Waveform Design: OFDM, OTFS, and FMCW for ISAC

Which Waveform Is Best at Both Jobs?

Section 24.3 designed the spatial covariance Rx\mathbf{R}_x but was silent about what actually occupies the time–frequency plane. That choice matters: a waveform excellent for communication may have a pathological ambiguity function that ruins sensing, and vice versa. This section evaluates the three main candidates — OFDM, OTFS, and FMCW — as ISAC waveforms, using the same criteria the radar and comm communities care about: ambiguity function shape, delay–Doppler resolution, robustness to high mobility, and peak-to-average ratio. Two of the three contenders — OFDM and OTFS — are the waveforms of 5G NR and the candidate waveform for 6G respectively, so the question is not academic.

Definition:

Ambiguity Function of a Waveform

The ambiguity function of a complex baseband waveform s(t)s(t) is χ(τ,ν)≜∫−∞∞s(t)s∗(t−τ)ej2πνt dt.\chi(\tau, \nu) \triangleq \int_{-\infty}^{\infty} s(t) s^*(t - \tau) e^{j2\pi\nu t}\,dt. It is the complex correlation of the waveform with a delayed and Doppler-shifted copy of itself. The magnitude ∣χ(τ,ν)∣|\chi(\tau, \nu)| is the matched-filter response to a target at delay τ\tau and Doppler ν\nu. The central lobe width determines the delay–Doppler resolution; the sidelobe structure determines the "clutter floor" against which weak targets compete.

OFDM as a Radar Waveform

OFDM has served as the baseline 5G NR waveform since 2018 and will likely remain in 6G. For ISAC it has three properties that are both strengths and weaknesses:

  1. Thumbtack ambiguity function. An OFDM symbol with random (data-dependent) subcarrier values has a pseudo-random, approximately delta-like ambiguity function in (τ,ν)(\tau, \nu), with sidelobes of order 1/N1/\sqrt{N} where NN is the number of subcarriers. Excellent for radar.
  2. CP-based coarse range. The cyclic prefix length sets the unambiguous range; typical 5G NR CP is 4.7 μs, giving 700 m unambiguous monostatic range.
  3. Doppler sensitivity. OFDM is fragile in high-mobility scenarios: Doppler shifts of order Δf/10\Delta f / 10 (where Δf\Delta f is the subcarrier spacing) cause inter-carrier interference. For vehicular targets at 250 km/h and mmWave carrier, Doppler can exceed 30 kHz — comparable to the NR subcarrier spacing itself.

Definition:

OFDM-ISAC Signal Model

In an OFDM-ISAC system, the transmitted signal for communication symbols Xn,kX_{n,k} on subcarrier kk of OFDM symbol nn is s(t)=∑n∑k=0N−1Xn,k ej2πkΔf(t−nT) rect((t−nT)/T).s(t) = \sum_{n} \sum_{k=0}^{N-1} X_{n,k}\,e^{j2\pi k \Delta f (t - nT)}\,\text{rect}((t-nT)/T). The radar processing at the monostatic receiver takes the received echo symbols Yn,kY_{n,k}, divides out the known Xn,kX_{n,k} (allowed because the BS is the transmitter), and applies a 2D FFT over (n,k)→(ν,τ)(n,k) \to (\nu, \tau) to obtain the range–Doppler map: R[ℓ,m]=∑n,kYn,kXn,ke−j2π(ℓk/N−mn/M).R[\ell, m] = \sum_{n,k} \frac{Y_{n,k}}{X_{n,k}} e^{-j2\pi(\ell k / N - m n / M)}. The peak at (ℓ⋆,m⋆)(\ell^\star, m^\star) gives the estimated range bin and Doppler bin.

The division Y/XY/X eliminates the data dependence of the probing signal — the communication payload acts like a pseudo-random code that is perfectly known to the monostatic transceiver. This is the reason OFDM-ISAC works "for free" on the NR downlink.

⚠️Engineering Note

OFDM-ISAC Parameters in 5G NR

5G NR gives three useful numerical benchmarks for OFDM-ISAC:

  • Range resolution: Δr=c/(2NΔf)\Delta r = c / (2 N \Delta f). For FR2 with N=3168N = 3168 allocated subcarriers and Δf=120\Delta f = 120 kHz, Δr≈0.4\Delta r \approx 0.4 m.
  • Maximum unambiguous range: rmax⁡=cTCP/2r_{\max} = c T_{\text{CP}} / 2. For FR2 numerology-3 with TCP≈0.59 μT_{\text{CP}} \approx 0.59\,\mus, rmax⁡≈88r_{\max} \approx 88 m.
  • Doppler resolution: Δν=1/(MTsym)\Delta \nu = 1/(M T_{\text{sym}}) where MM is the number of coherent OFDM symbols. For M=140M = 140 symbols (one slot at μ=3\mu = 3), Δν≈20\Delta \nu \approx 20 kHz, translating to ~40 m/s velocity resolution at 28 GHz.

These figures are better than many short-range automotive radars already in production. The main gap is rmax⁡r_{\max} at FR2: only 88 m is too short for long-range ISAC. Mitigation: use longer CP numerologies or multiple-slot coherent integration.

Practical Constraints
  • •

    NR numerology μ\mu = 0,1,2,3,4 gives Δf\Delta f = 15, 30, 60, 120, 240 kHz respectively.

  • •

    Normal CP overhead: ~7% in time; extended CP available at μ\mu = 2.

  • •

    Maximum slot ∼\sim 1 ms (numerology-0) / 0.0625 ms (numerology-4).

📋 Ref: 3GPP TS 38.211, Section 4.3.2

Definition:

OTFS as a Delay-Doppler Native Waveform

Orthogonal Time-Frequency-Space (OTFS) modulation places data symbols in the delay-Doppler domain (equivalently, the Zak domain). Each symbol is a 2D pulse γ(τ−τi,ν−νi)\gamma(\tau - \tau_i, \nu - \nu_i). The channel acts by convolution in delay-Doppler; for a target at delay τ0\tau_0, Doppler ν0\nu_0, the response is a pure shift γ(τ−τ0,ν−ν0)\gamma(\tau - \tau_0, \nu - \nu_0).

For ISAC, OTFS has three structural advantages over OFDM:

  1. The data lives in the same domain where targets appear — sensing and demodulation share a single transform.
  2. The ambiguity function is exactly the transmit pulse shape γ\gamma, giving predictable resolution without data-dependent sidelobes.
  3. Doppler robustness is native: a 30 kHz Doppler shifts the data by one Doppler bin rather than causing inter-carrier interference as in OFDM.

Theorem: Unambiguous Delay-Doppler Region of OTFS-ISAC

For an OTFS frame with NN delay bins (each of width Δτ=1/B\Delta\tau = 1/B where BB is bandwidth) and MM Doppler bins (each of width Δν=1/(NTsym)\Delta\nu = 1/(NT_{\text{sym}})), the unambiguous delay-Doppler region for target sensing is ∣τ∣≤NΔτ/2=N/(2B),∣ν∣≤MΔν/2=M/(2NTsym).|\tau| \leq N\Delta\tau / 2 = N/(2B),\qquad |\nu| \leq M\Delta\nu / 2 = M/(2NT_{\text{sym}}). Any target outside this rectangle aliases into it; any two targets inside whose delay-Doppler separation is smaller than one bin width are unresolvable. The "ISAC waveform budget" is the area NM/(2NTsym⋅B)=M/(2BTsym)N M / (2 N T_{\text{sym}} \cdot B) = M/(2 B T_{\text{sym}}).

OTFS is a rectangular tiling of the delay-Doppler plane with the waveform's ambiguity cell. Communication uses cells for data; sensing uses cells for target returns. The same rectangle bounds both resolutions simultaneously — there is no hidden sensing degradation relative to pure-radar OTFS. This is the exact sense in which OTFS is "native ISAC".

Show Hint

The delay-Doppler lattice of OTFS is orthogonal by construction; the ambiguity function of the frame equals the autocorrelation of the prototype pulse.

The unambiguous range in delay is NΔτN\Delta\tau because the lattice repeats periodically.

The unambiguous Doppler range is MΔνM\Delta\nu by the dual argument.

Proof
Delay lattice period

The OTFS receiver applies a discrete inverse Zak transform with period NΔτN\Delta\tau in delay. Targets at delay τ0+kNΔτ\tau_0 + k N\Delta\tau map to the same receive bin for integer kk. Hence the unambiguous delay range is ∣τ−τ0∣≤NΔτ/2|\tau - \tau_0| \leq N\Delta\tau / 2.

Doppler lattice period

Similarly, the receive Zak transform has Doppler period MΔνM\Delta\nu. Unambiguous Doppler: ∣ν−ν0∣≤MΔν/2|\nu - \nu_0| \leq M\Delta\nu / 2.

Joint region

The intersection defines a rectangle of area NMΔτΔν=M/(BTsym)N M \Delta\tau \Delta\nu = M/(B T_{\text{sym}}) (after substituting the bin widths). Targets outside alias into the rectangle; resolutions inside are (Δτ,Δν)(\Delta\tau, \Delta\nu). ■\blacksquare

🎓CommIT Contribution(2021)

OTFS-ISAC via Delay-Doppler Domain Signal Processing

W. Yuan, R. Schober, G. Caire — Proc. IEEE ICC Workshops, 2021

Yuan, Schober, and Caire formulated the first OTFS-ISAC transceiver in which sensing and demodulation share the same delay-Doppler processing pipeline. Their central observation: the OTFS receiver already estimates the delay-Doppler channel h(τ,ν)h(\tau, \nu) to perform equalization, and that channel estimate is exactly the sensing output. No additional radar processing is needed — the sensing and demodulation stages merge into one.

This gives OTFS-ISAC a structural efficiency that OFDM-ISAC cannot match: in OFDM, the radar range-Doppler map requires a separate 2D FFT on the received grid, and the comm equalizer handles ICI separately; in OTFS, a single delay-Doppler channel estimator delivers both the equalizer taps and the target list. The downside is implementation complexity: the OTFS transform is not a native operation in 5G NR hardware, and retrofitting it to legacy chipsets is nontrivial.

isacotfscairewaveformView Paper →
🎓CommIT Contribution(2020)

Effective Diversity and ISAC Performance of OTFS

L. Gaudio, M. Kobayashi, G. Caire — IEEE Transactions on Wireless Communications, vol. 19, no. 9

Gaudio, Kobayashi, and Caire established the performance of OTFS for joint radar parameter estimation and communication, proving that OTFS matches the CRB on target range and velocity while retaining OFDM-level communication rates under high-mobility channels. Two of their quantitative results are now textbook:

  1. CRB achievement. Under the OTFS grid assumption, the OTFS radar delay and Doppler estimators are asymptotically efficient: the estimation variance matches the CRB for the chosen lattice resolution.
  2. Dual-function robustness. In channels with Doppler spread up to the OTFS Doppler lattice, OFDM suffers from 3–5 dB of ICI loss while OTFS has zero ICI by construction. The comparison is decisive in the 100+ km/h regime.

Together with the Yuan–Schober–Caire paper above, these two CommIT contributions form the theoretical backbone for all subsequent OTFS-ISAC work at TU Berlin and in the 6G standardization community.

isacotfscairecrbhigh-mobilityView Paper →

Definition:

FMCW as the Automotive Radar Benchmark

Frequency-modulated continuous-wave (FMCW) radar transmits a linear chirp: s(t)=ejπBt2/Tc,0≤t≤Tc.s(t) = e^{j\pi B t^2 / T_c}, \quad 0 \leq t \leq T_c. Mixing the received echo with the transmitted chirp yields a beat frequency proportional to target range: fb=2rB/(cTc)f_b = 2 r B / (c T_c). A second dimension of fast-FFT over a sequence of chirps gives Doppler. Automotive radars at 77 GHz universally use FMCW because of its excellent range-Doppler decoupling and extremely low implementation cost.

FMCW is the ISAC purist's worst nightmare: it is a pure radar waveform with no natural data payload. Embedding bits requires phase/amplitude modulation of the chirp, which degrades the ambiguity function. Most ISAC work treats FMCW as the comparative baseline rather than a candidate waveform — the question is whether OFDM or OTFS can match FMCW's sensing performance while simultaneously carrying a serious data rate.

OFDM vs OTFS vs FMCW for ISAC

CriterionOFDM-ISACOTFS-ISACFMCW
Native domainTime-frequencyDelay-DopplerTime-frequency chirp
Ambiguity functionThumbtack (data-dependent)Prototype pulse (deterministic)Tilted ridge
Range resolutionc/(2NΔf)c/(2N\Delta f)c/(2B)c/(2B)c/(2B)c/(2B)
Doppler resolution1/(MTsym)1/(MT_{\text{sym}})1/(NMTsym)1/(NMT_{\text{sym}})1/(NTc)1/(NT_c)
High-mobility robustnessPoor (ICI)ExcellentExcellent
Data rate capabilityFull NR throughputFull 6G throughputNear zero
Hardware maturity5G NR mass-marketResearch/pre-stdAutomotive mass-market
PAPRHighModerateZero (constant envelope)

Ambiguity Function: OFDM vs OTFS

Comparison of the range-Doppler ambiguity surfaces of OFDM and OTFS under the same bandwidth and frame duration. OFDM shows data-dependent sidelobes; OTFS shows a clean rectangular ambiguity cell.

Parameters
64
32
120

Waveform Diversity Pareto: Sensing CRB vs Comm Rate

For a fixed total transmit power, the CRB on target range estimation versus the achievable communication rate, for OFDM, OTFS, and FMCW waveforms. Each curve is parameterized by the fraction of subcarriers (or chirps) dedicated to data vs sensing reference symbols.

Parameters
10
100
120

Example: Range CRB for OTFS-ISAC at 100 MHz

An OTFS-ISAC system at 28 GHz carrier uses B=100B = 100 MHz bandwidth, N=64N = 64 delay bins, M=32M = 32 Doppler bins, and integrated SNR per delay-Doppler cell of 20 dB. Compute the Cramér–Rao bound on the target range estimate.

Solution
Delay CRB

The CRB on delay estimation for a waveform of bandwidth BB and integrated SNR ρ\rho is CRB(τ)=18π2ρβrms2,\text{CRB}(\tau) = \frac{1}{8\pi^2 \rho \beta_{\text{rms}}^2}, where βrms\beta_{\text{rms}} is the rms bandwidth. For a rectangular OTFS pulse, βrms≈B/12\beta_{\text{rms}} \approx B/\sqrt{12}. With B=108B = 10^8 and ρ=102\rho = 10^2: βrms2≈8.3×1014\beta_{\text{rms}}^2 \approx 8.3 \times 10^{14}, CRB(τ)≈1/(8π2⋅100⋅8.3×1014)≈1.5×10−19\text{CRB}(\tau) \approx 1/(8\pi^2 \cdot 100 \cdot 8.3\times10^{14}) \approx 1.5\times 10^{-19} s2^2.

Range CRB

Range r=cτ/2r = c\tau/2, so CRB(r)=(c/2)2CRB(τ)≈(1.5×108)2⋅1.5×10−19≈3.4×10−3\text{CRB}(r) = (c/2)^2 \text{CRB}(\tau) \approx (1.5\times 10^8)^2 \cdot 1.5\times10^{-19} \approx 3.4 \times 10^{-3} m2^2. Hence σr≈5.8\sigma_r \approx 5.8 cm — sub-wavelength range precision.

Interpretation

At ρ=20\rho = 20 dB and 100 MHz, OTFS delivers centimeter-accuracy range estimates in a single dwell. That is well inside the performance envelope of automotive radars. Massive MIMO array gain on top of this lowers the required per-antenna SNR and extends the range; multistatic fusion (Section 24.4) adds diversity and resolves ambiguities. ■\blacksquare

⚠️Engineering Note

OTFS Implementation Complexity vs OFDM

OTFS requires an extra 2D transform stage compared to OFDM: symplectic finite Fourier transform (SFFT) or discrete Zak transform. The numerical complexity is O(NMlog⁡(NM))\mathcal{O}(NM\log(NM)) per OTFS frame — a constant factor above the OFDM cost and a small fraction of the total baseband budget. The bigger operational cost is the end-to-end latency: OTFS commits to a full MM-symbol dwell before delivering any demodulated data, whereas OFDM can process each symbol independently. For URLLC applications the latency tradeoff is severe; for eMBB and ISAC it is acceptable.

Other practical concerns: PAPR is moderately high (between OFDM and FMCW), the pilot overhead for delay-Doppler channel estimation is non-trivial, and the tight coupling between sensing resolution and frame size means the frame duration is now a sensing parameter, not just a communication parameter.

Practical Constraints
  • •

    OTFS SFFT complexity: O(NMlog⁡(NM))\mathcal{O}(NM\log(NM)) per frame.

  • •

    Minimum OTFS latency: M⋅TsymM \cdot T_{\text{sym}} before first decoded symbol.

  • •

    PAPR typical OTFS: 8–10 dB vs OFDM 10–12 dB vs FMCW 0 dB.

Common Mistake: OFDM at Vehicular Speeds Is Not Free

Mistake:

A designer picks OFDM-ISAC for a 6G vehicular sensing application, citing 5G NR compatibility.

Correction:

At 28 GHz carrier and 250 km/h, the Doppler shift is ≈6.5\approx 6.5 kHz. For FR2 with 120 kHz subcarrier spacing, ICI from Doppler contributes roughly (6.5/120)2≈0.3%(6.5/120)^2 \approx 0.3\% — manageable. But at 500 km/h (high-speed rail) the Doppler is 13 kHz, ICI rises to 1.2%, and sensing Doppler resolution becomes comparable to the subcarrier spacing — leading to Doppler-rate ambiguities. OTFS sidesteps this entirely because Doppler is a data-grid index, not an inter-carrier nuisance. For the high-mobility scenarios where sensing matters most (drones, aircraft, high-speed rail), OTFS has a fundamental advantage that is worth the extra implementation complexity.

Quick Check

For a 6G ISAC deployment targeting 300 km/h vehicular sensing at 28 GHz, which waveform has the best inherent tradeoff?

FMCW — it is the mature automotive choice.

OFDM with extended CP.

OTFS with delay-Doppler channel estimation.

Pulsed monostatic radar, with comm sent out-of-band.

Correction:
OTFS with delay-Doppler channel estimation.

OTFS is delay-Doppler native: it is robust to Doppler, delivers full comm rates, and gives CRB-optimal range-velocity estimation in a single transform. This is the core of the Yuan–Schober–Caire and Gaudio–Kobayashi–Caire CommIT contributions.

Historical Note: OTFS from Radar to 6G

2017–2024

OTFS was introduced by Hadani et al. (2017) as a new modulation for high-mobility communication. Its radar-like delay-Doppler domain operation immediately attracted the sensing community, and by 2020 papers linking OTFS directly to ISAC began to appear — notably the Gaudio–Kobayashi–Caire effectiveness paper and the Yuan–Schober–Caire transceiver. By 2024, OTFS was one of the two leading waveform candidates (alongside improved OFDM) in the ITU-R IMT-2030 preparatory studies, driven largely by its superior ISAC properties.

Key Takeaway

OFDM for backward compatibility, OTFS for ISAC-first design. OFDM-ISAC works well enough that 5G NR can be retrofitted for sensing without changing the waveform — the CommIT pipeline for ISAC in existing NR networks relies on this. For a greenfield 6G design where sensing is a native service and vehicular mobility is the target scenario, OTFS-ISAC is the theoretically-clean choice: delay-Doppler resolution, Doppler robustness, and CRB-achieving parameter estimation all in a single transform.

Why This Matters: OTFS, LEO Satellites, and the ISAC Connection

Chapter 23 discussed LEO massive MIMO, where OTFS earned its place as the waveform of choice for high-Doppler NTN links. The same Doppler robustness that makes OTFS good for LEO comm makes it excellent for LEO-based wide-area sensing: a single satellite pass can coherently integrate a target track across tens of seconds without suffering the ICI losses OFDM would incur. The 6G NTN-ISAC convergence — tracked targets from space, wide-area coverage — is one of the most active research areas at the intersection of cell-free, OTFS, and ISAC.

Cell-Free ISAC and Multistatic DetectionChapter Summary