Single carrier frequency division multiple access (SC-FDMA) continued to be the main multi-access technique adopted in the uplink of both 4G and 5G communication systems. SC-FDMA systems not only enjoys the same benefits of orthogonal frequency division multiplexing (OFDM) systems such as spectral efficiency and resistance to inter-symbol interference and multipath fading, it outperforms OFDM systems in terms of PAPR performance due to the discrete Fourier transform (DFT) precoding process that precedes the conventional OFDM modulation process. 

As the modern communication systems require more flexible modulation schemes that offers better tradeoff between bit rate and bandwidth, odd-bit cross QAM integration into adaptive QAM modulation had been widely suggested to support higher bit rates for an acceptable level of power and bandwidth. Therefore, studying the BER performance of cross odd-bit M-QAM schemes in communication systems under practical impairments such as additive white Gaussian noise and phase noise is crucial to assess the viability of their adoption despite their added complexity and implementation cost.

Many articles and researches studied and analyzed the effect of AWGN and phase noise on the BER performance of communication systems. In [1], the effects of phase noise and AWGN on high-order rectangular QAM systems were studied, with a focus on decision-directed carrier recovery. The authors derived a generalized BER expression for QAM under phase noise and proposed optimized decision regions to improve detection in the presence of frequency offsets. A closed-form BER expression was derived for interleaved SC-FDMA systems using BPSK and square M-QAM over fading channels in [2]. The analysis offers a mathematical foundation for BER estimation, with results confirmed through simulation. In [3], on the other hand, an exact closed-form SER expression was developed for square and rectangular QAM under AWGN with phase noise. The analysis captures RF impairments from local oscillators and models symbol error using a finite summation of 2D Q-functions, validated through simulation.

In [4], authors proposed a blind iterative method for compensating phase noise in SC-FDMA systems without using pilot signals or assuming a specific phase noise model. The approach improves detection accuracy with minimal iteration, showing strong performance in LTE uplink scenarios. A practical approach to modeling, estimating, and mitigating phase noise in SC-FDMA systems was presented in [5], with a focus on LTE uplink. The authors proposed a realistic phase noise model and developed an estimation algorithm tested under multipath fading and different oscillator types.

In [6], the impact of receiver phase noise on SC-FDMA was analyzed, showing that common phase error causes symbol rotation while higher frequency components induce inter-carrier interference (ICI). The study highlights that linear receiver performance degrades more in interleaved SC-FDMA than in localized schemes due to phase noise interacting with multi-user interference. While in [7], a new analytical expression for NMSE in asynchronous SC-FDMA was derived considering CFO and joint transmit-receive phase noise. The work revealed how subcarrier mapping affects robustness and introduced a pilot-free, iterative phase noise compensation method with low complexity, showing strong simulation performance.

In [8], a joint frequency-domain equalization and phase noise estimation method was proposed for single-carrier systems over highly time-dispersive channels. The approach uses iterative equalization combined with recursive filtering to track and compensate phase noise with minimum mean-squared error. SC-FDMA with rotated constellations was analyzed in [9] under phase noise and channel estimation errors. The study showed that this approach maintains its diversity and coding gains even under impairments, and can outperform OFDM in high-rate coded scenarios. 

Authors in [10] proposed a phase noise pre-correction method for SC-FDMA transmitters, using an adaptive polynomial fitting based prediction algorithm. The scheme pre-compensates for phase noise before transmission, offering low hardware complexity and showing strong performance improvements in simulations. On the other hand, the impact of phase noise on SC-FDMA-based massive MIMO uplink systems was analyzed in [11], considering both synchronous and non-synchronous oscillator scenarios. 

A new tracking reference signal (TRS) designs were proposed by authors to compensate phase noise in SC-FDMA waveforms, targeting 5G uplink scenarios. The work addresses the PAPR increase from traditional TRS methods and introduces FDM-based TRS patterns that effectively reduce phase noise impact with minimal PAPR growth. For [13], authors developed an SNR analysis and estimation method to mitigate phase noise in millimeter-wave SC-FDE systems. The proposed algorithm improves MMSE equalization performance without relying on iterative decoding. 

In [14], an iterative method for joint mitigation of both phase noise and deep fading in SC-FDMA systems was proposed. The approach leverages the characteristics of phase noise in both time and frequency domains within a frequency-domain equalization framework, showing strong performance in simulations. Whereas in [15], the SER performance of SC-FDMA was evaluated under various modulation schemes including BPSK, QPSK, and M-QAM. The study compared SC-FDMA to OFDM and showed that increased spreading factors reduce SER. Among mappings, localized SC-FDMA outperformed distributed mapping in terms of reliability.

Finally, in [16], an empirical phase noise model for OFDM systems was developed using real-world data. Analytical expressions for common phase error and ICI statistics were derived, with a compact covariance matrix representation. A Gibbs sampler using this model showed improved symbol estimation over traditional Wiener-based methods. 

In summary, even though those previous studies mentioned earlier had extensively analyzed SC-FDMA performance under phase noise, AWGN, and various QAM rectangular, most of them have focused on either standard modulation formats, simplified system configurations, or separate handling of impairments. Joint evaluation of BER performance for odd-bit cross-QAM in SC-FDMA systems, under combined phase noise and AWGN, caught no research interest. This work aims to fill that gap by providing a focused investigation on the BER performance of SC-FDMA system in the simultaneous presence of AWGN and phase noise while adopting odd-bit cross QAM modulation.

SC-FDMA System Model

Single-Carrier Frequency Division Multiple Access (SC-FDMA) is a transmission scheme adopted in many modern standards uplink like LTE, LTE-Advanced and 5G-NR due to its DFT spreading structure that enhances its Peak-to-Average Power Ratio (PAPR) performance compared to the conventional OFDM [17], [18]. While SC-FDMA shares many structural elements with OFDM, SC-FDMA inserts an additional DFT operation before subcarrier mapping, making it well-suited for power-sensitive uplink transmissions. Figure (1) shows the block diagram of SC-FDMA transceiver [19].

As shown in Figure (1), the data bits are grouped into K bits groups depending on the size of the QAM modulation and converted into QAM symbols vector (S). After converting the serial stream of symbols into M parallel streams, discrete Fourier transform (DFT) process of size (M) is applied to the parallel symbol streams using FFT block, this operation introduces correlation between symbols by spreading them across the assigned bandwidth giving the SC-FDMA its superior PAPR performance [20]. DFT precoding process of data symbols is expressed in equation 1.

The M-DFT output is mapped into N subcarriers using  either localized mapping method or interleaved

Figure 1: SC-FDMA System Model

Figure 2: Rectangular 32-QAM Constellation

Figure 3: Cross 32-QAM Constellation

Figure 4: BER vs Eb/No Curves at PhN

Figure 5: BER vs Eb/No Curves at PhN

Figure 6: BER vs Eb/No Curves at PhN

mapping method [17]. Pilot signals for channel estimation, phase noise tracking, and synchronization are also added at reserved subcarriers. The subcarrier mapping can generally be expressed as follows:

Figure 7: BER vs Eb/No Curves at PhN

Figure 8: BER vs Eb/No Curves at PhN

Table 1: Simulation Parameters

Table 2: Log. BER Gain for k Odd QAM vs k+1 Square QAM at PhN

Table 3: Log. BER Gain for k Odd QAM vs k+1 Square QAM at PhN

Table 4: Log. BER Gain for k Odd QAM vs k+1 Square QAM at PhN

Table 5: Log. BER Gain for k Odd QAM vs k+1 Square QAM at PhN

Table 6: Log. BER Gain for k Odd QAM vs k+1 Square QAM at PhN

Finally, N-point inverse DFT process is performed using IFFT block. This converts the frequency domain signal into time domain signal creating the transmitted SC-FDMA signal. A Cyclic prefix (CP) of length (L), which is the replication of SC-FDMA symbol tail, is used to prepend the SC-FDMA symbol in order to combat inter-symbol interference (ISI) [17]. IDFT process and cyclic prefix insertion are expressed in the following equations:

After CP insertion and transmission, the transmitted SC-FDMA signal passes through a fading channel h(n) and be corrupted by additive white gaussian noise and phase noise. The received signal is expressed as follows.

At the receiver side, the processes performed in transmission are essentially reversed. the cyclic prefix is removed and N-point FFT is applied to restore the frequency domain representation of the received subcarriers. After FFT, Original subcarriers of the user are extracted by subcarrier demapping while discarding the rest. Equalization and phase noise recovery are applied to mitigate the combined distortion of channel effects, additive white gaussian noise, and phase noise [21,22]. 

Finally, M-point DFT is applied to retrieve the original data symbols and QAM demodulation is performed to convert them back into bits.

Odd-bit QAM

In digital communication systems, even-bit QAM schemes like 16-QAM, 64-QAM, and 256-QAM are typically used due to their square constellation. they allow easy bit-symbol mapping and demapping compared to the odd-bit non-square QAM schemes [23]. However, odd-bit QAM schemes are increasingly needed to achieve specific spectral efficiencies like 5, 7, or 9 bits/symbols to enhance the performance of adaptive QAM.

The direct and easy way to support odd-bit QAM schemes is to use rectangular QAM constellations like 32-QAM (5 bits/symbol) shown in Figure 2. However, the uneven distribution of constellation points causes unbalanced vulnerability to noise between them worsening the BER performance. Moreover, uneven energy distribution lowers the power efficiency and degrades the PAPR performance [20].

On the other hand, cross QAM constellations provides an improved approach by balancing the symbol energy and minimum Euclidean distance while keeping a compact constellation. A cross 32-QAM shown in Figure 3 for example, is designed in a way that avoids placing constellation points far from origin by trimming the unused corner points forming a cross-constellation shape [24].

The enhanced distribution of constellation points in cross-QAM gives it a superior energy efficiency and consequently improved BER and PAPR performance. The mathematical proof written below proves how a cross 32-QAM outperforms a rectangular 32-QAM in terms of energy efficiency.

Assuming that constellation point of an M-QAM scheme is represented by the in-phase component (x) and quadrature component (y). the average energy is generally determined as follows:

By applying equation 6 to compute the average energy of both rectangular 32-QAM (32-RQAM) and the cross 32-QAM (32-CQAM) constellations shown in Figures 2 and 3, we find

E_{av (30-RQAM)} = 26

E_{av (30-CQAM)} = 20

From the calculations of average energy of both schemes and the energy penalty (∆E), we find that 32-RQAM requires a 1.14 dB more energy than 32-CQAM for the same Euclidean distance. This proves that odd-bit cross QAM constellations generally outperforms odd-bit rectangular QAM schemes in terms of energy efficiency [24].

AWGN and Phase Noise Models

In practice, transmitted signals are subject to a variety of impairments as they propagate through the channel. Two main sources of distortion are Additive White Gaussian Noise (AWGN) and Phase Noise (PN). Since these impairments can severely degrade system performance, accurate models are needed to study and mitigate their adverse effects [25].

AWGN channel model assumes that a noise term, which has a flat power spectral density, is linearly added to the signal. This noise term follows a Gaussian distribution with zero mean and variance (σ²) that controls the noise power. The received signal under AWGN model can be expressed as follows:   

r_k = s_k+n_k…

(7)

where, r_k is the received signal, s_k is the transmitted signal, and is the noise term [25].

Phase Noise, on the other hand, arises due to imperfections in the local oscillators of transmitters and receivers. These imperfections cause the carrier phase to drift over time, leading to random rotations in the constellation of the signal. Constellation rotation may cause Common Phase Error (CPE) and Inter-Carrier Interference (ICI) [26,27].

Unlike AWGN, Phase noise is commonly modeled as a multiplicative distortion. The signal being affected by both AWGN and phase noise is expressed as follows: 

Where, ∅_k represents the random phase noise sample at sample k.

One of the widely accepted models of phase noise is the Wiener process model. It models the phase noise sample as follows:     

∅_k = ∅_k-1+Δ_k …

(9)

Where Δ_k is phase deviation step. It is an independent and identically random variable that follows a gaussian distribution of N (0,σ_Δ²). The power spectral density of a Wiener process modeled phase noise is approximated to a Lorentzian shape since it is suitable for free-running oscillators [27,28,29].

Simulation Framework and Parameters

The simulation of M-QAM SC-FDMA system under the joint presence of AWGN and phase noise was carried 

out using MATLAB program. The simulation parameters are listed in Table 1.

In order to obtain the BER performance of the system, the following steps were performed.

• Data bits were generated randomly and transmitted via the system under the joint presence of AWGN and phase noise in form of 10000 packets, each packet contains (M×k) bits. Where M is the number of the allocated subcarriers for the user and k represents the number of QAM modulation order in bits/symbol. The packet size is made dependent on the modulation order (k) to make the symbol rate constant across all modulation sizes. This is crucial to ensure fair comparison between schemes of different constellation sizes. 
• After reception, the uncoded bit error rate (BER) is computed as follows:

     Where and refer to the transmitted and received ith bit of the jth packet. Pkt represents the number of packets sent (which is 10000 in our model).

• The uncoded BER performance was tested over wide range of signal to noise ratio (SNR) and phase noise standard deviation (σ_Δ) values. SNR was varied from 0 dB to 30 dB in steps of 1 dB while the test range of σ_Δ was [0^o – 20^o] in steps of 1^o
• BER vs Eb/No curves for both cross odd and even square modulation schemes were plotted to demonstrate the viability of adopting cross odd-bit QAM schemes in adaptive QAM rather than the relying solely on even square QAM schemes. Eb/No was used instead of SNR to ensure fair comparison among different modulation formats. Eb/No is obtained from the SNR using the following relation [30]:

Where η is the bit efficiency factor. It is used to ensure that BER comparisons are fairly done under the same net bit rate per symbol. The bit efficiency factor can be expressed in terms of modulation order (k) where:

In order to evaluate the viability and relative performance of cross odd-bit QAM quantitatively across various sizes, logarithmic BER gain in (dB) is computed by comparing the cross odd-bit 2^k-QAM with the square 2^k+1-QAM at common Eb/No value. The logarithmic BER gain of cross 2^k-QAM over square 2^k+1-QAM is determined as follows:

Simulation Results

The simulation model, whose framework and parameters explained earlier, was executed to assess the uncoded BER performance of cross odd-bit QAM scheme. Figures (4 – 8) show the BER vs Eb/No curves for different phase noise deviation (σ_Λ) values and modulation orders (k).

        Table 2 shows the log. BER gain for k odd-bit QAM schemes over k+1 square QAM at σ_Δ = 0°.

        Table 3 shows the log. BER gain for k odd-bit QAM schemes over k+1 square QAM at σ_Δ = 5°.

        Table 4 shows the log. BER gain for k odd-bit QAM schemes over k+1 square QAM at σ_Δ = 10°.

        Table 5 shows the log. BER gain for k odd-bit QAM schemes over k+1 square QAM at σ_Δ = 15°.

        Table 6 shows the log. BER gain for k odd-bit QAM schemes over k+1 square QAM at σ_Δ = 20°.

In this paper, the uncoded BER performance of cross odd-bit M-QAM schemes under the joint presence of AWGN and phase noise is investigated and compared with the square M-QAM schemes. The high logarithmic BER gain of low order cross odd-bit M-QAM schemes (8-QAM, 32-QAM, and 128-QAM) over their square M-QAM counterparts of higher order proves the viability of integrating them into adaptive QAM schemes. However, higher orders of cross-odd bit QAM (512-QAM and 1024-QAM) showed a marginal logarithmic BER gain over their square M-QAM counterparts of higher order rendering them less compelling relative to their implementation complexity and hardware cost.

1. Bougeard, S. et al. “Performance optimization of high order QAM in presence of phase noise and AWGN: Application to a decision directed frequency synchronization system.” Wireless Personal Communications, vol. 37, no. 1, 2006, pp. 123-138. https://doi.org/10.1007/s11277-006-8863-x.

2. Sanchez-Sanchez, J.J., U. Fernandez Plazaola, M.C. Aguayo-Torres, and J.T. Entrambasaguas. “Closed-form BER expression for interleaved SC-FDMA with M-QAM.” 2009 IEEE 70th Vehicular Technology Conference Fall, Anchorage, AK, USA, 2009, pp. 1-5. https://doi.org/10.1109/VETECF.2009.5378758.

3. Lari, M. et al. “SER computation in M-QAM systems with phase noise.” Wireless Personal Communications, vol. 70, no. 1, 2013, pp. 1575-1587. https://doi.org/10.1007/ s11277-012-0766-4.

4. Gomaa, A. and N. Al-Dhahir. “Blind phase noise compensation for SC-FDMA with application to LTE-uplink.” 2012 IEEE International Conference on Communications (ICC), Ottawa, ON, Canada, 2012, pp. 4305-4309. https://doi.org/10.1109/ICC.2012.6363796.

5. Syrjälä, V. “Modelling and practical iterative mitigation of phase noise in SC-FDMA.” 2012 IEEE 23rd International Symposium on Personal, Indoor and Mobile Radio Communications - (PIMRC), Sydney, NSW, Australia, 2012, pp. 2395-2400. https://doi.org/10.1109/PIMRC.2012. 6362758.

6. Sridharan, G. and T.J. Lim. “Performance analysis of SC-FDMA in the presence of receiver phase noise.” 2011 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications, Toronto, ON, Canada, 2011, pp. 1978-1982. https://doi.org/10.1109/PIMRC.2011.6 139857.

7. Gomaa, A. and N. Al-Dhahir. “Phase noise in asynchronous SC-FDMA systems: Performance analysis and data-aided compensation.” IEEE Transactions on Vehicular Technology, vol. 63, no. 6, July 2014, pp. 2642-2652. https://doi.org/10.1109/TVT.2013.2294014.

8. Pedrosa, P. et al. “Joint frequency domain equalisation and phase noise estimation for single-carrier modulations in doubly-selective channels.” IET Communications, vol. 9, no. 9, 2015, pp. 1138-1146. https://doi.org/10.1049/iet-com.2014.0900.

9. Ng, B.K. “The performance of SC-FDMA systems with rotated constellation in the presence of phase noise and channel estimation error.” 2015 International Conference on Wireless Communications & Signal Processing (WCSP), Nanjing, China, 2015, pp. 1-5. https://doi.org/10. 1109/WCSP.2015.7341025.

10. Xu, Z., Ren, G., and Z. Zhang. “Phase noise pre-correction scheme for SC-FDMA signals in LTE-uplink.” Chinese Journal of Electronics, vol. 26, no. 3, 2017, pp. 634-639. https://doi.org/10.1049/cje.2017.03.005.

11. Ng, B.K., T. Choi, and C.T. Lam. “Performance of SC-FDMA-based multiuser massive MIMO system in the presence of phase noise.” 2017 IEEE 17th International Conference on Communication Technology (ICCT), Chengdu, China, 2017, pp. 645-649. https://doi.org/10.1109/ICCT.20 17.8359716.

12. Guo, Y., H. Yang, and Y. Zhang. “Tracking reference signal design for phase noise compensation for SC-FDMA waveform.” 2017 IEEE 17th International Conference on Communication Technology (ICCT), Chengdu, China, 2017, pp. 319-323. https://doi.org/10.1109/ICCT.2017 .8359653.

13. Yoon, J., O. Jo, J.-W. Choi, S. Lee, J. Choi, and S.-C. Kim. “SNR analysis and estimation for efficient phase noise mitigation in millimetre-wave SC-FDE systems.” IET Communications, vol. 12, no. 10, 2018, pp. 2347-2356. https://doi.org/10.1049/iet-com.2018.5306.

14. Anbar, M., N. Iqbal et al. “Iterative SC-FDMA frequency domain equalization and phase noise mitigation.” 2018 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS), Ishigaki, Japan, 2018, pp. 91-95. https://doi.org/10.1109/ISPACS.2018.8923193.

15. Roy, J.S. and S.S. Mishra. “Performance of SC-FDMA for LTE uplink under different modulation schemes.” 2019 International Conference on Mechatronics, Robotics and Systems Engineering (MoRSE), Bali, Indonesia, 2019, pp. 202-206. https://doi.org/10.1109/MoRSE48060.2019.8998652.

16. Duan, Q. et al. “Empirical modelling and analysis of phase noise in OFDM systems.” IET Communications, vol. 18, no. 6, 2024, pp. 1358-1370. https://doi.org/10.1049/cmu2.12832.

17. Cox, C. An Introduction to LTE: LTE, LTE-Advanced, SAE and 4G Mobile Communications, 2nd ed., Wiley, 2014.

18. Zarrinkoub, H. Understanding LTE with MATLAB, John Wiley & Sons Inc, 2014.

19. Sana, A.M., et al. “Performance analysis of circular 16-QAM constellation for single-carrier-frequency division multiple access systems.” Journal of Computational and Theoretical Nanoscience, vol. 16, no. 3, 2019, pp. 1019-1022.

20. Holma, H. and A. Toskala. LTE for UMTS: Evolution to LTE-Advanced, 2nd ed., Wiley, 2011.

21. Dahlman, E. et al. 4G: LTE/LTE-Advanced for Mobile Broadband, 2nd ed., Academic Press, 2014.

22. Tse, D. and P. Viswanath. Fundamentals of Wireless Communication, Cambridge Univ. Press, 2005.

23. Rappaport, T.S. Wireless Communications: Principles and Practice, 2nd ed., Prentice Hall, 2002.

24. Benedetto, S. and E. Biglieri. Principles of Digital Transmission: With Wireless Applications, Springer, 1999.

25. Proakis, J.G. and M. Salehi. Digital Communications, 5th ed., McGraw-Hill, 2008.

26. Costa, E. and S. Pupolin. “M-QAM-OFDM system performance in the presence of a nonlinear amplifier and phase noise.” IEEE Transactions on Communications, vol. 50, no. 3, Mar. 2002, pp. 462-472. https://doi.org/10. 1109/26.989015.

27. Demir, A., A. Mehrotra, and J. Roychowdhury. “Phase noise in oscillators: A unifying theory and numerical methods for characterization.” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 5, May 2000, pp. 655-674. https://doi.org/ 10.1109/81.847869.

28. Ghassemi, A. and T.A. Gulliver. “Phase noise modeling and mitigation in OFDM systems.” Proc. IEEE International Conference on Communications (ICC), Dresden, Germany, 2009, pp. 1-5.

29. Lundén, A. et al. “Oscillator phase-noise effects in 5G-level circularly pulse-shaped OFDM systems.” *EURASIP Journal on

30. Advances in Signal Processing*, vol. 2017, no. 1, 2017, pp. 1-17. https://doi.org/10.1186/s13634-017-0500-0.

31. MathWorks. “convertSNR.” MATLAB Documentation, MathWorks, Natick, MA, USA. [Online]. Available: https://www.mathworks.com/help/comm/ref/convertsnr.htm.