High-Resolution Range Profiles in Radar

• High-Resolution Range Profiles (HRRPs) are one-dimensional radar signatures that detail target scattering structures via wideband or stepped-frequency processing.
• Modern HRRP synthesis employs sparse recovery and subspace methods to reconstruct profiles robustly even in the presence of missing or jammed data.
• Recent advances integrate super-resolution, generative, and graph-based deep learning techniques to enhance automatic target recognition and overcome traditional diffraction limits.

High-Resolution Range Profiles (HRRPs) are one-dimensional radar signatures that represent the resolved scattering structure of a target along the line-of-sight, typically generated via wideband or stepped-frequency signal processing. HRRPs capture the superposed echoes from dominant scattering centers and serve as critical data structures in radar automatic target recognition (RATR), low-observability detection, and super-resolved ranging. Their formation, signal models, theoretical limits, and recent algorithmic innovations span sparse recovery, subspace methods, generative models, and graph-based deep learning.

1. HRRP Signal Models and Range Resolution

HRRPs emerge from wideband radar operations, most commonly with stepped-frequency waveforms. For $N$ pulses at carrier frequencies $f_n = f_0 + n\Delta f$, the total RF bandwidth $B = N\Delta f$ determines the nominal range resolution:

$\Delta R = \frac{c}{2B}$

where $c$ is the speed of light. The unambiguous range is set by the frequency step, $R_{\max} = c/(2\Delta f)$. The measured baseband sequence for point scatterer targets is

$$y_n = \sum_{k=1}^K \alpha_k \exp(-j2\pi f_n \tau_k) + e_n$$

where $K$ is number of scatterers, $\alpha_k$ are complex amplitudes, $\tau_k$ delays, and $e_n$ additive noise (Hu et al., 2014, Liu et al., 2013).

True range resolution is not solely bandwidth-limited. Information-theoretic bounds incorporate SNR and amplitude/phase effects, such that the minimal bin width

$$\Delta r_{\min} \geq \frac{c}{2B} \frac{\log_2 L}{\log_2 (1+\mathrm{SNR})}$$

where $L$ is the number of distinguishable amplitude levels; accordingly, exploiting phase and amplitude can surpass the canonical limit in moderate-SNR regimes (Fuller et al., 19 Mar 2025).

2. Sparse Recovery and Subspace Profiling

Modern HRRP synthesis leverages sparse recovery and subspace decomposition. The radar observation equation for $M$ measured frequencies and $N$ range bins is

$\mathbf{y} = \Phi \mathbf{x} + \mathbf{u}$

where $\Phi$ is the measurement (dictionary) matrix, $\mathbf{x}$ the unknown range profile, and $\mathbf{u}$ noise.

Sparse representation (compressed sensing):

$$\underset{\tilde{h}}{\min} \|\tilde{h}\|_1 \quad \text{s.t.} \quad \|y - \Phi \tilde{h}\|_2 \leq \varepsilon$$

Under missing data (interference/jamming), the $\ell_1$ approach reconstructs full range profiles with robustness to lost pulses, eliminating sidelobes and maintaining resolution (Hu et al., 2010).

Subspace estimation (Missing-MUSIC): Covariance is estimated from all valid pairs, forming a Toeplitz matrix from autocovariance lags:

$$\hat{R}[h] = \frac{1}{Q(h)} \sum_{n=0}^{N-1-h} I(n)I(n+h)y_{n+h}y_n^*$$

Eigen-decomposition yields signal/noise subspaces, with Root-MUSIC providing delay estimation. Missing-MUSIC exhibits strong performance under arbitrary missing bands, preserving grid-free resolution and low sidelobes (Hu et al., 2014).

3. Adaptive and Cognitive Step-Frequency HRRP

RaSSteR and similar random-sparse SF radars index frequency bands randomly to mitigate range-Doppler coupling and operate robustly under spectrum interference. The measurement is

$$y_n = \beta\, e^{-j\frac{4\pi}{c}(f_c + d_n \Delta f)(R + v n T)}$$

with $d_n$ randomly chosen frequency steps. Sparse $\ell_1$ recovery enables high-resolution profile and Doppler estimation while concentrating transmit energy in clean spectral regions, boosting hit-rate under jamming (Mishra et al., 2020).

4. Super-Resolution Techniques Beyond Rayleigh Limit

Classical HRRP formation is diffraction-limited ($\Delta R = c/(2B)$). Recent advances integrate sparse coding, subspace algorithms (MUSIC, OMP), and deep learning to break the Rayleigh barrier.

DSSR-Net employs dimension scaling, lifting the received signal into a high-dimensional tensor, alternating model-guided data consistency via adaptive weight maps $\beta$ and neural U-Net based shrinkage in a proximal feature extraction stage:

$X_{k+1} = \beta \odot X_f + (1-\beta) \odot T(Z_k)$

Post-iteration dimension reduction yields the super-resolved profile. DSSR-Net demonstrates target separation below $0.6\Delta R_\mathrm{Rayleigh}$ and operates at SNRs down to $8$–$10$ dB, outperforming conventional spectral and sparse methods (Wang et al., 6 Apr 2025).

5. Generative and Graph-Based HRRP Models

Deep generative and graph-based methods deliver highly discriminative and interpretable HRRP representations for RATR.

Recurrent Gamma Belief Networks (rGBN): HRRP sequences are encoded by hierarchical gamma-distributed latent states, optimized with stochastic-gradient MCMC and recurrent variational inference. Supervised rGBN jointly models HRRP and class labels for robust, interpretable classification, with ELBO as the training objective (Guo et al., 2020).

GraphNet architectures: HRRPGraphNet recasts HRRP range cells as nodes of an undirected graph with adjacency reflecting amplitude and bin proximity:

$E_{ij} = \frac{r_i r_j}{|i-j| + 1}$

Global context is fused via graph convolution and attention pooling, which raises robustness under limited samples and noise compared to sequence models (CNN, RNN). This facilitates feature learning relevant for physical scattering structures and inter-class separation (Chen et al., 11 Jul 2024).

6. HRRP Recognition and Few-Shot ATR

In RATR scenarios, HRRP inputs suffer distortion from electronic countermeasures such as interrupted-sampling repeater jamming (ISRJ). Robust recognition integrates:

• Prior knowledge of jamming-induced point-spread functions (PSF)
• Dual-attention feature fusion between clean HRRP and PSF priors
• Hybrid loss (cross-entropy and supervised contrastive components) for invariant, discriminative feature learning

This approach, as shown in recent work, increases OOD generalization capacity relative to state-of-the-art CNN, transformer, and attention-based baselines (Sun et al., 28 Nov 2025).

Aspect sensitivity of HRRPs (variation due to target orientation) challenges few-shot ATR. The Aspect-Distributed Prototype (ADP) framework utilizes aspect-aware class prototypes for improved matching in large-language-model-in-context learning paradigms, significantly enhancing robustness under limited support samples (Bi et al., 7 Dec 2025).

7. Practical Implementation, Limitations, and Future Directions

HRRP synthesis and recognition algorithms are implemented across classic convex solvers (SPGL1, ADMM, OMP), hybrid MCMC/VAE frameworks (TLASGR-MCMC, RVI), model-guided neural networks (DSSR-Net), and graph/deep learning architectures (HRRPGraphNet). These methods trade off resolution, sidelobe suppression, robustness to missing/jammed data, and computational cost.

Key limitations include increased required SNR for super-resolution techniques, computational burden of fully connected graph representations, residual model mismatch in mechanism-dependent dictionaries, and efficiency scalability in hardware-constrained embedded platforms.

Ongoing extensions target structured/bayesian sparse priors, adaptive graph construction, unified generative and discriminative learning, and integrated fusion of HRRP with Doppler or multipolarization data. Domain-specific waveform design and joint amplitude-phase processing remain crucial for pushing range resolution beyond spectral diffraction bounds (Fuller et al., 19 Mar 2025, Wang et al., 6 Apr 2025, Chen et al., 11 Jul 2024).