CRLB for Sensing: Theory, Applications & Achievability

• CRLB for sensing is a fundamental result that establishes a lower bound on the variance of unbiased estimators using Fisher information.
• The framework applies classical and Bayesian formulations to address challenges in quantized, privacy-perturbed, and nonstandard sensing environments.
• It informs the design of optimal sensor arrays, sensor localization, and distributed algorithms, ensuring precise and efficient parameter estimation.

The Cramér-Rao Lower Bound (CRLB) is a fundamental result in statistical estimation theory, establishing a lower bound on the covariance of any unbiased estimator for parameters of interest. In sensing applications—spanning compressed sensing, sensor localization, signal processing with quantized or privacy-perturbed data, integrated sensing/communications, privacy-preserving networks, and beyond—CRLB formulates a sharp and often attainable metric for performance limits. This article details the theory, specialized forms, achievability, and modern applications of the CRLB in state-of-the-art sensing problems, with an emphasis on rigorous mathematical and algorithmic perspectives.

1. Fundamentals of the Cramér–Rao Lower Bound for Sensing

Let random vector $y$ be observed according to $p(y;\theta)$ with $\theta\in\mathbb{R}^d$ unknown. Under standard regularity, the Fisher information matrix (FIM),

$$I_{ij}(\theta) = \mathbb{E}_{y|\theta} \left[ \frac{\partial \log p(y;\theta)}{\partial \theta_i} \frac{\partial \log p(y;\theta)}{\partial \theta_j} \right],$$

quantifies the sensitivity of the likelihood to $\theta$. The CRLB asserts,

$\mathrm{Cov}[\hat\theta(y)] \succeq I^{-1}(\theta)$

for any unbiased estimator $\hat\theta(y)$. For scalar $\theta$, $\mathrm{Var}[\hat\theta] \ge 1/I(\theta)$. The geometric view frames the FIM as a Riemannian metric ("Fisher–Rao metric") on the statistical manifold, with the bound set by the squared norm of the gradient of the function of interest under this metric (Blaom, 2017).

If $\theta$ is not globally identifiable, $I(\theta)$ may be singular. A unified treatment uses the Moore–Penrose pseudoinverse, with

$\mathrm{Cov}[\hat\theta] \succeq I^{+}(\theta)$

attainable by unbiased estimators after suitable reduction to a full-rank subspace (Namkung et al., 2024).

2. Optimal Sensing and CRLB in Linear-Gaussian Models

In canonical linear sensing models,

$$y = A x + w, \quad w \sim \mathcal{N}(0, \sigma^2 I),$$

the FIM reduces to $I(x) = (1/\sigma^2) A^\top A$, so the CRLB is $$\mathrm{Cov}[\hat x] \succeq \sigma^2 (A^\top A)^{-1}$$ (Niazadeh et al., 2010). For compressed sensing with sparse $x$, under a measurement matrix $A$ whose distribution satisfies a concentration-of-measure property, the joint-typicality estimator achieves the CRLB asymptotically as $N\to\infty$, provided the sparsity ratio $\alpha=K/N$ and measurement redundancy $\beta=M/K$ remain in the admissible regime (Niazadeh et al., 2010).

The Bayesian CRLB (BCRB) incorporates prior information, leading to additive "data" and "prior" Fisher information terms, often yielding closed-form expressions for both non-blind (known basis) and blind compressed sensing (Zayyani et al., 2010). In such cases, the BCRB provides a meaningful performance bound even for biased estimators or when enforcing sparsity via priors.

3. CRLB in Quantized, Privacy-Perturbed, and Nonstandard Sensing

When observation models depart from full-precision or additive-Gaussian cases—e.g., networks of one-bit (binary) sensors, or privacy-perturbed/released data—the Fisher information, and thus the CRLB, are fundamentally altered:

Quantized/One-Bit Sensors: The Fisher information matrix for estimating a parameter vector from sign-quantized or binary observations is reduced compared to the analog case. For random binary measurements,

$$J_{mn} = \sum_{i=1}^S \frac{f^2(\tau-C_i(\theta))}{F(\tau-C_i(\theta))[1-F(\tau-C_i(\theta))]} \frac{\partial C_i}{\partial \theta_m} \frac{\partial C_i}{\partial \theta_n}$$

($f$ is Gaussian PDF, $F$ is CDF, $C_i(\theta)$ is the physical model at sensor $i$) (Ristic et al., 2014). The bound quantifies loss of information due to quantization and provides a rigorous design rule.

Privacy-Preserving Sensing: Injecting obfuscation noise (randomized mechanisms) for privacy purposes leads to a privacy-constrained CRLB. The Fisher information available to an adversary is capped (via matrix $S$), and the minimal attainable MSE is characterized by the privacy-preserving CRLB

$$\Sigma_{PPCR} = [ H^\top S^{1/2}(S^{1/2} I_y(H\theta)^{-1} S^{1/2} + I)^{-1} S^{1/2} H ]^{-1}$$

with identifiability under privacy satisfied iff $H^\top S H \succ 0$ (Ke et al., 7 Nov 2025).

Dynamic/Distributed State Estimation under Privacy: Recurrent and distributed settings (multi-sensor, multi-time) admit analogous privacy-preserving CRLB formulations, with an additivity principle for block-wise Fisher information and explicit distributed algorithms attaining these bounds (Ke et al., 7 Nov 2025, Guo et al., 2024).

Parameter-Dependent Likelihood Support: When the support of the likelihood $p(y;\theta)$ depends on $\theta$, the classical CRLB fails. The Cramér-Rao–Leibniz lower bound (CRLLB) introduces boundary correction terms, ensuring validity in these cases and unifying traditional and boundary-influenced estimation theory (Lu et al., 2 Aug 2025).

4. Structure and Optimization of CRLB in Modern Sensing Architectures

4.1 Sensor Localization

For localization of a sensor using measurements from randomly placed anchors, the CRLB for unbiased estimators of position can be analyzed via the trace of the CRLB matrix, $T = \mathrm{tr}(\mathrm{CRLB})$. The distribution of $T$ is characterized as normal (RSS/bearing) or affine-chi-square (TOA) in the large-$N$ (number of anchors) regime; closed-form means and variances reveal the direct scaling of accuracy with anchor density, noise, channel model, and geometry (Huang et al., 2011).

4.2 Adaptive/Programmable Arrays and IRS-Aided Sensing

Large Intelligent Surfaces (LIS): For positioning via LIS, the CRLB in the lateral directions decays quadratically with surface area, and in the depth dimension, either linearly (on the central perpendicular line) or quadratically depending on terminal offset. Distributed deployment (partitioning total area into sub-apertures) modifies the area gain via additional geometric factors (Hu et al., 2017).

IRS-Aided Radar and Sensing: Joint optimization of AP and IRS beamforming using the CRLB as an explicit cost function enables superior DoA/MIMO parameter estimation. Alternating optimization, semidefinite programming, and SCA-based surrogates are employed to tackle the non-convexity (Song et al., 2022). When multiple IRS platforms are available, Doppler-aware IRS phase shift optimization using the scalarized CRLB or A-optimality criterion yields significant accuracy gains in NLoS scenarios (Esmaeilbeig et al., 2022). Similar methodology applies in integrated sensing-communication (ISAC) systems, where beamforming is designed to optimize a CRLB–sum-rate trade-off (Fang et al., 5 Feb 2025).

4.3 Sensing with Pinching-Antenna Arrays

The Bayesian CRLB (BCRB) serves as a lower bound for random-parameter estimation in pinching-antenna assisted sensing (PASS). In such systems, dynamic spatial reconfiguration of antenna locations enables the BCRB to be minimized with respect to both location and power allocation variables. Optimality often mandates dynamic repositioning schemes, for which efficient KKT-based and coordinate search algorithms are developed (Jiang et al., 10 Oct 2025).

5. Achievability, Robustness, and Generalizations

Achievability: For classical and many privacy-constrained or structured settings, there exist explicit unbiased (or privacy-constrained efficient) estimators that asymptotically attain the CRLB (e.g., joint-typical estimators in compressed sensing, LSEs in privacy-preserving linear models, and consensus-based distributed algorithms in distributed privacy-preserving estimation) (Niazadeh et al., 2010, Ke et al., 7 Nov 2025, Guo et al., 2024).

Robust and Generalized CRLBs: Under outliers or heavy-tailed noise, the classical CRLB may lose relevance. Information-geometric generalizations based on alternative divergence functions (e.g., Basu-Harris-Hjort-Jones, BHHJ divergence) produce “$\alpha$-CRLBs” that benchmark robustness/efficiency trade-offs. These incorporate escort distributions and modify the Fisher information accordingly, thus extending CRLB theory to robust statistical estimation (Dhadumia et al., 28 Jul 2025).

Boundary and Non-Gaussian Effects: When regularity conditions fail (e.g., parameter-dependent support), only CRLLB-type bounds incorporating boundary correction guarantee valid error lower bounds (Lu et al., 2 Aug 2025). The achievable covariance (efficiency) depends on both model structure and correct modeling of support.

6. Practical and Computational Considerations

Low-Complexity Algorithms: Direct minimization or constraint of the CRLB typically results in non-convex or large-scale SDPs. Modern approaches, including SCA, shifted generalized power iteration (SGPI), relaxation of growing block matrices, and recursive covariance updates, enable tractable online implementation with order-of-magnitude speedups, making explicit CRLB optimization viable in high-dimensional/practical ISAC systems (Fang et al., 5 Feb 2025, Guo et al., 2024).

Privacy and Differential Privacy: The CRLB forms a lower bound not just on estimator MSE but also on what any adversary can infer in privacy-constrained state estimation frameworks. Algorithms that enforce a CRLB-based privacy constraint inherently provide (quantifiable) differential privacy guarantees under the Gaussian mechanism (Guo et al., 2024).

7. Limitations, Open Directions, and Summary Table

The table summarizes principal CRLB forms and their distinguishing features:

Setting	Fisher Information Structure	Achievability Notes
Classical linear-Gaussian	$I = (1/\sigma^2)A^\top A$	Efficient (MLE/LSE)
Quantized/binary sensing	$J = M \Lambda M^\top$ (matrix-perturbed)	Efficiency lost at low noise (Ristic et al., 2014)
Privacy-preserving sensing	$I_z(θ)$ upper-bounded by privacy S	Gaussian mechanism optimal (Ke et al., 7 Nov 2025)
Structured/Robust (e.g. BHHJ)	$\alpha$-FIM; escort measures	Tight at $\alpha\to 0$ only (Dhadumia et al., 28 Jul 2025)
Parameter-dependent support (CRLLB)	FIM plus boundary correction	Tight only if collinearity holds (Lu et al., 2 Aug 2025)

In modern sensing, CRLB analysis provides a rigorous and often computationally tractable framework for performance benchmarking, system design, and privacy quantification. Extensions of the bound to accommodate Bayesian priors, quantization, structural or privacy constraints, and nonstandard statistical models continue to drive advanced sensing research and applications.