K–R Residual Analysis for OFDM ISAC: Low-Rank Structure and Sensing Feasibility

We investigate the structural properties of minimum mean-square error (MMSE) equalization residuals in orthogonal frequency-division multiplexing (OFDM) integrated sensing and communication (ISAC) systems to determine whether communication processing byproducts contain exploitable sensing information. Through singular value decomposition (SVD) analysis across seven distinct channel models, nine target configurations, and system signal-to-noise ratios (SNR) from 0 to 30 dB, we establish three principal findings. First, the MMSE residual artifact arising from pilot-based channel estimation exhibits a universally low-rank structure, with dominant-to-second singular value ratios (σ₁/σ₂) ranging from 1.5 to 4.6 across all tested channel conditions. The first singular value captures 28–45% of total residual energy, originating from the structured interpolation error inherent in pilot-based estimation. Second, the separability between this artifact and target-dependent structure is channel-specific: rank-1 SVD removal yields statistically significant positive differential power (+ 0.5 to + 2.5 dB) at target locations under urban micro (UMi) propagation, but the optimal removal rank varies across channel conditions—ranging from PCA-0 (no removal) for rich multipath to PCA-5 for weak non-line-of-sight environments. Third, under favorable conditions, the method reveals multi-target structure (vehicle at 30 m, pedestrian at 15 m, and unmanned aerial vehicle at 8 m simultaneously visible) from a single OFDM frame. However, we critically demonstrate that while target-dependent structure is statistically detectable across multiple frames, single-frame constant false alarm rate (CFAR) detection at practical false alarm rates is not achievable due to variance dominance, with hypothesis test statistic ratios (H1/H0) near unity. These findings establish the feasibility boundary for residual-based ISAC sensing and identify the quantified performance gap—measurable and specific—that multiple-input multiple-output (MIMO) antenna arrays, wider bandwidth, or multi-frame coherent accumulation must bridge for practical deployment. To our knowledge, this represents the first structural characterization of MMSE residuals for ISAC purposes.