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,

$\Pi_T$0

Zero-cyclic autocorrelation of $\Pi_T$1 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 $\Pi_T$2 such that its autocorrelation $\Pi_T$3 forms a Nyquist pulse. For square-root raised-cosine (SRRC) pulses with symbol interval $\Pi_T$4 and roll-off factor $\Pi_T$5, $\Pi_T$6 becomes the classic raised-cosine,

$\Pi_T$7

The Fourier transform $\Pi_T$8 satisfies $\Pi_T$9 as a raised-cosine spectrum, concentrating energy within $T$0, 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 $T$1 dB (versus $T$2 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 $T$3). The DD-SRN-FMCW frame is defined as

$T$4

where $T$5 is chirp power. The ODDM modulator processes $T$6 by taking the IDFT along Doppler, vectorizing, and pulse-shaping with $T$7, yielding $T$8. At the receiver, following DD-domain matched filtering, a simple cyclic correlation (length-$T$9) with $f_c$0 achieves "DD chirp compression":

$f_c$1

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 $f_c$2 and approximately Gaussian ODDM data $f_c$3 combine to a Rician sum $f_c$4. The complementary CDF of the PAPR for frame $f_c$5 is approximated as

$f_c$6

where $f_c$7 is the chirp-to-data power ratio. Increasing $f_c$8 rapidly suppresses high-PAPR events, so even $f_c$9 reduces PAPR several dB below ODDM-only or DDIP-pilot cases.

Spectral Characteristics:

The total spectrum sums the $\epsilon$0 Doppler-tone of $\epsilon$1 and spectra of $\epsilon$2 Doppler tones with data. The chirp's power spectral density is

$\epsilon$3

with $\epsilon$4 the length-$\epsilon$5 Dirichlet kernel and $\epsilon$6 flat for the zero-cyclic autocorrelation chirp.

Ambiguity Function and Resolution:

The extended cross-ambiguity $\epsilon$7 satisfies

$\epsilon$8

conferring uncoupled range and velocity mainlobes. Resolutions are $\epsilon$9 (range) and $mT/M$0 (velocity). Side-lobe levels remain $mT/M$1 dB below the mainlobe along delay and $mT/M$2 dB below along Doppler.

Cramér–Rao Bound (CRB):

For $mT/M$3 point scatterers at parameters $mT/M$4, the Fisher information matrix yields the CRB:

$mT/M$5

Delay and Doppler CRB for DD-SRN-FMCW lie within $mT/M$6–$mT/M$7 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 $mT/M$8 is superimposed onto an ODDM data frame $mT/M$9:

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

with $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$1 for $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$2 carrying QAM data. The time-domain waveform

$c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$3

propagates through the doubly-selective channel. At a co-located radar receiver, $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$4 is recovered, $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$5 is subtracted, and $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$6 is computed as above. Delay-Doppler estimation employs a super-resolution OMP (Orthogonal Matching Pursuit) algorithm with grid evolution. For communications, the receiver treats $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$7 as a known superimposed pilot, applies OMP-based channel estimation, and uses soft SIC-MMSE turbo equalization for reliable $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$8 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 $c[m]=e^{j\pi m^2/M}, \quad m=0,\ldots,M-1,$9 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).