Chirp Multicarrier Waveform: Principles & Performance

• Chirp multicarrier waveforms are signaling schemes that use linearly or affinely varying chirp subcarriers instead of static sinusoids, enhancing resilience against multipath and Doppler disturbances.
• They integrate transforms like DAFT and DFnT in place of FFT/IFFT, thereby providing efficient modulation, pilot-aided channel estimation, and low-complexity receiver designs.
• Their superior performance in doubly dispersive channels and dual functionality for communications and sensing make them a promising candidate for next-generation (6G) ISAC applications.

Searching arXiv for recent and foundational papers on chirp multicarrier waveforms, including MCDM/OCDM/AFDM and ISAC-related developments. Chirp multicarrier waveform denotes a class of multicarrier signaling schemes in which the elementary subcarriers are chirps rather than static sinusoids. Across the literature, this class includes multicarrier chirp-division multiplexing (MCDM), orthogonal chirp division multiplexing (OCDM), and affine frequency division multiplexing (AFDM), as well as related hybrids and delay-Doppler-domain extensions. The common rationale is that chirp signals exhibit robustness to multipath, Doppler shifts, and time-frequency selectivity, while multicarrier architectures preserve flexible symbol-domain processing, pilot insertion, and high-order modulation. Recent work frames these waveforms as a technically motivated alternative to sinusoidal OFDM in doubly dispersive channels and as a natural substrate for integrated sensing and communications (ISAC) because chirp parameters and ambiguity properties can be tuned for both data transmission and parameter estimation (Huang et al., 2018, Bemani et al., 2021, Rou et al., 30 Apr 2026).

1. Definition and waveform family

Chirp multicarrier waveforms replace the fixed-frequency tones of OFDM with linearly or affinely frequency-varying basis functions. In MCDM, the $k$-th orthogonal chirp waveform is written as

$\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$

with chirp rate $\mu = \frac{B_c}{T}$ and orthogonality when $\Delta f=\frac{1}{T}$ (Huang et al., 2018). The corresponding transmitted symbol takes the form

$$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$

This construction already contains the essential idea of chirp multicarrier signaling: data symbols modulate a bank of orthogonal chirp subcarriers rather than a bank of pure exponentials.

OCDM uses the inverse discrete Fresnel transform (IDFnT) as the modulation operator, with the baseband symbol expressed as

$\mathbf{S} = \mathbf{\Phi}_M^H \mathbf{X},$

where $\mathbf{X}$ contains communication symbols in the Fresnel domain (Bhattacharjee et al., 2021). AFDM generalizes this direction further through the discrete affine Fourier transform (DAFT), introducing tunable quadratic-phase parameters. In matrix form,

$$\mathbf{S} = \mathbf{A} \mathbf{s}, \quad \mathbf{A} = \mathbf{\Lambda}_{c_2} \mathbf{F} \mathbf{\Lambda}_{c_1},$$

and the inverse modulation step is

$$\mathbf{s} = \mathbf{\Lambda}_{c_1}^H \mathbf{F}^H \mathbf{\Lambda}_{c_2}^H \mathbf{x}$$

(Bemani et al., 2021). Within the DAFT parameterization, OFDM appears as a special case with $c_1=c_2=0$, OCDM is obtained for a particular nonzero choice, and AFDM occupies the more general parameter regime (Zhang et al., 2024, Sui et al., 8 Aug 2025).

A later unifying treatment shows that OCDM and AFDM fall within the conventional Weyl-Heisenberg multicarrier framework, where the root chirp corresponds directly to the prototype pulse. In that view, chirp-domain waveforms can be interpreted as pulse-shaped OFDM with a constant-envelope chirp prototype pulse (Jiang et al., 15 Mar 2026). This suggests that “chirp multicarrier waveform” is less a single modulation and more a design family characterized by chirped subcarrier bases, transform-domain realizations, and parameterized time-frequency spreading.

2. Orthogonality, transforms, and signal-space structure

A defining technical property is that chirp subcarriers can remain orthogonal even though their instantaneous frequencies vary over time. For MCDM, the cross-correlation satisfies

$\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$0

when $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$1 (Huang et al., 2018). This preserves the main algebraic advantage of OFDM—subcarrier decoupling—while changing the subcarrier geometry in the time-frequency plane.

To implement such systems, the literature introduces chirp-adapted transform pairs. For MCDM, the orthogonal chirp transform (OCT) and inverse OCT play the role analogous to FFT/IFFT: $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$2 and

$\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$3

(Huang et al., 2018). OCDM analogously relies on the DFnT/IDFnT pair (Bhattacharjee et al., 2021), whereas AFDM uses DAFT/IDAFT with two chirp parameters (Bemani et al., 2021).

The DAFT-based unified framework introduces the chirp slope factor $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$4 in the time-frequency representation. In that formulation, OFDM corresponds to $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$5, OCDM to $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$6, and AFDM to $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$7 (Zhang et al., 2024). The chirp slope determines the orientation of subcarrier “strips” in the time-frequency plane, and adjusting it can enhance BER when SNR is sufficiently high (Zhang et al., 2024). A broader 6G-oriented perspective describes this as a transition from “static sinusoidal subcarriers” to parameterizable chirps that spread each symbol’s energy across both time and frequency in a sheared fashion (Rou et al., 30 Apr 2026).

A continuous-time interpretation refines this picture further. The AFDM basis can be written as a root chirp multiplied by frequency shifts, and the chirp phase is transparent to subcarrier orthogonality: $\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$8 (Jiang et al., 15 Mar 2026). That result places chirp-domain waveforms inside orthodox multicarrier theory rather than outside it.

3. Channel interaction, diversity, and high-mobility operation

The principal motivation for chirp multicarrier waveform design is operation in doubly dispersive channels. In underwater acoustic, RF indoor, vehicular, THz, and general high-mobility settings, OFDM’s sinusoidal subcarriers suffer from Doppler-induced inter-carrier interference and loss of orthogonality, whereas chirp subcarriers are used specifically because they are robust to multipath and Doppler (Huang et al., 2020, Huang et al., 2018, Rou et al., 30 Apr 2026).

In AFDM, the design objective is to set DAFT parameters so that channel paths with distinct delays or Doppler frequency shifts do not overlap in the DAFT domain. The resulting DAFT-domain impulse response conveys a full delay-Doppler representation of the channel, and AFDM can achieve the full diversity of linear time-varying channels (Bemani et al., 2021). A sufficient condition is stated through the parameter choice

$\psi_k(t) = \sqrt{\frac{1}{T} e^{j\left(2\pi k\Delta f t + \pi \mu t^2\right)},\qquad 0\leq t\leq T$9

with the channel-spread condition

$\mu = \frac{B_c}{T}$0

in the unified DAFT framework (Zhang et al., 2024). The 2021 AFDM analysis states that with appropriate $\mu = \frac{B_c}{T}$1 and $\mu = \frac{B_c}{T}$2, the diversity order equals the number of channel paths $\mu = \frac{B_c}{T}$3 (Bemani et al., 2021).

By contrast, MCDM and OCDM emphasize robustness through orthogonal chirps and diagonal or nearly diagonal effective channel structure. In MCDM, after OCT demodulation the system model reduces to

$\mu = \frac{B_c}{T}$4

with a diagonal channel matrix, enabling per-subcarrier detection and low detection complexity (Huang et al., 2018). In underwater acoustic extensions, MU-MCDM adapts preamble length, number of subcarriers, pilot portion, guard period, and user allocation to time-variant channels (Huang et al., 2020).

A recent perspective article contrasts two waveform philosophies for high mobility: 2D isotropic spreading in the delay-Doppler domain, exemplified by OTFS, and sheared spreading via parameterizable chirps, exemplified by AFDM (Rou et al., 30 Apr 2026). This suggests that chirp multicarrier waveforms are not merely “more robust OFDM,” but a distinct method of encoding diversity by embedding delay and Doppler structure into the waveform itself.

4. Receiver architectures, estimation, and practical implementation

Chirp multicarrier systems preserve much of the standard multicarrier receiver pipeline—synchronization, compensation, channel estimation, equalization, and symbol detection—but substitute chirp-domain transforms and chirp-aware estimation blocks. In MCDM, a low-complexity receiver is described with packet synchronization, carrier frequency offset compensation, channel estimation, and symbol detection (Huang et al., 2018). Packet synchronization uses a pseudorandom training sequence and maximum correlation, while carrier frequency offset is estimated from two identical PN sequences (Huang et al., 2018).

In MU-MCDM for underwater acoustic communications, the receiver is cast as a maximum likelihood design performing synchronization, channel estimation, and symbol detection. The pilot-aided model is

$\mu = \frac{B_c}{T}$5

with ML estimate

$\mu = \frac{B_c}{T}$6

followed by linear interpolation and symbol detection through

$\mu = \frac{B_c}{T}$7

(Huang et al., 2020).

OCDM automotive ISAC adopts pilot-aided channel estimation and then decouples communication decoding and radar parameter extraction through the SUNDAE algorithm (Bhattacharjee et al., 2021). In THz automotive radar, wideband OCDM runs over multiple subbands and combines subband estimates with weights based on the Cramér-Rao lower bound: $\mu = \frac{B_c}{T}$8 (Bhattacharjee et al., 2023).

For AFDM, implementation is repeatedly described as close to OFDM. The modulation and demodulation use an FFT/IFFT core surrounded by chirp multiplications (Bemani et al., 2021), and a 6G perspective article states that AFDM requires only two per-symbol phase rotation blocks sandwiching the standard IFFT/FFT core (Rou et al., 30 Apr 2026). However, a later continuous-time analysis qualifies several idealized assumptions. Practical implementations rely on sub-Nyquist discrete-time samples and therefore exhibit frequency aliasing; the aliased chirps are conditionally orthogonal, and an appropriate sample-wise pulse-shaping filter is required to maintain mutual orthogonality (Jiang et al., 15 Mar 2026). That paper further derives an exact input-output relation over delay-Doppler channels and shows that the effective channel at a practical receiver does not, in general, admit the simple DD spreading-function model commonly assumed in the literature (Jiang et al., 15 Mar 2026). This is a notable corrective to simplified discrete models.

5. Communication performance and comparison with OFDM

The literature consistently evaluates chirp multicarrier waveforms against OFDM in terms of BER, complexity, diversity, and tolerance to frequency-selective or Doppler-rich channels. For MCDM in wireless communications, simulations and indoor RF experiments show that the system outperforms OFDM, with simulation gains at BER $\mu = \frac{B_c}{T}$9 of $\Delta f=\frac{1}{T}$0 dB for BPSK, $\Delta f=\frac{1}{T}$1 dB for QPSK, and $\Delta f=\frac{1}{T}$2 dB for 16-QAM (Huang et al., 2018). The same study reports greater improvements in indoor multipath-rich RF experiments, including a $\Delta f=\frac{1}{T}$3 dB lower BER for BPSK at $\Delta f=\frac{1}{T}$4 dB SNR and, for 32-QAM, a $\Delta f=\frac{1}{T}$5 dB gain and up to 1.39 Mbps throughput with BER $\Delta f=\frac{1}{T}$6 (Huang et al., 2018).

For underwater acoustic MU-MCDM, BER decreases with higher subcarrier count, and the summary states that at SNR $\Delta f=\frac{1}{T}$7 dB, $\Delta f=\frac{1}{T}$8 achieves BER $\Delta f=\frac{1}{T}$9 (Huang et al., 2020). The same work reports that higher pilot density improves BER through better CSI and that comb-type user allocation offers better BER performance in frequency-selective scenarios (Huang et al., 2020).

AFDM is repeatedly positioned as a high-mobility waveform with full diversity and BER superior to conventional multicarrier schemes (Bemani et al., 2021, Zhang et al., 2024). The DAFT-based unified framework states that BER can be enhanced, at sufficiently high SNR, by adjusting the chirp slope factor $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$0 (Zhang et al., 2024). A broader survey of chirp signaling for next-generation multi-carrier networks states that AFDM offers $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$1 dB SNR gains over OFDM/OTFS in the figures discussed there, though that statement is presented at survey level rather than as a standalone derivation (Sui et al., 8 Aug 2025).

Not all performance claims concern BER alone. Some work emphasizes peak-to-average power ratio. A JRC-oriented chirp-convolved data transmission waveform shows that transmit sequences can be selected to achieve $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$2 dB PAPR (Berggren et al., 2022). DFT-s-OFDM with chirp modulation claims that chirp modulation does not degrade the PAPR and that, at identical spectral efficiency, splitting information bits into constellation and chirp streams can yield up to $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$3 dB BER advantage (Liu et al., 6 Aug 2025). These results indicate that chirp multicarrier design is also used to balance spectral efficiency, diversity, and transmitter efficiency rather than only raw error performance.

6. Sensing, ISAC, and ambiguity-function engineering

Chirp multicarrier waveforms are closely linked to sensing because chirps already underpin pulse compression and FMCW radar. In joint radar and communication settings, the ambiguity function becomes a central design object. For the multicarrier chirp-based JRC waveform CCDT, the time-discrete periodic ambiguity function is derived as

$$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$4

and the chirp rate together with the transmit sequence can shape the ambiguity function to be thumbtack-like or ridge-like along either the delay or Doppler axis (Berggren et al., 2022). The same paper reports better signal detection performance than OFDM and DFT-s-OFDM on channels with large Doppler frequency (Berggren et al., 2022).

OCDM-based ISAC for automotive settings uses Fresnel-domain signaling and the SUNDAE receiver to obtain both radar and communication performance, with RMSE for range and velocity converging to the CRLB at high SNR and BER close to OTFS and better than OFDM under Doppler (Bhattacharjee et al., 2021). Wideband THz automotive radar further exploits multi-carrier OCDM over multiple subbands and reports that the optimal weighted combining scheme achieves sub-millimeter range accuracy at SNR $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$5 dB and improves velocity RMSE by $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$6 dB relative to single-subband estimates (Bhattacharjee et al., 2023).

AFDM has increasingly been repurposed for ISAC-specific designs. One study analyzes the AFDM ambiguity function under different built-in parameters and finds that the waveform with specific parameter $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$7 owns the near-optimal time-domain ambiguity function, enabling a low-complexity matched-filtering range estimator with the same BER performance as OTFS using simple LMMSE equalization (Zhu et al., 2023). Another frame-based AFDM-ISAC design allocates a single chirp subcarrier for sensing and channel estimation inside each ISAC symbol and uses analog-domain dechirping to exploit chirp compression gains without requiring full-duplex hardware (Luo et al., 1 Jul 2026).

Hybrid designs also extend beyond pure chirp-domain modulation. AAC-OFDM forms

$$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$8

an affine addition of OFDM and chirp signals for ISAC, and reports improved autocorrelation properties, improved RMSE of range and velocity, and lower PAPR, with up to about $$x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},\qquad 0\leq t\leq T.$$9 dB PAPR reduction for $\mathbf{S} = \mathbf{\Phi}_M^H \mathbf{X},$0 (Kumar et al., 22 Jan 2026). This suggests that chirp multicarrier waveform design can operate either by replacing sinusoidal carriers or by superimposing chirp structure onto legacy waveforms.

7. Variants, extensions, and emerging directions

The recent literature broadens chirp multicarrier waveform design in several directions. AFDM has become a platform for index modulation and permutation-domain signaling. Pre-chirp-domain index modulation assigns distinct pre-chirp parameters to different subcarriers while preserving orthogonality, thereby embedding extra information bits in the pre-chirp pattern without extra energy consumption (Liu et al., 2024). Multiple-mode AFDM with index modulation extends this by conveying additional information through constellation mode selection and chirp activation patterns, with more than $\mathbf{S} = \mathbf{\Phi}_M^H \mathbf{X},$1 dB SNR improvement at $\mathbf{S} = \mathbf{\Phi}_M^H \mathbf{X},$2 over several AFDM-IM benchmarks at the same spectral efficiency (Liu et al., 17 Jul 2025). Chirp-permuted AFDM introduces a chirp-permutation domain that preserves AFDM’s core characteristics while enhancing ambiguity function resolution and peak-to-sidelobe ratio in the Doppler domain (Rou et al., 28 Jul 2025).

Another line of work merges chirp signaling with delay-Doppler-domain multiplexing more explicitly. Orthogonal Chirp Delay-Doppler Division Multiplexing (CDDM) spreads data symbols across the DD domain through an orthogonal chirp-Zak transform and reports significant BER improvements over existing schemes under perfect CSI, superior out-of-band emissions, and lower-complexity pilot-based estimation than ODDM in the imperfect-CSI case (Bai et al., 22 Nov 2025).

Superposition-based variants show that chirp multicarrier ideas are also relevant in low-power and coexistence-oriented systems. For LoRa, chirp-layered superposition coding places a high-spreading-factor chirp on top of a low-SF transmission so that the interference at the legacy dechirp-and-DFT demodulator appears nearly flat across bins, improving spectral efficiency while keeping the standard LoRa demodulation process largely intact (Huang et al., 7 Apr 2026). For SWIPT, superimposed chirp waveforms over selected subbands with a diplexer-based integrated receiver provide a 30% improvement in average harvested energy compared with multisine waveforms in the considered system setup (Roy et al., 2023).

The overall research direction has been summarized as a shift “from sinusoids to chirps” for 6G and beyond (Rou et al., 30 Apr 2026). A plausible implication is that chirp multicarrier waveform research is converging toward a unified design space in which communication robustness, sensing capability, spectral efficiency, and implementation reuse are jointly optimized rather than treated as separate objectives.