Constant-Envelope OFDM (CE-OFDM)

• CE-OFDM is a modulation technique that uses phase modulation to embed OFDM structure, achieving a constant envelope over the waveform.
• It enables closed-form spectrum, ambiguity, and autocorrelation analyses by converting amplitude variation into phase information.
• The design supports power-efficient transmission and radar/communications applications via gradient-based sidelobe control and precise parameter tuning.

Constant-Envelope OFDM (CE-OFDM) is an OFDM-derived modulation class in which the complex baseband waveform is constructed to have constant, or approximately constant, modulus in time, so that information is carried through instantaneous phase or frequency rather than through complex amplitude. In the radar-communications literature, CE-OFDM is commonly modeled as a baseband FM waveform with rectangular support and unit energy, and under PSK symbolization it is also a special case of the Multi-Tone Sinusoidal FM (MTSFM) waveform. This dual interpretation is central: it explains both the transmitter-side appeal of CE-OFDM for nonlinear, power-efficient amplifiers and the availability of closed-form spectrum, ambiguity-function, and autocorrelation analyses, as well as gradient-based waveform design procedures for range-sidelobe control (Felton et al., 2023, Felton et al., 2023).

1. Signal model and modulation principle

A standard continuous-time CE-OFDM model writes the baseband signal as

$$s(t)=\frac{\operatorname{rect}(t/T)}{\sqrt{T}}\,e^{j\varphi(t)},$$

with waveform duration $T$, unit-energy normalization, and instantaneous phase $\varphi(t)$. Because the waveform is of the form $e^{j\varphi(t)}$ multiplied by a real window, the envelope is constant on the support, ignoring the window edges. In this formulation, CE-OFDM is not a linear superposition of subcarrier amplitudes in the usual OFDM sense; rather, an OFDM-like Fourier series is embedded in the phase law itself (Felton et al., 2023).

One representation used for CE-OFDM phase is

$$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$

with complex PSK symbols $c_\ell$, modulation index $h$, and $L$ positive-frequency subcarriers. After conversion to a real trigonometric series and, under PSK, $|\Gamma_\ell|=1$, the same model becomes

$$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$

The corresponding instantaneous frequency is

$T$0

Hence CE-OFDM is an FM waveform whose modulation function is a smooth multi-tone sinusoidal series, with PSK symbol phases entering as phase offsets of the sinusoidal components (Felton et al., 2023).

This immediately distinguishes CE-OFDM from conventional OFDM. Conventional OFDM forms a linear sum of subcarriers with complex amplitudes and therefore exhibits large envelope fluctuations and high PAPR. CE-OFDM instead places the OFDM-like structure into $T$1, producing a constant-envelope signal that is markedly more compatible with highly efficient nonlinear RF power amplifiers. A common source of confusion is to treat CE-OFDM as merely “low-PAPR OFDM”; more precisely, it is an FM-type, phase-domain realization of OFDM structure (Felton et al., 2023).

The MTSFM connection is mathematically exact under PSK. General MTSFM uses

$T$2

where $T$3 are arbitrary real amplitudes. CE-OFDM corresponds to the special case $T$4, and under PSK, $T$5 for all $T$6. This identification allows CE-OFDM to inherit generalized-Bessel-function and Ellipse-of-Ambiguity tools previously developed for MTSFM analysis (Felton et al., 2023).

2. Closed-form spectrum, autocorrelation, and ambiguity structure

Because the CE-OFDM phase is multi-tone sinusoidal, the waveform admits a generalized-Bessel-function expansion,

$T$7

where $T$8 denotes an $T$9-dimensional generalized Bessel function. This yields a compact exact spectrum,

$\varphi(t)$0

so the spectrum is expressed as a weighted sum of sinc lines at harmonics of $\varphi(t)$1. The weights encode the modulation index, number of tones, and PSK phase pattern (Felton et al., 2023).

The radar ambiguity function is

$\varphi(t)$2

and the autocorrelation function (ACF) is $\varphi(t)$3. For CE-OFDM, the closed-form ambiguity function is a double generalized-Bessel sum with a triangular delay-support factor $\varphi(t)$4 and sinc kernels in Doppler. The corresponding ACF inherits a structured multi-lobed form whose sidelobes depend strongly on the PSK code, even though key mainlobe parameters do not (Felton et al., 2023).

Near the origin, the mainlobe is characterized by the Ellipse of Ambiguity (EOA),

$\varphi(t)$5

For rectangular CE-OFDM, the closed-form parameters are

$\varphi(t)$6

and

$\varphi(t)$7

Under PSK, $\varphi(t)$8, so the RMS bandwidth depends only on $\varphi(t)$9, $e^{j\varphi(t)}$0, and $e^{j\varphi(t)}$1, not on the particular PSK phases. This exact invariance is one of the defining structural properties of PSK-encoded CE-OFDM (Felton et al., 2023).

The normalized range-Doppler coupling factor is

$e^{j\varphi(t)}$2

with maximum

$e^{j\varphi(t)}$3

Since $e^{j\varphi(t)}$4 decays approximately as $e^{j\varphi(t)}$5, PSK CE-OFDM “will almost always possess a thumbtack-like AF shape,” especially for practical values such as $e^{j\varphi(t)}$6. In the example with $e^{j\varphi(t)}$7, time-bandwidth product $e^{j\varphi(t)}$8, and random $e^{j\varphi(t)}$9 symbols, the waveform had $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$0, PSLR $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$1 dB, and ISL $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$2 dB (Felton et al., 2023).

These results imply a clean separation between design variables. The parameters $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$3, $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$4, and $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$5 fix bandwidth and mainlobe scale, whereas the PSK phases $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$6 primarily control sidelobes and range-Doppler coupling. This separation underlies later optimization-based CE-OFDM synthesis.

3. Autocorrelation sidelobes and GISL-based waveform synthesis

For matched-filter radar reception, CE-OFDM’s ACF governs range resolution and sidelobe masking. CE-OFDM with PSK has been shown to exhibit thumbtack-like ambiguity functions, but also a pedestal of ACF sidelobes. In this setting, the conventional metrics are the peak-to-sidelobe level ratio (PSLR) and integrated sidelobe level (ISL), defined over delay regions outside the first-null mainlobe interval $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$7. To obtain a differentiable objective that interpolates between ISL and PSLR, CE-OFDM sidelobe design can be formulated with the generalized integrated sidelobe level,

$$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$8

where $$\varphi(t)=2\pi h\sum_{\ell=-L}^{L} c_\ell e^{j\frac{2\pi \ell t}{T}},$$9 reduces to ISL and $c_\ell$0 approaches PSLR. Large but finite $c_\ell$1 gives a smooth yet PSLR-like objective (Felton et al., 2023).

The gradient-descent GISL (GD-GISL) method parameterizes the CE-OFDM waveform entirely by the continuous PSK phases $c_\ell$2, corresponding to the idealized case $c_\ell$3. After time sampling, the discrete phase vector is expressed through sampled cosine and sine basis matrices, and the ACF samples are computed by FFT-based convolution: $c_\ell$4 with $c_\ell$5 the zero-padded waveform. The discrete GISL objective becomes

$c_\ell$6

where $c_\ell$7 and $c_\ell$8 select sidelobe and mainlobe regions. The full gradient is derived by Wirtinger calculus, and each iteration uses only a small number of FFT/IFFT operations, yielding $c_\ell$9 complexity per iteration (Felton et al., 2023).

The update law is a gradient-descent step with heavy-ball momentum,

$h$0

together with a reset to steepest descent whenever the momentum direction ceases to be a descent direction. Step sizes are selected by Armijo backtracking, and the algorithm terminates when the gradient norm falls below a threshold or a maximum iteration count is reached (Felton et al., 2023).

For $h$1, $h$2, time-bandwidth product $h$3, $h$4, $h$5, and $h$6 iterations, the algorithm produced two representative outcomes. When the entire sidelobe region was penalized, GISL improved from $h$7 dB to $h$8 dB and PSLR from $h$9 dB to $L$0 dB. When optimization was restricted to the sub-region $L$1, GISL improved from $L$2 dB to $L$3 dB and PSLR from $L$4 dB to $L$5 dB. In both cases the mainlobe width and spectral extent were unchanged, because for fixed $L$6 and $L$7 the RMS bandwidth is fixed under PSK encoding (Felton et al., 2023).

The significance is methodological as well as numerical. CE-OFDM phases $L$8 are natural continuous design variables; GISL provides a differentiable sidelobe objective; and FFT-based gradients make high-dimensional CE-OFDM waveform optimization computationally feasible.

4. Alphabet constraints, phase quantization, and discrete implementation

The most favorable optimization setting for CE-OFDM is the continuous-phase idealization $L$9, where each $|\Gamma_\ell|=1$0 can take any value in $|\Gamma_\ell|=1$1. This permits smooth gradient descent, fine-grained control of the phase modulation function, and strong local sidelobe suppression. Practical CE-OFDM systems, however, usually employ finite $|\Gamma_\ell|=1$2-ary PSK, so the optimized continuous phases must be quantized to

$|\Gamma_\ell|=1$3

That post-optimization truncation introduces phase perturbations $|\Gamma_\ell|=1$4, which alter $|\Gamma_\ell|=1$5, $|\Gamma_\ell|=1$6, and the resulting ACF (Felton et al., 2023).

The empirical effect is systematic. Larger $|\Gamma_\ell|=1$7 means smaller phase perturbations and better preservation of the optimized ACF. Smaller $|\Gamma_\ell|=1$8 produces increasingly severe sidelobe growth in the delay region targeted by the GISL objective. In the reported simulations, $|\Gamma_\ell|=1$9-PSK preserved most of the continuous-phase sidelobe suppression, whereas lower-order PSK substantially degraded it. This does not contradict the constant-envelope property: modulus remains constant because all PSK symbols stay on the unit circle, but the finely optimized autocorrelation structure is sensitive to phase quantization (Felton et al., 2023).

This sensitivity should not be confused with loss of bandwidth control. Under PSK, the exact RMS bandwidth expression contains no dependence on $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$0, so symbol quantization does not alter $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$1; what changes is the detailed interference structure represented by sidelobes and range-Doppler coupling (Felton et al., 2023).

From an implementation perspective, the discrete-time CE-OFDM synthesis route is unusually favorable. The waveform is parameterized only by subcarrier phases; ACF and gradient evaluations are FFT-based; and the per-iteration cost is $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$2. This makes CE-OFDM tractable even for large numbers of samples or long waveforms. A plausible implication is that direct optimization over discrete phase alphabets, rather than optimize-then-quantize, is a natural next step when finite-PSK sidelobe fidelity is critical.

5. Multiantenna, OFDMA, and quantized-precoding extensions

CE-OFDM is often discussed alongside constant-envelope precoding, but the two are not identical. CE-OFDM designs a single-transmitter or per-antenna waveform whose time-domain samples satisfy a constant-modulus constraint while retaining OFDM structure. CE precoding in massive MU-MIMO instead enforces constant modulus at each antenna and symbol time in order to suppress multiuser interference. The underlying feasible sets are closely related, but the optimization targets differ: CE-OFDM emphasizes waveform shape, spectral behavior, demodulability, or sensing metrics, whereas CE precoding emphasizes MUI suppression and downlink performance (He et al., 2020).

The multiuser OFDM extension of this idea appears explicitly in nonlinear massive MU-MIMO-OFDM precoding. SQUID-OFDM adapts squared-infinity-norm Douglas-Rachford splitting to frequency-selective massive MU-MIMO-OFDM and generates per-antenna time-domain transmit signals constrained to constant envelope; subsequent phase quantization produces finite-cardinality transmit alphabets suitable for low-resolution DACs, including 1-bit implementations (Jacobsson et al., 2017). In a cell-free massive MIMO-OFDM setting, quantized CE precoding has been combined with AP-level power control, where some APs are attenuated or shut off if they are “harmful” under coarse phase quantization; the reported result is a substantial uncoded BER improvement relative to the maximum-antenna-power baseline for both 1-bit and 2-bit DACs (Demir et al., 22 May 2026).

A different line of work focuses on compatibility with the prevailing CP-OFDMA framework. A CP-OFDMA-compatible CE waveform has been constructed from a general CE FDMA signal model, an optimized CE-constrained pulse-shaping filter, time-domain binary pilot optimization for frequency-domain CE pilot behavior, and a receiver that combines delay-domain denoising, power-delay-profile estimation, reduced-dimension LMMSE channel estimation, and an MRC-aided LMMSE equalizer exploiting periodicity and conjugate symmetry. In that construction, strict CE and near-CE versions reduce PAPR dramatically while preserving FFT-based transceiver structure and New Radio compatibility (Zhu et al., 28 Oct 2025).

These developments broaden the meaning of CE-OFDM from a single waveform formula to a family of architectures. Some instances are phase-series FM waveforms analyzed via ambiguity theory; others are nonlinear OFDM precoders with CE time-domain constraints; others retrofit CE behavior into CP-OFDMA-compatible stacks. The common denominator is the deliberate conversion of OFDM’s high-envelope-fluctuation problem into a phase-structured constant-modulus design problem.

6. Applications, related variants, and design trade-offs

The most fully developed application domain for CE-OFDM in the cited literature is radar and dual-function radar/communications. The combination of constant envelope, compact spectrum, and analyzable ambiguity structure makes CE-OFDM attractive when a waveform must simultaneously satisfy transmitter efficiency constraints and range-Doppler requirements. The ability to optimize ACF sidelobes in selected delay intervals, without changing mainlobe width or spectral extent, is particularly relevant to near-zero-delay interference control in DFRC overlays (Felton et al., 2023).

Related ISAC work has also explored waveforms that are not strictly CE-OFDM in the phase-series sense but are clearly part of the same design space. The affine addition of a chirp and an OFDM signal produces a near constant-envelope OFDM waveform, termed AAC-OFDM, with an explicit PAPR bound and a threshold on the chirp weight $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$3 guaranteeing PAPR reduction. In the reported FR2 example with $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$4 and QPSK, the PAPR at CCDF $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$5 decreased from about $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$6 dB for OFDM to about $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$7 dB for AAC-OFDM at $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$8, while the waveform also exhibited better autocorrelation properties and improved range and velocity RMSE trade-offs (Kumar et al., 22 Jan 2026). Because this construction is near constant-envelope rather than strictly constant-envelope, it should be regarded as adjacent to CE-OFDM rather than identical to it.

Another related direction is FM-OFDM, which the literature positions as a constant-envelope waveform for high-mobility ISAC. There the OFDM sequence modulates instantaneous frequency, producing $$\varphi(t)=2\pi h\sum_{\ell=1}^{L} |\Gamma_\ell| \cos\!\Big(\frac{2\pi \ell t}{T}+\phi_\ell\Big), \qquad \Gamma_\ell=e^{j\phi_\ell}.$$9 with 0 dB PAPR. In doubly dispersive channels, FM-OFDM is compared against both CP-OFDM and phase-only CE-OFDM; the reported conclusion is superior range and velocity estimation accuracy at high SNR and under high mobility, with improved robustness relative to phase-unwrapping-based CE-OFDM baselines (Bouziane et al., 22 Aug 2025). This suggests that “constant-envelope OFDM” is best understood as a broader waveform-design paradigm containing several concrete realizations.

Across these realizations, the central trade-offs are consistent. Constant envelope improves PA efficiency and hardware friendliness, but amplitude freedom is sacrificed. In PSK CE-OFDM, bandwidth and mainlobe scale become deterministic functions of $T$00, $T$01, and $T$02, while sidelobe control migrates to the discrete phase code. Continuous-phase optimization can produce strong ACF improvements, but finite-alphabet quantization perturbs the optimized solution. CP-OFDMA-compatible CE schemes preserve implementation continuity, but typically consume additional spectral support or structural redundancy. Multiantenna CE precoding can exploit spatial degrees of freedom to compensate for constant-modulus constraints, but then CE-OFDM becomes intertwined with quantization-aware precoding and power allocation.

In that sense, CE-OFDM is not a single modulation recipe but a technically coherent research area spanning FM-type phase-domain waveform synthesis, ambiguity-function theory, nonlinear FFT-based optimization, quantized massive-MIMO precoding, and standard-compatible OFDMA implementations. The unifying objective is unchanged: retain OFDM’s multicarrier flexibility while replacing its unfavorable envelope statistics with a constant-envelope transmit structure.