A Novel Frame Structure for Joint Blind Channel and CFO Estimation for Mobile OMA/NOMA OFDM

This article introduces a novel channel and carrier frequency offset (CFO) estimation scheme for orthogonal frequency-division multiplexing (OFDM) transmission over time-varying channels. The proposed design is highly flexible and compatible with orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) transmission. The new estimator performs the channel and CFO estimation jointly and blindly, highlighting the efficiency of the proposed estimator. The blind CFO and channel estimation are realized by developing a new OFDM frame layout, where phase shift keying (PSK) and amplitude shift keying (ASK) symbols are used to modulate specific subcarriers over successive OFDM frames. An arbitrary modulation scheme can be used to modulate all other subcarriers. The new frame layout enables the use of amplitude coherent detection (ACD) and Viterbi-and-Viterbi (VAV), which are used to perform blind channel and CFO estimation and compensation. A closed-form analytical formula is derived for the exact symbol error rate (SER) of the ASK symbols, which is then used to derive a precise formula for the mean squared error (MSE) for the OMA case. For NOMA, the same approach can be used and the analysis is generally similar, hence, Monte Carlo simulation is invoked to evaluate the performance for the NOMA case. The obtained results show that the new frame layout can improve the spectral efficiency while enabling accurate channel and CFO estimation for both OMA and NOMA with computational complexity analogous to pilot-assisted schemes. The improved spectral efficiency is due to the replacement of pilot symbols with data-bearing symbols. The performance of the system is evaluated in terms of MSE and SER for a wide range of operating scenarios, and the results confirm the robustness and reliability of the proposed scheme for both OMA and NOMA.

A Novel Frame Structure for Joint Blind Channel and CFO Estimation for Mobile OMA/NOMA OFDM Lina Bariah , Member, IEEE, Arafat Al-Dweik , Senior Member, IEEE, Mahmoud Aldababsa , Senior Member, IEEE, and Sami Muhaidat , Senior Member, IEEE Abstract-This article introduces a novel channel and carrier frequency offset (CFO) estimation scheme for orthogonal frequency-division multiplexing (OFDM) transmission over timevarying channels.The proposed design is highly flexible and compatible with orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) transmission.The new estimator performs the channel and CFO estimation jointly and blindly, highlighting the efficiency of the proposed estimator.The blind CFO and channel estimation are realized by developing a new OFDM frame layout, where phase shift keying (PSK) and amplitude shift keying (ASK) symbols are used to modulate specific subcarriers over successive OFDM frames.An arbitrary modulation scheme can be used to modulate all other subcarriers.The new frame layout enables the use of amplitude coherent detection (ACD) and Viterbi-and-Viterbi (VAV), which are used to perform blind channel and CFO estimation and compensation.A closed-form analytical formula is derived for the exact symbol error rate (SER) of the ASK symbols, which is then used to derive a precise formula for the mean squared error (MSE) for the OMA case.For NOMA, the same approach can be used and the analysis is generally similar, hence, Monte Carlo simulation is invoked to evaluate the performance for the NOMA case.The obtained results show that the new frame layout can improve the spectral efficiency while enabling accurate channel and CFO estimation for both OMA and NOMA with computational complexity analogous to pilot-assisted schemes.The improved spectral efficiency is due to the replacement of pilot symbols with data-bearing symbols.The performance of the system is evaluated in terms of MSE and SER for a wide range of operating scenarios, and the results confirm the robustness and reliability of the proposed scheme for both OMA and NOMA.Index Terms-Blind, CFO, channel estimation, OFDM, timevarying channels, SER, NOMA.

I. INTRODUCTION
O RTHOGONALfrequency-division multiplexing (OFDM)   is a widely recognized multi-carrier transmission approach that offers a high information rate, high spectral efficiency, and robustness against channel frequency selectivity for current and future broadband wireless communications.OFDM is extensively exploited in many standards such as digital video broadcasting (DVB), digital audio broadcasting (DAB), asymmetric digital subscriber line (ADSL), wireless local area networks (LAN), worldwide interoperability for microwave access (WiMAX), long term evolution (LTE) and LTE-Advanced [1].Moreover, OFDM is a potential candidate for the upcoming fifth generation (5G) wireless communication standard [2].To further improve the spectral efficiency of OFDM, several researchers proposed integrating OFDM and NOMA [3], [4], [5], [6].In such scenarios, multiple users can share each subcarrier simultaneously.
Although OFDM is immune to the frequency selectivity nature of the wireless channels, its performance is highly susceptible to CFO, which is caused by the mismatch between the carrier frequencies in the local oscillators.In OFDM systems, CFO induces a loss of orthogonality between subcarriers, leading to inter-carrier interference (ICI) and consequently SER performance degradation.In addition to the effect of CFO, channel estimation is another crucial issue for OFDM.Most of the reported work about OFDM assumes perfect knowledge of channel state information (CSI).However, CSI must be estimated in practical scenarios and then used for coherent detection.Therefore, accurate estimation and compensation of the channel and CFO are required for reliable detection at the receiver side.There are two main approaches for channel and CFO estimation, namely, blind and pilot-assisted.Generally speaking, blind approaches enjoy high spectral efficiency; however, they suffer from high computational complexity, slow convergence, and identification ambiguities.Pilot-assisted systems have low complexity and higher accuracy compared to blind methods; nevertheless, pilotassisted estimation has low spectral efficiency.It is worth mentioning that the channel and CFO estimation can be performed implicitly and explicitly.
Many algorithms are designed to estimate channel gains under different frequency and time selectivity scenarios without considering CFO estimation [7], [8], [9], [10].In these approaches, CFO can be considered as a Doppler shift and, therefore, can be estimated implicitly.However, if the CFO value is greater than the channel Doppler spread, then explicit CFO estimation is required.In fact, although the explicit channel and CFO estimation have generally higher computational complexity, they can provide more accurate estimates [11].The expectationmaximization (EM) algorithm was adopted by many researchers as an efficient approach for joint channel and CFO estimation [12], [13], [14], [15].Although the EM algorithm provides relatively accurate channel and CFO estimates, most of the reported work in the EM algorithm assumes time-invariant channels.Therefore, they are not suitable for high mobility scenarios.By adopting the EM algorithm, which supports high mobility, joint channel and CFO estimation is proposed in [16].The proposed estimator exploits the basis expansion model (BEM) to reduce the estimation complexity by reducing the required number of estimated gains.Another BEM-dependent joint estimator was proposed in [17] for doubly-selective channels to reduce complexity and avoid the possible identifiability problem associated with some estimators [18].In [19], an maximum likelihood (ML)-based algorithm was utilized for joint estimation.The proposed approach in [19] provides accurate estimation; however, this comes at the expense of high complexity.We highlight that the aforementioned schemes are data-aided; therefore, they suffer from low spectral efficiency.In [20], a joint channel and CFO estimator is proposed where the joint estimation is performed blindly.Although the proposed estimator provides good performance, its main drawbacks are the high computational complexity and the scaling ambiguity that requires the transmission of at least one pilot.

A. Motivation and Contributions
Despite the fact that joint CFO and channel estimation is a well-investigated research topic, to the best of our knowledge, there is no approach available yet that allows blind joint estimation for channel and CFO while maintaining low complexity, high spectral efficiency, accuracy, and short convergence time over time-varying frequency-selective channels.Particularly for NOMA-OFDM systems [21].Therefore, this article presents a spectrally efficient blind joint channel and CFO estimation algorithm that is comparable to pilot-assisted approaches in terms of complexity, accuracy, and convergence time.The main contributions of this work are as follows.
1) Proposes a new OFDM frame layout in which the pilot symbols are replaced with ASK modulated informationbearing symbols, and hence it improves the spectral efficiency.The symbols with the same subcarriers' indices in the consecutive OFDM frame are modulated using PSK.2) Utilizes the ASK symbols for blind CFO estimation using the iterative VAV algorithm for both OMA and NOMA.
3) The proposed frame layout is used for blind detection of the ASK symbols, which bear the information in their amplitudes.Therefore, the ACD can be integrated because it requires only the channel gain to detect the ASK symbols, which can be extracted from the consecutive constant modulus PSK symbols.4) The initial estimates of the ASK symbols are exploited to estimate the channel gain in a decision-directed manner using least-square estimation (LSE).Therefore, the ASK and PSK symbols cooperate to detect the data symbols of each other.5) The accuracy of the initial channel estimates is improved using noise filtering (NF).6) The proposed scheme is applied to the OMA and NOMA configurations.7) The performance of the proposed CFO estimator is evaluated in terms of MSE where an accurate analytical expression is derived.Furthermore, closed-form expressions are derived for the SER for the ASK symbols.8) The obtained analytical results, verified by Monte-Carlo simulations, demonstrate that the performance of the proposed approach is comparable to pilot-assisted LSE, however, with better spectral efficiency.

B. Paper Organization
The rest of the article is organized as follows.Section II discusses the OFDM system and channel models for OMA and NOMA.Section III presents CFO estimation and compensation for OMA.The channel estimation approach for OMA is proposed in Section IV followed by SER and MSE analysis.Section V presents the CFO and channel estimation for NOMA.Section VI introduces the noise reduction method.The simulation and numerical results are presented in Section VII, and the article is concluded in Section VIII.The list of acronyms is given in Appendix A, and the proofs of Lemma 1 and Theorem 2 are given in Appendices B and C, respectively.

A. OMA
In a single-input singl-output (SISO) OFDM system, a sequence of independent information symbols d = [d 0 , d 1 , . . ., d N −1 ] is transmitted by digitally modulating N subcarriers.The system designer can select a certain modulation scheme such as quadrature amplitude modulation (QAM), ASK or PSK with modulation orders M Q , M A and M P , respectively.For typical OFDM-based wireless communications, the entries of the vector d are classified into two types or sets, reference Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.symbols (I P ), which is also denoted as the pilot symbol set and the information symbols set (I Q ).Both sets are mutually exclusive, and their subcarriers within each resource block are usually preallocated and fixed.The subcarrier d k in d is considered a pilot if k ∈ I P , and an information symbol if k ∈ I Q , The information vector d is multiplied by an N -point inverse discrete Fourier transform (IDFT) matrix to generate the timedomain vector x = F H d, where F is the discrete Fourier transform (DFT) matrix.The nth sample in x, during the signaling period τ , can be expressed as, In multipath channels, the delayed versions of multipath components cause inter-symbol interference (ISI).Therefore, to mitigate ISI, cyclic prefix (CP) is created by appending the last N cp as a preamble for x to form a sequence x of size The CP length should be selected to be greater than the maximum delay-spread of channel L h , i.e., N CP ≥ L h .In this work, the link between the transmitter and receiver is modeled as a frequency-selective multipath fading channel composed of L h mutually independent multipath signals.The considered channel is quasi-static where the gains of the multipath signals h τ l , l = 0, 2,..., L h − 1, are modeled to be fixed within one OFDM symbol duration, but they change according to Jakes model [22] over consecutive OFDM symbols.Consequently, after the CP is removed, the received signal at the receiver can be written as where n is additive white Gaussian noise (AWGN) with zero mean and variance σ 2 n = E{|z| 2 } and H τ is circulant channel matrix.It's worth noting that H τ can be diagonalized by the IDFT/DFT matrices, H τ = FH τ F H , where H τ is diagonal channel matrix with diagonal elements is the channel frequency response at subcarrier k and τ is the time index.Additionally, is the normalized CFO, which is uniformly distributed over [− min , max ].C matrix denotes the time domain samples of the accumulated phase shift caused by the CFO [23], Therefore, the received signal can be represented as After that, the received sequence is multiplied by C * (ˆ ) to eliminate the CFO effect, and then a N -point DFT is performed to generate the frequency domain samples, where ˆ is the estimate of .Assuming a very accurate CFO estimation, the information symbols can be detected coherently using maximum likelihood detector (MLD) [24], given that the channel gains H τ k ∀k ∈ I P are known at the receiver side.In practical scenarios, ML detection is performed using estimated versions of the channel gains Ĥτ k , which can be achieved using one of the several existing algorithms [25], [26], [27].Pilotassisted channel estimation is the most widely adopted approach for most communications systems such as WiMAX, DVB-T2, LTE, and IEEE 802.11.Given that LSE [28] is used, the channel estimates can be obtained at the pilots positions, Then, Ĥτ k ∀k ∈ I Q is obtained using linear interpolation [29], least-square-fitting [30], or using any other appropriate interpolation scheme.

B. NOMA
At the transmitter side, the OFDM-NOMA generally follows the OMA case except that some or all of the information symbols in d are formed by superposition modulation (SM) of multiple user' symbols.In this work, we consider a downlink powerdomain NOMA system serving two users, U 1 and U 2 .Therefore, users share the K ≤ N subcarriers in each OFDM symbol.We consider K = N unless otherwise mentioned.The superposed baseband NOMA symbol can be expressed as (2) , (8) where T represents the modulated information sequence of user n, the transmit power at the base station (BS) is denoted by P T , and the power allocated for the nth user is ϕ n ∈ [0, 1] is.Without loss of generality, the transmit power P T is normalized to unity, i.e., ϕ 1 + ϕ 2 = 1.The information symbols d (n) k ∀{k, n} have unit average power, i.e., E{|d For proper detection of NOMA symbols, the power allocation factors must satisfy the following constraint [31], At the receiver, the CP is removed and DFT is used for demodulation.For the nth user, the DFT output can be represented as where {r (n) , w ), and G (n) ∈ C K×K is the nth user channel frequency response matrix, which is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and where d n is the free-space distance between the nth user and BS, ξ is the pathloss exponent, g is the ith multipath element value, and Q (n) + 1 is the total number of multipath reflections.In the NOMA design adopted throughout this article, we consider the user with a lower average channel gain, i.e., E{ Consequently, the power factors are typically assigned such that the user with a lower channel gain is allocated more power than a user with a higher channel gain, that is, β 1 > β 2 .It is worth noting that the coherence time of the channel is assumed to be larger than the OFDM symbol period.At the receiver side, after applying the DFT to separate the subcarriers, the detection process of the information symbols can be achieved using the successive interference cancellation (SIC) [32] or joint multi-user detection (JMuD) [33].It is sensible to mention that both SIC and JMuD have equal bit error rate (BER) performance, while their computational complexity and processing delay properties are different [33].More specifically, the SIC has lower complexity because it is simply an efficient implementation of the JMuD.For the delay, the SIC performs the detection sequentially while the JMuD performs it in parallel, and thus requires less time.

III. CFO ESTIMATION AND COMPENSATION FOR OMA
In this work, we propose a new frame layout to allow joint blind channel and CFO estimation, where pilot subcarriers are replaced with ASK modulated information symbols.Furthermore, in the consecutive OFDM frame, the information subcarriers at the same locations as the ASK subcarriers are PSK modulated.The new frame layout is applied to the conventional LTE OFDM frame to evaluate the proposed approach, where an ASK modulated symbol is located every six subcarriers and every two OFDM symbols.The modified frame OFDM is shown in Fig. 1.From the figure, it is noticeable that information symbols over consecutive OFDM frames can be classified into three groups; namely, ASK symbols located at the pilot locations and their subcarriers' indices belong to I A , the consecutive The ASK symbols are utilized in the iterative VAV algorithm [23] for CFO estimation, where VAV is widely considered in many wireless applications as a highly robust CFO estimator with reasonable computational and implementation complexities.Therefore, the CFO estimates over the OFDM frames at time indices τ 1 and τ 2 can be expressed as where where N A is the number of ASK symbols used for the estimation and K denotes the number of possible phase angles in the constellation.In this work, since we only consider the ASK symbols to perform the CFO estimation, then K = 1.Consequently, the computational complexity will be reduced.Because the performance of the proposed CFO estimator is influenced by the initial value of , which is uniformly distributed over [− min , max ], the CFO estimation in ( 13) can be performed iteratively to provide more accurate estimates as described in Algorithm 1.The proposed CFO estimation is also shown in Fig. 2 along with channel estimation that is presented in the following section.As can be seen from the figure, the proposed and pilot-based CFO estimators have the same structure.The only difference is that r τ 1 in the proposed scheme consists of ASK symbols, while for the pilot-based one it consists of pilot symbols.For channel estimation, the structure is different, where ACD is not required for the pilot scenario.The main advantage of using the iterative VAV algorithm is that it is robust and can provide an accurate estimate of CFO.It is worth mentioning that the observation period depends on the ASK subcarriers' locations.Therefore, the observation period equals to (τ 2 − τ 1 )T t , where τ 2 and τ 1 represent the time indices of the consecutive OFDM symbols that have ASK symbols and T t is the total OFDM symbol duration.
To evaluate the performance of the introduced CFO estimator, we calculate the variance of the estimated CFO value ˆ , After some iterations, the difference between the estimated ˆ and the actual will become approximately zero, that is, ˆ ≈ .Consequently, considering that the variance of the real and imaginary parts of Φ( ) is very small compared to the squared real mean, the variance of ˆ can be approximated as follows [34], where Φ( ) I and Φ( ) R represent the imaginary and real parts of Φ( ), respectively.The variance of ˆ can be obtained from ( 16) as follows, where σ 2 d is the variance of the data symbols, N A is the number of ASK symbols in each OFDM symbol, Γ is the signalto-noise ratio (SNR), and ρ H is the correlation factor, It should be noted that Cramer-Rao bound (CRB) for the case of unmodulated subcarriers with fixed amplitude in AWGN channels is given by [23], [34] IV. PROPOSED BLIND CHANNEL ESTIMATOR FOR OMA ACD has attracted noticeable attention [35], [36], [37], [38], [39], [40] due to its capability to bridge the gap between coherent and non-coherent detection.In ACD, the information symbols, which are modulated using ASK can be detected using only the envelope, amplitude, of the channel response |H| = α.Therefore, no channel phase information is needed for the detection process.ACD is highly flexible where the error rate performance can be traded off versus the complexity using optimal, suboptimal, or heuristic detectors.A critical superiority of ACD is that the channel envelope α can be achieved blindly if another constant-envelope symbol, such as PSK symbols, has approximately the same value of α.For OFDM, ACD with fully bind detection can be achieved by controlling the modulation type for certain subcarriers over adjacent OFDM symbols in the time or frequency domain.
For example, we may constrain the modulation types for certain subcarriers to ASK and PSK, with modulation orders M A and M P , respectively.All other subcarriers can be modulated using any modulation type and order such as QAM with modulation order M Q .The PSK symbols' envelopes are exploited to detect the ASK symbols blindly as their envelope is generally equal to the channel envelope.Then, the detected ASK information symbols are applied to a LSE to generate the initial channel estimates, as described in (7), which corresponds to the decision-directed detection process.Next, the channel gains computed for all ASK symbols are interpolated to generate the channel gains for all subcarriers so that the consecutive data frames can be extracted.Therefore, all ASK, PSK, and QAM symbols can be detected coherently using the complex channel gains achieved using the proposed blind channel estimation process.In other words, ASK and PSK symbols assist each other, cooperate, to replace the pilot symbols while being able to transfer information.As can be noted from the proposed algorithm design, it can be concluded that the computational complexity of the proposed channel estimator is equivalent to conventional pilot-assisted LSE channel estimators.The computation performed to detect the ASK symbols is needed anyway for the ACD process.
Given that the kth symbols in the time slots τ 1 and τ 2 are an ASK and PSK symbols, respectively, the DFT operations outcome for the two time slots after CFO compensation are given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Algorithm 2: Channel Estimation.1: for ∀k ∈ I P do 3: Compute dk,τ 1 using (23) Detect all ASK symbols at the pilot positions using ACD 4: Compute Ĥτ 1 k ∀k ∈ I P using (7) Use LSE to estimate the channel gain at the ASK symbols 5: end for 6: Use interpolation to find Ĥk ∀k ∈ I Q Channel estimation for all PSK and QAM symbols 7: Output: Ĥk ∀k and where d k,τ 1 and d k,τ 2 are the ASK and PSK modulated symbols with modulation orders M A and M P , respectively, and n (.) ∼ CN(0, σ 2 n ) represents the AWGN.δ = − ˆ and Ň = (N − 1)/N .Additionally, ψ τ k represents the ICI term due to CFO at time index τ , Considering N is large and based on the Central Limit Theorem, ψ τ k can be considered as a Gaussian random variable with zero mean and variance where Using the fact that H τ 1 k ≈ H τ 2 k in our system, and assuming that σ 2 n E m (•) ∀m enable the construction of the following heuristic blind detector, Without loss of generality, the transmitted energies, Once dk,τ 1 is obtained, the channel estimate at the kth subcarrier can be calculated using (7).As can be noted from the channel estimation process, the pilot symbols are replaced by information bearing ASK symbols, and the modulation scheme for the consecutive subcarrier is PSK.Therefore, the average spectral efficiency of the proposed system over the corresponding two subcarriers is 1  2 (log 2 M A + log 2 M P ).On the other hand, in conventional pilot-based frame design, the average spectral efficiency over the same two subcarriers is 1  2 log 2 M Q .Consequently, the proposed scheme may improve the spectral efficiency if In most wireless standards such as fourth generation (4G) and 5G, the values of M Q are generally adaptively selected based on channel conditions and quality of service (QoS) requirements [41], [42], and the overall system spectral efficiency is calculated by averaging the total number of bits over all subcarriers.

A. SER Analysis
To simplify the analysis, we initially evaluate the correct decision probability given that d (m) τ 1 is transmitted, which can be written as which after some manipulations can be simplified to [35], where the decision variable To find a closed-form for the SER, the CDF of ζ k,τ 1 is required.
Lemma 1: The CDF of ζ k,τ 1 is given by where ρ is the correlation factor between α 2 and can be calculated by ρ = E{α 2 τ 1 α 2 τ 2 }.Since it is assumed that the multipath components are mutually independent, then ρ can be expressed as, Additionally, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. and where are the variances of the non-central chi-squared variables |r τ 1 k | 2 and |r τ 2 k | 2 , respectively.
Proof: See Appendix B. Theorem 1: The exact closed-form expression for the probability of correct detection is given by In (32), F m and D m are expressed, respectively as and where and In (33) and (34), γ m and w m are given by, and Proof: Using Lemma 1, PC τ 1 can be calculated as Therefore, PC τ 1 can be computed by substituting ( 27) into (39), which after simplification gives Theorem 1, and then the proof is completed.

B. MSE Analysis
After obtaining the initial estimates of the transmitted ASK symbols dk,τ 1 , by performing ACD, the channel gains Hτ 1 k k ∈ I P can be computed as Hτ 1 k = r τ 1 k / dk,τ 1 .Therefore, MSE is calculates as, Theorem 2: The MSE is given by where C denotes the correct decision and C denotes the incorrect decision.
and MSE| C and MSE| C are given in ( 42) and (43) shown at the bottom of the next page, respectively.
Proof: See Appendix C.

V. CHANNEL AND CFO ESTIMATION FOR NOMA
A. NOMA-SISO 1) QAM symbols will be replaced by NOMA symbols.Any modulation orders can be used.2) ASK and PSK symbols can be assigned to both users.For the ASK, we can use the detector described in [38].
3) The PSK symbols should have a constant envelope to enable channel amplitude estimation.We can use binary PSK with phase rotation and equal power.

B. NOMA-MIMO
For the multiple-input multiple-output (MIMO) case, we can generally follow the same approach used in LTE, where only one antenna is activated during pilot transmission.However, for the proposed system, we null two adjacent resource elements as depicted in Fig. 3.

VI. CHANNEL ESTIMATION ERROR REDUCTION
After finding the initial estimates for the channel gains at the ASK subcarriers, channel gains for the rest of the subcarriers Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.can be obtained using different interpolation approaches, such as linear interpolation, least-square fitting, or spline interpolation.The estimated channel gains are then used to coherently detect the ASK, PSK, and QAM symbols.It is worth noting that these interpolation techniques do not change the values of the initial channel estimates, ∀k ∈ I P , consequently, the error rate at the ASK subcarriers after the coherent detection will be similar to the ACD.Therefore, to enhance the channel estimates at the ASK, PSK, and QAM symbols, the proposed approach in [43] is adopted.The NF algorithm is based on the maximum delayspread not exceeding the CP length, and the channel impulse response is fixed during one OFDM symbol.Hence, to get the time-domain representation of the channel at the receiver side, N -point IDFT is applied to the interpolated channel vector Hτ 1 , As a consequence of the estimation error and the Gaussian noise, hτ 1 might have N non-zero elements, therefore, to reduce the estimation error, we drop the samples that have indices larger than N cp − 1, Then, to get the fully enhanced estimated channel frequency response for coherent detection, an N -point DFT is applied,

VII. NUMERICAL AND SIMULATION RESULTS
This section evaluates the system performance numerically for a broad range of operating scenarios.The proposed framework is applied to an OFDM with a frame layout with specifications similar to the LTE downlink resource block.The DFT is set to N = 256, N cp = 18 samples, the sampling rate f s = 5.6 MHz, and the subcarrier frequency separation Δf = 15 kHz.The pilot symbols are replaced by ASK data symbols that have modulation order M A , and the successive information symbols Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.As can be noted from the figure, the theoretical and simulation results show a perfect match for all the considered values of M A .Furthermore, the figure shows that at high SNRs, the SER exhibits an error floor, which is caused by the estimation error, The SER degradation due to the imperfect CSI is inversely proportional to the modulation order because higher order modulations are more sensitive to the AWGN, which is common for all values of M A .
Figs. 5 and 6 evaluate the mobility impact on the SER of the new estimator using various SNR values for = 0 and 0.1.The figures show that in the low SNR range, the SER is mostly dominated by the AWGN.Therefore, the effect of the channel variation on the SER is insignificant.On the other hand, in the high SNR region, the channel variations caused by mobility have a substantial consequence.As anticipated, at high SNR values, as velocity increases, the SER remarkably increases.Furthermore, it can be seen from the figures that at very high velocities, v ≥ 100 km/hr, SER for different SNRs converges to the value, which is due to the error floor at high velocities caused by the channel and CFO estimation errors, which are caused  by the ICI and channel envelope difference between successive OFDM frames.Furthermore, from Fig. 6, it is perceptible that the difference between the case where = 0 and = 0.1 is generally insignificant, where at low SNRs, that is, SNR ≤ 30 dB, there is no difference between the two scenarios.At high values of SNRs, SNR ≥ 30 dB, the difference is in the range of 0.5 to 1 dB.This implies that using the iterative VAV algorithm with the proposed frame layout, as shown in Section III compensates the effect of CFO perfectly.
In Fig. 7, the performance of the presented CFO is evaluated for a different number of iterations.From the figure we can notice that, as we increase the number of iterations, we get a wider range in which can be estimated accurately.For example, for  it can be observed that for | | ≤ 0.22 the estimated ˆ matches the real ; consequently, the effect of CFO can be totally eliminated after a few iterations.
To evaluate the performance of the CFO estimation and compensation, the variance of the estimated CFO is provided in Fig. 8 at different velocities.The number of ASK symbols used is 48.In the figure, it can be seen that the performance of the CFO estimator is susceptible to velocity.However, the impact of mobility becomes significant at high SNRs where the variance exhibits an error floor.However, the estimator can still provide reliable estimates at moderate SNR and high mobility conditions.For example, at a variance of 2 × 10 −6 , the degradation between the cases of v = 0 km/hr and v = 70 km/hr is less than 2 dB, and the difference almost vanishes for all speeds when the variance is about 10 −5 .Furthermore, the performance of the proposed CFO estimator is compared to the CRB where the figure shows that the difference at high SNRs is about 4.5 dB, which demonstrates the efficiency of the proposed estimator.
Fig. 9 shows the MSE of the final channel estimates before and after NF in various mobility scenarios.The figure shows that the NF process can provide a noticeable improvement to channel estimates, where the NF can reduce the estimation error caused by channel variation over consecutive OFDM symbols.For example, for various velocities at MSE = 10 −2 , the NF can provide an improvement in performance of approximately 5 dB.Consequently, the corresponding error rate will be reduced by enhancing the final channel estimates, allowing more accurate detection of the ASK, PSK, and QAM symbols.The figure also shows the MSE of pilot-based channel estimation using LSE.The figure shows that the pilot-based estimation outperforms the blind approach by about 3 dB at MSE of 10 −2 .The observed degradation is mostly due to the detection errors of the ASK symbols, which are used in the channel estimation process.Nevertheless, the accuracy obtained is sufficient to provide a reliable coherent detection.The impact of channel estimation errors is shown in Fig. 10.It can be seen from the figure that the NF managed to effectively improve the channel estimation errors caused by the mobile channel where the SER for all velocity values are approximately equal for SER for SER 25 dB.
Fig. 11 compares the BER of the proposed system under various spectral efficiencies for the SISO frame.In Fig. 11(a), log 2 M A + log 2 M P = 3, while log 2 M Q = 2. Therefore, the proposed system provides an average 1.5 bits for the two considered subcarriers while the pilot-based system provides an average of 1 b per subcarrier.However, the BER of the  proposed system has a degradation of about 2 dB at BER of 10 −3 .Therefore, a certain cost is associated with the achieved spectral efficiency improvement.Fig. 11(b) shows the BER for the case of equal spectral efficiency of 2 bits per subcarrier.For this scenario, the proposed scheme managed to provide a SNR advantage of about 2.5 dB.It should be highlighted that the overall spectral efficiency should consider the data symbols over all subcarriers, and hence it also depends on the density of pilot symbols in the frame structure.In addition, blind channel estimation schemes may vary depending on SNR and other operating conditions [42].
Fig. 12 shows the BER of the NOMA users (U 1 and U 2 ).The results are generated for M (1) A = 2 using several values of v. Here, the power allocation factors for users are assumed to be ϕ 1 = 0.6 and ϕ 2 = 0.4, and the path loss exponent is given as ξ = 4.It is noticed that the error performance for both users is different due to the interference between them.The user U 1 has the worst BER because it suffers from interference from U 2 .
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
On the other hand, U 2 achieves the best BER because it does not suffer from any interference.
13 the impact of mobility on the BER performance of the proposed CFO estimator.It can be observed that mobility remarkably affects the BER of both users.As the velocity increases, the BER performance goes worse.In addition, it converges to close values because of the error floor at high velocities.
Fig. 14 plots SER for the NOMA users versus SNR with different velocity values.The error rate for both users deteriorates as the velocity increases and converges to close values at high speeds.

VIII. CONCLUSION AND FUTURE WORK
This work introduced an efficient blind joint channel and CFO estimation scheme based on a hybrid frame layout where the information subcarriers are modulated using ASK, PSK, and QAM modulation schemes.The ASK and PSK symbols may belong to a single user, as in the case of OMA, or multiuser as in the case of NOMA.These modulation schemes collaborate to estimate and correct the CFO and then detect the ASK symbols blindly.The detected symbols are then used to obtain the channel estimates in a decision-directed fashion.Closed-form expressions for the SER and MSE were derived to evaluate the performance of the proposed approach for OMA while Monte Carlo simulation is used for NOMA since the analyses for both systems are generally similar.Analytical results were validated by Monte Carlo simulation for different scenarios.The simulation and analytical results showed that the proposed approach can provide a robust and accurate joint estimation.The complexity of the proposed estimator is comparable to the LSE using pilot symbols.In addition, the NF method is introduced to improve the final estimates of the channels.The simulation results showed a significant improvement when using the NF method.
In future work, the spectral efficiency of the proposed system will be evaluated with adaptive bit loading to holistically evaluate the potential of the proposed system in improving the spectral efficiency.

APPENDIX B PROOF OF LEMMA 1
The CDF of ζ can be computed as follows, Nevertheless, if r τ 2 k ω τ 2 is considered as a constant, the conditional CDF can be written as Moreover, by conditioning on α τ 1 , then Hence, the conditional CDF of ω τ 1 is given by [44] where Q(a, b) is the Marcum Q-function of first order [45].Accordingly, the CDF of ζ τ 1 is . Therefore, the PDF of ω τ 2 is given by [44] Substituting ( 49) and (51) in (50), and using change of variables, x = √ ω τ 2 , the integral in (50) becomes, The integral in (52) can be solved using [46, eq. 46], which gives, The conditioning on α τ 1 and α τ 2 can be removed by averaging over their joint PDF.Because E{|H H , the joint PDF can be written as [44], Therefore, the CDF of ζ τ 1 can be evaluated as follows Solving the integral in (55) will result in Lemma 1, and the proof is completed.

APPENDIX C PROOF OF THEOREM 2
In (40), dk,τ 1 is random, and hence, the MSE can be written as which can be expressed as, Expanding the first part of (57) yields to, where k and AWGN n τ 1 k , respectively.Therefore, to find ) of the sampled noise should be derived.Using Bayes' rule for mixed distributions, the desired conditional PDF can be written as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
It is worth mentioning that E{|n τ 1 k | C τ 1 | 2 } < E{|n τ 1 k | 2 } because the noise samples with high amplitudes are dropped by the conditioning process.At high SNRs, the effect of n τ 1 on P C τ 1 becomes negligible and hence E{|n Considering the case where dk,τ 1 = d k,τ 1 , the MSE can be expressed as shown in (43), where E m τ 1 = Ẽm τ 1 .The conditional channel variance σ 2 By using Bayes rule we note that and thus, Evaluating the double integrals in (64) gives where A 1 is given in (66) shown at the top of this page, and and 2 F 1 (•, •; •; •) is the Gaussian hypergeometric function [47].
And finally, the conditioning on E m τ 1 and Ẽm τ 1 can be eliminated by averaging over their joint probabilities, which gives Theorem 2, and then the proof is completed.

P
r (•) symbolizes probability; C m×n denotes the set of matrices with dimension m × n.F X (x) and f X (x) denote the cumulative distribution function (CDF) and probability density function (PDF) of a random variable X, respectively; (•) T and (•) H denote the transpose and Hermitian transpose operations, respectively.E{•} and V{•} represent the expectation and variance, respectively, and M{•} is the MSE, and μ = m + 1.Λ m τ and Δ m τ denote E m τ + E m+1 τ and E m−1 τ + E m τ , respectively.| • | represents the absolute value; CN(μ, σ 2 ) stands for the complex Gaussian distribution with mean μ and variance σ 2 .Q(•) and Q(•, •) denote the Q-function and the Marcum Q-function of first order, respectively.2 F 1 (•, •; •; •), J 0 (•), and I 0 (•) are the Gaussian hypergeometric function, the Bessel function of the first kind, and zero-order modified Bessel function of the first kind, respectively.χ 2 (•, • , •) represents the chi-squared distribution.

τ 1 and n τ 1 k
| C τ 1 are the channel frequency response, CFO and AWGN given that the detection is correct.Therefore, ψ τ 1 k | C τ 1 and n τ 1 k | C τ 1 are sampled versions of the CFO ψ τ 1