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Characterizing the Narrowband Ambiguity Function of Frequency Modulated OFDM Waveforms

David A. Hague

arXiv:2610.10869v1eess.SP

TL;DR

FM-OFDM’s sensing properties are less understood than its communications applications, despite its potential relevance to ISAC waveform design. The paper characterizes its spectral and AF properties using the EOA model and demonstrates pilot-based ACF sidelobe optimization; it reports fixed RMS bandwidth for given waveform parameters and a Thumbtack-Like AF for practical designs.

  • Problem

    FM-OFDM’s sensing characteristics are little known, although understanding them could inform sensing-centric and joint ISAC waveform design.

    Source: §I · p. 1

  • Method

    The paper uses the EOA model to characterize FM-OFDM’s AF mainlobe and optimizes ACF sidelobes with a subset of sub-carrier symbols reserved as sensing pilots.

    Source: §I · p. 2

  • Results

    FM-OFDM has fixed RMS bandwidth for given h, T, and L, and practical designs almost always have a Thumbtack-Like AF; the paper also reports greater spectral efficiency than CE-OFDM.

    Source: §III · pp. 3–4, §V · p. 5

  • Takeaways & Limitations

    Pilot-based sidelobe optimization trades sensing refinement against communications throughput and is a preliminary step toward FM-OFDM optimization for SCWD and JWOD.

    Source: §V · p. 5

Abstract

from arXiv · show

This paper characterizes the spectral and Ambiguity Function (AF) properties of Frequency Modulated Orthogonal Frequency Division Multiplexing (FM-OFDM) waveforms employing Phase-Shift Keying (PSK). The mainlobe structure of the AF is well approximated by the Ellipse of Ambiguity (EOA) model. The EOA produces exact closed-form expressions that describe the AF mainlobe width in time-delay and Doppler as well as the degree of coupling between them. These expressions yield three fundamental insights into the spectral and AF shapes of FM-OFDM waveforms employing PSK: (1) these waveforms possess a fixed Root Mean Square (RMS) bandwidth for a given modulation index $h$, pulse-length $T$, and number of sub-carriers $L$, (2) they are roughly twice as spectrally efficient as Constant Envelope OFDM (CE-OFDM) waveforms with the same RMS bandwidth, and (3) for practical waveform design parameters will always possess a ``Thumbtack-Like'' AF shape. Finally, this paper demonstrates a waveform optimization technique that minimizes the FM-OFDM waveform's Auto-Correlation Function (ACF) sidelobes using only a subset of the sub-carrier symbols as sensing ``pilot'' symbols leaving the rest for communications. This pilot optimization technique is then demonstrated via two illustrative design examples.

I. INTRODUCTION

ISAC waveform design must account for both sensing and communications, but FM-OFDM’s sensing characteristics have received little attention. This paper characterizes its AF and spectral properties and demonstrates pilot-based ACF sidelobe optimization.

I. INTRODUCTION

  • ISAC waveform design spans communication-centric, sensing-centric, and joint optimization approaches, each accommodating both functions to different degrees.Joint design aims to optimize sensing and communications together for a potentially better tradeoff.Source: §I · p. 1
  • Conventional multicarrier waveforms can have high PAPR, requiring transmitter power backoff, while CE-OFDM encodes symbols in instantaneous phase to maintain a constant envelope.The source links high PAPR to reduced transmitter energy efficiency and describes CE-OFDM as better suited to efficient transmission on practical electronics,.Source: §I · p. 1
  • FM-OFDM is presented as a constant-envelope ISAC waveform whose sensing characteristics are little known, despite prior work emphasizing communications applications.The paper positions better understanding of its sensing properties as relevant to sensing-centric and joint waveform design.Source: §I · p. 1
  • FM-OFDM is described as more spectrally efficient than CE-OFDM while retaining constant modulus, and prior FM-OFDM work has focused mainly on communications.The cited communications topics include receiver design [13], [14], channel estimation, and time-varying-channel robustness.Source: §I · p. 1
  • The paper uses PSK-encoded FM-OFDM and the Ellipse of Ambiguity model to characterize its AF mainlobe and derive spectral and AF insights.The EOA characterization leverages results from MTSFM and radar literature –.Source: §I · pp. 1–2
  • The paper’s pilot optimization reduces FM-OFDM ACF sidelobes using a subset of sub-carrier symbols, leaving the remaining symbols for communications.The number of sensing symbols trades off optimization performance against communications throughput.Source: §I · p. 2

II. THE FM-OFDM WAVEFORM MODEL

The section introduces the FM-OFDM waveform model and describes the AF’s mainlobe and sidelobe structure.

II. THE FM-OFDM WAVEFORM MODEL

  • The section formally defines the FM-OFDM waveform model.Source: §II · p. 2
  • The section describes the ambiguity function’s mainlobe structure.Source: §II · p. 2
  • The section also describes the ambiguity function’s sidelobe structure.Source: §II · p. 2

A. The Structure of the Ambiguity Function

The section establishes the FM pulse, ambiguity function, and autocorrelation function definitions used to describe waveform response and AF/ACF shape.

A. The Structure of the Ambiguity Function

1) Mainlobe Structure:

The EOA approximates the AF mainlobe with a coupled ellipse whose parameters characterize range and Doppler sensitivity and coupling. The section gives closed-form quantities for FM-OFDM and notes that discrete modulation coefficients can shape the mainlobe.

1) Mainlobe Structure:

  • The EOA is the fixed-height contour of a second-order AF mainlobe approximation and is a coupled ellipse.The AF mainlobe relates to single-target range and Doppler estimation and multiple-target resolution.Source: 1) Mainlobe Structure · p. 2
  • In the EOA expression, RMS bandwidth determines time-delay sensitivity, RMS pulse-length determines Doppler sensitivity, and ρ is the range-Doppler coupling factor.Source: 1) Mainlobe Structure · p. 2
  • The RMS bandwidth, RMS pulse-length, and range-Doppler coupling factor are expressed using waveform quantities including its spectrum, time support, phase derivative, and first time moment.The passage states that the RMS pulse-length depends solely on pulse length T, while bandwidth and coupling are largely dependent on the modulation function.Source: 1) Mainlobe Structure · p. 2
  • The FM-OFDM EOA parameters are available in exact closed form, and discrete modulation coefficients can be selected to design specific EOA characteristics and shape the AF mainlobe.Source: 1) Mainlobe Structure · p. 2

2) Sidelobe Structure:

The section uses the Generalized Integrated Sidelobe Ratio (GISR) to evaluate and optimize the ACF sidelobe structure. Its finite-p form approximates PSLR while retaining a smooth objective for gradient-descent optimization.

2) Sidelobe Structure:

  • GISR evaluates the ratio of ℓp-norms over the ACF sidelobe and mainlobe regions.The paper uses it as the primary metric for evaluating and optimizing ACF sidelobes.Source: 2) Sidelobe Structure · p. 3
  • For integer p ≥ 2, the ACF mainlobe spans 2∆τ between its first nulls, while Ωτ excludes the mainlobe.At p = 2, GISR is the standard ISR; as p approaches infinity, it approaches PSLR.Source: 2) Sidelobe Structure · p. 3
  • A large but finite p approximates PSLR with a smooth objective that can be traversed using gradient-descent waveform optimization.The max-norm can produce a discontinuous objective that prevents efficient use of gradient-descent methods.Source: 2) Sidelobe Structure · p. 3

B. The FM-OFDM Waveform

The FM-OFDM waveform is an FM waveform whose instantaneous frequency function takes an OFDM modulation form. The paper uses M-PSK symbols and relates this model to CE-OFDM and MTSFM.

B. The FM-OFDM Waveform

  • FM-OFDM is realized as an FM waveform whose instantaneous frequency function is a form of OFDM modulation.Source: B. The FM-OFDM Waveform · p. 3
  • The waveform has L sub-carriers, modulation index h controlling operational bandwidth, and in-phase and quadrature symbol values aℓ and bℓ.Source: B. The FM-OFDM Waveform · p. 3
  • With M-PSK, sub-carrier magnitudes satisfy |Γℓ| = 1; FM-OFDM is closely related to CE-OFDM and is a special case of MTSFM.Unlike FM-OFDM's PSK symbols, the MTSFM model allows aℓ and bℓ to vary in magnitude.Source: B. The FM-OFDM Waveform · p. 3

III. FM-OFDM AF MAINLOBE PROPERTIES

The EOA model characterizes FM-OFDM AF mainlobe properties and yields closed-form RMS bandwidth and coupling results. The analysis finds similar PSD shapes but roughly twice the sub-carriers of CE-OFDM at the same RMS bandwidth, and near-zero range-Doppler coupling for practical designs.

III. FM-OFDM AF MAINLOBE PROPERTIES

  • The EOA parameters characterize the AF mainlobe, including the relationship between time-delay and Doppler.The EOA-derived expressions are presented as closed-form results for the waveform’s spectral and AF characteristics.Source: §III · p. 3
  • The RMS bandwidth expression is fixed for all real h, T, and L, regardless of the encoded PSK sequence.RMS bandwidth measures both time-delay resolution and spectral spread.Source: §III · p. 3
  • FM-OFDM and CE-OFDM have nearly identical PSD shapes at the same RMS bandwidth, while FM-OFDM uses twice as many sub-carriers.The PSDs are well modeled by a Gaussian with variance given by the RMS bandwidth.Source: §III · pp. 3–4, Figure 1 · p. 4
  • The normalized RDCF behaves like a correlation coefficient, and its distribution tightens around zero as the number of sub-carriers increases.The violin plots show normalized RDCF values across PSK sequences for varying L.Source: §III · pp. 3–4
  • Practical FM-OFDM designs with tens of sub-carriers have nearly zero range-Doppler coupling and a “Thumbtack-Like” AF shape.As L increases, the distribution of normalized coupling concentrates near zero.Source: §III · p. 4, Figure 2 · p. 4

IV. CONTROLLING THE ACF SIDELOBES OF FM-OFDM WAVEFORMS

The paper optimizes FM-OFDM ACF sidelobes by adjusting a subset of sub-carrier symbols while reserving the rest for communications. The examples show lower sidelobes with more optimized symbols, at the cost of communications throughput.

IV. CONTROLLING THE ACF SIDELOBES OF FM-OFDM WAVEFORMS

  • Earlier CE-OFDM sidelobe optimization used continuous-valued M-PSK coefficients and removed communications throughput.The FM-OFDM approach instead reserves a subset of symbols for communications.Source: §IV · pp. 4–5
  • The routine optimizes ACF sidelobes over a subset of P sub-carrier symbols, reserving the remaining Q = L − P symbols for communications.The sensing symbols act analogously to pilot symbols, and changing P and Q trades sensing refinement against communications throughput.Source: §IV · p. 5
  • Using more optimized symbols P produced noticeably lower ACF sidelobes, but reduced communications throughput.The examples compare P = 16, 32, and 64 for two delay regions; the routine finds only a local minimum and is not computationally streamlined.Source: §IV · p. 5

V. CONCLUSION

The paper derives closed-form characterizations of the spectral and AF shapes of PSK FM-OFDM and demonstrates pilot-symbol optimization of its ACF sidelobes. The examples show a tradeoff between sensing performance and communications throughput.

V. CONCLUSION

  • The derived closed-form expressions characterize the spectral and AF shapes of PSK FM-OFDM waveforms.The paper concludes that these waveforms are more spectrally efficient than CE-OFDM and almost always have a “Thumbtack-Like” AF.Source: §V · p. 5
  • Increasing the number of sensing symbols P reduces ACF sidelobes in the prescribed delay region, at the cost of communications throughput.Figure 3 compares optimization over all time-delays with optimization over a sub-region.Source: Figure 3 · p. 5, §V · p. 5
  • The pilot optimization technique minimizes FM-OFDM ACF sidelobes using a subset of sub-carrier symbols as sensing pilots, leaving the rest for communications.The illustrative examples show a tradeoff between refining sensing characteristics and communications throughput.Source: §V · p. 5

APPENDIX DERIVING THE FM-OFDM’S EOA PARAMETERS

The appendix derives the RMS bandwidth and range-Doppler coupling factor for the rectangularly windowed FM-OFDM waveform.

APPENDIX DERIVING THE FM-OFDM’S EOA PARAMETERS

A. RMS Bandwidth

The appendix derives RMS bandwidth using the waveform expressions and Parseval’s theorem, and derives the range-Doppler coupling factor by evaluating integrals.

A. RMS Bandwidth

  • The RMS bandwidth derivation inserts the waveform expression into equation (7), then simplifies an integral using Parseval’s theorem for Fourier series.The supplied passages do not state the resulting expression.Source: A. RMS Bandwidth · pp. 5–6

B. Range-Doppler Coupling Factor

  • The range-Doppler coupling derivation expands an expression using a trigonometric identity; its first integral is zero by odd symmetry, and integration by parts evaluates the second.The resulting RDCF uses |Γℓ| = 1.Source: B. Range-Doppler Coupling Factor · p. 6
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