Research article Topical Sections

BER performance analysis of polar-coded FBMC/OQAM in the presence of AWGN and Nakagami-m fading channel


  • Offset quadrature amplitude modulation–based filter bank multicarrier (FBMC-OQAM) method is a promising technology for future wireless communication systems. It offers several advantages over traditional orthogonal frequency-division multiplexing (OFDM) modulation, including higher spectral efficiency, lower out-of-band emission, and improved robustness to time-frequency selective channels. Polar codes, a new class of error-correcting codes, have received much attention recently due to their ability to achieve the Shannon limit with practical decoding complexity. This paper analyzed and investigated the error rate performance of polar-coded FBMC-OQAM systems. Our results show that applying polar codes to FBMC-OQAM systems significantly improves the error rate. In addition, we found that employing random code interleavers can yield additional coding gains of up to 0.75 dB in additive white Gaussian noise (AWGN) and 2 dB in Nakagami-m fading channels. Our findings suggest that polar-coded FBMC-OQAM is a promising combination for future wireless communication systems. We also compared turbo-coded FBMC-OQAM for short code lengths, and our simulations showed that polar codes exhibit comparable error-correcting capabilities. These results will be of interest to researchers and engineers working on the advancement of future wireless communication systems.

    Citation: Tadele A. Abose, Fanuel O. Ayana, Thomas O. Olwal, Yihenew W. Marye. BER performance analysis of polar-coded FBMC/OQAM in the presence of AWGN and Nakagami-m fading channel[J]. AIMS Electronics and Electrical Engineering, 2024, 8(3): 311-331. doi: 10.3934/electreng.2024014

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  • Offset quadrature amplitude modulation–based filter bank multicarrier (FBMC-OQAM) method is a promising technology for future wireless communication systems. It offers several advantages over traditional orthogonal frequency-division multiplexing (OFDM) modulation, including higher spectral efficiency, lower out-of-band emission, and improved robustness to time-frequency selective channels. Polar codes, a new class of error-correcting codes, have received much attention recently due to their ability to achieve the Shannon limit with practical decoding complexity. This paper analyzed and investigated the error rate performance of polar-coded FBMC-OQAM systems. Our results show that applying polar codes to FBMC-OQAM systems significantly improves the error rate. In addition, we found that employing random code interleavers can yield additional coding gains of up to 0.75 dB in additive white Gaussian noise (AWGN) and 2 dB in Nakagami-m fading channels. Our findings suggest that polar-coded FBMC-OQAM is a promising combination for future wireless communication systems. We also compared turbo-coded FBMC-OQAM for short code lengths, and our simulations showed that polar codes exhibit comparable error-correcting capabilities. These results will be of interest to researchers and engineers working on the advancement of future wireless communication systems.



    Stochastic nonlinear control is a hot topic because of its wide application in economic and engineering fields. The pioneer work is [1,2,3,4,5,6,7,8]. Specifically, [1,2,3] propose designs with quadratic Lyapunov functions coupled with weighting functions and [4,5,6,7,8] develop designs with quartic Lyapunov function, which are further developed by [9,10,11,12]. Recently, a class of stochastic systems (SSs) whose Jacobian linearizations may have unstable modes, has received much attention. Such systems are also called stochastic high-order nonlinear systems (SHONSs), which include a class of stochastic benchmark mechanical systems [13] as a special case. In this direction, [14] studies the state-feedback control with stochastic inverse dynamics; [15] develops a stochastic homogeneous domination method, which completely relaxes the order restriction required in [13], and investigates the output-feedback stabilization for SHONSs with unmeasurable states; [16] investigates the output-feedback tracking problem; [17] studies the adaptive state-feedback design for state-constrained systems. It should be emphasized that [13,14,15,16,17], only achieve stabilization in asymptotic sense (as time goes to infinity). However, many real applications appeals for prescribed-time stabilization, which permits the worker to prescribe the convergence times in advance.

    For the prescribed-time control, [18,19] design time-varying feedback to solve the regulation problems of normal-form nonlinear systems; [20] considers networked multi-agent systems; [21,22] investigates the prescribed-time design for linear systems in controllable canonical form; [23] designs the output-feedback controller for uncertain nonlinear strict-feedback-like systems. It should be noted that the results in [18,19,20,21,22,23] don't consider stochastic noise. For SSs, [24] is the first paper to address the stochastic nonlinear inverse optimality and prescribed-time stabilization problems; The control effort is further reduced in [25]; Recently, [26] studies the prescribed-time output-feedback for SSs without/with sensor uncertainty. It should be noted that [24,25,26] don't consider SHONSs. From practical applications, it is important to solves the prescribed-time control problem of SHONSs since it permits the worker to set the convergence times in advance.

    Motivated by the above discussions, this paper studies the prescribed-time design for SSs with high-order structure. The contributions include:

    1) This paper proposes new prescribed-time design for SHONSs. Since Jacobian linearizations of such a system possibly have unstable modes, all the prescribed-time designs in [24,25,26] are invalid. New design and analysis tools should be developed.

    2) This paper develops a more practical design than those in [13,14,15,16,17]. Different from the designs in [13,14,15,16,17] where only asymptotic stabilization can be achieved, the design in this paper can guarantee that the closed-loop system is prescribed-time mean-square stable, which is superior to those in [13,14,15,16,17] since it permits the worker to prescribe the convergence times in advance without considering the initial conditions.

    The remainder of this paper is organized as follows. In Section 2, the problem is formulated. In Section 3, the controller is designed and the stability analysis is given. Section 4 uses an example to explain the validity of the prescribed-time design. The conclusions are collected in Section 6.

    Consider the following class of SHONSs

    dx1=xp2dt+φT1(x)dω, (2.1)
    dx2=udt+φT2(x)dω, (2.2)

    where x=(x1,x2)TR2 and uR are the system state and control input. p1 is an odd integral number. The functions φi:R2Rm are smooth in x with φi(0)=0, i=1,2. ω is an m-dimensional independent standard Wiener process.

    The assumptions we need are as follows.

    Assumption 1. There is a constant c>0 such that

    |φ1(x)|c|x1|p+12, (2.3)
    |φ2(x)|c(|x1|p+12+|x2|p+12). (2.4)

    We introduce the function:

    μ(t)=(Tt0+Tt)m,t[t0,t0+T), (2.5)

    where m2 is an integral number. Obviously, the function μ(t) is monotonically increasing on [t0,t0+T) with μ(t0)=1 and limtt0+Tμ(t)=+.

    Our control goal is to design a prescribed-time state-feedback controller, which guarantees that the closed-loop system has an almost surely unique strong solution and is prescribed-time mean-square stable.

    In the following, we design a time-varying controller for system (2.1)–(2.2) step by step.

    Step 1. In this step, our goal is to design the virtual controller x2.

    Define V1=14ξ41, ξ1=x1, from (2.1), (2.3) and (2.5) we have

    LV1(ξ1)=ξ31xp2+32ξ21|φ1|2ξ31xp2+32c2ξp+31ξ31(xp2xp2)+ξ31xp2+32c2μpξp+31. (3.1)

    Choosing

    x2=μ(2+32c2)1/pξ1μα1ξ1, (3.2)

    which substitutes into (3.1) yields

    LV1(ξ1)2μpξp+31+ξ31(xp2xp2), (3.3)

    where α1=(2+32c2)1/p.

    Step 2. In this step, our goal is to design the actual controller u.

    Define ξ2=xkx2, from (3.2) we get

    ξ2=x2+μα1ξ1. (3.4)

    By using (2.1)–(2.2) and (3.4) we get

    dξ2=(u+mTμ1+1/mα1ξ1+μα1xp2)dt+(φT2+μα1φT1)dω. (3.5)

    We choose the following Lyapunov function

    V2(ξ1,ξ2)=V1(ξ1)+14ξ42. (3.6)

    It follows from (3.3), (3.5)–(3.6) that

    LV22μpξp+31+ξ31(xp2xp2)+ξ32(u+mTμ1+1/mα1ξ1+μα1xp2)+32ξ22|φ2+μα1φ1|2. (3.7)

    By using (3.4) we have

    ξ31(xp2xp2)|ξ1|3|ξ2|(xp12+xp12)|ξ1|3|ξ2|((β1+1)μp1αp11ξp11+β1ξp12)=(β1+1)μp1αp11|ξ1|p+2|ξ2|+β1|ξ1|3|ξ2|p, (3.8)

    where

    β1=max{1,2p2}. (3.9)

    By using Young's inequality we get

    (β1+1)μp1αp11|ξ1|p+2|ξ2|13μpξp+31+13+p(3+p3(2+p))(2+p)((β1+1)αp11)p+3μ3ξp+32 (3.10)

    and

    β1|ξ1|3|ξ2|p13μpξ3+p1+pp+3(3+p9)3/pβ(3+p)/p1μ3ξp+32. (3.11)

    Substituting (3.10)–(3.11) into (3.8) yields

    ξ31(xp2xp2)23μpξp+31+(pp+3(p+39)3/pβ(p+3)/p1+1p+3(p+33(p+2))(p+2)((β1+1)αp11)p+3)μ3ξp+32. (3.12)

    From (2.3), (2.4), (3.2) and (3.4) we have

    |φ2+μα1φ1|22|φ2|2+2μ2α21|φ1|24c2(|x1|1+p+|x2|1+p)+2c2α21μ2|x1|1+pβ2μ1+p|ξ1|p+1+4c22p|ξ2|1+p, (3.13)

    where

    β2=4c2(1+2pαp+11)+2c2α21. (3.14)

    By using (3.13) we get

    32ξ22|φ2+μα1φ1|213μpξ3+p1+(23+p(p+33(1+p))(1+p)/2(32β2)(3+p)/2+6c22p)μ3(1+p)/2ξp+32. (3.15)

    It can be inferred from (3.7), (3.12) and (3.15) that

    LV2μpξp+31+ξ32(u+mTμ1+1/mα1ξ1+μα1xp2)+β3μ3(p+1)/2ξp+32, (3.16)

    where

    β3=p3+p(3+p9)3/pβ(3+p)/p1+13+p(p+33(p+2))(p+2)((β1+1)αp11)3+p+23+p(p+33(1+p))(1+p)/2(32β2)(3+p)/2+6c22p. (3.17)

    We choose the controller

    u=mTμ1+1/mα1ξ1μα1xp2(1+β3)μ3(p+1)/2ξp2, (3.18)

    then we get

    LV2μpξp+31μ3(p+1)/2ξp+32. (3.19)

    Now, we describe the main stability analysis results for system (2.1)–(2.2).

    Theorem 1. For the plant (2.1)–(2.2), if Assumption 1 holds, with the controller (3.18), the following conclusions are held.

    1) The plant has an almost surely unique strong solution on [t0,t0+T);

    2) The equilibrium at the origin of the plant is prescribed-time mean-square stable with limtt0+TE|x|2=0.

    Proof. By using Young's inequality we get

    14μξ41μpξp+31+β4μ3, (3.20)
    14μξ42μ3(p+1)/2ξp+32+β4μ3, (3.21)

    where

    β4=p1p+3(3+p4)4/(p1)(14)(3+p)/(p1). (3.22)

    It can be inferred from (3.19)–(3.21) that

    LV214μξ4114μξ42+2β4μ3=μV2+2β4μ3. (3.23)

    From (2.1), (2.2) and (3.18), the local Lipschitz condition is satisfied by the plant. By (3.23) and using Lemma 1 in [24], the plant has an almost surely unique strong solution on [t0,t0+T), which shows that conclusion 1) is true.

    Next, we verify conclusion 2).

    For each positive integer k, the first exit time is defined as

    ρk=inf{t:tt0,|x(t)|k}. (3.24)

    Choosing

    V=ett0μ(s)dsV2. (3.25)

    From (3.23) and (3.25) we have

    LV=ett0μ(s)ds(LV2+μV2)2β4μ3ett0μ(s)ds. (3.26)

    By (3.26) and using Dynkin's formula we get

    EV(ρkt,x(ρkt))=Vn(t0,x0)+E{ρktt0LV(x(τ),τ)dτ}Vn(t0,x0)+2β4tt0μ3eτt0μ(s)dsdτ. (3.27)

    Using Fatou Lemma, from (3.27) we have

    EV(t,x)Vn(t0,x0)+2β4tt0μ3eτt0μ(s)dsdτ,t[t0,t0+T). (3.28)

    By using (3.25) and (3.28) we get

    EV2ett0μ(s)ds(Vn(t0,x0)+2β4tt0μ3eτt0μ(s)dsdτ),t[t0,t0+T). (3.29)

    By using (3.6) and (3.29) we obtain

    limtt0+TE|x|2=0. (3.30)

    Consider the prescribed-time stabilization for the following system

    dx1=x32dt+0.1x21dω, (4.1)
    dx2=udt+0.2x1x2dω, (4.2)

    where p1=3, p2=1. Noting 0.2x1x20.1(x21+x22), Assumption 1 is satisfied.

    Choosing

    μ(t)=(11t)2,t[0,1), (4.3)

    According to the design method in Section 3, we have

    u=3μ1.5x11.5μx3257μ6(x2+1.5μx1)3 (4.4)

    In the practical simulation, we select the initial conditions as x1(0)=6, x2(0)=5. Figure 1 shows the responses of (4.1)–(4.4), from which we can obtain that limtt0+TE|x1|2=limtt0+TE|x2|2=0, which means that the controller we designed is effective.

    Figure 1.  The responses of closed-loop system (4.1)–(4.4).

    This paper proposes a new design method of prescribed-time state-feedback for SHONSs. the controller we designed can guarantee that the closed-loop system has an almost surely unique strong solution and the equilibrium at the origin of the closed-loop system is prescribed-time mean-square stable. The results in this paper are more practical than those in [13,14,15,16,17] since the design in this paper permits the worker to prescribe the convergence times in advance without considering the initial conditions.

    There are some related problems to investigate, e.g., how to extend the results to multi-agent systems [27], impulsive systems [28,29,30] or more general high-order systems [31,32,33,34].

    This work is funded by Shandong Provincial Natural Science Foundation for Distinguished Young Scholars, China (No. ZR2019JQ22), and Shandong Province Higher Educational Excellent Youth Innovation team, China (No. 2019KJN017).

    The authors declare there is no conflict of interest.



    [1] Bizaki HK (2016) Towards 5G wireless networks: a physical layer perspective. BoD–Books on Demand. https://doi.org/10.5772/63098
    [2] Demir AF, Elkourdi M, Ibrahim M, Arslan H (2019) Waveform design for 5G and beyond. arXiv preprint arXiv: 1902.05999. https://doi.org/10.1002/9781119333142.ch2
    [3] Chang RW (1966) High-speed multichannel data transmission with bandlimited orthogonal signals. Bell Syst Tech J 45: 1775‒1796. https://doi.org/10.1002/j.1538-7305.1966.tb02435.x doi: 10.1002/j.1538-7305.1966.tb02435.x
    [4] Saltzberg B (1967) Performance of an efficient parallel data transmission system. IEEE Transactions on Communication Technology 15: 805‒811. https://doi.org/10.1109/TCOM.1967.1089674 doi: 10.1109/TCOM.1967.1089674
    [5] Jiang T, Chen D, Ni C, Qu D (2017) OQAM/FBMC for future wireless communications: Principles, technologies and applications. Academic Press. https://doi.org/10.1016/B978-0-12-813557-0.00010-3
    [6] Nissel R, Rupp M (2018) Pruned DFT-spread FBMC: Low PAPR, low latency, high spectral efficiency. IEEE T Commun 66: 4811‒4825. https://doi.org/10.1109/TCOMM.2018.2837130 doi: 10.1109/TCOMM.2018.2837130
    [7] Wang Y, Guo Q, Xiang J, Wang L, Liu Y (2024) Bi-orthogonality recovery and MIMO transmission for FBMC Systems based on non-sinusoidal orthogonal transformation. Signal Processing 109427. https://doi.org/10.1016/j.sigpro.2024.109427 doi: 10.1016/j.sigpro.2024.109427
    [8] Arikan E (2009) Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE T Inform Theory 55: 3051‒3073. https://doi.org/10.1109/TIT.2009.2021379 doi: 10.1109/TIT.2009.2021379
    [9] Ali MH, Al-Rubaye GA (2024) Performance Evaluation of 5G New Radio Polar Code over Different Multipath Fading Channel Models. International Journal of Intelligent Engineering & Systems 17. https://doi.org/10.22266/ijies2024.0430.36 doi: 10.22266/ijies2024.0430.36
    [10] Meenalakshmi M, Chaturvedi S, Dwivedi VK (2024) Enhancing channel estimation accuracy in polar-coded MIMO–OFDM systems via CNN with 5G channel models. AEU-Int J Electron Commun 173: 155016. https://doi.org/10.1016/j.aeue.2023.155016 doi: 10.1016/j.aeue.2023.155016
    [11] Vangala H, Hong Y, Viterbo E (2015) Efficient algorithms for systematic polar encoding. IEEE Commun Lett 20: 17‒20. https://doi.org/10.1109/LCOMM.2015.2497220 doi: 10.1109/LCOMM.2015.2497220
    [12] Hall EK, Wilson SG (1998) Design and analysis of turbo codes on Rayleigh fading channels. IEEE J Sel Areas Commun 16: 160‒174. https://doi.org/10.1109/49.661105 doi: 10.1109/49.661105
    [13] Liu L, Ling C (2016) Polar codes and polar lattices for independent fading channels. IEEE T Commun 64: 4923‒4935. https://doi.org/10.1109/TCOMM.2016.2613109 doi: 10.1109/TCOMM.2016.2613109
    [14] Si H, Koyluoglu OO, Vishwanath S (2014) Polar coding for fading channels: Binary and exponential channel cases. IEEE T Commun 62: 2638‒2650. https://doi.org/10.1109/TCOMM.2014.2345399 doi: 10.1109/TCOMM.2014.2345399
    [15] Trifonov P (2015) Design of polar codes for Rayleigh fading channel. 2015 international symposium on wireless communication systems (ISWCS) 331‒335. IEEE. https://doi.org/10.1109/ISWCS.2015.7454357
    [16] Rafik Z, Hmaied S, Daniel R, Yahia M (2017) BER analysis of FBMC-OQAM systems with Phase Estimation Error. IET Communications.
    [17] Marina P, Isnawati AF, Afandi MA (2020) Performance analysis of FBMC O-QAM system using varied modulation level. Jurnal Infotel 12: 45‒51. https://doi.org/10.20895/infotel.v12i2.482 doi: 10.20895/infotel.v12i2.482
    [18] BS R (2021) Performance Analysis of OFDM, FBMC and UFMC Modulation Schemes for 5G Mobile Communication MIMO systems. Proceedings of the International Conference on IoT Based Control Networks & Intelligent Systems-ICICNIS.
    [19] Hassan ES (2024) Performance enhancement and PAPR reduction for MIMO based QAM-FBMC systems. Plos one 19: e0296999. https://doi.org/10.1371/journal.pone.0296999 doi: 10.1371/journal.pone.0296999
    [20] Al-Amaireh H, Kollár Z (2022) Low complexity PPN-FBMC receivers with improved sliding window equalizers. Phys Commun 54: 101795. https://doi.org/10.1016/j.phycom.2022.101795 doi: 10.1016/j.phycom.2022.101795
    [21] Wang Y, Guo Q, Xiang J, Liu Y (2023) Doubly selective channel estimation and equalization based on ICI/ISI mitigation for OQAM-FBMC systems. Phys Commun 59: 102120. https://doi.org/10.1016/j.phycom.2023.102120 doi: 10.1016/j.phycom.2023.102120
    [22] Malkamaki E (1992) Binary and multilevel offset QAM, spectrum efficient modulation schemes for personal communications. In [1992 Proceedings] Vehicular Technology Society 42nd VTS Conference-Frontiers of Technology 325‒328. IEEE.
    [23] Bellanger M, Le Ruyet D, Roviras D, Terré M, Nossek J, Baltar L, et al. (2010) FBMC physical layer: a primer. PHYDYAS, January 25: 7‒10.
    [24] Choi JM, Oh Y, Lee H, Seo JS (2017) Pilot-aided channel estimation utilizing intrinsic interference for FBMC/OQAM systems. IEEE T Broadcast 63: 644‒655. https://doi.org/10.1109/TBC.2017.2711143 doi: 10.1109/TBC.2017.2711143
    [25] Yu B, Hu S, Sun P, Chai S, Qian C, Sun C (2016) Channel estimation using dual-dependent pilots in FBMC/OQAM systems. IEEE Commun Lett 20: 2157‒2160. https://doi.org/10.1109/LCOMM.2016.2599882 doi: 10.1109/LCOMM.2016.2599882
    [26] Bedoui A, Et-tolba M (2020) A neuro-fuzzy based detection approach for HARQ-CC in FBMC-OQAM systems. 2020 9th IFIP international conference on performance evaluation and modeling in wireless networks (PEMWN) 1‒7. IEEE. https://doi.org/10.23919/PEMWN50727.2020.9293073
    [27] Doré JB, Gerzaguet R, Cassiau N, Ktenas D (2017) Waveform contenders for 5G: Description, analysis and comparison. Phys Commun 24: 46‒61. https://doi.org/10.1016/j.phycom.2017.05.004 doi: 10.1016/j.phycom.2017.05.004
    [28] Abose TA, Olwal TO, Mohammed MM, Hassen MR (2024) Performance analysis of insertion loss incorporated hybrid precoding for massive MIMO. AIMS Electronics and Electrical Engineering 8: 187‒210. https://doi.org/10.3934/electreng.2024008 doi: 10.3934/electreng.2024008
    [29] Abose TA, Mariye YW, Demissie AM, Olwal TO (2023) Energy Efficiency Enhancement of Ultra Dense Multiuser MIMO System by Using Wyner Model. 2023 14th International Conference on Computing Communication and Networking Technologies (ICCCNT) 1‒6. IEEE. https://doi.org/10.1109/ICCCNT56998.2023.10308201
    [30] TS 38.212 NR; Multiplexing and channel coding, 3GPP, 2018.
    [31] Balatsoukas-Stimming A, Parizi MB, Burg A (2015) LLR-based successive cancellation list decoding of polar codes. IEEE T Signal Process 63: 5165‒5179. https://doi.org/10.1109/TSP.2015.2439211 doi: 10.1109/TSP.2015.2439211
    [32] Afisiadis O, Balatsoukas-Stimming A, Burg A (2014) A low-complexity improved successive cancellation decoder for polar codes. 2014 48th Asilomar Conference on Signals, Systems and Computers 2116‒2120. IEEE. https://doi.org/10.1109/ACSSC.2014.7094848
    [33] Arıkan E (2010) Polar codes: A pipelined implementation. Proc 4th ISBC 2010: 11‒14.
    [34] Kahraman S, Ç elebi ME (2012) Code based efficient maximum-likelihood decoding of short polar codes. 2012 IEEE International Symposium on Information Theory Proceedings 1967‒1971. IEEE. https://doi.org/10.1109/ISIT.2012.6283643
    [35] Berrou C, Glavieux A, Thitimajshima P (1993) Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1. Proceedings of ICC'93-IEEE International Conference on Communications 2: 1064‒1070. IEEE.
    [36] Benedetto S, Divsalar D, Montorsi G, Pollara F (1996) A soft-input soft-output maximum a posteriori (MAP) module to decode parallel and serial concatenated codes. TDA progress report 42: 1‒20. https://doi.org/10.1109/4234.552145 doi: 10.1109/4234.552145
    [37] Viterbi AJ (1998) An intuitive justification and a simplified implementation of the MAP decoder for convolutional codes. IEEE J Sel Areas Commun 16: 260‒264. https://doi.org/10.1109/49.661114 doi: 10.1109/49.661114
    [38] Nakagami M (1960) The m-distribution—A general formula of intensity distribution of rapid fading. Statistical methods in radio wave propagation 3‒36. Pergamon. https://doi.org/10.1016/B978-0-08-009306-2.50005-4
    [39] Abose TA, Megersa KT, Jember KA, Kejela DC, Daka ST, Dinagde MB (2023) Performance Comparison of M-ary Phase Shift Keying and M-ary Quadrature Amplitude Modulation Techniques Under Fading Channels. International Conference on Communication and Intelligent Systems 235‒245. Singapore: Springer Nature Singapore. https://doi.org/10.1007/978-981-97-2079-8_19
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