Square-Root Nyquist FMCW Waveform

• The square-root Nyquist FMCW waveform is an integrated sensing and communication design that embeds a Nyquist-shaped chirp into an ODDM framework for precise range-Doppler resolution.
• It utilizes square-root Nyquist filtering to sharply control spectral leakage and lower the PAPR, mitigating issues inherent in conventional FMCW signals.
• The waveform supports joint radar and communication functionalities with low-complexity receiver processing, achieving performance within 2 dB of ideal benchmarks.

The square-root-Nyquist-filtered FMCW (SRN-FMCW) waveform is an integrated sensing and communication (ISAC) signal design that embeds a Nyquist-shaped chirp into the orthogonal delay-Doppler (DD) division multiplexing (ODDM) framework. It addresses long-standing limitations of conventional linear frequency-modulated continuous-wave (FMCW) waveforms in ISAC systems, such as high peak-to-average power ratio (PAPR), spectral inefficiencies, and sub-optimal range-Doppler ambiguity properties. SRN-FMCW achieves a controlled spectral envelope, dramatically reduces out-of-band emission, and enables near-ideal range and velocity resolutions while integrating seamlessly into ODDM for low-complexity receiver operation (Huang et al., 2 Feb 2026).

1. Mathematical Formulation of SRN-FMCW

The SRN-FMCW transmit waveform is based on a linear-FMCW chirp, expressed as

$$c(t)=e^{j2\pi(f_c+\frac{1}{2}\epsilon t)t}\cdot\Pi_T(t-T/2),$$

where $\Pi_T$ is a rectangular window of duration $T$, $f_c$ the carrier frequency, and $\epsilon$ the chirp rate. The infinite chirp is sampled at $mT/M$, forming the quadratic phase sequence

$c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$

subject to integer-rationality conditions. The SRN-filtered chirp subpulse is constructed as

$c_a(t)=\sum_{m=0}^{M-1} c[m]\cdot a(t-mT/M),$

where $a(t)$ is a square-root-Nyquist (SRN) pulse with support $\approx T/M$ (e.g., square-root raised cosine, SRRC). The full SRN-FMCW waveform utilizes 100% duty cycle,

$s_{c_a}(t)=\sum_{n=0}^{N-1} c_a(t-nT).$

Zero-cyclic autocorrelation of $c[m]$ ensures that the matched-filter output against itself yields a Nyquist-shaped delay response, mitigating Fresnel ripples characteristic of finite rectangular chirps.

2. Properties of SRN Filtering

SRN filtering leverages a pulse $a(t)$ such that its autocorrelation $g(\tau)=\int a(t)a^*(t-\tau)dt$ forms a Nyquist pulse. For square-root raised-cosine (SRRC) pulses with symbol interval $T/M$ and roll-off factor $\beta$, $g(\tau)$ becomes the classic raised-cosine,

$$g(\tau) = \begin{cases} \frac{\pi}{4}\,\mathrm{sinc}(1/(2\beta)) & |\tau|=T/(2\beta M), \ \frac{\cos(\pi\beta M\tau/T)}{1-(2\beta M\tau/T)^2}\,\mathrm{sinc}(M\tau/T) & \text{otherwise}. \end{cases}$$

The Fourier transform $A(f)$ satisfies $|A(f)|^2$ as a raised-cosine spectrum, concentrating energy within $|f| \leq M/(2T)$, thus sharply controlling out-of-band emission. When applied to the chirp, this spectral shaping reduces sidelobe energy and outer-bandwidth spectral spill, outperforming unshaped chirps by suppressing out-of-band emission to $-40$ dB (versus $-10$ dB for rectangular pulses).

3. Delay-Doppler Embedded SRN-FMCW (DD-SRN-FMCW) Frame

The SRN-FMCW can be embedded into an ODDM transmitter by populating symbols only along the delay axis (Doppler index $n=0$). The DD-SRN-FMCW frame is defined as

$$X_c[m,n] = \begin{cases} \sqrt{N E_c} \, c[m], & n=0, \ 0, & \text{otherwise}, \end{cases}$$

where $E_c=E\{|s_{c_a}(t)|^2\}$ is chirp power. The ODDM modulator processes $X_c$ by taking the IDFT along Doppler, vectorizing, and pulse-shaping with $a(t)$, yielding $s_{c_a}(t)$. At the receiver, following DD-domain matched filtering, a simple cyclic correlation (length-$M$) with $c[\cdot]^*$ achieves "DD chirp compression":

$$D_c[\ell,n] = \sum_{m=0}^{M-1} c^*[(m-\ell)_M] \cdot Y_c[m,n],$$

producing the channel's delay-Doppler response with a Nyquist-shaped mainlobe (delay) and Dirichlet-kernel mainlobe (Doppler).

4. Performance Metrics and Analysis

Peak-to-Average Power Ratio (PAPR):

The near-constant envelope $c_a(t)$ and approximately Gaussian ODDM data $s_d(t)$ combine to a Rician sum $s(t)=s_{c_a}(t)+s_d(t)$. The complementary CDF of the PAPR for frame $\gamma$ is approximated as

$$P(\gamma > \gamma_0) \approx 1-\left[1-Q_1\left(\sqrt{2\rho}, \sqrt{2\gamma_0(1+\rho)}\right)\right]^{MN},$$

where $\rho = E_c/E_s$ is the chirp-to-data power ratio. Increasing $\rho$ rapidly suppresses high-PAPR events, so even $\rho \approx -8\,\mathrm{dB}$ reduces PAPR several dB below ODDM-only or DDIP-pilot cases.

Spectral Characteristics:

The total spectrum sums the $n=0$ Doppler-tone of $s_{c_a}(t)$ and spectra of $N-1$ Doppler tones with data. The chirp's power spectral density is

$$|S_{c_a}(f)|^2 = E_c\,N\,\omega(f)|A(f)|^2 |\phi(-NTf)|^2,$$

with $\phi(\nu)$ the length-$N$ Dirichlet kernel and $\omega(f)$ flat for the zero-cyclic autocorrelation chirp.

Ambiguity Function and Resolution:

The extended cross-ambiguity $A(\tau,\nu)$ satisfies

$$A(0,\nu)=N\,\phi(-\nu NT) \,\,\, \text{(Doppler cut)},\ A(\tau,0)=M\,g(\tau) \,\,\, \text{(delay cut)},$$

conferring uncoupled range and velocity mainlobes. Resolutions are $\Delta\tau = T/M$ (range) and $\Delta\nu = 1/(NT)$ (velocity). Side-lobe levels remain $>30$ dB below the mainlobe along delay and $>30$ dB below along Doppler.

Cramér–Rao Bound (CRB):

For $P$ point scatterers at parameters $\{\lvert h_p\rvert,\angle h_p, l_p, k_p\}$, the Fisher information matrix yields the CRB:

$$\text{MSE}(\hat{\theta}_i) \geq [\mathcal{F}^{-1}]_{i,i},\ [\mathcal{F}]_{i,j} = \frac{2}{\sigma_z^2} \text{Re}\sum_{m,n} \left(\frac{\partial Y[m,n]}{\partial \theta_i}\right)^* \left(\frac{\partial Y[m,n]}{\partial \theta_j}\right)$$

Delay and Doppler CRB for DD-SRN-FMCW lie within $1$–$2$ dB of the ideal impulse pilot and linear FMCW, supporting its superresolution sensing capability.

5. ODDM-FMCW ISAC Waveform Construction

To realize joint sensing and communication, the DD-SRN-FMCW frame $X_c$ is superimposed onto an ODDM data frame $X_d$:

$X = X_c + X_d,$

with $X_d[m,n]\in\mathcal{A}$ for $n=1,\ldots,N-1$ carrying QAM data. The time-domain waveform

$s(t) = s_{c_a}(t) + s_d(t)$

propagates through the doubly-selective channel. At a co-located radar receiver, $Y$ is recovered, $X_d$ is subtracted, and $D_c$ is computed as above. Delay-Doppler estimation employs a super-resolution OMP (Orthogonal Matching Pursuit) algorithm with grid evolution. For communications, the receiver treats $X_c$ as a known superimposed pilot, applies OMP-based channel estimation, and uses soft SIC-MMSE turbo equalization for reliable $X_d$ recovery.

6. Numerical Performance Highlights

The following table summarizes key numerical results of the ODDM-FMCW scheme:

Metric	ODDM-FMCW (with DD-SRN-FMCW)	Comparison
PAPR (M=256, N=64, ρ=-8 dB)	6 dB lower than ODDM-DDIP, 4 dB lower than ODDM data (CCDF=10⁻³)	—
BER (communications)	≤0.5 dB from ODDM with perfect CSI (JCEDD)	ODDM-DDIP lags by >3 dB at BER=10⁻⁴
NRMSE (sensing)	Delay/Doppler NRMSE within 2 dB of CRBs (for SNR ≥10 dB)	Matches pure DDIP pilots
Sensing-Comms Trade-off	Optimal $\rho\approx-10$ dB with fixed SNR minimizes BER and NRMSE	Demonstrates flexible resource allocation

This demonstrates that ODDM-FMCW, via SRN-FMCW, achieves significant reduction in PAPR, controlled spectral occupancy, and simultaneous high-performance communication and sensing under integrated DD-domain processing for ISAC (Huang et al., 2 Feb 2026). A plausible implication is that SRN-FMCW waveforms enable high spectral efficiency and low-complexity hardware implementation in next-generation joint radar–communication systems.

7. Integration and Significance in ISAC Systems

SRN-FMCW constitutes a robust ISAC primitive: it minimizes transmitter complexity by utilizing 100% duty-cycle Nyquist-filtered chirps, sharply reduces PAPR, and ensures spectral containment compatible with regulatory spectral emission limits. The Nyquist-shaped delay response removes Fresnel ripple artifacts, while the ODDM embedding enables seamless coexistence with large-scale communication signaling. The framework provides a practical, easily realizable pathway for ISAC deployment, supporting both superresolution sensing and reliable data transmission under standard low-complexity signal processing pipelines (Huang et al., 2 Feb 2026).