Novel four-dimensional direct sequence spread spectrum waveform design based on prolate spheroidal wave functions

Abstract

With its excellent anti-interference capability, low probability of interception, and code-division multiple access capability, direct sequence spread spectrum plays a crucial role in unmanned aerial vehicle swarm communications, high-precision ranging, and massive connectivity in beyond fifth-generation and sixth-generation networks. However, low spectral efficiency and poor energy concentration of conventional rectangular chip waveforms used in spread spectrum modulation make it difficult to meet the high-rate and high-sensitivity reception requirements of future systems. To address this issue, this study proposes a spread spectrum chip waveform design method based on prolate spheroidal wave functions (PSWFs). By constructing a set of four orthogonal PSWF-based chip waveforms, the proposed method enhances signal dimensionality and spectral efficiency under finite time-bandwidth product constraints. Furthermore, an equivalent low-pass filter structure is integrated to reduce implementation complexity. Theoretical analysis and simulation results demonstrate that the proposed waveform achieves approximately double the spectral efficiency of conventional waveforms while maintaining comparable error performance. It also exhibits superior anti-interference and anti-interception capabilities, providing a viable waveform design approach for next-generation spread spectrum communication systems.

Keywords

Direct sequence spread spectrum / Prolate spheroidal wave functions / Chip waveform / Spectral efficiency / Anti-interference

References

[1] Cai XM , Huang CW , Chen PP , et al., 2025. Design of non-coherent RIS-empowered DCSK with two-level nested index modulation. IEEE Trans Wirel Commun, 24 (4): 3044- 3058.

[2] Dai XM , Huang ZY , Luo YQ , et al., 2025. A low-complexity algorithm for designing waveforms with low weighted integrated sidelobe levels. Wirel Pers Commun, 140 (1): 453- 466.

[3] De Sanctis M , Cianca E , Rossi T , et al., 2015. Waveform design solutions for EHF broadband satellite communications. IEEE Commun Mag, 53 (3): 18- 23.

[4] Dogukan AT , Basar E , 2026. Reconfigurable intelligent surface code index modulation with quadrature receive spatial modulation. IEEE Trans Green Commun Netw, 10: 522- 532.

[5] Du J , Chen C , Zeng ZH , et al., 2025. Flexible waveform design for RO-ISAC based on OFDM-embedded maximum length sequence. IEEE Wirel Commun Lett, 14 (10): 3334- 3338.

[6] Duan XD , Huang ZL , Liang SY , et al., 2025. AI-agent communication network for 6G:vision, architecture, and key technologies. Front Inform Technol Electron Eng, 26 (11): 2065- 2080.

[7] Han K , Meng KT , Masouros C , et al., 2026. MIMO-OFDM signaling design for non-coherent distributed ISAC systems. IEEE Trans Wirel Commun, 25: 7792- 7808.

[8] He CL , Xiao LX , Li S , et al., 2025. Index modulation aided OTFS for SCMA system. China Commun, 22 (8): 19- 28.

[9] Huang GJ , Zhang SZ , Jin SG , et al., 2025. A novel multi-beam directional modulation OFDM waveform design for integrated sensing and communication. IEEE Trans Veh Technol, 74 (11): 18220- 18225.

[10] Hui Y , Tang KH , Liu KM , et al., 2026. Power and signal dual modulation in DC/DC converters using FSK-based DSSS for high anti-interference performance. IEEE Trans Power Electron, 41 (3): 3712- 3725.

[11] Jin XP , Zhang P , Wan C , et al., 2023. RIS assisted dual-function radar and secure communications based on frequency-shifted chirp spread spectrum index modulation. China Commun, 20 (10): 85- 99.

[12] Jin Y , Ji HB , 2011. A novel cyclic-moment based method for DS-SS QPSK signal blind parameter estimation. Chin J Electron, 20 (4): 686- 690.

[13] Jin Y , Deng ZL , Zhang H , et al., 2026. A review of UAV positioning in LAIN:toward a 5G-core "space-air-ground" integrated and cooperative architecture. ENGINEERING Inform Technol Electron Eng, 27 (1): 1- 16.

[14] Liu JC , Zhang JT , Li T , et al., 2025. Unintended interference suppression based on de-cision feedback adaptive cancellation for DSSS satellite communication. Chin J Electron, 34 (2): 673- 682.

[15] Liu MZ , Liu J , Peng Z , et al., 2025. Efficient MMSE equalization for direct-sequence spread-spectrum underwater communications. IEEE Internet Things J, 12 (11): 16325- 16335.

[16] Lu FP , Zhang ZL , An Q , et al., 2025. A simplified fast Fourier transform algorithm. Proc 8th World Conf on Computing and Communication Technologies, p.176- 179.

[17] Lu ZW , Jiao YW , Chen YD , 2024. High stability code tracking for band-limited DSSS systems. IEICE Trans Commun, E107-B (11): 706- 718.

[18] Murugesan PPK , Sivabalin VPG , Nanjan SM , et al., 2023. Orthogonal modes using prolate spheroidal wave function in holographic communication systems. Proc Int Conf on Signal Processing, Computation, Electronics, Power and Telecommunication, p.1- 6.

[19] Ni ZT , Zhang JA , Liu RP , et al., 2026. Ambiguity function-based filter designs for OFDM ISAC systems. IEEE Wirel Commun Lett, 15: 181- 185.

[20] Oh S , Moon H , 2024. Novel algorithm for joint channel estimation and spreading se-quence detection in DSSS systems. IEEE Access, 12: 128694- 128701.

[21] Osipov A , Rokhlin V , Xiao H , 2013. Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation.Springer, New York, USA.

[22] Qian M , Ji F , Ge Y , et al., 2025. Generalized code index modulation aided AFDM for spread spectrum systems. IEEE Wirel Commun Lett, 14 (10): 3229- 3233.

[23] Sarwar MS , Ramatryana INA , Budiman G , et al., 2025. An uplink frequency-time index modulation multiple access for 6G networks. IEEE Trans Veh Technol, 74 (5): 7866- 7880.

[24] Shao YZ , Zhang TQ , Yue PY , et al., 2026. Direct joint detection and localization of weak DSSS signals via cooperative LEO satellites. IEEE Trans Commun, 74: 6571- 6583.

[25] Slepian D , Pollak HO , 1961. Prolate spheroidal wave functions, Fourier analysis and uncertainty-I. Bell Syst Techn J, 40 (1): 43- 63.

[26] Su ZP , Zhang GF , Zhou XX , et al., 2021. Segment-adaptive spread spectrum audio wa-termarking using NSGA-II. Chin J Electron, 30 (4): 727- 735.

[27] Todorović BM , Stojanović NM , Velikić GS , et al., 2024. Is direct sequence spread spectrum modulation promising for AI-based design of 6G networks? IEEE Consum Electron Mag, 13 (5): 7- 10.

[28] Vangelista L , Jechoux B , Canonici JX , et al., 2024. Golden modulation:a new and ef-fective waveform for massive IoT. IEEE Trans Commun, 72 (4): 1938- 1948.

[29] Waheed A , Mohammady S , 2024. Spread spectrum OFDM-IM with efficient precoded matrix and low complexity detector. IEEE Access, 12: 192293- 192300.

[30] Wang S , Song Z , Hua ZZ , et al., 2026. GaussMask-DSSS:enhancing covert spread spectrum communication with Gaussian cloaking and deep learning-aided syn-chronization. IEEE J Sel Areas Commun, 44: 1495- 1511.

[31] Xu YZ , Peng YY , Al-Hazemi F , et al., 2026. Dual-functional radar and communication system based on hybrid index modulation. IEEE Wirel Commun Lett, 15: 1886- 1890.

[32] Xu ZC , Lu FP , Zhang LF , et al., 2025. Generalized multiple-mode prolate spheroidal wave functions multi-carrier waveform with index modulation. J Syst Eng Elec-tron, 36 (2): 311- 322.

[33] Xue R , Xie MM , 2023. Partial response continuous phase modulation with a prolate spheroidal wave function for compass system in the S-band. EURASIP J Adv Sig-nal Process, 2023 (1): 88.

[34] Yan WH , An JP , Feng YB , et al., 2026. Coherent acquisition of MC-DSSS signal for LEO satellite communication. IEEE Trans Commun, 74: 6395- 6408.

[35] Yang YX , Gao P , 2023. Design method of multi-layer multi-parameter multi-term weighted-type fractional Fourier transform composite modulation communication signal. J Electron Inform Technol, 45 (4): 1192- 1200 (in Chinese).

[36] Yang ZJ , Su W , Zhang HK , 2026. Overview of heterogeneous AMN architectures inte-grating satellite and terrestrial networks:a communication-sensing-computing perspective. Chin J Aeronaut, 39 (5): 103918.

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