Near Constant-Envelope OFDM Waveform

• Near constant-envelope OFDM waveforms are designed to minimize envelope fluctuations, reducing PAPR and enabling efficient nonlinear power amplifier operation.
• They employ diverse synthesis methods—from FM-based modulation and convex optimization to deep learning—to maintain spectral efficiency and robust autopilot characteristics.
• Applications include power-efficient wireless transmission and integrated sensing and communications, offering improved radar ambiguity and enhanced channel performance.

A near constant-envelope Orthogonal Frequency-Division Multiplexing (OFDM) waveform is an OFDM-based signal engineered to have minimal instantaneous envelope fluctuations, enabling highly efficient operation of nonlinear power amplifiers and mitigating peak-to-average power ratio (PAPR) drawbacks of conventional OFDM. The practical motivation is to facilitate power-efficient, robust communications—and, in radar and integrated sensing and communications (ISAC) scenarios, to yield waveforms with desirable ambiguity and autocorrelation properties—while retaining the spectral efficiency and implementation flexibility of OFDM. A diverse set of algorithmic and architectural strategies exists to synthesize such waveforms, ranging from FM/phase-modulation schemes to deep learning, convex optimization, and signal-invariant filter design.

1. Theoretical Formulation: Constant-Envelope OFDM and Related Models

The constant-envelope OFDM (CE-OFDM) model is defined by synthesizing the transmit waveform as a sum of orthogonal subcarriers, phase-modulated such that the composite envelope is strictly (or nearly) invariant. Let $L$ denote the number of subcarriers and $h$ the modulation index. Then, with $T$ the pulse duration and $\{\phi_\ell\}_{\ell=1}^L$ phase codes (commonly PSK),

$$s(t) = \frac{1}{\sqrt{T}}\,\text{rect}\left(\frac{t}{T}\right)\exp\left(j\,2\pi h \sum_{\ell=1}^L \left|\Gamma_\ell\right| \cos\left(2\pi \ell t/T + \phi_\ell\right)\right)$$

where $\left|\Gamma_\ell\right|=1$, and $\{\phi_\ell\}$ take PSK values. This phase law is a special case of the multi-tone sinusoidal frequency modulated (MTSFM) model constrained such that each Fourier coefficient is on the unit circle (Felton et al., 2023, Felton et al., 2023).

Alternative approaches include direct time-domain FM modulations (FM-OFDM (Bouziane et al., 22 Aug 2025)), phase quantization in massive MIMO precoding (e.g., SQUID-OFDM (Jacobsson et al., 2017)), and affine combinations with chirp signals (as in “amalgamated chirp–OFDM” (Kumar et al., 22 Jan 2026)). The category also encompasses near constant-envelope waveforms obtained via constrained optimization, neural network learning, or tone-reservation (Huleihel et al., 2020, Huang et al., 2015, Lu et al., 2023, Li et al., 17 Mar 2025).

2. Synthesis Methodologies

2.1 Phase-Constrained and FM-based Designs

Classical CE-OFDM creates constant-envelope waveforms by modulating the instantaneous phase according to a (possibly optimized) sum of sinusoids weighted by PSK codebooks, as in the canonical formula above (Felton et al., 2023). FM-OFDM (Frequency Modulated OFDM) applies an FM nonlinearity to the time-domain OFDM symbol:

$$f[n] = m f_\Delta x[n],\quad \phi[n] = \phi_0 + 2\pi T_s \sum_{u=0}^n f[u],\quad s[n] = A e^{j\phi[n]}$$

This approach achieves strict constant envelope ($|s[n]| = A$) with tunable spectral occupancy and resilience to amplifier nonlinearities (Bouziane et al., 22 Aug 2025).

2.2 Optimization-Based Envelope Equalization

Many strategies formulate envelope flattening as an optimization problem. Common variants include:

• Tone reservation with least-squares fitting: Partitioning the frequency-domain symbol vector into a reserved set (free variables) and an informative set (fixed), with the optimization minimizing the coefficient of variation of the envelope (CVE). See (Huang et al., 2015):

$$\min_{\mathbf{b},\,\beta,\,\boldsymbol\theta}\; \left\|\,\mathbf{c}+\mathbf{B}\,\mathbf{b} -\beta\,e^{j\boldsymbol\theta}\right\|_2^2$$

Iterative schemes guarantee monotonic decrease in PMEPR (peak-to-mean envelope power ratio).

• Phase Difference Constrained PAPR Minimization: As in the ADMM-PLPOI scheme, impose constraints in the frequency domain: unimodularity ($|x_n|=1$), limited phase deviation from original symbols ($|\arg x_n - \arg c_n| < \theta$), and explicit PAPR constraint. Solved efficiently via the alternating direction method of multipliers (Li et al., 17 Mar 2025).
• Two-stage Resource/Phase Optimization for ISAC: First optimize subcarrier power and assignment (possibly imposing minimum distance between active comm subcarriers for interference-robustness), then phase-align the remaining “sensing” carriers to minimize envelope fluctuations via least-squares (Lu et al., 2023).

2.3 Machine Learning Approaches

Deep learning-based architectures (notably convolutional autoencoders) operate end-to-end, mapping input symbol vectors to near-constant-envelope time-domain waveforms while jointly minimizing mean-square error, PAPR, and spectral regrowth measures such as adjacent channel power ratio (ACPR). These models include differentiable amplifier nonlinearities as layers during training and employ schedule-based multi-objective learning (Huleihel et al., 2020).

2.4 DFT-Spread and CPM-Inspired Modulation

Waveform design based on continuous-phase modulation (CPM) or its hybridizations with DFT-spread-OFDM (e.g., 3MSK) constrains phase transitions to small steps (e.g., $\pm\pi/2$), with or without excess-bandwidth allocation. These realize near-constant envelope and strong phase noise robustness (Renfors et al., 2021).

2.5 Implementation in CP-OFDMA and FDMA

Autonomous constant-envelope signaling can be integrated into standardized OFDMA frameworks via pulse-shaping filters obeying the strict envelope constraint in time. Optimized (possibly parametric plus windowed) FIR prototype filters yield both spectral containment and the CE property. Binary pilot and multi-stage estimation methods ensure robust channel estimation and equalization under these modulations (Zhu et al., 28 Oct 2025).

3. Mathematical Properties: PAPR, Ambiguity Function, and Sidelobe Control

A key metric is the PAPR (or PMEPR), defined as

$$\mathrm{PAPR} = \frac{\max_{n}|x[n]|^2}{\frac{1}{N}\sum_{n=0}^{N-1}|x[n]|^2}$$

Ideal CE-OFDM/FM-OFDM/FDM/CPM-based models achieve PAPR = 1 (0 dB); near-constant-envelope optimizations deliver 3–6 dB PAPR depending on constraint tightness and codebook size (Bouziane et al., 22 Aug 2025, Li et al., 17 Mar 2025, Huleihel et al., 2020, Huang et al., 2015).

Radar/ISAC utility is evaluated via the ambiguity function (AF), especially the shape and sidelobe structure:

• Closed-form mainlobe characterization (EOA model): For CE-OFDM, the AF mainlobe forms an ellipse with analytically derived RMS bandwidth and negligible range-Doppler coupling for large subcarrier count, yielding a “thumbtack-like” AF (Felton et al., 2023).
• Generalized Integrated Sidelobe Level (GISL): Gradient-descent-based CE-OFDM design can further suppress ACF sidelobes in user-specified windows while maintaining (near-)constant envelope (Felton et al., 2023).

4. Empirical Performance: Quantitative Benchmarks

Comparative quantitative results are as follows:

Waveform / Method	Typical PAPR CCDF ($10^{-4}$)	Side/Autocorrelation	BER Penalty / Trade-off	Special Features
Standard OFDM	11–13 dB	Tall sidelobes	Baseline	High dynamic range, robust demodulation
CE-OFDM/FM-OFDM	0–0.4 dB	Thumbtack/“good”	None/Negligible	Strict CE, ideal for efficient PA
ADMM-PLPOI ($\theta=0.6$)	4.5 dB	Low autocorr. sidelobes	$<$1 dB QPSK loss	Real-time feasible, trade-off via $\theta$
CAE autoencoder	5.5 dB	Controlled by $\lambda_2$/$\lambda_3$	$\lesssim$2 dB	Deep learning, supports nonlinear PA models
3MSK/CPM Hybrids	1.3–4.9 dB	CPM-like, low OOB	0–1.2 dB in strong PN	Noncoherent detection, high PA output
TR-CVE LS (tone reservation)	1.05 dB	Sidelobes as unmod.	None	No data loss if S$^I$ $\gg$ S$^R$
NCE-CP-OFDMA (Zhu et al., 28 Oct 2025)	0.37 dB	Sidelobes $<$ –30dB	$<$0.2 dB vs QPSK	3GPP-compliant, mult-user, simple Rx
Hybrid Chirp–OFDM ($\alpha=0.5$)	9.0 dB	Mixed thumbtack/sinc	Moderate ($\sim$ few dB)	Sensing/comm trade-off

Trade-offs exist: overly tight envelope constraints can degrade BER (esp. for QAM), setpoint selection (e.g., $\theta$ in PLPOI) is crucial, and tone-reservation reduces net rate. Hybrid methods (e.g., affine chirp-OFDM combinations) offer nontrivial PAPR reduction at moderate BER penalty and improved range/Doppler estimation for ISAC (Kumar et al., 22 Jan 2026, Li et al., 17 Mar 2025, Huang et al., 2015, Zhu et al., 28 Oct 2025).

5. Receiver Architectures and Channel Estimation

Constant-envelope modulations frequently necessitate receiver modifications:

• PLL/Discriminator Receivers: FM-OFDM and CE-OFDM variants often require robust phase tracking/discrimination front-ends for demodulation and range/Doppler estimation (Bouziane et al., 22 Aug 2025).
• ESPRIT-aided Channel Estimation: Customized estimators for binary CE pilot sequences (instead of conventional flat pilots) have been developed, exploiting the structure for efficient, almost-ideal channel recovery (Zhu et al., 28 Oct 2025).
• Deep Receivers: In end-to-end learned systems, the receiver is a trainable neural network tuned to invert both the transmitter and PA nonlinearity, often including Bussgang linearization (Huleihel et al., 2020).
• Noncoherent Trellis Decoding: In CPM-based (e.g., 3MSK) systems, trellis-based noncoherent detection is preferred, with built-in phase noise immunity and self-tracking (Renfors et al., 2021).

6. Applications and Implementation Considerations

Applications include:

• Power-efficient wireless transmission: By enforcing (near-)constant envelope, transmitters can operate deep into PA saturation with no DPD, boosting battery life and deployment flexibility (Jacobsson et al., 2017, Zhu et al., 28 Oct 2025).
• Integrated sensing and communications (ISAC): CE-OFDM and its hybrids support joint radar-communications, achieving near thumbtack AF for accurate range/Doppler, low spectral leakage, and strong interference robustness (Kumar et al., 22 Jan 2026, Lu et al., 2023).
• Massive MIMO and mmWave base stations: Phase-quantized constant-envelope waveforms reduce RF front-end complexity and data rates, especially in large array deployments (Jacobsson et al., 2017).
• IoT and coverage-limited scenarios: CPM-inspired and FDMA CE designs are advantageous under severe phase noise, low SNR, or NTN/LPWAN constraints (Renfors et al., 2021, Zhu et al., 28 Oct 2025).

Implementation issues include phase-quantized DAC mapping, computational cost (often managed by FFT-based algorithms or low-complexity iterative schemes), and coding/mapping for multi-user support. Strict envelope preservation can be relaxed (near-CE) to further suppress autocorrelation/range sidelobes at negligible BER penalty (Zhu et al., 28 Oct 2025).

7. Trade-Offs, Parameter Selection, and Guidelines

Optimal operation hinges on trade-offs between envelope flatness (PAPR/CVE), spectral containment, autocorrelation properties, BER, throughput, and complexity:

• Parameter selection: Tightening phase-difference or amplitude constraints enhances envelope flatness at the cost of BER/reliability—selecting, e.g., $\theta\sim0.5$–$0.6$ radians in PD-based designs gives strong PAPR suppression with minimal BER loss (Li et al., 17 Mar 2025).
• Tone-reservation schemes: Trading a fraction of subcarriers for envelope control requires balancing data/pilot overhead and desired PMEPR (Huang et al., 2015).
• Spectral shape and regulatory compliance: Optimal filter design under the CE constraint can achieve $>$30 dB sidelobe rejection without distorting orthogonality (Zhu et al., 28 Oct 2025).
• Multi-objective learning: Gradual loss scheduling in deep models is recommended for first ensuring low symbol error, then squeezing envelope variations with explicit PAPR/ACLR/ACPR penalties (Huleihel et al., 2020).

Empirical guidelines emphasize moderate constraint tightness, careful pilot/sequence design, oversampling for PAPR capture, and exploiting structure (e.g., periodicity, conjugate symmetry in FDMA) for receiver efficiency.

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A near constant-envelope OFDM waveform is thus the result of diverse engineering and mathematical principles, with its realization context-dependent on system goals—be it high amplifier efficiency, robust ISAC performance, or compatibility with standardized frameworks. Algorithmic flexibility—from parametric phase law synthesis and convex optimization to machine learning—underpins current and emerging solutions in this area (Felton et al., 2023, Bouziane et al., 22 Aug 2025, Li et al., 17 Mar 2025, Huleihel et al., 2020, Huang et al., 2015, Zhu et al., 28 Oct 2025, Lu et al., 2023, Kumar et al., 22 Jan 2026, Felton et al., 2023, Renfors et al., 2021, Jacobsson et al., 2017).