Noise-Normalized Squared Correlation Metric

• The noise-normalized squared correlation metric is a dimensionless measure that extends classical correlation and SNR concepts to noisy, high-dimensional environments.
• It leverages covariance theory, random matrix theory, and cross-correlation analysis to accurately assess signal association and detection performance.
• Its practical applications span radar, imaging, quantum sensing, and multivariate inference, offering enhanced reliability over traditional metrics.

The noise-normalized squared correlation metric provides a unified, dimensionless measure of signal association or detection capability, accounting for additive noise and background fluctuations in a normalized fashion. It extends classical notions of correlation and signal-to-noise ratio (SNR) to modern statistical signal processing, high-dimensional inference, radar detection, and uncertainty quantification. The metric is tailored to performance prediction in noisy environments, integrating structure from covariance theory, random matrix theory, and cross-correlation analysis.

1. Definition and Mathematical Formulation

The canonical noise-normalized squared correlation is a normalized quantity, typically denoted as $\rho^2$, which quantifies the fraction of variance or signal power that is preserved against independent noise contributions. In the context of coherent noise radar, for received and reference baseband voltages $z_1(t) = I_1(t) + j Q_1(t)$ and $z_2(t) = I_2(t) + j Q_2(t)$, the squared correlation metric is defined as

$$\rho^2 = \frac{P}{P+P_{n1}} \cdot \frac{P}{P+P_{n2}}$$

where $P$ is the “perfectly correlated” signal power, and $P_{n1}$, $P_{n2}$ are noise powers in the receive and reference channels, respectively. Equivalently, $\rho = E[I_1 I_2] / \sqrt{P_1 P_2}$ when phase alignment $\phi = 0$ and $P_{1,2}$ are total powers in each channel (Luong et al., 2020).

In multivariate and high-dimensional settings, noise normalization incorporates spectral structure. Given an $n\times n$ sample correlation matrix $C$ with largest eigenvalue $\lambda_{\max}$ and Marchenko–Pastur upper edge $\lambda_+ = (1 + \sqrt{n/T})^2$ for $T$ samples, the noise-normalized squared correlation [Editor's term: NN-SQCOR] is

$$R^2_{NN} = \max \left\{0, \frac{\lambda_{\max} - \lambda_+}{n - \lambda_+} \right\}$$

ensuring $R^2_{NN} \in [0,1]$ for non-negative-definite matrices (Salimi et al., 2024).

In detection under noise covariance uncertainty,

$$NSNR(v;\hat\Sigma, \Sigma) = \frac{(v^T \hat\Sigma^{-1} v)^2}{(v^T \Sigma^{-1} v) \cdot (v^T \hat\Sigma^{-1} \Sigma \hat\Sigma^{-1} v)}$$

interpreted as the squared cosine of the angle between “whitened” signal vectors under true and estimated covariances (Diskin et al., 2024).

2. Operational Contexts and Use Cases

Radar and Quantum Sensing

In coherent noise radar and quantum two-mode squeezing radars, the metric $\rho^2$ directly quantifies detectability: the higher $\rho^2$, the greater the statistical distinguishability of a signal from noise. Detection performance, such as the receiver operating characteristic (ROC), can be expressed in closed form as a function of $\rho^2$ (or integrated as $\lambda = N \rho^2$ for $N$ samples), revealing a direct “noise-normalized” analog to classical SNR (Luong et al., 2020, Luong et al., 2022, Luong et al., 2019).

Imaging and Uncertainty Quantification

In particle image velocimetry (PIV), the “peak-to-root-mean-square ratio” (PRMSR) is a variant of noise-normalized squared correlation: $$\mathrm{PRMSR} = \frac{|C_{\text{peak}}|^2}{\sigma_{\text{noise}}^2}$$ with $C_{\text{peak}}$ as the primary correlation peak and $\sigma_{\text{noise}}^2$ the variance in a sidelobe (“noise”) region. PRMSR is tightly coupled to empirical uncertainty estimates, outperforming simpler SNR metrics for coverage and accuracy (Xue et al., 2014).

Multivariate Inference and Feature Selection

Using spectral methods, $R^2_{NN}$ enables robust estimation of “true” signal associations in high-dimensional correlation matrices, accounting for random matrix fluctuations. This is instrumental in multivariate feature selection, portfolio risk assessment, and denoising in high-dimensional statistics. Only associations above the noise-induced eigenvalue threshold are considered structurally meaningful (Salimi et al., 2024).

Covariance Estimation and Adaptive Detection

The NSNR metric connects SNR degradation due to covariance estimation error with a normalized measure bounded in $[0,1]$. It admits an explicit lower bound in terms of the Kullback-Leibler divergence, linking estimation, detection, and information-theoretic criteria (Diskin et al., 2024).

3. Statistical Properties and Assumptions

The metric’s statistical robustness is rooted in well-understood noise models:

• Zero-mean Gaussianity is assumed for all signal and noise components.
• Additive noise in both target and reference arms is explicit in construction.
• Signal covariance fit to sample moments (via Frobenius norm minimization) renders the metric adaptive to empirical realities.
• Spectrally, noise normalization leverages random matrix results (Marchenko–Pastur law) to separate structure from noise in finite-sample regimes (Salimi et al., 2024).

Critical operational regimes are:

• $\rho^2 \rightarrow 0$ as signal is overwhelmed by independent noise.
• $\rho^2 \rightarrow 1$ in the limit of negligible noise and perfect correlation.
• Large $N$ and small $\rho$ asymptotics: Detection performance depends only on the product $\lambda = N \rho^2$ (Luong et al., 2022).

4. Relation to Conventional SNR and Detection Theory

A key result is the explicit mapping between noise-normalized squared correlation and SNR in radar: $$\rho^2(R) = \frac{1}{1 + (R/R_c)^4} = \frac{\mathrm{SNR}}{1 + \mathrm{SNR}}$$ for negligible reference noise, where $R_c$ is the characteristic range. For small SNR, $\rho^2 \approx \mathrm{SNR}$; for high SNR, $\rho^2$ saturates at unity, rendering it the natural, bounded generalization of SNR (Luong et al., 2020).

The ROC for a noise-radar detector is given by

$$p_D(p_{FA}| \rho,N) = Q_1 \left( \sqrt{2 N \rho^2 / (1-\rho^2)}, \sqrt{ -2 \ln p_{FA} / (1-\rho^2) } \right)$$

where $Q_1$ is the Marcum Q-function. Substituting the functional dependence $\rho^2(R)$ yields range-dependent performance prediction. In the large-sample, low-correlation regime, the ROC depends only on $\lambda = N \rho^2$, mirroring SNR-based frameworks (Luong et al., 2022, Luong et al., 2019).

5. Computation, Implementation, and Pseudocode

For each domain, computation is straightforward and well-specified:

• Radar/Detection: Estimate channel variances, compute sample cross-correlations, normalize according to total (signal + noise) power (Luong et al., 2020, Luong et al., 2019).
• PIV (PRMSR):

1. Subtract correlation plane minimum.
2. Locate maximal peak $C_{\text{peak}}$.
3. Define sidelobe $\Omega$ as all points with $<0.5\times C_{\text{peak}}$.
4. Compute $$\sigma^2_{\text{noise}} = \frac{1}{|\Omega|} \sum_{(s,t)\in\Omega} \tilde{C}(s,t)^2$$.
5. Output $$\mathrm{PRMSR} = C_{\text{peak}}^2 / \sigma^2_{\text{noise}}$$ (Xue et al., 2014).

• High-Dimensional Inference:

1. Center and scale variables.
2. Compute empirical correlation $C$.
3. Obtain $\lambda_{\max}$.
4. Compute $R^2_{NN}$ using the Marchenko–Pastur noise edge $\lambda_+$ (Salimi et al., 2024).

6. Impact, Limitations, and Applications

Impact

The noise-normalized squared correlation metric underlies current approaches to performance prediction, uncertainty quantification, and feature selection across radar, imaging, quantum information, and high-dimensional statistics. It produces normalized, bounded, noise-aware measures that outperform or generalize traditional SNR- and Pearson-correlation-based quantities in practical, noise-rich settings.

Limitations

• Assumption of joint Gaussianity and white noise may not hold in heavy-tailed or correlated environments.
• At large $\rho$ or small $N$, higher-order or finite-sample corrections are non-negligible (Luong et al., 2022).
• Practical implementation in environments with structured non-Gaussian interference or model mismatch requires modifications.
• In multivariate settings, finite-sample noise can inflate eigenvalues, motivating the explicit random-matrix-based noise normalization (Salimi et al., 2024).

Applications Table

Domain	Metric Name/Formula	Key Reference
Coherent Radar	$\rho^2 = (P/(P+P_{n1}))\cdot(P/(P+P_{n2}))$	(Luong et al., 2020)
PIV	$\mathrm{PRMSR}=C_{\text{peak}}^2/\sigma^2$	(Xue et al., 2014)
Cov Estimation	$NSNR$ as squared “whitened” correlation	(Diskin et al., 2024)
High-dim Assoc.	$R^2_{NN}=(\lambda_{\max}-\lambda_+)/\ldots$	(Salimi et al., 2024)
Quantum Sensing	$\mathcal{N}^2 = C_{12}^2 / (V_1 V_2)$	(Inui et al., 2020)

7. Connections to Related Metrics and Extensions

The noise-normalized squared correlation metric unifies and generalizes several classic measures:

• Classical SNR: Bounded, normalized, and noise-aware form.
• Pearson’s $r^2$: Generalized to multivariate and high-dimensional settings, noise-corrected via random-matrix theory (Salimi et al., 2024).
• Information-theoretic loss: Linked to KL divergence bounds in covariance estimation (Diskin et al., 2024).
• Empirical uncertainty quantification: PRMSR underlying statistical uncertainty models (Xue et al., 2014).

The metric’s adaptability—embodying both detection performance and uncertainty quantification under minimal assumptions—positions it as a fundamental tool in modern signal processing, statistical inference, and high-dimensional statistics.