Orthogonal Least Squares with Integrated Information Theoretic Criteria for Joint Number of Targets and DoA Estimation

Technical Summary: Orthogonal Least Squares with Integrated Information Theoretic Criteria for Joint Number of Targets and DoA Estimation

Problem Statement
In radar applications utilizing antenna arrays, estimating the direction-of-arrival (DoA) of multiple targets is a fundamental task. While Maximum Likelihood (ML) estimators can theoretically achieve the Cramér-Rao Lower Bound (CRLB), their computational complexity grows exponentially with the number of targets, rendering them unsuitable for real-time operation. Greedy methods, such as Orthogonal Least Squares (OLS), offer a computationally efficient alternative by iteratively selecting DoAs. However, standard OLS implementations typically require the number of targets ($K$) to be known a priori. In practice, $K$ is an unknown parameter that must be estimated from data.

Existing approaches to estimate $K$ often rely on fixed-threshold stopping rules based on residual energy, which are difficult to tune as optimal thresholds depend on varying noise and signal levels. Alternatively, Information Theoretic Criteria (ITC), such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), provide a principled framework for model-order selection by balancing a likelihood term against a complexity penalty. However, direct application of ITC requires ML parameter estimates for every candidate model order, which reintroduces the prohibitive computational cost of multidimensional searches.

Methodology
This paper proposes a framework that integrates ITC model-order selection directly into the OLS greedy estimation procedure to jointly estimate the number of targets and their DoAs. The authors develop three distinct algorithmic strategies:

1. Disjoint Rank-based ITC-OLS: This approach decouples the tasks. First, the number of targets is estimated using a rank-based ITC applied to the sample covariance matrix of the received signals. This estimate determines the number of iterations for a subsequent standard OLS procedure to estimate the DoAs.
2. Joint Selection-based ITC-OLS: This method integrates the selection process directly. Instead of requiring full ML estimates for every candidate $k$, the algorithm uses the greedy OLS estimates at each step. To compensate for the fact that OLS estimates are not ML estimates, a modified ITC objective function is introduced. This function includes an "ML-correction parameter" ($\sigma^2_c$), interpreted as a minimum noise-variance level. The algorithm iteratively adds a new target angle only if it reduces the ITC objective value, effectively using the ITC as a dynamic stopping criterion.
3. Hybrid Rank-and-Selection-based ITC-OLS: This approach combines the strengths of the previous two. It first estimates a preliminary target count ($\hat{K}$) using the rank-based ITC. The OLS procedure is then forced to select at least $\hat{K}$ targets. For iterations beyond $\hat{K}$, the algorithm switches to the joint selection-based ITC logic, adding further targets only if they improve the objective function. This design aims to leverage the robustness of rank-based methods at low Signal-to-Noise Ratios (SNR) while utilizing the precision of selection-based methods at higher SNRs.

All three algorithms are derived under both AIC and BIC frameworks. The system model assumes a passive radar setup using unknown Orthogonal Frequency-Division Multiplexing (OFDM) data frames from an access point, where the receiver does not decode the transmitted symbols.

Key Contributions
The paper makes the following specific contributions:

• Novel Algorithms: The proposal of three algorithms (disjoint, joint, and hybrid) that successfully combine ITC target-number estimation with OLS DoA estimation.
• ML-Correction Parameter: The introduction of a parameter ($\sigma^2_c$) within the ITC objective function. This allows the use of greedy OLS estimates in place of computationally expensive ML estimates. The parameter controls the trade-off between hit rate and false-alarm rate, particularly at high SNRs where greedy methods might otherwise overfit by detecting spurious targets.
• Practical Application: The application of these methods to a passive radar scenario using unknown OFDM data, demonstrating feasibility without symbol decoding.
• Performance Analysis: A comprehensive numerical evaluation showing that the hybrid algorithm consistently outperforms the other variants and a baseline fixed-threshold method.

Results
Numerical simulations, conducted over 10,000 Monte-Carlo runs, evaluated the algorithms using Youden's J statistic (the difference between hit rate and false-alarm rate). Key findings include:

• Superiority of Hybrid Approach: The hybrid ITC-OLS algorithm achieved the best overall performance across varying SNR levels, numbers of targets, and bandwidths (number of subcarriers). It successfully leveraged the complementarity of rank-based and selection-based strategies.
• AIC vs. BIC: In the evaluated scenario, the AIC-based variants consistently outperformed the BIC-based variants.
• SNR Regimes:
  • At low SNR, the disjoint rank-based method was superior because the selection-based method tended to stop too early due to noise dominance in the objective function.
  • At medium to high SNR, the joint and hybrid methods improved performance as the quality of OLS angle estimates increased.
• Parameter Tuning: The ML-correction parameter $\sigma^2_c$ was found to be critical at high SNRs to prevent overestimation. Optimal values were identified around $-35$ dB for AIC and $-45$ dB for BIC. These values were found to be robust across variations in the number of symbols, subcarriers, and targets, though they may require tuning for different antenna array sizes.

Significance
The paper claims that its primary significance lies in bridging the gap between the computational efficiency of greedy OLS methods and the statistical rigor of ITC model-order selection. By circumventing the need for exhaustive ML searches, the proposed methods enable real-time joint estimation of target count and DoA. The hybrid approach, in particular, offers a robust solution that adapts to different noise regimes without requiring manual threshold tuning, addressing a known limitation of fixed-threshold stopping rules in practical radar systems.