devices.

To facilitate these features, Wi-Fi 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]. MIMO-OFDM 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 MIMO-OFDM 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 peak-to-average 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 inter-carrier-interference (ICI) issues because of the unused subcarriers. Thus MIMO-OFDM-IM stands out as an endearing candidate for meeting the challenging needs of next-generation networks  [31–37].

MIMO-OFDM/MIMO-OFDM-IM 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 next-generation 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 10⁶ 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 well-established fact, and it is extensively studied leading to an abundance of literature. The CFO estimation is carried out using data-aided techniques like utilizing the training sequences which are pre-appended 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 OFDM-IM 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 OFDM-IM.

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 OFDM-based systems toward the carrier frequency offset which reduces the system performance. They have come up with the closed-form 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 MIMO-OFDM 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 MIMO-OFDM 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 ML-based 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 (OFDM-GIM). In this technique, initially the receiver computes and corrects the offset error with the help of the pre-allocated 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 signal-to-noise 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 ESIM-OFDM with well-established 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 OFDM-IM 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 OFDM-IM, wherein depending on the channel conditions, the transmission mode can be alternately switched between the OFDM and OFDM-IM. 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 OFDM-IM. In the article [57], authors have given a closed-form expression for the OFDM-IM system using modulation schemes such as M-ary quadrature amplitude modulation and M-ary phase shift keying under the influence of Nakagami-m fading channel. Their simulation outcomes demonstrate that OFDM-IM systems provide better performance in high-data-rate 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 two-level efficient computation and correction method that effectively eliminates the frequency error in MIMO-WLAN 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 quasi-static channel conditions and packet-based transmission. Also, as MIMO configuration increases the synchronization structure also needs to be enhanced in data-aided 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 data-aided 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 MIMO-OFDM WLAN and MIMO-OFDM-IM WLAN systems, although the IM-based WLAN system gives a better performance as compared to normal MIMO-OFDM 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/MIMO-OFDM/OFDM-IM-based 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 two-level 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 MIMO-OFDM-IM 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.

Table 1 Notations

4 MIMO-OFDM-IM system model

A MIMO-OFDM-IM 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 pre-appended, and thus a cyclic structure is created. This process is repeated in all the N_t streams.

Fig. 1

MIMO-OFDM-IM Block Schematic. It is a block diagram of the MIMO-OFDM-IM system with N_t transmit antennas. A data of dN_t bits are 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. 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 is pre-appended and this process is repeated in all the N_t streams

Table 2 Subcarriers selection criteria

The nth MIMO-OFDM-IM symbol transmitted at the mth transmit antenna may be represented as in Eq. (1) where m = {1, 2, …., N_t}

$${{\varvec{t}}}_{n}^{m}= \left[\begin{array}{c}{{\varvec{x}}}_{n,L}^{m}\\ {{\varvec{x}}}_{n}^{m}\end{array}\right]$$

(1)

where

$${{\varvec{x}}}_{n}^{m}=\left[{x}_{n}^{m}\left(0\right),{x}_{n}^{m}\left(1\right),{x}_{n}^{m}\left(2\right),{\dots \dots \dots x}_{n}^{m}\left(N-1\right)\right]^{T}$$

(2)

$${{{\varvec{x}}}_{n,L}^{m}=[x}_{n}^{m}\left(N-L\right),\dots .,{x}_{n}^{m}\left(N-2\right),{x}_{n}^{m}\left(N-1\right)]^{T}$$

(3)

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)

$${{\varvec{y}}}_{n}^{p}=\sum_{m=1}^{{N}_{t}}{e}^{j2\pi \frac{n\left(N+L\right)\phi }{N} }{{\varvec{\phi}}}_{0}{{\varvec{H}}}^{p,m}\left[\begin{array}{c}{{\varvec{x}}}_{n-1,L}^{m}\\ {{\varvec{x}}}_{n}^{m}\end{array}\right]+{{\varvec{w}}}_{n}^{p}$$

(4)

where

$${{\varvec{\phi}}}_{0}=diagonal\{ 1,{e}^{\frac{j2\pi \phi }{N} },\dots \dots \dots ,{e}^{\frac{j2\pi \left(N+L-1\right)\phi }{N} }\}$$

(5)

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.

$${{\varvec{H}}}^{p,m}=\left[\begin{array}{ccc}\begin{array}{ccc}{h}^{p,m}(L)& \cdots & {h}^{p,m}\left(0\right)\\ \vdots & \ddots & \ddots \\ 0& \cdots & {h}^{p,m}\left(L\right)\end{array}& \begin{array}{c}\cdots \\ \ddots \\ \dots \end{array}& \begin{array}{c}0\\ \vdots \\ {h}^{p,m}\left(0\right)\end{array}\end{array}\right]$$

(6)

The nth AWGN signal with ${\sigma }_{n}^{2}$ variance on the pth receiving stream can be represented as

$${{\varvec{w}}}_{n}^{p}={[w}_{n}^{p}\left(0\right),{w}_{n}^{p}\left(1\right),..{.. w}_{n}^{p}(N+L-1)]^{\text{T}}$$

(7)

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 inter-symbol-interference (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).

$${\widetilde{{\varvec{y}}}}_{n}^{p}\triangleq {\left[{y}_{n-1}^{p}\left(L\right), \dots {y}_{n-1}^{p}\left(N+L-1\right), {y}_{n}^{p}\left(0\right),\dots {y}_{n}^{p}\left(L-1\right)\right]}^{T}$$

(8)

The nth constructed remodulation vector on the pth receiver antenna ${{\varvec{y}}}_{n}^{p}$ is represented as

$${\widetilde{{\varvec{y}}}}_{n}^{p}=\sum_{m=1}^{{N}_{t}}{e}^{j2\pi \phi \frac{n\left(N+L\right)-N}{N} }{{\varvec{\phi}}}_{0}{{\varvec{H}}}^{p,m}\left[\begin{array}{c}{{\varvec{x}}}_{n-1}^{m}\\ {{\varvec{x}}}_{n,L}^{m}\end{array}\right]+{\widetilde{{\varvec{w}}}}_{n}^{p}$$

(9)

where the noise component ${\widetilde{{\varvec{w}}}}_{n}^{p}$ is:

$${\widetilde{{\varvec{w}}}}_{n}^{p}={[w}_{n-1}^{p}\left(L\right),\dots {w}_{n-1}^{p}\left(N+L-1\right) , {w}_{n}^{p}\left(0\right),{w}_{n}^{p}\left(L-1\right)]^{\text{T}}$$

(10)

Considering the difference vector of nth received vector and the remodulation vector at the pth antenna

$${{\varvec{r}}}_{n }^{p}\left(\uptheta \right)= \sum_{m=1}^{{N}_{t}}{{\varvec{y}}}_{n}^{m}- {e}^{j2\pi \theta } {\widetilde{{\varvec{y}}}}_{n}^{m}$$

(11)

where the variable θ is used to estimate the offset introduced, substituting Eqs. (4) and (9) in Eq. (11) results into:

$${{\varvec{r}}}_{n}^{p}\left(\uptheta \right)=\sum_{m=1}^{{N}_{t}}{e}^{j2\pi n\frac{\left(N+L\right)\theta }{N}}{{\varvec{\phi}}}_{0} {{\varvec{H}}}^{p,m}\left(\left[\begin{array}{c}{{\varvec{x}}}_{n-1,L}^{p}\\ {{\varvec{x}}}_{n}^{p}\end{array}\right]-{e}^{j2\pi \left(\theta -\phi \right)}\left[\begin{array}{c}{{\varvec{x}}}_{n-1}^{p}\\ {{\varvec{x}}}_{n,L}^{p}\end{array}\right]\right)+ \underbrace {{\left( {\varvec{w}_{n}^{p} - e^{{j2\pi \theta }} \varvec{\tilde{w}}_{n}^{p} } \right)}}_{n}$$

(12)

Computing the autocorrelation of Eq. (11), R_rr(θ) can be mathematically represented as in (13), considering the signal and noise to be uncorrelated.

$${{\varvec{R}}}_{rr}^{p}\left(\theta \right)=E\left\{\left({{\varvec{y}}}_{n}^{p}-{e}^{j2\pi \theta } {\widetilde{{\varvec{y}}}}_{n}^{p}\right){\left({{\varvec{y}}}_{n}^{p}-{e}^{j2\pi \theta } {\widetilde{{\varvec{y}}}}_{n}^{p}\right)}^{\ddagger }\right\}$$

(13)

Equation (13) can be alternately represented as

$${{\varvec{R}}}_{rr}^{p}\left(\theta \right)= {\sigma }_{x}^{2}{{\varvec{\phi}}}_{0}{\varvec{H}}\boldsymbol{\varpi }{{\varvec{H}}}^{\ddagger }{{{\varvec{\phi}}}_{0}}^{\ddagger }+ {\sigma }_{\mathcalligra{n}}^{2}{{\varvec{R}}}_{\mathcalligra{n}}^{p}\left(\theta \right)$$

(14)

where ${\sigma }_{x}^{2}$ is the mean power of the signal at the pth antenna and ϖ is a (N + 2L) × (N + 2L) matrix:

$$\boldsymbol{\varpi }=\left[\begin{array}{ccccc} 2\left(1-{\alpha }_{1}\right){I}_{L}& 0& 0& {\alpha }_{2}^{*} {I}_{L}& 0\\ 0& 2{I}_{L}& 0& 0& {\alpha }_{2}^{*} {I}_{L}\\ 0& 0& 2{I}_{N-2L}& 0& 0\\ {\alpha }_{2}{I}_{L}& 0& 0& 2{I}_{L}& 0\\ 0& {\alpha }_{2}{I}_{L}& 0& 0& 2\left(1-{\alpha }_{1}\right){I}_{L}\end{array}\right]$$

(15)

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):

$${{\varvec{R}}}_{\mathcalligra{n}}^{p}\left(\theta \right)=\left[\begin{array}{ccc}2{I}_{L}& 0& -{e}^{-j2\uppi \theta }{I}_{L}\\ 0& 2{I}_{N-L}& 0\\ -{e}^{-j2\uppi \theta }{I}_{L}& 0& 2{I}_{L}\end{array}\right]$$

(16)

From Eq. (14), it can be concluded that the diagonal values of ${{\varvec{R}}}_{rr}^{p}\left(\theta \right)$ are

$${\left[{{\varvec{R}}}_{rr}^{p}\left(\theta \right)\right]}_{i.i}=\left\{\begin{array}{c}2\left({\zeta }^{p}-{\alpha }_{1}{\zeta }_{i}^{p}\right) if i \in S\\ 2{\zeta }^{p}+{2({\sigma }_{\mathcalligra{n}}^{p})}^{2} otherwise\end{array}\right.$$

(17)

where the sequence S = {0, 1, …, L − 1, N, N + 1 …N + L − 1}

$${\zeta }^{p} \triangleq \sum_{m=1}^{{N}_{t}}{{ (\sigma }_{x}^{p})}^{2}\sum_{l=0}^{L}{\left|{h}^{\left(m,p\right)}\left(l\right)\right|}^{2}$$

(18)

and,

$$\triangleq \left\{\begin{array}{c}\sum_{m=1}^{{N}_{t}}{{ (\sigma }_{x}^{p})}^{2}\sum_{l=i+1}^{L}{\left|{h}^{\left(m,p\right)}\left(l\right)\right|}^{2} \text{if} 0\le i\le L-1\\ \sum_{m=1}^{{N}_{t}}{{ (\sigma }_{x}^{p})}^{2}\sum_{l=0}^{i-N}{\left|{h}^{\left(m,p\right)}\left(l\right)\right|}^{2} if N\le i\le N+L-1\end{array}\right.$$

(19)

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.

$$J(\theta )=\sum_{p=1}^{{N}_{r}}\sum_{i\in s}{\left[{{\varvec{R}}}_{rr}^{p}\left(\theta \right)\right]}_{i,i}$$

(20)

Expanding Eq. (20) using Eq. (13):

$$J\left(\theta \right)\approx \sum_{p=1}^{{N}_{r}}\sum_{n=1}^{K-1}\sum_{i\in s}\left({{\varvec{y}}}_{n}^{p}(i)-{e}^{j2\pi \theta } {\widetilde{{\varvec{y}}}}_{n}^{p}(i)\right)\left({\left({{\varvec{y}}}_{n}^{p}(i)-{e}^{j2\pi \theta } {\widetilde{{\varvec{y}}}}_{n}^{p}(i)\right)}^{*}\right)$$

(21)

$$\widehat{\phi }=\text{arg}\underset{\theta \in (-0.5 ,0.5)}{\text{min}}\widetilde{J }\left(\theta \right)$$

(22)

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.

$${\widehat{\phi }}_{L1}=\sum_{p=1}^{Nr} \frac{1}{2\pi } angle\left[\sum_{n=1}^{K-1}\sum_{i\in s}{{\varvec{y}}}_{n}^{p}\left(i\right) {\left({\widetilde{{\varvec{y}}}}_{n}^{p}\left(i\right)\right)}^{*} \right]$$

(23)

Once the L1 estimate is computed using Eq. (23), the mean square error (MSE) for ${\widehat{\phi }}_{L1}$ exhibits error flooring even as the signal-to-noise 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:

$${\widehat{\phi }}_{L2}=\sum_{p=1}^{Nr} \frac{1}{2\pi }angle\left[\sum_{n=1}^{K-1}\sum_{i\in {s}_{\text{min}}}{{\varvec{y}}}_{n}^{p}\left(i\right) {\left({\widetilde{{\varvec{y}}}}_{n}^{p}\left(i\right)\right)}^{*} \right]$$

(24)

$$\text{MSE}(\widehat{\phi }) = \frac{1}{R} \sum_{i=1}^{R}{ \left|\widehat{\phi }\left(i\right)- \phi \right|}^{2}$$

(25)

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 N-FFT length is 64 with a CP period of 25% is pre-appended 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 MIMO-OFDM and MIMO-OFDM-IM 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.

Fig. 2

Comparison plot of MIMO-OFDM-IM and MIMO-OFDM for 2 × 2 under TGn channels. It is a plot of mean square error comparison between the MIMO-OFDM and MIMO-OFDM-IM system for a 2 × 2 MIMO 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 for both the cases of MIMO-OFDM and MIMO-OFDM-IM. 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. 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 MIMO-OFDM 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, for TGn F the MSE crosses 10^–5 in both cases of MIMO-OFDM, MIMO-OFDM-IM and they both overlap

Fig. 3

Comparison plot of MIMO-OFDM-IM and MIMO-OFDM for 3 × 3 under TGn channels. Figure 3 is a comparison plot of MSE of the frequency error estimated with respect to varying SNR for MIMO-OFDM-IM and MIMO-OFDM with 3 × 3 antenna configuration. The characteristics of Fig. 3 follow with Fig. 2 with a slight improvement over 2 × 2 antenna configuration. The improvement can be seen in the lower SNR regions MIMO-OFDM and MIMO-OFDM-IM MSE overlap but still at higher SNR regions IM has marginal better performance

Fig. 4

Comparison plot of MIMO-OFDM-IM and MIMO-OFDM for 4 × 4 under TGn channels. Figure 4 is a comparison plot of MIMO-OFDM and MIMO-OFDM-IM with 4 × 4 antenna configuration. In this plot, it is clearly seen that MSE of both the MIMO-OFDM and MIMO-OFDM-IM 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. It can also be observed that at low SNR, the IM performs better compared to Non-IM system and as the MIMO configuration increases this difference diminishes and for Model E and F, they completely overlap

Fig. 5

Comparison plot of MIMO-OFDM-IM with proposed algorithm and using inactive tones. 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 at the high SNR region but it is marginal as compared to our proposed algorithm

Fig. 6

Comparison plot of OFDM-IM and OFDM for SISO under TGn channels. Figure 6 is a comparison plot of OFDM and OFDM-IM with SISO configuration, the MSE varies between 10^–9 and 10^–5 for A to F models and the pattern continues similar to MIMO. Reassuring that the proposed algorithm perfectly works in OFDM/OFDM-IM/MIMO-OFDM/MIMO-OFDM-IM

Fig. 7

Comparison plot of MIMO-OFDM-IM and MIMO-OFDM with TGn channels at ±25 ppm for 3 × 3 configuration. Figure 7 is the plot of MIMO-OFDM and MIMO-OFDM-IM with frequency error further increased to ±25 ppm for 3 × 3 configuration, and our proposed algorithm equally works well and the result pattern continues here too

Fig. 8

Comparison plot of SISO-OFDM-IM and MIMO-OFDM-IM 4 × 4 with TGn channels at ±30 ppm. 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 and 10^–8 and 10^–5 and 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

Table 3 Configurations

The best MSE performance can be observed for TGn A in both the cases of MIMO-OFDM and MIMO-OFDM-IM. 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 non-frequency 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 MIMO-OFDM-IM compared to MIMO-OFDM 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 MIMO-OFDM 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 MIMO-OFDM and MIMO-OFDM-IM, and their performance overlaps.

Figure 3 is a comparison plot of the MSE of the frequency error estimated with respect to varying SNR for MIMO-OFDM-IM and MIMO-OFDM 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 MIMO-OFDM and MIMO-OFDM-IM MSE overlap. Still in high SNR regions, IM has marginally better performance.

Figure 4 is a comparison plot of MIMO-OFDM and MIMO-OFDM-IM with a 4 × 4 antenna configuration. In this plot, it is clearly seen that the MSE of both the MIMO-OFDM and MIMO-OFDM-IM 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 non-IM 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 OFDM-IM 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/OFDM-IM/MIMO-OFDM/MIMO-OFDM-IM modulation methods.

Figure 7 is the plot of MIMO-OFDM and MIMO-OFDM-IM 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 high-efficiency WLANs that are based on the MIMO-OFDM-IM 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 MIMO-OFDM-IM technique but can be used with the same efficiency for MIMO-OFDM/OFDM/OFDM-IM 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.