- 1Department of Electronic and Electrical Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria
- 2School of Electronic Engineering, Dublin City University, Dublin, Ireland
- 3School of Computing, Engineering, and Built Environment, Glasgow Caledonian University, Glasgow, United Kingdom
The non-orthogonal multiple access (NOMA) scheme has been recognized as a promising candidate for future generation wireless communication networks that require a high data rate and spectral efficiency. In order to achieve a superior spectral efficiency, the orthogonal frequency division multiplexing (OFDM) technique with the robust channel estimation algorithm can be combined with the NOMA scheme. However, due to the inherent problem of inter-carrier interference in the OFDM-based systems, channel estimation using training symbols are not usually perfect. Furthermore, the successive interference cancellation (SIC) process commonly carried out in NOMA systems to address the interference among users is also imperfect under practical scenarios. Thus, a theoretical framework of the OFDM-based NOMA system to analyze the effect of both the imperfect SIC and channel estimation errors is presented in this paper. In particular, the effect of both the channel estimation errors and imperfect SIC, introduced at the receiver, on the bit error rate (BER) performance metric is analyzed under Rayleigh fading channel conditions. Our analytical results are compared to the corresponding simulation results for different design parameters. It is shown that the analytical and simulation results are in good agreement. Moreover, while revealing the importance of CSI and SIC, the results equally show that the assumption of perfect CSI and SIC is not practical, hence the need for accurate schemes for improved BER performance.
1 Introduction
Future wireless networks are expected to support higher data rates, very large network capacity, high spectral efficiency, and more mobility. The non-orthogonal multiple access (NOMA) scheme is seen as a very viable and attractive scheme that can effectively meet the demands of future radio access networks and applications. Earlier multiple access schemes have focused on the orthogonal techniques (Gilhousen et al., 1991; Brannstrom et al., 2002; Liu et al., 2006; Wang et al., 2006; Liu et al., 2017). Although orthogonal techniques ensure the avoidance of intra-cell interference, the techniques usually involve stringent requirements. Orthogonal techniques also require a sufficient distance between re-used channels, which results in a reduced cellular spectral efficiency. On the other hand, the NOMA scheme ensures a high spectral efficiency and mass connectivity. The NOMA scheme is able to accommodate concurrent users through non-orthogonal resource allocation (Liu et al., 2019; Jain et al., 2020; Li et al., 2019; Bariah et al., 2020). Although this comes with an increased inter-cell interference (ICI), the successive interference cancellation (SIC) technique (Liu et al., 2017; Liu et al., 2019) is exploited to mitigate the ICI at the receiver.
In order to achieve a higher spectral efficiency and a better throughput, the NOMA scheme can be combined with the orthogonal frequency division-multiplexing (OFDM) scheme (Li et al., 2016; Moose, 1994; Chen, 2002; Morelli and Mengali, 2000; Hilario-Tacuri et al., 2021). The combination of the NOMA and the OFDM scheme, however, comes with challenges such as the peak-to-average power ratio (PAPR) and channel estimation problems (Mohsan et al., 2023; Kay, 1995). In Liu et al. (2019); Jain et al. (2020), the BER analysis of NOMA-enabled visible light communication systems was studied. Furthermore, the study in Li et al. (2019) provided a comprehensive performance analysis for spatial modulation-aided cooperative NOMA, while Bariah et al. (2020) carried out a study on error performance of NOMA-based systems with power constraints. In Hilario-Tacuri et al. (2021), authors focused on deriving the analytical expression of BER for the downlink NOMA-OFDM system in the presence of a high-power amplifier (HPA) with memory. Nevertheless, it is commonly assumed in the literature (Liu et al., 2019; Jain et al., 2020; Bariah et al., 2020; Hilario-Tacuri et al., 2021) that the successive interference cancellation method commonly carried out in the NOMA system to address the interference among users is perfect. However, this is not always true in practice, and the inaccurate interference assessment of the multiple users using SIC detection can lead to more prominent errors and affect the bit error rate (BER) performance of the system. So far in the literature, not so much has been done to mitigate the effect of the inaccurate interference assessment and to achieve some improvement in providing more accurate reconstruction in using SIC detection. In Ouaissa et al. (2022), the authors considered mitigating the effect of imperfection in NOMA by introducing a cancellation error in the receiver, while Mohsan et al. (2023) proposed a recurrent neural network-based guided frequency interference coefficient estimation algorithm in a NOMA visible light communication (VLC) system.
This work focuses on the theoretical BER analysis by derivations of closed-form analytical expressions to analyze the performance of the OFDM-based NOMA system. Unlike Li et al. (2016); Hilario-Tacuri et al. (2021) who investigated a downlink NOMA-OFDM system, our paper considers an uplink system. The consideration for the uplink introduces more challenges which include the need for accurate CSI for both users and potential error propagation on the weaker signal due to imperfect SIC. Consequent upon this, we studied a system that is plagued with channel estimation or channel state information (CSI) errors and successive interference cancellation (SIC) errors. To date, the BER derivation and analysis for an OFDM-based NOMA system with both channel estimation and SIC errors have not been comprehensively addressed in the literature. This gap is filled by deriving and analyzing the closed-form BER expressions using binary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK), and 16-QAM modulation schemes in a multipath-fading channel. Notably, the approach employed for the BER derivations and analysis in this work can be easily extended to other modulation sizes. The derived BER expressions are in a simple form that requires no numerical integration, thus providing a practical and efficient tool for performance evaluation. Table 1 outlines the main differences between our work and the previous studies on the BER of the NOMA system.
Moreover, the results obtained in this study can significantly benefit research involving machine learning (ML) techniques. The closed-form BER expressions and insights into the impact of channel estimation and SIC errors can serve as a valuable benchmark for training and validating ML models. These models can be designed to optimize power allocation, improve the channel estimation accuracy, and mitigate the effects of SIC errors in OFDM-based NOMA systems. By integrating the analytical results from this work, future research direction can enhance the robustness and efficiency of ML-driven solutions, fostering advancements in adaptive and intelligent communication systems. By bridging the gap between theoretical analysis and practical implementation, this work not only advances the understanding of OFDM-based NOMA systems but also lays the groundwork for future ML-driven optimization in fifth generation (5G) and beyond wireless communications.
The main contributions in this paper, therefore, include the following:
1. The closed-form bit error expressions for OFDM-based NOMA systems are derived using BPSK, QPSK, and 16-QAM modulation schemes. This is different from the works in Li et al. (2016); Hilario-Tacuri et al. (2021). While Li et al. (2016) merely present the BER results obtained using computation and simulation with the assumption of perfect SIC and without consideration for channel estimation, the authors in Hilario-Tacuri et al. (2021),, although analytically investigated the BER performance of the OFDM-based NOMA system, did not consider channel estimation, and the work equally assumes a perfect SIC at the receiver in the analysis.
2. The impact of channel estimation errors on the BER performance of OFDM-based NOMA systems is investigated. In addition, this work theoretically shows the importance of power allocation and channel estimation in OFDM-based NOMA systems.
3. The effect of SIC error on the BER performance of OFDM-based NOMA systems is investigated. The effect of the SIC error is practically considered during the modeling and derivation of the closed-form BER expressions for the various modulation schemes considered.
The rest of this paper is organized as follows: Section 2 presents the NOMA-OFDM system model. Section 3 presents the closed-form BER derivations and analysis. Section 4 presents the simulation results, while Section 5 presents the conclusion.
2 The NOMA-OFDM system model
An uplink OFDM-based NOMA system with
For ease of notation, the superscript
where in Equation 2, the parameter
where
which can be, otherwise, rewritten as Equation 4b
where
3 The BER analysis for NOMA-OFDM
In this section, the OFDM-based NOMA system is analyzed to consider the effect of both the imperfect SIC and channel estimation errors on the bit error performance metric.
3.1 BPSK derivation
The BER expressions for an uplink OFDM-based NOMA system is derived and analyzed in this section. The BER expression is derived using BPSK signaling in a multipath fading channel. It is assumed that BS has the knowledge of the relative distance of the users in the OFDM-based NOMA system. Thus, the user with a shorter relative distance to the BS is taken as the stronger user, while the farther user is taken as the weaker user. For the purpose of this work,
and the covariance of
Thus, from Equation 6, a more compact expression can be obtained as in Equation 7
where
Therefore, the received data symbol after FFT can be expressed as in Equation 8
where
3.1.1 Impact of SIC and channel estimation errors
For BPSK modulation, the impact of SIC imperfection and channel estimation errors on the error probability can be expressed in Equation 9 as (Savaux et al., 2016)
In Kara and Kaya (2018), to obtain the error probability, a joint distribution for the sum of the independent non-identical Rayleigh channel distributions is obtained. Furthermore, the probability density function for the difference of two independent non-identical Rayleigh distributions is derived. These joint distributions are then used to obtain the bit error probability of the NOMA system. However, a much simpler method is to obtain the error probability using the approach in Proakis (2001), which can be derived in such a way that numerical integration is not required. Hence, the derivation of
where
The average bit error can be expressed in Equation 17 as
Since the SIC is only performed on user 2, it is assumed that there is no SIC error on user 1. Therefore, the impact of CSI errors only on the BER performance of user 1 can be considered by rewriting
such that Equation 10 can be written as Equation 21:
where
If we assume
Now for the second user, the received data symbol before cancellation can be expressed in Equation 23 as
After cancellation, we can obtain Equation 24 and, subsequently Equation 25 as follows:
where
The expression for channel estimation is obtained in Equation 26 as
where
To obtain
Therefore,
where Equations 29, 31, and 34 are obtained from Equations 28, 30, and 33 respectively. In a similar fashion to Equation 10, the probability of bit error on user 2 can be written as in Equation 35:
The average bit error can be expressed in Equation 36 as
The impact of SIC errors and the CSI errors on the BER performance can be considered under the following two scenarios as
Under the scenario of imperfect CSI but a perfect interference assessment at the receiver,
such that Equation 35 can be written as Equation 40:
where
Analogous to Equations 37–39, the effect of the imperfection on SIC only (while it is assumed that there is perfect channel estimation) on the parameters
where the expression in Equation 35 can now be written as in Equation 44:
The expression in Equation 44 shows that error probability increases with increasing imperfection in the interference assessment by the SIC at the receiver. As the component of the SIC, i.e.,
However, for a system with perfect channel estimation and a perfect interference assessment at the receiver, the expressions in Equations 40, 44 reduce to Equation 45 as
If we assume
3.2 QPSK derivation
The closed-form BER expressions for the BPSK-modulated OFDM-based NOMA have been derived in Section 3.1. Here, the BER expression for an OFDM-based NOMA, using QPSK signaling, is considered. Generally, the constellation of QPSK can be expressed mathematically as in Equation 46
As shown in Figure 1, using Gray encoding, two information bits are mapped into a QPSK constellation symbol. In a scenario where perfect channel estimation is assumed, one constellation symbol can be sent as a result of the symmetry of the constellations and decision boundary. However, when considering an OFDM-based NOMA system with channel estimation errors, the constellations of the demodulated signals are scaled and rotated. Therefore, in this case, two constellation symbols are sent and the BER of the most significant bit (MSB) and the BER of the least significant bit (LSB) are then computed. Now, for the first user, considering two constellation symbols
For the BER of the LSB for the first user, the decision boundary is the imaginary axis. Thus, considering two constellation symbols
The average bit error can then be expressed in Equation 49 as
The expressions in Equations 47, 48 are computed using Lemma 1 (See Supplementary Appendix S1), while
respectively, while
3.3 16-QAM derivation
Here, the 16-QAM constellation, which is generally denoted by
Figure 2. 16-QAM constellation with Gray encoding: (A) 16-QAM constellation. (B) 16-QAM constellation (bit-by-bit demapping).
Next is to obtain the decision boundary for the LSB bit, which are
where
Hence,
Now, based on Equation 55, the LSB can be obtained using Lemma I. Therefore,
while
Therefore,
Then, the average BER for the 16-QAM constellation can be expressed in Equation 59 as
Now, to obtain the BER for the second user, the error probability of the MSB bit of the I-components can be expressed in Equation 60 as
In order to obtain the error probability of the LSB for the second user, a new variable
Let
Thus,
Based on Equation 63,
while
4 Numerical results
In this section, the BER performances of the different modulation schemes considered are evaluated using the expressions derived in Section 3. The results are validated using simulations. The NOMA-OFDM system, in the presence of SIC and channel estimation errors, is implemented with
4.1 Impact of the modulation order on BER performance
Figures 3, 4 illustrate the BER performances of the NOMA-OFDM system for user 1 and user 2, respectively, for BPSK and QPSK modulation schemes after channel estimation and SIC. The solid lines are obtained from simulations, while the dotted lines represent the plots from the theoretical computation. From these results, the following can be observed:
4.2 Impact of channel estimation errors on BER performance
In Figures 5, 6, the BER versus transmit SNR
Figures 7, 8 show the relevance of CSI and the impact of CSI errors on the BER performance of user 1 for QPSK and 16-QAM constellations. The plots show the performance when the channel is not estimated and the performance after channel estimation. These figures, therefore, illustrate the significance of channel estimation and equally reveal the need for accurate CSI for improved BER performance. Although performance shows marked improvement after channel estimation, nevertheless, these results show that the assumption of perfect CSI is not practical as it can easily be seen that the plot for the estimated channel does not agree with the plot for perfect CSI. In Figure 12, the BER performance of user 2 is shown. The result compares the system performance under three scenarios:
Figure 7. Impact of the channel estimation error on the BER of QPSK-modulated NOMA-OFDM signals for user 1.
Figure 8. Impact of the channel estimation error on the BER of 16-QAM-modulated NOMA-OFDM signals for user 1.
4.3 Impact of SIC imperfection on BER performance
Figures 9, 10 show the impact of SIC imperfections together with CSI errors on the BER performance of user 2 for QPSK and 16-QAM constellations. These figures illustrate the degrading impact of SIC imperfection on the BER performance. In particular, when compared with the plots in Figures 7, 8, the impact of the SIC error becomes more obvious. In Figures 11, 12, the BER performance of user 2 is as shown. As earlier stated, the result compares the system performance under three scenarios:
Figure 9. BER sensitivity of QPSK-modulated NOMA-OFDM signals in the presence of SIC and channel estimation errors for user 2.
Figure 10. BER sensitivity of 16-QAM-modulated NOMA-OFDM signals in the presence of SIC and channel estimation errors for user 2.
Figure 13 compares the SIC scheme in Im and Lee (2019), where the interference term is modeled as a Gaussian random variable
4.4 Comparative analysis of the impact of CSI errors and SIC errors on BER performance
Revisiting Figure 12, the result further compares the impact of channel estimation, channel estimation errors, and SIC errors on the BER performance. The plot reveals that the impact of channel estimation errors is seen to be more significant that the impact of the SIC errors. This is due to the fundamental performance limit introduced by the CSI error, regardless of the SNR level stated as discussed. On the other hand, SIC errors decrease with increasing SNR. It is noteworthy to know that the worse-case scenario occurs when there is SIC imperfection and channel estimation is not carried out, while the best-case scenario happens with perfect channel estimation and perfect SIC, highlighting the critical role of channel estimation and successive interference cancellation in the NOMA-OFDM system.
5 Conclusion
In this paper, closed-form BER expressions for an uplink OFDM-based NOMA have been derived and analyzed. The BER expressions are derived while considering the degrading effects of imperfect SIC and channel estimation errors. The closed-form expressions are initially derived using BPSK signaling. Then, BER expressions for QPSK and 16-QAM constellations are derived and analyzed. Obtained results show agreement between the analytical and simulated results. It is also noteworthy to know that the impact of the SIC error on the overall performance wanes in higher SNR regions, while the influence of the SIC error is better characterized by the proposed model, as against the existing model in the literature. Moreover, it is equally important to note that CSI errors cause more degradation on the BER performance than the SIC imperfections particularly, at a high SNR level due to the fundamental performance limit introduced by the channel estimation errors. The obtained results while revealing the importance of CSI and SIC equally show that the assumption of perfect CSI and SIC is not practical, hence the need for accurate schemes for improved BER performance. Therefore, this work can provide insights into a general behavior of NOMA-OFDM systems in the presence of SIC imperfection and the channel estimation error with a view to understanding the system’s performance boundaries and limitations. The BER derivation approach utilized in this work can be extended easily for other high-order modulation schemes. Furthermore, the BER expressions obtained, which are dependent on the training symbols, are in simple forms that require no numerical integration.
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
AO: writing–review and editing, writing–original draft, and conceptualization. MB: writing–review and editing, writing–original draft, software, and conceptualization. FO: writing–review and editing and writing–original draft. EO: writing–review and editing and writing–original draft. AF: writing–review and editing and writing–original draft.
Funding
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
Acknowledgments
The authors would like to thank the editor and the reviewers for their constructive comments and suggestions, which improve the quality of this paper.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frcmn.2024.1472624/full#supplementary-material
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Keywords: OFDM-based NOMA, orthogonal frequency division multiplexing, non-orthogonal multi-access, bit error rate, successive interference cancelation, channel estimation
Citation: Olawole AA, Balogun MB, Offiong FB, Obayiuwana E and Fisusi A (2024) Average BER evaluation of the uplink OFDM-based NOMA system under SIC and channel estimation error. Front. Comms. Net 5:1472624. doi: 10.3389/frcmn.2024.1472624
Received: 29 July 2024; Accepted: 23 October 2024;
Published: 07 November 2024.
Edited by:
Qingchun Chen, Guangzhou University, ChinaReviewed by:
Hongjiang Lei, Chongqing University of Posts and Telecommunications, ChinaMilica Pejanovic-Djurisic, University of Montenegro, Montenegro
Copyright © 2024 Olawole, Balogun, Offiong, Obayiuwana and Fisusi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Akinbode A. Olawole, YWxleF9vbGF3b2xlQG9hdWlmZS5lZHUubmc=