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Orthogonal Chirp Division Multiplexing

Xing Ouyang, Jian Zhao

arXiv:1601.06576v1cs.IT

TL;DR

The paper tackles the loss of spectral efficiency and interference problems that arise when chirp waveforms are multiplexed. It proposes OCDM, implemented with the discrete Fresnel transform and supported by single-tap equalization for linear or quasi-static channels. The reported results show that OCDM exploits multipath diversity and, with MMSE equalization, outperforms OFDM under multipath fading.

  • Problem

    Chirp systems sacrifice spectral efficiency, while multiple chirps within the same period and bandwidth produce inter-chirp interference.

  • Method

    The paper orthogonally multiplexes modulated chirps using OCDM, implements it with the discrete Fresnel transform, and develops eigen-decomposition-based single-tap equalization.

  • Results

    OCDM exploits multipath diversity with linear equalizers, and its MMSE-equalized form outperforms OFDM under multipath fading channels.

  • Takeaways & Limitations

    OCDM provides an orthogonal chirp-multiplexing approach that uses the same bandwidth while supporting channel compensation and multipath-diversity exploitation.

Abstract

from arXiv · show

Chirp waveform plays a significant role in radar and communication systems for its ability of pulse compression and spread spectrum. This paper presents a principle of orthogonally multiplexing a bank of linear chirp waveforms within the same bandwidth. The amplitude and phase of the chirps are modulated for information communication. As Fourier trans-form is the basis for orthogonal frequency division multiplexing (OFDM), Fresnel transform underlies the proposed orthogonal chirp division multiplexing (OCDM). Digital implementa-tion of the OCDM system using discrete Fresnel transform is proposed. Based on the con-volution theorem of the Fresnel transform, the transmission of the OCDM signal is analyzed under the linear time-invariant or quasi-static channel with additive noise, which can gener-alize typical linear transmission channels. Based on the eigen-decomposition of Fresnel transform, efficient digital signal processing algorithm is proposed for compensating chan-nel dispersion by linear single- tap equalizers. The implementation details of the OCDM system is discussed with emphasis on its compatibility to the OFDM system. Finally, simula-tion are provided to validate the feasibility of the proposed OCDM under wireless channels. It is shown that the OCDM system is able to utilize the multipath diversity and outperforms the OFDM system under the multipath fading channels.

I. INTRODUCTION

The paper addresses the spectral-efficiency and inter-chirp-interference limits of conventional chirp communication by introducing orthogonal chirp division multiplexing (OCDM), which overlaps multiple modulated chirps within the same bandwidth without interference. It develops a discrete Fresnel-transform implementation, channel analysis, and efficient equalization for OCDM.

  • Motivation: Chirp communication uses wideband transmission that resists channel noise, multipath fading, and Doppler effects, but sacrifices spectral efficiency for processing gain and multipath resolution.This makes chirps attractive when reliable communication is prioritized, particularly for low-rate applications.
  • Motivation: Multiple chirps sharing the same period and bandwidth conventionally cause inter-chirp interference, limiting straightforward multiplexing.Multi-code UWB instead divides the spectrum into spectrally separated chirps and recovers information through orthogonal codes.
  • OCDM principle: OCDM orthogonally multiplexes a bank of chirp waveforms within the same bandwidth, with chirp amplitude and phase carrying information.The modulated chirps overlap temporally and spectrally while remaining orthogonal along the chirp dimension.
  • OCDM principle: The Fresnel transform provides the fundamental mechanism for OCDM, analogous to the Fourier transform in OFDM.The transmitter uses the inverse discrete Fresnel transform, while the receiver uses the discrete Fresnel transform to recover the OCDM signal.
  • Channel processing: Fresnel-transform convolution properties support OCDM transmission analysis over linear time-invariant or quasi-static channels and compensation with time- or frequency-domain equalizers.An eigen-decomposition-based single-tap frequency-domain equalizer is proposed as more efficient than the two earlier equalization approaches.
  • Evaluation and scope: Simulations evaluate OCDM under wireless multipath fading channels, while the paper also discusses digital implementation and compatibility with OFDM.The introduction states that the OCDM system can exploit multipath diversity and that its MMSE-equalized form outperforms OFDM.

II. FRESNEL TRANSFORM

The Fresnel transform, originating in classical optics, is introduced as the mathematical basis for OCDM. Its convolution property enables the subsequent model of OCDM signal transmission under linear time-invariant channels.

  • Connection to OCDM: The Fresnel transform’s linear and circular convolution properties support the mathematical model of OCDM signal transmission under linear time-invariant channels.The paper introduces the discrete Fresnel transform for use in the following OCDM analysis.
  • Transform basis: The Fresnel transform is an integral transformation originating from classical optics and is also a special case of the linear canonical transform.It describes near-field diffraction phenomena and is presented here as the basis for OCDM.
  • Optical interpretation: Near-field diffraction occurs when a monochromatic wave encounters an aperture comparable in size to its wavelength, producing a pattern at a distance z.The paper expresses this diffraction behavior using the Fresnel transform of the aperture’s complex transmittance.
  • Convolution property: The Fresnel transform can be expressed in convolution form, and its transform of a convolution equals one function convolving with the Fresnel transform of the other.This convolution property differs from the Fourier-transform theorem in which convolution becomes multiplication.

B. Discrete Fresnel Transform

The discrete Fresnel transform (DFnT) provides a unitary, convolution-compatible representation linked to Talbot effects, while a revised formulation removes earlier matrix-size degeneracy.

  • The DFnT matrix represents the optical field of light spots at a fraction ZT ⁄ N of the Talbot distance.
  • Earlier DFnT formulations had degeneracy: matrix size was N ⁄ 2 for N ≡ 0 and 2 (mod 4), but N for N ≡ 1 and 3 (mod 4).This degeneracy hindered use of the DFnT as a general mathematical tool.
  • A recent DFnT derivation removes this degeneracy, with representations differing for even and odd N.
  • The DFnT is unitary and has an eigen-decomposition, supporting signal-processing operations in the OCDM framework.
  • The DFnT maps circular convolution into multiplication of the corresponding DFnT-domain sequences.This is the discrete counterpart of the Fresnel-transform convolution property.

III. ORTHOGONAL CHIRP DIVISION MULTIPLEXING

OCDM multiplexes mutually orthogonal chirps within one bandwidth and period, using amplitude or phase modulation to improve the spectral efficiency of chirp spread-spectrum systems.

  • Chirp Spread Spectrum: Conventional CSS uses processing gain with B >> Rs and traditionally relies on analog modulation, limiting use of advanced formats such as QAM.
  • Waveform construction: The root chirp has chirp rate α = N ⁄ T2 and time-bandwidth product about BT = N.
  • Principle of OCDM: OCDM constructs N mutually orthogonal chirp waveforms within a given bandwidth and period.The waveforms are orthogonal in the chirp dimension rather than the frequency dimension used by OFDM.
  • Principle of OCDM: The spectral efficiency of OCDM increases by N over the CSS system.
  • Modulation: OCDM modulates each chirp's amplitude and phase, allowing PAM, PSK, and QAM symbols selected from a codebook χ.
  • Transmission and recovery: A bank of modulated chirps is transmitted in blocks, and each symbol can be recovered with a matched filter for its corresponding chirp.

C. Digital Implementation of OCDM

OCDM has a digital implementation based on the discrete Fresnel transform: IDFnT synthesizes the modulated chirps, while DFnT recovers transmitted symbols.

  • The continuous-time OCDM signal is sampled to obtain its discrete representation, with separate DFnT forms for even and odd N.
  • The proposed digital OCDM transceiver includes time-domain and frequency-domain equalizer options.
  • The discrete modulated-chirp synthesis equations are exactly the IDFnT, enabling matrix-based digital implementation.
  • Because the DFnT matrix is unitary, the receiver recovers the transmitted information symbols by applying the inverse operation, DFnT.

IV. OCDM SIGNAL UNDER LTI SYSTEMS

The paper models OCDM transmission over static or quasi-static LTI channels with AWGN and guard intervals, then derives linear equalization using time- or frequency-domain processing.

  • The channel model assumes a static or quasi-static LTI channel with AWGN whose response remains constant within one OCDM block.
  • The receiver is assumed to have channel information, perfect timing, and perfect frequency synchronization.
  • Guard intervals prevent inter-symbol interference and may use either zero padding or a cyclic prefix.
  • Under CP or ZP conditions, the OCDM signal can be represented with a circulant channel impulse-response matrix and additive noise.
  • Channel compensation: The Fresnel-transform convolution property makes the chirp waveforms transparent to the channel after receiver DFnT, while unitary transformation preserves AWGN.
  • Channel compensation: Channel distortion can be compensated using multi-tap time-domain equalization or more efficient single-tap frequency-domain equalization.
  • Equalizer scope: The paper considers linear equalizers because nonlinear DFE and ML methods increase computational and hardware complexity.

B. Proposed Equalization Algorithm for OCDM

The paper proposes an efficient OCDM frequency-domain equalization algorithm based on Fresnel-transform eigen-decomposition. It uses phase cancellation and linear single-tap equalizers to compensate channel dispersion, while MMSE balances compensation against noise enhancement.

  • The proposed OCDM receiver uses single-tap frequency-domain equalization to compensate channel dispersion.The algorithm is based on the eigen-decomposition properties of the Fresnel transform and is illustrated in Fig. 7.
  • DFT-domain processing represents the received OCDM signal using diagonal channel-response and Fresnel-transform coefficient matrices.The channel frequency response and Fresnel-transform eigenvalues become diagonal in the DFT domain.
  • The receiver first cancels the phase induced by the Fresnel-transform coefficient matrix, then applies a diagonal single-tap equalizer.The equalizer coefficients can follow either zero-forcing or minimum mean square error criteria.
  • After equalization, inverse DFT processing recovers the transmitted information.The recovered signal differs according to whether ZF or MMSE equalization is used.
  • ZF completely removes channel distortion but enhances noise, whereas MMSE balances noise enhancement and channel compensation.
  • The OCDM implementation can use existing OFDM-compatible transform operations without significantly increasing computational complexity.The discrete Fresnel transform can be realized with fast Fourier transform algorithms.

A. Relation between the Fourier and Fresnel transforms

The discrete Fresnel transform differs from the DFT by additional quadratic phase factors. This relationship enables OCDM generation and recovery through FFT-based operations compatible with OFDM architectures.

  • The Fresnel-transform and DFT kernels are related through additional quadratic phases.Both transforms are trigonometric linear canonical transforms, but the Fresnel kernel contains quadratic phase terms beyond the Fourier kernel.
  • The discrete Fresnel transform can be implemented by FFT in three steps.
  • The proposed receiver architecture requires only a single-tap equalizer.An alternative receiver can use transform operations with either time-domain or frequency-domain equalization.
  • OCDM and OFDM transmit modulated waveforms in blocks with guard intervals to avoid intersymbol interference.Both cyclic prefix and zero padding can fill the OCDM guard interval.
  • The OCDM signal structure is compatible with OFDM and can be integrated into an OFDM system.
  • Relative to OFDM, OCDM adds phase rotations and an additional inverse DFT at the receiver.The receiver architecture can otherwise retain OFDM processing elements, including DFT-based equalization.

C. Arithmetic Complexity of OCDM

The paper compares OCDM and OFDM arithmetic complexity using complex multiplications per subcarrier or chirp. OCDM incurs modest additional processing, with receiver choice determining the overhead.

  • The comparison excludes compulsory operations such as synchronization and channel estimation because their complexity varies with the adopted algorithms.
  • For time-domain equalization, OCDM complexity is L per symbol when the transverse filter has L taps.The stated case assumes L is larger than the channel impulse-response length.
  • Frequency-domain equalization adds log2N complex multiplications per symbol beyond the single-tap equalizer.The FDE scheme also requires two additional DFT operations.
  • FDE is preferable to TDE in applications with relatively large channel delay spreads because of computational complexity.
  • Arithmetic complexity is measured by complex multiplications per subcarrier or chirp, with L denoting TDE taps and N denoting chirps or subcarriers.
  • With receiver scheme #2, OCDM complexity is only slightly increased relative to OFDM.The OCDM overhead depends on which receiver scheme is adopted.

VI. SIMULATION

Simulations evaluate OCDM against OFDM under 10-ray multipath Rayleigh and EVA channels using ZF and MMSE equalizers. OCDM benefits from multipath and spatial diversity, but low-SNR behavior depends on equalization and modulation level.

  • With ZF equalization in the 10-ray channel, OCDM requires higher SNR than OFDM for the same BER at low SNR.The BER curves approach OFDM as SNR increases because ZF noise enhancement becomes smaller.
  • With MMSE equalization, OCDM outperforms OFDM under multipath fading because it balances channel compensation and noise enhancement.The paper attributes the superior performance to OCDM’s use of multipath diversity.
  • OCDM with MMSE is inferior to OFDM at low SNR, with degradation becoming more pronounced from 4-QAM to 64-QAM.Noise enhancement remains under MMSE, and higher-order modulation is more sensitive to detrimental effects.
  • Receive diversity significantly improves both systems, and two-antenna OCDM with ZF becomes much better than OFDM in the EVA channel.Spatial diversity suppresses noise enhancement and improves OCDM performance in the low-SNR region.
  • The simulations conclude that spatial diversity suppresses noise enhancement while multipath diversity dominates performance for OCDM with linear equalizers.

VII. CONCLUSION

The paper establishes OCDM as an orthogonal chirp-multiplexing system based on the Fresnel transform, with a DFnT implementation and efficient single-tap equalization. Simulations show compatibility with OFDM, exploitation of multipath diversity, and improved performance over OFDM with both MMSE and, with receive diversity, ZF equalization.

  • System principle: OCDM orthogonally multiplexes a bank of chirp waveforms, using their amplitude and phase for modulation within the same bandwidth.The principle is based on the Fresnel transform rather than the Fourier transform underlying OFDM.
  • System principle: The Fresnel convolution theorem models OCDM transmission in LTI channels, while DFnT-based processing provides a digital implementation.The analysis covers transmission in linear time-invariant channels, and the implementation shows chirp waveforms are transparent to the dispersive channel.
  • Channel compensation: Eigen-decomposition of the DFnT matrices enables a simpler and more efficient single-tap equalization algorithm for channel compensation.Both time-domain and frequency-domain equalizers can be applied.
  • Implementation: The OCDM system is compatible with existing OFDM systems without significant modification.The paper discusses implementation compatibility with the OFDM system.
  • Evaluation: Simulations under wireless multipath channels show that OCDM exploits multipath diversity and that OCDM with MMSE equalization outperforms OFDM.Receive diversity also suppresses linear-equalizer noise enhancement and enables notable improvement for OCDM with ZF equalization compared with OFDM.
  • Conclusion: The paper presents OCDM as an attractive alternative for high-speed communication systems because of its OFDM compatibility and ability to counteract detrimental channel effects.The stated channel effects include noise, multipath fading, and Doppler effects.
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