DFT-p-FDMA Based Chirp Transmission in CP-OFDM for Unified ISAC Waveform Design

Abstract: We propose an integrated sensing and communications (ISAC) framework that supports chirp signal transmission in CP-OFDM-based multiple access communication systems, enabling efficient coexistence of communication and sensing capabilities. Our framework employs the discrete Fourier transform phase rotated and permuted frequency division multiple access (DFT-p-FDMA) waveform to transmit chirp signals using a portion of the frequency resources, while ensuring interference-free concurrent CP-OFDM data transmissions on other bands. We analyze the effective channel behavior under the DFT-p-FDMA waveform, characterizing how delays and Doppler shifts impact radar target echoes. We also show how processing multiple received symbols improves Doppler resolution in practical scenarios. Our framework allows flexible adjustment of range-Doppler resolution through optimized time-frequency resource allocation, offering a versatile solution for ISAC applications. Simulation results validate the framework's performance in delay and Doppler estimation, highlighting its potential to support ISAC in next-generation wireless networks.

• The paper introduces a DFT-p-FDMA preprocessing method that embeds constant-envelope chirps into CP-OFDM while allowing concurrent data transmission on separate frequency resources.
• The proposed receiver forms a two-dimensional DAFT–Doppler map across multiple OFDM symbols, achieving Doppler resolution of 1/[L(1+Ncp/N)] and separating closely spaced targets as integration length increases.
• Simulations show delay and Doppler estimation remains robust under RCS fluctuations, with performance degradation below 1.7 dB at the tested operating points, while coherence time limits the gains from longer integration.

Overview and Motivation

This paper, authored by Carpi, Cho, Kim, and Zhang of Samsung Research America (2607.15575), proposes an integrated sensing and communications (ISAC) framework that embeds chirp-based radar signaling within a CP-OFDM multiple access system using the discrete Fourier transform phase rotated and permuted frequency division multiple access (DFT-p-FDMA) waveform. The central design goal is backward compatibility: chirp transmission occupies a portion of the frequency resources while concurrent CP-OFDM data transmissions proceed on other sub-bands without mutual interference. This addresses a practical constraint for 6G ISAC deployment, where sensing must coexist with legacy 4G/5G air interfaces rather than require dedicated radar spectrum or hardware.

The choice of a chirp waveform is motivated by two properties well established in the radar literature: the decoupling of pulse duration from range resolution, which permits high-energy long pulses with fine range resolution, and the constant-envelope characteristic yielding low PAPR and improved power amplifier efficiency. The framework builds on the DAFT-domain lineage that includes AFDM, but differs from prior AFDM-based ISAC studies in that it operates inside the standard CP-OFDM transceiver chain via a pre-processing stage consisting of an $M$-point DFT followed by a phase-rotated permutation matrix.

System Model and Effective Channel Analysis

The transmitter applies $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$ to the DAFT-domain input sequence, where the unitary permutation matrix $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$ is parameterized by an integer chirp rate $c$. Subcarrier mapping and an $N$-point IDFT then produce the OFDM symbol. A key structural result is that when $c$ yields valid permutation matrices, the cyclic prefix is equivalent to a chirp-periodic prefix (CPP), which both combats multipath and guarantees non-interference with concurrent CP-OFDM sub-bands.

The paper's principal analytical contribution is a closed-form expression for the effective channel,

$$\bar{\mathbf{H}}_l[n,i] = \frac{1}{M}\sum_{k,q} e^{j2\pi\frac{nk-iq}{M}} e^{j[\theta(\pi_P(q))-\theta(\pi_P(k))]} \sum_p \gamma(d_p,\nu_p,l)\, e^{-j2\pi\,\pi_P(q)\frac{d_p}{N}} D_N(\nu_p+\pi_P(q)-\pi_P(k)),$$

where $D_N(x)$ is a Dirichlet-type kernel capturing Doppler-induced frequency spreading and $\gamma(d_p,\nu_p,l)$ is the spreading function incorporating CP-induced phase rotation across symbols. For a constant-pulse input $\mathbf{u}_l = [\sqrt{M},0,\dots,0]^T$—i.e., single-chirp transmission—the post-processing signal takes the form $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$0 with $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$1.

Two structural insights follow directly from this expression. First, pure delays (zero Doppler) place echo peaks at DAFT indices $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$2, so echoes are separated by multiples of the chirp rate $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$3; this makes delay estimation structurally simple. Second, fractional Doppler values manifest as leakage and peak shifts relative to these no-Doppler indices, meaning Doppler information is encoded in the distortion pattern of the DAFT-domain profile rather than as clean peaks—an effect illustrated with four-echo examples showing integer Doppler producing one-sample shifts and fractional Doppler ($\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$4, $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$5) producing multi-peak leakage.

Receiver Processing and Doppler Resolution

Because Doppler estimation from a single OFDM symbol relies on ambiguous leakage patterns—and degrades further when multiple targets overlap in the post-processing signal—the paper proposes collecting the $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$6 post-processing signals within one TTI into a matrix $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$7 and applying a row-wise $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$8-point DFT. The resulting 2D map $\mathbf{x}_l = \mathbf{P}\mathbf{F}_M\mathbf{u}_l$9 spans the DAFT domain and the Doppler domain, analogous to the range-Doppler map of classical pulse-Doppler radar. For a single target, the index pair $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$0 maximizing $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$1 constitutes the maximum likelihood estimate of the quantized delay-Doppler pair.

The Doppler resolution is

$$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$2

so doubling the number of processed OFDM symbols halves the Doppler bin width. A notable subtlety the paper identifies is Doppler ambiguity: normalized Doppler values differing by integer multiples of $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$3 map to the same bin index via the modulo operation, so estimates are constrained to $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$4. Fractional-multiple Dopplers produce inter-bin leakage, and the factor $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$5 must be adjusted if non-consecutive symbols are used. An illustrative example with two targets at delays $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$6 and closely spaced Dopplers $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$7 shows that they are unresolvable at $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$8 but separate cleanly onto distinct Doppler bins at $$\mathbf{P} = \mathbf{F}_M\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H\boldsymbol{\Lambda}_c^H\mathbf{F}_M^H$$9—a concrete demonstration that time-frequency resource allocation directly trades off range-Doppler resolution.

Simulation Results Under RCS Fluctuations

The evaluation uses a point scatterer model with channel gain derived from the radar range equation, and models temporal RCS fluctuation via a first-order auto-regressive process controlled by a persistence parameter $c$0, where $c$1 denotes constant RCS. The scenario is a monostatic radar observing a single drone target: $c$2 GHz, $c$3 kHz, $c$4, $c$5, average RCS $c$6, target at 83 m range moving at 43 m/s (within the FAA's 45 m/s drone speed limit), evaluated over $c$7 TTIs.

The headline quantitative findings are:

Metric	Operating point	Gap, $c$8 vs. $c$9
Delay error rate	0.01	$N$0 dB ($N$1); $N$2 dB ($N$3)
Doppler RMSE	0.01	$N$4 dB ($N$5); $N$6 dB ($N$7)

In the fluctuation-free case, doubling $N$8 yields the expected ~3 dB gain. However, under RCS fluctuations the benefit of longer integration shrinks: for $N$9, the gap between $c$0 and $c$1 reduces to approximately 2 dB. With $c$2, cumulative RCS phase variation reaches up to 56% of the Doppler shift over a $c$3 TTI, causing energy dispersion across multiple delay-Doppler components. The implication is that RCS coherence time, not merely SNR or integration length, becomes a binding constraint on achievable Doppler resolution—a limitation inherent to any coherent integration approach, not specific to this waveform.

Limitations and Open Questions

Several assumptions bound the generality of the results. The closed-form effective channel derivation assumes $c$4 (full-band allocation, identity subcarrier mapping) and, in the appendix, permutation symmetry $c$5; the behavior under partial-band allocation with concurrent CP-OFDM traffic—which is the framework's stated motivation—is analyzed only qualitatively through the CPP/CP equivalence argument, not quantified. The simulations are restricted to a single point scatterer with Swerling-like RCS fluctuations; multi-target performance under noise, clutter, and realistic UAV micro-Doppler dynamics remains unexamined. The Doppler ambiguity constraint $c$6 is assumed rather than resolved algorithmically. Finally, no communication-side results (e.g., throughput or error rates on the multiplexed CP-OFDM data) are reported, leaving the communications-sensing trade-off of resource splitting unquantified.

Conclusion

The paper establishes DFT-p-FDMA as a viable mechanism for chirp-based ISAC within standard CP-OFDM infrastructure, supported by a closed-form effective channel characterization, a multi-symbol receiver that achieves Doppler resolution scaling as $c$7, and simulation evidence of robust delay-Doppler estimation under RCS fluctuations with degradation gaps below 1.7 dB at the tested operating points. The open questions it leaves—partial-band interference quantification, multi-target performance, ambiguity resolution, and joint communication-sensing trade-offs—define the immediate agenda for maturing this waveform toward practical 6G ISAC deployment.