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

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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. Cell Communication and Signaling Integrated sensing and communication (ISAC) OFDM MMSE equalization singular value decomposition residual analysis low-rank structure 6G feasibility characterization Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 I. Introduction A. Motivation and Context Sixth-generation (6G) wireless systems are expected to fundamentally integrate sensing and communication capabilities into a single unified infrastructure. This paradigm, known as integrated sensing and communication (ISAC), promises to transform cellular base stations from pure data conduits into distributed environmental sensors capable of detecting, tracking, and classifying objects in their surroundings using the same radio-frequency waveforms that carry communication data [1]–[3]. The momentum behind ISAC is substantial. At the 3GPP RAN #108 plenary meeting in June 2025, ISAC was formally included in the scope of study for 6G radio, establishing it as a “Day 1” feature for the next-generation standard [4]. Standardization activities are progressing through Release 20, with the ETSI ISAC Industry Specification Group developing channel models and evaluation methodologies [5], [6]. Industry leaders including Samsung, Ericsson, Nokia, and Huawei are investing heavily in ISAC research, recognizing its potential to create new revenue streams through sensing-as-a-service capabilities [7], [8]. Current ISAC receiver architectures typically process the received signal through parallel communication and sensing processing chains, each operating on the full received waveform [9], [10]. The communication chain performs channel estimation, equalization, and data detection, while the sensing chain performs matched filtering, range-Doppler processing, and target detection. These parallel chains share the same received signal but process it independently, each with its own computational cost and hardware requirements. B. The Residual Question We pose a fundamentally different question: does the communication receiver itself produce byproducts that contain sensing-relevant information? Specifically, after MMSE equalization and data detection, the receiver computes a residual signal—the difference between the received signal and the reconstructed signal based on the estimated channel and detected symbols. This residual is conventionally discarded or, at best, used for iterative communication refinement. We investigate whether it contains exploitable structure for sensing. The motivation for this question is both practical and theoretical. Practically, if sensing information survives the equalization process and resides in the residual, it could be extracted without additional waveform design, dedicated spectrum allocation, or separate sensing hardware—making ISAC deployment significantly simpler and cheaper. Theoretically, understanding what the MMSE equalizer preserves and destroys provides insight into the fundamental information flow within ISAC receivers. The answer, as we shall demonstrate, is nuanced. The residual does contain target-dependent structure, but it is buried beneath a stronger structured artifact arising from imperfect channel estimation. Separating these two components is possible under specific conditions but not universally achievable. This characterization—establishing both what is present and what is missing—constitutes the core contribution of this work. C. Related Work The most closely related work is the LISAC framework by Bian et al. [11], which employs a “residual-assisted MMSE equalizer” where a neural network calibrates the coarse MMSE channel estimate using the equalization residual. LISAC demonstrates that residual processing can improve communication bit error rate (BER), but it does not investigate the residual’s sensing content—the residual is treated as an error signal for communication refinement, not as an information carrier for sensing. Our work differs fundamentally in three respects: we characterize the residual’s sensing content (not its communication utility), we use interpretable SVD decomposition (not neural networks), and we identify the artifact structure and its channel-dependent separability, which LISAC does not address. Data-aided iterative ISAC receivers, such as the bistatic framework by Temiz et al. [12], perform joint sensing and data demodulation by iteratively refining channel and symbol estimates. These approaches process the full received signal through dedicated sensing algorithms, rather than exploiting the communication receiver’s residual as we propose. The fundamental difference is that our approach seeks to extract sensing information from a byproduct of the communication process, not from the signal itself. In the broader ISAC literature, the foundational survey by Liu et al. [1] established the joint radar-communication design framework, while Xiong et al. [13] derived the fundamental Cramér–Rao bound (CRB)-rate tradeoff for ISAC under Gaussian channels. Signal processing overviews by Zhang et al. [14] and Koivunen et al. [15] provide comprehensive treatments of multicarrier ISAC techniques. The seminal work by Sturm and Wiesbeck [16] established the theoretical foundation for OFDM-based radar sensing. None of these works, however, specifically characterize the structure of the MMSE equalization residual or its potential for sensing. D. Contributions This paper makes three specific contributions: C1: Universal low-rank artifact structure. We demonstrate through SVD analysis that the MMSE residual artifact—caused by pilot interpolation and channel estimation error—exhibits a dominant low-rank structure across all seven tested channel models. The first singular value captures 28–45% of total residual energy, with σ₁/σ₂ ratios from 1.5 to 4.6. This finding is channel-independent and constitutes a new structural characterization of MMSE residuals. C2: Channel-dependent separability. We show that while the artifact is universally low-rank, its separability from target-dependent structure is channel-specific. Under UMi-like propagation, rank-1 SVD removal exposes target signatures with +0.8 to +2.5 dB differential power. Under strong non-line-of-sight (NLOS) or rich multipath conditions, the optimal removal rank changes—sometimes higher-rank removal helps, sometimes no removal is best. We characterize this dependence systematically. C3: Quantified detection gap. We establish that the target-dependent structure exposed by SVD processing is statistically detectable across many frames but insufficient for single-frame CFAR detection at practical false alarm rates. We quantify this gap precisely: for a −10 dB target at SNR = 20 dB, the mean differential is +0.89 dB with standard deviation ±1.40 dB, yielding a signal-to-variability ratio of approximately 0.6. We identify MIMO beamforming, wider bandwidth, and multi-frame accumulation as the specific mechanisms that can bridge this gap. E. Paper Organization The remainder of this paper is organized as follows. Section II presents the system model including the OFDM ISAC signal model, channel estimation pipeline, and residual construction. Section III provides the core residual structure analysis via SVD, establishing the universal low-rank finding and channel-dependent separability. Section IV evaluates sensing performance across multiple target scenarios, SNR conditions, and detection metrics. Section V presents the artifact suppression investigation and its results. Section VI discusses the findings, their implications, and limitations. Section VII concludes the paper. II. System Model A. OFDM ISAC Signal Model We consider an OFDM monostatic ISAC base station operating at carrier frequency fc = 28 GHz with Nsc = 256 active subcarriers, Nsym = 64 OFDM symbols per frame, and subcarrier spacing Δf = 120 kHz, corresponding to the 3GPP NR FR2 (Frequency Range 2) numerology [2]. The effective signal bandwidth is Nsc × Δf = 30.72 MHz, providing a range resolution of ΔR = c/(2B) = 4.88 m, where c = 3 × 10⁸ m/s is the speed of light. The received signal at subcarrier k ∈ {0, 1, ..., Nsc−1} and OFDM symbol index l ∈ {0, 1, ..., Nsym−1} is modeled as: Y[k,l] = (Hcomm[k,l] + Htarget[k,l]) · X[k,l] + W[k,l] (1) where Hcomm[k,l] represents the communication channel comprising the line-of-sight (LOS) path and multipath components that the receiver’s channel estimator is designed to capture; Htarget[k,l] represents weak reflections from sensing targets (vehicles, pedestrians, unmanned aerial vehicles) that are 20–40 dB below the communication channel power and are NOT modeled by the channel estimator; X[k,l] denotes the transmitted symbol at the (k,l)-th resource element, which is either a known pilot symbol or a 16-QAM data symbol; and W[k,l] ∼ CN(0, σ²) is additive white Gaussian noise (AWGN). The communication channel is constructed as a superposition of multipath components: Hcomm[k,l] = Σp αp · exp(j(-2πkΔfτp + 2πlTsymνp)) (2) where αp, τp, and νp are the complex gain, propagation delay, and Doppler frequency of the p-th path, and Tsym is the total OFDM symbol duration including cyclic prefix. An identical expression holds for Htarget[k,l] with target-specific parameters. The key assumption underlying this work is that the channel estimator captures Hcomm (the strong communication paths) but treats Htarget as noise. This is realistic because sensing target reflections are typically 20–40 dB below the communication channel—well below the estimator’s noise floor. This assumption is what creates the possibility of target information surviving into the residual. TABLE I: SYSTEM PARAMETERS Parameter Value Justification Carrier frequency fc 28 GHz 3GPP NR FR2 Subcarrier spacing Δf 120 kHz Standard FR2 numerology Subcarriers Nsc 256 Effective BW = 30.72 MHz OFDM symbols Nsym 64 Frame duration Modulation 16-QAM Moderate spectral efficiency Pilot spacing Every 4th subcarrier 25% pilot overhead Channel estimation LS + interp. + LMMSE Standard pipeline Range resolution 4.88 m c/(2·Nsc·Δf) SNR range 0–30 dB Full operating range Monte Carlo trials 15–80 per condition Statistical significance B. Channel Estimation and MMSE Equalization The channel estimation pipeline follows the standard three-stage approach used in 3GPP NR receivers. First, least-squares (LS) estimation is performed on pilot subcarriers, spaced every 4th subcarrier (25% pilot density). Pilots are known QPSK symbols with a fixed random seed for reproducibility. Second, the LS estimates are linearly interpolated to data subcarriers in the frequency domain. Third, a moving-average LMMSE smoothing filter (kernel size 7) is applied to reduce noise in the estimates. Per-subcarrier MMSE equalization then produces symbol estimates: X̂[k,l] = Ĥ*[k,l] / (|Ĥ[k,l]|² + σ²) · Y[k,l] (3) followed by hard decision (nearest 16-QAM constellation point) to produce X̂hard[k,l]. C. Residual Construction The communication residual is defined as the difference between the received signal and the reconstructed signal: r[k,l] = Y[k,l] − Ĥ[k,l] · X̂hard[k,l] (4) To enable sensing processing, the data symbol modulation must be removed from the residual. We perform regularized division: rs[k,l] = r[k,l] · X̂*hard[k,l] / (|X̂hard[k,l]|² + ε) (5) where ε = 0.01 is a regularization parameter to avoid noise amplification near zero-crossings of the QAM constellation. This sensing residual rs decomposes into three components: rs = rs,artifact + rs,target + rs,noise (6) where rs,artifact arises from the channel estimation error (Ĥ − Hcomm), rs,target carries sensing information from Htarget, and rs,noise is thermal noise. The central question of this paper is whether rs,target can be separated from rs,artifact. III. Residual Structure Analysis A. Naive Residual Sensing Fails Before presenting the SVD analysis, we first establish a critical negative result: direct application of standard sensing processing to the residual does not work. Applying a 2D-FFT to the sensing residual rs to produce a range-Doppler map (RDM) yields no detectable target signatures. The RDM with targets present and without targets present are visually and statistically indistinguishable. The differential power at true target locations—defined as Δ = 10·log10(Pwith/Pwithout) where Pwith and Pwithout are the peak RDM powers with and without targets present—is consistently negative (−0.5 to −4 dB) across all target gains from −5 to −30 dB. This means the target signal is not merely weak; it is actually less visible than background noise fluctuations at the target location. This negative result is itself significant: it establishes that MMSE residuals cannot be naively repurposed for sensing. The residual is dominated by structured channel estimation artifacts, not target echoes. Understanding and overcoming this artifact is the subject of the following analysis. B. Universal Low-Rank Artifact Structure We decompose the sensing residual matrix Rs ∈ C^(Nsc × Nsym) via singular value decomposition (SVD): Rs = Σi σi ui viᴴ (7) where σ₁ ≥ σ₂ ≥ ... are the ordered singular values, and ui, vi are the corresponding left and right singular vectors. We perform this analysis with no targets present (Htarget = 0) to characterize the pure artifact structure. Table II presents the singular value analysis across seven channel models at SNR = 20 dB, averaged over 20 independent noise realizations. TABLE II: SINGULAR VALUE ANALYSIS OF THE MMSE RESIDUAL ACROSS CHANNEL MODELS (NO TARGETS, SNR = 20 dB) Channel Model Paths σ₁/σ₂ σ₁/σ₃ E(σ₁) Interpretation UMi default 3 4.47 4.61 43.5% Strong rank-1 Strong NLOS 3 2.79 4.36 39.7% Moderate rank-1 Weak NLOS 3 4.52 4.71 44.1% Strong rank-1 Many paths 5 4.37 4.66 44.4% Strong rank-1 High Doppler comm 3 1.51 2.17 28.3% Weak rank-1 LOS only 1 4.63 4.78 44.8% Strong rank-1 Rician K=10 3 4.62 4.77 44.6% Strong rank-1 The key observation is that σ₁/σ₂ exceeds 1.5 for all seven channel models. This confirms that the artifact possesses a universally dominant rank-1 component. The first singular value alone captures 28–45% of total residual energy. The physical origin of this low-rank structure is the frequency-domain interpolation used in channel estimation: interpolating between pilot subcarriers produces a smooth estimation error that is highly correlated across subcarriers and OFDM symbols, concentrating energy in the first singular vector. C. Channel-Dependent Separability Having established the low-rank artifact structure, we now investigate whether removing this artifact exposes target-dependent structure. We define the cleaned residual after removing k singular values: Rs,clean(k) = Σi=k+1 σi ui viᴴ (8) We measure target visibility using the differential power metric: Δ = 10·log10(Pwith/Pwithout) in dB, where Pwith and Pwithout are the peak range-Doppler map powers at the true target location with and without targets present, averaged over 30 independent Monte Carlo trials. Targets are placed at 30 m range with 1500 Hz Doppler, at −22 dB gain relative to the communication channel. TABLE III: DIFFERENTIAL POWER (dB) AT TARGET LOCATION VS. PCA REMOVAL RANK (SNR = 20 dB, TARGET GAIN = −22 dB) Channel PCA-0 PCA-1 PCA-2 PCA-3 PCA-5 Best UMi default −0.01 +1.26 −0.78 −0.57 +0.49 PCA-1 Strong NLOS −0.10 −0.36 −0.93 −0.57 +0.04 PCA-5 Weak NLOS −0.07 −0.05 +0.43 −0.27 +0.59 PCA-5 Many paths +0.25 −0.41 −0.62 −0.92 −0.99 PCA-0 High Doppler −0.17 −0.23 −0.10 −0.36 +0.27 PCA-5 LOS only −0.05 −0.03 −0.75 −0.34 +0.05 PCA-5 Rician K=10 +0.02 −0.20 −0.56 −0.45 +0.33 PCA-5 Three distinct regimes emerge from Table III: Regime 1 — Clean separation (UMi): The artifact concentrates tightly in the first singular vector, and PCA-1 cleanly exposes target structure with +1.26 dB differential. This is the ideal case where the artifact and target subspaces are nearly orthogonal. Regime 2 — Distributed artifact (Weak NLOS, Rician): The artifact energy is distributed across multiple singular values, requiring PCA-5 for positive differential (+0.59 dB and +0.33 dB respectively). The artifact subspace is multi-dimensional, and partial removal is needed before target structure becomes visible. Regime 3 — Subspace entanglement (Many paths): The artifact and target subspaces are completely entangled. PCA-0 (no removal at all) gives the best result (+0.25 dB), because any SVD removal also removes target energy. This represents a fundamental limit of subspace-based separation methods. IV. Sensing Performance Evaluation A. Multi-Target Visibility Under UMi Under the UMi channel with PCA-1 at SNR = 20 dB, we test nine distinct target scenarios to evaluate the robustness of residual-based sensing across diverse target configurations. Each scenario is evaluated over 25 independent Monte Carlo trials. TABLE IV: PCA-1 DIFFERENTIAL ACROSS TARGET SCENARIOS (UMi CHANNEL, SNR = 20 dB) Scenario Range Velocity Gain (dB) Diff (dB) Status Close fast car 10 m 120 km/h −20 +1.03 ± 2.46 ✓ Detected Two vehicles 20+40 m 77+58 km/h −22/−24 +1.36, +1.30 ✓ Both Car+Ped+UAV 30+15+8 m 48+4+77 −22/−30/−26 +0.76, +1.28, +1.63 ✓ All three High-speed 25 m 300 km/h −22 +1.08 ± 1.65 ✓ Detected Original pair 30+12 m 29+6 km/h −22/−30 +0.89, +1.26 ✓ Both Stationary 40 m 0 km/h −20 −0.41 ± 1.34 ✗ FAILS Very weak 35 m 19 km/h −40 +0.04 ± 0.57 ~ Marginal Far slow ped 50 m 5 km/h −28 +0.08 ± 0.87 ~ Marginal The three-target scenario (vehicle at 30 m, pedestrian at 15 m, and UAV at 8 m simultaneously visible) represents the strongest positive result. All three targets show positive differential despite having different ranges, velocities, and radar cross-sections. This demonstrates multi-target capability from a single OFDM residual frame—a key result for practical ISAC deployment. The stationary target failure (−0.41 dB) provides an equally important negative result. Targets at zero Doppler produce residual structure at DC (zero frequency offset), which is precisely the strongest component of the channel estimation artifact. Since PCA-1 removes the dominant singular vector—which is dominated by the DC artifact—it inadvertently removes the stationary target signature along with the artifact. This is a fundamental limitation of subspace-based artifact removal for near-zero-Doppler targets and cannot be resolved without external information (e.g., a target-free calibration frame). B. SNR Dependence The differential power under UMi/PCA-1 increases monotonically with system SNR, confirming physical correctness of the approach: TABLE V: DIFFERENTIAL POWER VS. SNR (UMi/PCA-1, TARGET GAIN = −10 dB) SNR (dB) 5 10 15 20 25 30 Diff (dB) +0.31 +0.56 +0.69 +0.92 +1.46 +2.55 This monotonic increase confirms the expected physics: higher SNR yields more accurate channel estimation, which produces a cleaner artifact (closer to pure rank-1), enabling more effective PCA-based separation and stronger target visibility. The improvement from +0.31 dB at 5 dB SNR to +2.55 dB at 30 dB SNR represents an 8× improvement in linear scale. C. The Detection Gap Despite statistically positive differentials, single-frame CFAR detection is not achievable at practical false alarm rates. We evaluate CFAR detection by computing the test statistic (cell-under-test power divided by estimated noise power from surrounding training cells) for 80 independent trials at SNR = 20 dB under UMi/PCA-1. TABLE VI: CFAR DETECTION PERFORMANCE (SNR = 20 dB, UMi, PCA-1) Target Gain H1/H0 Ratio Pd @ Pfa=0.01 Pd @ Pfa=0.05 −10 dB 0.97 0.000 0.025 −15 dB 1.07 0.013 0.050 −20 dB 1.03 0.000 0.037 The H1/H0 ratios near unity indicate that per-frame variance dominates the target-induced power shift. The mean differential of +0.89 dB for −10 dB targets has a standard deviation of ±1.40 dB, yielding a signal-to-variability ratio of approximately 0.6. For reliable single-frame detection (Pd > 0.9 at Pfa = 0.01), this ratio must exceed approximately 3.0, implying a performance gap of approximately 7 dB (14 dB in power). D. Communication Performance The K–R iterative receiver, which performs 3 iterations of residual-based symbol refinement (each iteration computes the residual, adds it back to the equalized signal scaled by the inverse channel estimate, and re-decides), provides a consistent communication symbol error rate (SER) improvement of +0.1 to +0.2 dB across all tested SNR values from 0 to 30 dB. This confirms that the R-step sensing extraction does not degrade communication performance—the two outputs (decoded bits and range-Doppler map) are complementary. V. Artifact Suppression Investigation Recognizing that the raw PCA-1 approach achieves only modest differentials (+0.5 to +2.5 dB) and fails for CFAR detection, we investigated multiple artifact suppression strategies to determine whether additional processing could improve target visibility. A. Template Subtraction We constructed an artifact template by averaging the residual RDM over 100 target-free trials, then subtracted this template (with adaptive scaling) from target-bearing RDMs. This approach failed: the artifact varies per noise realization and per data symbol pattern, making a static template inaccurate. Template subtraction typically made the differential worse (−10 to −100 dB), not better, by introducing new artifacts where the template did not match the current realization. B. Pilot-Domain Residual We also investigated constructing the residual from pilot observations only (Y_p/X_p − Ĥ at pilot positions), avoiding data decision errors entirely. While pilot-domain PCA-1 achieved positive differential at strong target gains (+2.50 dB at −5 dB), it was outperformed by data-domain PCA-1 at moderate and weak gains. The data domain provides 4× more observations (256 vs. 64 subcarriers), enabling more accurate SVD-based separation despite the additional decision noise. C. Channel Robustness We tested PCA-1 across four alternative communication channel configurations (strong NLOS, weak NLOS, many-path, and high-Doppler) using the original target set. Results show that PCA-1 gives positive differential only for the default UMi channel (+0.67 dB). The strong NLOS channel gives −0.85 dB, weak NLOS gives −0.35 dB, and many-path gives −0.60 dB. This confirms that PCA-1 is not a universal solution—it is the correct answer specifically for UMi-like channels where the artifact is cleanly rank-1 and well-separated from the target subspace. VI. Discussion A. Signal Existence vs. Detectability Our results reveal a fundamental gap between two distinct questions. The existence question—“Does target-dependent structure persist in the MMSE residual?”—is answered affirmatively: PCA-based artifact removal consistently exposes statistically significant target-dependent power under UMi conditions, across diverse target configurations including multiple simultaneous targets. The detectability question—“Can a single OFDM frame reliably detect targets at practical false alarm rates?”—is answered negatively at the tested parameters, with H1/H0 ratios near unity. This distinction is important for the ISAC research community. Many ISAC papers implicitly assume that signal existence implies detectability. Our results quantify the gap: approximately 7 dB of additional processing gain is needed to bridge from the current signal-to-variability ratio of 0.6 to the threshold of 3.0 required for reliable detection. B. Bridging the Gap: Three Mechanisms MIMO beamforming: With Nt transmit and Nr receive antennas, the residual gains Nt · Nr spatial diversity. An 8×2 array configuration would provide approximately 12 dB additional gain on the residual, well exceeding the 7 dB gap. This is the most promising path to practical deployment. Bandwidth increase: Expanding from 30.72 MHz to 400 MHz (the maximum 3GPP FR2 component carrier bandwidth) improves range resolution from 4.88 m to 0.375 m. This physically separates target range bins from artifact regions, reducing subspace overlap and enabling cleaner separation. Multi-frame accumulation: Coherent integration over M frames provides approximately 10·log10(M) dB processing gain. Integrating 50 frames (≈17 dB gain) would comfortably exceed the detection threshold, at the cost of requiring target stationarity over the integration window. C. Why the Artifact Is Low-Rank The pilot-based channel estimation pipeline with frequency-domain interpolation produces estimation errors that are smooth functions of subcarrier index. The interpolation kernel acts as a low-pass filter on the estimation error, suppressing high-frequency error components and concentrating energy in the first few spatial frequencies—which correspond to the first few singular values. This smoothness is independent of the communication channel itself, which explains the universality of the low-rank finding across all seven tested channel models. D. Why Separability Is Channel-Dependent Although the artifact is universally low-rank, target energy may project onto the same singular vectors when: (i) target delays are similar to communication path delays, producing similar frequency-domain oscillation patterns; (ii) the communication channel itself has rich multipath structure that spreads estimation error across multiple singular vectors, overlapping with the target subspace; or (iii) target Doppler shifts are near zero, aligning with the artifact’s dominant DC component. E. Comparison with LISAC The LISAC framework [11] and our work represent complementary perspectives on MMSE residuals in ISAC. LISAC treats the residual as an error signal and trains a neural network to correct communication symbol estimates. We treat the residual as an information carrier and use SVD to characterize its structure for sensing. LISAC has no sensing output; we produce a range-Doppler map. LISAC requires training data and neural network inference; we use a single SVD operation. The two approaches could potentially be combined: LISAC’s neural network could be augmented with a sensing output head trained on the residual’s target-dependent subspace. F. Limitations Several limitations should be noted. First, the differentials (+0.5 to +2.5 dB) are small, representing a proof-of-concept rather than a detection-ready system. Second, the channel robustness is limited—PCA-1 works well only for UMi-like channels. Third, all results are simulation-based with simplified multipath models; full 3GPP TR 38.901 stochastic channel models and hardware-in-the-loop validation would strengthen the findings. Fourth, the SISO configuration provides no spatial diversity; MIMO is essential for practical performance. Fifth, the number of Monte Carlo trials (15–80 per condition) should be increased to 500+ for publication-quality statistical confidence. VII. Conclusion We have established three findings regarding the structure of MMSE communication residuals in OFDM ISAC systems. First, the residual artifact from pilot-based channel estimation exhibits a universally low-rank structure across all tested channel conditions, with σ₁/σ₂ ratios from 1.5 to 4.6. The first singular value captures 28–45% of total residual energy. This structural finding is independent of the communication channel and represents a new characterization of MMSE residuals. Second, target-dependent structure can be exposed via SVD-based artifact removal, but the optimal removal rank is channel-specific—ranging from rank-1 for UMi channels to rank-5 or no removal for rich multipath environments. This channel dependence arises from the alignment between artifact and target subspaces in the SVD domain. Third, while this target structure is statistically detectable across multiple trials (positive differential in 6 of 9 target scenarios, monotonically increasing with SNR), the per-frame variance prevents single-frame CFAR detection at practical false alarm rates. We quantify this gap precisely: approximately 7 dB of additional processing gain is needed, achievable through MIMO beamforming, wider bandwidth, or multi-frame accumulation. These findings define the feasibility boundary for residual-based ISAC sensing: the signal exists, the gap to detection is measurable and specific, and the mechanisms to bridge it are identified. This characterization informs the design of future K–R receivers that combine model-based artifact removal with additional gain mechanisms to achieve practical sensing from communication residuals without waveform modification or dedicated sensing resources. References F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” IEEE Trans. Commun., vol. 68, no. 6, pp. 3834–3862, Jun. 2020. GPP, “Study on channel model for frequencies from 0.5 to 100 GHz,” 3GPP TR 38.901, v17.0.0, Mar. 2022. GPP, “Study on integrated sensing and communication (Release 19),” 3GPP TR 22.837, v19.2.0, Jun. 2024. GPP RAN #108, “New SID: Study on Integrated Sensing And Communication (ISAC) for NR,” RP-251861, Jun. 2025. GPP, “Study on channel modelling for ISAC for NR,” 3GPP TR 38.765, Release 19, 2025. ETSI ISAC ISG, “ISAC channel modelling — Perspectives from ETSI,” arXiv:2505.10275, May 2025. 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Temiz et al., “Bridging the gap via data-aided sensing: Can bistatic ISAC converge to genie performance?” arXiv:2505.01280, May 2025. Y. Xiong, F. Liu, Y. Cui, W. Yuan, T. X. Han, and G. Caire, “On the fundamental tradeoff of integrated sensing and communications under Gaussian channels,” IEEE Trans. Inf. Theory, vol. 69, no. 9, pp. 5723–5751, Sep. 2023. F. Liu and C. Masouros, “A tutorial on joint radar and communication transmission for vehicular networks,” IEEE Commun. Lett., vol. 25, no. 2, pp. 322–336, Feb. 2021. Z. Wei, H. Qu, W. Jiang, et al., “Integrated sensing and communication: Enabling techniques, applications, tools and data sets, standardization, and future directions,” IEEE Internet Things J., vol. 10, no. 13, pp. 11621–11666, Jul. 2023. C. Sturm and W. Wiesbeck, “Waveform design and signal processing aspects for fusion of wireless communications and radar sensing,” Proc. IEEE, vol. 99, no. 7, pp. 1236–1259, Jul. 2011. F. Liu, Y.-F. Liu, A. Li, C. Masouros, and Y. C. Eldar, “Cramér–Rao bound optimization for joint radar-communication beamforming,” IEEE Trans. Signal Process., vol. 70, pp. 240–253, Jan. 2022. H. Hua, T. X. Han, and J. Xu, “MIMO integrated sensing and communication: CRB-rate tradeoff,” IEEE Trans. Wireless Commun., vol. 23, no. 3, pp. 2839–2854, Mar. 2024. Z. Wei, J. Piao, X. Yuan, et al., “Waveform design for MIMO-OFDM integrated sensing and communication system: An information theoretical approach,” IEEE Trans. Commun., vol. 72, no. 1, pp. 496–509, Jan. 2024. W. Yuan, L. Zhou, S. K. Dehkordi, et al., “From OTFS to DD-ISAC: Integrating sensing and communications in the delay Doppler domain,” IEEE Wireless Commun., vol. 31, no. 6, pp. 152–160, Dec. 2024. O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Börjesson, “OFDM channel estimation by singular value decomposition,” IEEE Trans. Commun., vol. 46, no. 7, pp. 931–939, Jul. 1998. C. Baquero Barneto, T. Riihonen, M. Turunen, et al., “Full-duplex OFDM radar with LTE and 5G NR waveforms: Challenges, solutions, and measurements,” IEEE Trans. Microw. Theory Techn., vol. 67, no. 10, pp. 4042–4054, Oct. 2019. GPP, “Release 20 — 5G Advanced,” Work Plan SP-260360, presented at TSGs#111, Mar. 2026. S. Mura, D. Tagliaferri, M. Mizmizi, U. Spagnolini, and A. Petropulu, “Optimized waveform design for OFDM-based ISAC systems,” IEEE Trans. Wireless Commun., vol. 24, no. 6, pp. 5241–5257, Jun. 2025. R. Bomfin and M. Chafii, “On the performance analysis of zero-padding OFDM for monostatic ISAC systems,” IEEE Trans. Commun., vol. 72, no. 10, pp. 6304–6319, Oct. 2024. Z. Wei, H. Qu, and W. Jiang, “Iterative signal processing for integrated sensing and communication systems,” IEEE Trans. Green Commun. Netw., vol. 7, no. 1, pp. 401–412, Mar. 2023. B. Li, W. Yuan, F. Liu, N. Wu, and S. Jin, “OTFS-based ISAC: How delay Doppler channel estimation assists environment sensing?” IEEE Wireless Commun. Lett., vol. 13, no. 10, pp. 2843–2847, Oct. 2024. Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9299929","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":616441656,"identity":"346c5fa6-527b-4cff-9010-c2ca89e82147","order_by":0,"name":"RamaKrishna Pasupuleti","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9ElEQVRIiWNgGAWjYHACZoYEBjkgfbD9wwcgxcZOnBZjBgbGw22MM0BamInRwgDSwny8jZkHxscHDI73PjZ48McgcTvbwbbHNr+2yfMxMzB++JiDR8uZ48YJiW0GiTt7DrYb5/bdNmxjZmCWnLkNtxazG2nMBxIb/iRuuHGwQTq35zYjUAsbMy8hLQlAh224/7BB2rLntj1RWhIS2IBaDhxsk2b4cTuRoBb7M8eYDYB+MQZqaTbsbbid3MbM2IzXL5LtbcySP/4YyG44cPzhgx9/btvOb28++OEjHi2ogLENTDYQqx4E/pCieBSMglEwCkYKAAAdgVk9CekE7wAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0009-0008-8418-1430","institution":"Independent Resercher","correspondingAuthor":true,"prefix":"","firstName":"RamaKrishna","middleName":"","lastName":"Pasupuleti","suffix":""}],"badges":[],"createdAt":"2026-04-02 07:57:39","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9299929/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9299929/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106071963,"identity":"8f9383ba-500b-4550-8f23-6302cdbd183e","added_by":"auto","created_at":"2026-04-03 06:44:53","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":661796,"visible":true,"origin":"","legend":"\u003cp\u003eDiagnostic comparison of residual range-Doppler maps. Left: with targets present. Center: without targets (pure artifact). Right: noise only. The with-targets and without-targets maps are visually identical, confirming that naive residual sensing fails. White triangles mark true target locations (30 m/1500 Hz and 12 m/300 Hz).\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/6bc8f457b87051ff4d0169f1.png"},{"id":106401990,"identity":"1dbad656-956f-4a9a-a1c5-ee7b2def49e3","added_by":"auto","created_at":"2026-04-08 09:10:33","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":266639,"visible":true,"origin":"","legend":"\u003cp\u003eSingular value analysis across channel models. Top-left: normalized singular value decay showing universally rapid drop-off. Top-right: σ₁/σ₂ ratio per channel (all exceed 1.5 threshold). Bottom-left: PCA-k differential heatmap showing channel-dependent optimal rank. Bottom-right: summary statistics.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/243b6fed528e8ccb9b967249.png"},{"id":106095067,"identity":"f144b13f-3fba-4276-911b-a75586c36573","added_by":"auto","created_at":"2026-04-03 11:44:09","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":519117,"visible":true,"origin":"","legend":"\u003cp\u003eMulti-target detection results across scenarios. Left: PCA-1 differential for each scenario (green = positive, red = negative). Center: SNR sweep for select scenarios. Bottom: example PCA-1 cleaned range-Doppler maps showing target markers at true locations.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/3cc124b86f8882b3bed563a7.png"},{"id":106071965,"identity":"beec79fd-3d9c-4641-9c75-7b7f5c1d20ac","added_by":"auto","created_at":"2026-04-03 06:44:53","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":279723,"visible":true,"origin":"","legend":"\u003cp\u003ePilot-domain vs. data-domain residual comparison. Top-left: gain sweep showing PCA-1 advantage. Top-right: SNR sweep with monotonically increasing differential. Bottom-left: example pilot residual RDM. Bottom-right: summary statistics showing data-domain PCA-1 outperforms pilot-domain.\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/ced1d08176d3fc102acbb5a9.png"},{"id":106071966,"identity":"283935a4-2572-4ea2-8a4a-79db439384d1","added_by":"auto","created_at":"2026-04-03 06:44:53","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":272389,"visible":true,"origin":"","legend":"\u003cp\u003eDetection gap analysis. Top-left: gain sweep showing differential and target SNR (red curve stays near zero). Top-center: SNR sweep showing inconsistent detection. Top-right: coherent accumulation showing limited improvement. Bottom-left: communication K–R improvement (+0.1–0.2 dB). Bottom-right: summary statistics.\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/f41992b8cece6e7ff07ab70b.png"},{"id":106095074,"identity":"f0f1bfa9-3bd9-453c-adf1-b52a19d2d64d","added_by":"auto","created_at":"2026-04-03 11:44:10","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":59380,"visible":true,"origin":"","legend":"\u003cp\u003eCommunication performance: MMSE-only vs. K–R iterative receiver. The K–R receiver provides consistent +0.1 to +0.2 dB SER improvement across all SNR values, confirming that sensing does not degrade communication.\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/03aca74ea969a81dc7d58369.png"},{"id":106071968,"identity":"98126f96-c5d4-4449-8d4a-605cd5156a26","added_by":"auto","created_at":"2026-04-03 06:44:53","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":322980,"visible":true,"origin":"","legend":"\u003cp\u003eArtifact suppression investigation. Left: artifact template (RDM with no targets). Center: RDM after PCA + template subtraction (gain = −10 dB). Right: raw residual RDM for comparison. Template subtraction does not reliably improve detection because artifacts vary per realization.\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/fa56b1314df5e46cc3d5c2f9.png"},{"id":106071970,"identity":"da4fb958-f1f2-419b-b5b2-880c2c17967d","added_by":"auto","created_at":"2026-04-03 06:44:53","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":33072,"visible":true,"origin":"","legend":"\u003cp\u003ePhase correction impact on detection. An initial hypothesis was that MMSE-induced phase distortion degrades sensing—in fact, phase correction was found to hurt detection, not help it. The uncorrected residual preserves more target structure.\u003c/p\u003e","description":"","filename":"image8.png","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/3fdeeacf82112aebc2f466c7.png"},{"id":106960039,"identity":"797f73b5-9c8a-49a6-a0bb-b0e59b268a52","added_by":"auto","created_at":"2026-04-15 09:18:09","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3636359,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9299929/v1/2accd115-ab85-4e6a-98a2-7b58d4e414d3.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eK–R Residual Analysis for OFDM ISAC: Low-Rank Structure and Sensing Feasibility\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"I. Introduction","content":"\u003ch2\u003eA. Motivation and Context\u003c/h2\u003e\n\u003cp\u003eSixth-generation (6G) wireless systems are expected to fundamentally integrate sensing and communication capabilities into a single unified infrastructure. This paradigm, known as integrated sensing and communication (ISAC), promises to transform cellular base stations from pure data conduits into distributed environmental sensors capable of detecting, tracking, and classifying objects in their surroundings using the same radio-frequency waveforms that carry communication data [1]\u0026ndash;[3].\u003c/p\u003e\n\u003cp\u003eThe momentum behind ISAC is substantial. At the 3GPP RAN #108 plenary meeting in June 2025, ISAC was formally included in the scope of study for 6G radio, establishing it as a \u0026ldquo;Day 1\u0026rdquo; feature for the next-generation standard [4]. Standardization activities are progressing through Release 20, with the ETSI ISAC Industry Specification Group developing channel models and evaluation methodologies [5], [6]. Industry leaders including Samsung, Ericsson, Nokia, and Huawei are investing heavily in ISAC research, recognizing its potential to create new revenue streams through sensing-as-a-service capabilities [7], [8].\u003c/p\u003e\n\u003cp\u003eCurrent ISAC receiver architectures typically process the received signal through parallel communication and sensing processing chains, each operating on the full received waveform [9], [10]. The communication chain performs channel estimation, equalization, and data detection, while the sensing chain performs matched filtering, range-Doppler processing, and target detection. These parallel chains share the same received signal but process it independently, each with its own computational cost and hardware requirements.\u003c/p\u003e\n\u003ch2\u003eB. The Residual Question\u003c/h2\u003e\n\u003cp\u003eWe pose a fundamentally different question: does the communication receiver itself produce byproducts that contain sensing-relevant information? Specifically, after MMSE equalization and data detection, the receiver computes a residual signal\u0026mdash;the difference between the received signal and the reconstructed signal based on the estimated channel and detected symbols. This residual is conventionally discarded or, at best, used for iterative communication refinement. We investigate whether it contains exploitable structure for sensing.\u003c/p\u003e\n\u003cp\u003eThe motivation for this question is both practical and theoretical. Practically, if sensing information survives the equalization process and resides in the residual, it could be extracted without additional waveform design, dedicated spectrum allocation, or separate sensing hardware\u0026mdash;making ISAC deployment significantly simpler and cheaper. Theoretically, understanding what the MMSE equalizer preserves and destroys provides insight into the fundamental information flow within ISAC receivers.\u003c/p\u003e\n\u003cp\u003eThe answer, as we shall demonstrate, is nuanced. The residual does contain target-dependent structure, but it is buried beneath a stronger structured artifact arising from imperfect channel estimation. Separating these two components is possible under specific conditions but not universally achievable. This characterization\u0026mdash;establishing both what is present and what is missing\u0026mdash;constitutes the core contribution of this work.\u003c/p\u003e\n\u003ch2\u003eC. Related Work\u003c/h2\u003e\n\u003cp\u003eThe most closely related work is the LISAC framework by Bian et al. [11], which employs a \u0026ldquo;residual-assisted MMSE equalizer\u0026rdquo; where a neural network calibrates the coarse MMSE channel estimate using the equalization residual. LISAC demonstrates that residual processing can improve communication bit error rate (BER), but it does not investigate the residual\u0026rsquo;s sensing content\u0026mdash;the residual is treated as an error signal for communication refinement, not as an information carrier for sensing. Our work differs fundamentally in three respects: we characterize the residual\u0026rsquo;s sensing content (not its communication utility), we use interpretable SVD decomposition (not neural networks), and we identify the artifact structure and its channel-dependent separability, which LISAC does not address.\u003c/p\u003e\n\u003cp\u003eData-aided iterative ISAC receivers, such as the bistatic framework by Temiz et al. [12], perform joint sensing and data demodulation by iteratively refining channel and symbol estimates. These approaches process the full received signal through dedicated sensing algorithms, rather than exploiting the communication receiver\u0026rsquo;s residual as we propose. The fundamental difference is that our approach seeks to extract sensing information from a byproduct of the communication process, not from the signal itself.\u003c/p\u003e\n\u003cp\u003eIn the broader ISAC literature, the foundational survey by Liu et al. [1] established the joint radar-communication design framework, while Xiong et al. [13] derived the fundamental Cram\u0026eacute;r\u0026ndash;Rao bound (CRB)-rate tradeoff for ISAC under Gaussian channels. Signal processing overviews by Zhang et al. [14] and Koivunen et al. [15] provide comprehensive treatments of multicarrier ISAC techniques. The seminal work by Sturm and Wiesbeck [16] established the theoretical foundation for OFDM-based radar sensing. None of these works, however, specifically characterize the structure of the MMSE equalization residual or its potential for sensing.\u003c/p\u003e\n\u003ch2\u003eD. Contributions\u003c/h2\u003e\n\u003cp\u003eThis paper makes three specific contributions:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC1: Universal low-rank artifact structure.\u0026nbsp;\u003c/strong\u003eWe demonstrate through SVD analysis that the MMSE residual artifact\u0026mdash;caused by pilot interpolation and channel estimation error\u0026mdash;exhibits a dominant low-rank structure across all seven tested channel models. The first singular value captures 28\u0026ndash;45% of total residual energy, with \u0026sigma;₁/\u0026sigma;₂ ratios from 1.5 to 4.6. This finding is channel-independent and constitutes a new structural characterization of MMSE residuals.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC2: Channel-dependent separability.\u0026nbsp;\u003c/strong\u003eWe show that while the artifact is universally low-rank, its separability from target-dependent structure is channel-specific. Under UMi-like propagation, rank-1 SVD removal exposes target signatures with +0.8 to +2.5 dB differential power. Under strong non-line-of-sight (NLOS) or rich multipath conditions, the optimal removal rank changes\u0026mdash;sometimes higher-rank removal helps, sometimes no removal is best. We characterize this dependence systematically.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC3: Quantified detection gap.\u0026nbsp;\u003c/strong\u003eWe establish that the target-dependent structure exposed by SVD processing is statistically detectable across many frames but insufficient for single-frame CFAR detection at practical false alarm rates. We quantify this gap precisely: for a \u0026minus;10 dB target at SNR = 20 dB, the mean differential is +0.89 dB with standard deviation \u0026plusmn;1.40 dB, yielding a signal-to-variability ratio of approximately 0.6. We identify MIMO beamforming, wider bandwidth, and multi-frame accumulation as the specific mechanisms that can bridge this gap.\u003c/p\u003e\n\u003ch2\u003eE. Paper Organization\u003c/h2\u003e\n\u003cp\u003eThe remainder of this paper is organized as follows. Section II presents the system model including the OFDM ISAC signal model, channel estimation pipeline, and residual construction. Section III provides the core residual structure analysis via SVD, establishing the universal low-rank finding and channel-dependent separability. Section IV evaluates sensing performance across multiple target scenarios, SNR conditions, and detection metrics. Section V presents the artifact suppression investigation and its results. Section VI discusses the findings, their implications, and limitations. Section VII concludes the paper.\u003c/p\u003e"},{"header":"II. System Model","content":"\u003ch2\u003eA. OFDM ISAC Signal Model\u003c/h2\u003e\n\u003cp\u003eWe consider an OFDM monostatic ISAC base station operating at carrier frequency fc = 28 GHz with Nsc = 256 active subcarriers, Nsym = 64 OFDM symbols per frame, and subcarrier spacing \u0026Delta;f = 120 kHz, corresponding to the 3GPP NR FR2 (Frequency Range 2) numerology [2]. The effective signal bandwidth is Nsc \u0026times; \u0026Delta;f = 30.72 MHz, providing a range resolution of \u0026Delta;R = c/(2B) = 4.88 m, where c = 3 \u0026times; 10⁸ m/s is the speed of light.\u003c/p\u003e\n\u003cp\u003eThe received signal at subcarrier k \u0026isin; {0, 1, ..., Nsc\u0026minus;1} and OFDM symbol index l \u0026isin; {0, 1, ..., Nsym\u0026minus;1} is modeled as:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eY[k,l] = (Hcomm[k,l] + Htarget[k,l]) \u0026middot; X[k,l] + W[k,l] \u0026nbsp; \u0026nbsp; (1)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003ewhere Hcomm[k,l] represents the communication channel comprising the line-of-sight (LOS) path and multipath components that the receiver\u0026rsquo;s channel estimator is designed to capture; Htarget[k,l] represents weak reflections from sensing targets (vehicles, pedestrians, unmanned aerial vehicles) that are 20\u0026ndash;40 dB below the communication channel power and are NOT modeled by the channel estimator; X[k,l] denotes the transmitted symbol at the (k,l)-th resource element, which is either a known pilot symbol or a 16-QAM data symbol; and W[k,l] \u0026sim; CN(0, \u0026sigma;\u0026sup2;) is additive white Gaussian noise (AWGN).\u003c/p\u003e\n\u003cp\u003eThe communication channel is constructed as a superposition of multipath components:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eHcomm[k,l] = \u0026Sigma;p \u0026alpha;p \u0026middot; exp(j(-2\u0026pi;k\u0026Delta;f\u0026tau;p + 2\u0026pi;lTsym\u0026nu;p)) \u0026nbsp; \u0026nbsp; (2)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u0026alpha;p, \u0026tau;p, and \u0026nu;p are the complex gain, propagation delay, and Doppler frequency of the p-th path, and Tsym is the total OFDM symbol duration including cyclic prefix. An identical expression holds for Htarget[k,l] with target-specific parameters.\u003c/p\u003e\n\u003cp\u003eThe key assumption underlying this work is that the channel estimator captures Hcomm (the strong communication paths) but treats Htarget as noise. This is realistic because sensing target reflections are typically 20\u0026ndash;40 dB below the communication channel\u0026mdash;well below the estimator\u0026rsquo;s noise floor. This assumption is what creates the possibility of target information surviving into the residual.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTABLE I:\u0026nbsp;\u003c/strong\u003eSYSTEM PARAMETERS\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"624\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 233px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eParameter\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 195px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eValue\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 195px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eJustification\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eCarrier frequency fc\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e28 GHz\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e3GPP NR FR2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eSubcarrier spacing \u0026Delta;f\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e120 kHz\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eStandard FR2 numerology\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eSubcarriers Nsc\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e256\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eEffective BW = 30.72 MHz\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eOFDM symbols Nsym\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e64\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eFrame duration\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eModulation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e16-QAM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eModerate spectral efficiency\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003ePilot spacing\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eEvery 4th subcarrier\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e25% pilot overhead\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eChannel estimation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eLS + interp. + LMMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eStandard pipeline\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eRange resolution\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e4.88 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003ec/(2\u0026middot;Nsc\u0026middot;\u0026Delta;f)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eSNR range\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e0\u0026ndash;30 dB\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eFull operating range\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 233px;\"\u003e\n \u003cp\u003eMonte Carlo trials\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003e15\u0026ndash;80 per condition\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 195px;\"\u003e\n \u003cp\u003eStatistical significance\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003ch2\u003eB. Channel Estimation and MMSE Equalization\u003c/h2\u003e\n\u003cp\u003eThe channel estimation pipeline follows the standard three-stage approach used in 3GPP NR receivers. First, least-squares (LS) estimation is performed on pilot subcarriers, spaced every 4th subcarrier (25% pilot density). Pilots are known QPSK symbols with a fixed random seed for reproducibility. Second, the LS estimates are linearly interpolated to data subcarriers in the frequency domain. Third, a moving-average LMMSE smoothing filter (kernel size 7) is applied to reduce noise in the estimates.\u003c/p\u003e\n\u003cp\u003ePer-subcarrier MMSE equalization then produces symbol estimates:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eX̂[k,l] = Ĥ*[k,l] / (|Ĥ[k,l]|\u0026sup2; + \u0026sigma;\u0026sup2;) \u0026middot; Y[k,l] \u0026nbsp; \u0026nbsp; (3)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003efollowed by hard decision (nearest 16-QAM constellation point) to produce X̂hard[k,l].\u003c/p\u003e\n\u003ch2\u003eC. Residual Construction\u003c/h2\u003e\n\u003cp\u003eThe communication residual is defined as the difference between the received signal and the reconstructed signal:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003er[k,l] = Y[k,l] \u0026minus; Ĥ[k,l] \u0026middot; X̂hard[k,l] \u0026nbsp; \u0026nbsp; (4)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eTo enable sensing processing, the data symbol modulation must be removed from the residual. We perform regularized division:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003ers[k,l] = r[k,l] \u0026middot; X̂*hard[k,l] / (|X̂hard[k,l]|\u0026sup2; + \u0026epsilon;) \u0026nbsp; \u0026nbsp; \u0026nbsp;(5)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u0026epsilon; = 0.01 is a regularization parameter to avoid noise amplification near zero-crossings of the QAM constellation.\u003c/p\u003e\n\u003cp\u003eThis sensing residual rs decomposes into three components:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003ers = rs,artifact + rs,target + rs,noise \u0026nbsp; \u0026nbsp; (6)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003ewhere rs,artifact arises from the channel estimation error (Ĥ \u0026minus; Hcomm), rs,target carries sensing information from Htarget, and rs,noise is thermal noise. The central question of this paper is whether rs,target can be separated from rs,artifact.\u003c/p\u003e"},{"header":"III. Residual Structure Analysis","content":"\u003ch2\u003eA. Naive Residual Sensing Fails\u003c/h2\u003e\n\u003cp\u003eBefore presenting the SVD analysis, we first establish a critical negative result: direct application of standard sensing processing to the residual does not work. Applying a 2D-FFT to the sensing residual rs to produce a range-Doppler map (RDM) yields no detectable target signatures. The RDM with targets present and without targets present are visually and statistically indistinguishable.\u003c/p\u003e\n\u003cp\u003eThe differential power at true target locations\u0026mdash;defined as \u0026Delta; = 10\u0026middot;log10(Pwith/Pwithout) where Pwith and Pwithout are the peak RDM powers with and without targets present\u0026mdash;is consistently negative (\u0026minus;0.5 to \u0026minus;4 dB) across all target gains from \u0026minus;5 to \u0026minus;30 dB. This means the target signal is not merely weak; it is actually less visible than background noise fluctuations at the target location.\u003c/p\u003e\n\u003cp\u003eThis negative result is itself significant: it establishes that MMSE residuals cannot be naively repurposed for sensing. The residual is dominated by structured channel estimation artifacts, not target echoes. Understanding and overcoming this artifact is the subject of the following analysis.\u003c/p\u003e\n\u003ch2\u003eB. Universal Low-Rank Artifact Structure\u003c/h2\u003e\n\u003cp\u003eWe decompose the sensing residual matrix Rs \u0026isin; C^(Nsc \u0026times; Nsym) via singular value decomposition (SVD):\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eRs = \u0026Sigma;i \u0026sigma;i ui viᴴ \u0026nbsp; \u0026nbsp; (7)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u0026sigma;₁ \u0026ge; \u0026sigma;₂ \u0026ge; ... are the ordered singular values, and ui, vi are the corresponding left and right singular vectors. We perform this analysis with no targets present (Htarget = 0) to characterize the pure artifact structure.\u003c/p\u003e\n\u003cp\u003eTable II presents the singular value analysis across seven channel models at SNR = 20 dB, averaged over 20 independent noise realizations.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTABLE II:\u0026nbsp;\u003c/strong\u003eSINGULAR VALUE ANALYSIS OF THE MMSE RESIDUAL ACROSS CHANNEL MODELS (NO TARGETS, SNR = 20 dB)\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"624\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 187px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eChannel Model\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 67px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePaths\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026sigma;₁/\u0026sigma;₂\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026sigma;₁/\u0026sigma;₃\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eE(\u0026sigma;₁)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 131px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eInterpretation\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eUMi default\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.61\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e43.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eStrong rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eStrong NLOS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e2.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e39.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eModerate rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eWeak NLOS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e44.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eStrong rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eMany paths\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e44.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eStrong rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eHigh Doppler comm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e1.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e2.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e28.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eWeak rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eLOS only\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.63\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e44.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eStrong rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 187px;\"\u003e\n \u003cp\u003eRician K=10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e4.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e44.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 131px;\"\u003e\n \u003cp\u003eStrong rank-1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eThe key observation is that \u0026sigma;₁/\u0026sigma;₂ exceeds 1.5 for all seven channel models. This confirms that the artifact possesses a universally dominant rank-1 component. The first singular value alone captures 28\u0026ndash;45% of total residual energy. The physical origin of this low-rank structure is the frequency-domain interpolation used in channel estimation: interpolating between pilot subcarriers produces a smooth estimation error that is highly correlated across subcarriers and OFDM symbols, concentrating energy in the first singular vector.\u003c/p\u003e\n\u003ch2\u003eC. Channel-Dependent Separability\u003c/h2\u003e\n\u003cp\u003eHaving established the low-rank artifact structure, we now investigate whether removing this artifact exposes target-dependent structure. We define the cleaned residual after removing k singular values:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eRs,clean(k) = \u0026Sigma;i=k+1 \u0026sigma;i ui viᴴ \u0026nbsp; \u0026nbsp; (8)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eWe measure target visibility using the differential power metric: \u0026Delta; = 10\u0026middot;log10(Pwith/Pwithout) in dB, where Pwith and Pwithout are the peak range-Doppler map powers at the true target location with and without targets present, averaged over 30 independent Monte Carlo trials. Targets are placed at 30 m range with 1500 Hz Doppler, at \u0026minus;22 dB gain relative to the communication channel.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTABLE III:\u0026nbsp;\u003c/strong\u003eDIFFERENTIAL POWER (dB) AT TARGET LOCATION VS. PCA REMOVAL RANK (SNR = 20 dB, TARGET GAIN = \u0026minus;22 dB)\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"624\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 133px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eChannel\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 91px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eBest\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eUMi default\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+1.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003eStrong NLOS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.04\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003ePCA-5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003eWeak NLOS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003ePCA-5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eMany paths\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.99\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePCA-0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003eHigh Doppler\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003ePCA-5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003eLOS only\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003ePCA-5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 133px;\"\u003e\n \u003cp\u003eRician K=10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e\u0026minus;0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 91px;\"\u003e\n \u003cp\u003ePCA-5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eThree distinct regimes emerge from Table III:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eRegime 1 \u0026mdash; Clean separation (UMi):\u0026nbsp;\u003c/strong\u003eThe artifact concentrates tightly in the first singular vector, and PCA-1 cleanly exposes target structure with +1.26 dB differential. This is the ideal case where the artifact and target subspaces are nearly orthogonal.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eRegime 2 \u0026mdash; Distributed artifact (Weak NLOS, Rician):\u0026nbsp;\u003c/strong\u003eThe artifact energy is distributed across multiple singular values, requiring PCA-5 for positive differential (+0.59 dB and +0.33 dB respectively). The artifact subspace is multi-dimensional, and partial removal is needed before target structure becomes visible.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eRegime 3 \u0026mdash; Subspace entanglement (Many paths):\u0026nbsp;\u003c/strong\u003eThe artifact and target subspaces are completely entangled. PCA-0 (no removal at all) gives the best result (+0.25 dB), because any SVD removal also removes target energy. This represents a fundamental limit of subspace-based separation methods.\u003c/p\u003e"},{"header":"IV. Sensing Performance Evaluation","content":"\u003ch2\u003eA. Multi-Target Visibility Under UMi\u003c/h2\u003e\n\u003cp\u003eUnder the UMi channel with PCA-1 at SNR = 20 dB, we test nine distinct target scenarios to evaluate the robustness of residual-based sensing across diverse target configurations. Each scenario is evaluated over 25 independent Monte Carlo trials.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTABLE IV:\u0026nbsp;\u003c/strong\u003ePCA-1 DIFFERENTIAL ACROSS TARGET SCENARIOS (UMi CHANNEL, SNR = 20 dB)\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"624\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 160px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eScenario\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eRange\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 93px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eVelocity\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 73px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eGain (dB)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 113px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eDiff (dB)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 104px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eStatus\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003eClose fast car\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e10 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e120 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+1.03 \u0026plusmn; 2.46\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e✓ Detected\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003eTwo vehicles\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e20+40 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e77+58 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;22/\u0026minus;24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+1.36, +1.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e✓ Both\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eCar+Ped+UAV\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e30+15+8 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e48+4+77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;22/\u0026minus;30/\u0026minus;26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+0.76, +1.28, +1.63\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e✓ All three\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003eHigh-speed\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e25 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e300 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+1.08 \u0026plusmn; 1.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e✓ Detected\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003eOriginal pair\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e30+12 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e29+6 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;22/\u0026minus;30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+0.89, +1.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e✓ Both\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eStationary\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e40 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e0 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e\u0026minus;0.41 \u0026plusmn; 1.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e✗ FAILS\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003eVery weak\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e35 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e19 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+0.04 \u0026plusmn; 0.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e~ Marginal\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 160px;\"\u003e\n \u003cp\u003eFar slow ped\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e50 m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 93px;\"\u003e\n \u003cp\u003e5 km/h\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 73px;\"\u003e\n \u003cp\u003e\u0026minus;28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e+0.08 \u0026plusmn; 0.87\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e~ Marginal\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eThe three-target scenario (vehicle at 30 m, pedestrian at 15 m, and UAV at 8 m simultaneously visible) represents the strongest positive result. All three targets show positive differential despite having different ranges, velocities, and radar cross-sections. This demonstrates multi-target capability from a single OFDM residual frame\u0026mdash;a key result for practical ISAC deployment.\u003c/p\u003e\n\u003cp\u003eThe stationary target failure (\u0026minus;0.41 dB) provides an equally important negative result. Targets at zero Doppler produce residual structure at DC (zero frequency offset), which is precisely the strongest component of the channel estimation artifact. Since PCA-1 removes the dominant singular vector\u0026mdash;which is dominated by the DC artifact\u0026mdash;it inadvertently removes the stationary target signature along with the artifact. This is a fundamental limitation of subspace-based artifact removal for near-zero-Doppler targets and cannot be resolved without external information (e.g., a target-free calibration frame).\u003c/p\u003e\n\u003ch2\u003eB. SNR Dependence\u003c/h2\u003e\n\u003cp\u003eThe differential power under UMi/PCA-1 increases monotonically with system SNR, confirming physical correctness of the approach:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTABLE V:\u0026nbsp;\u003c/strong\u003eDIFFERENTIAL POWER VS. SNR (UMi/PCA-1, TARGET GAIN = \u0026minus;10 dB)\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"624\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 104px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eSNR (dB)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 87px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 87px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e10\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 87px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e15\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 87px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e20\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 87px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e25\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 80px;\"\u003e\n \u003cp\u003e\u003cstrong\u003e30\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 104px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eDiff (dB)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 87px;\"\u003e\n \u003cp\u003e+0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 87px;\"\u003e\n \u003cp\u003e+0.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 87px;\"\u003e\n \u003cp\u003e+0.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 87px;\"\u003e\n \u003cp\u003e+0.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 87px;\"\u003e\n \u003cp\u003e+1.46\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 80px;\"\u003e\n \u003cp\u003e+2.55\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eThis monotonic increase confirms the expected physics: higher SNR yields more accurate channel estimation, which produces a cleaner artifact (closer to pure rank-1), enabling more effective PCA-based separation and stronger target visibility. The improvement from +0.31 dB at 5 dB SNR to +2.55 dB at 30 dB SNR represents an 8\u0026times; improvement in linear scale.\u003c/p\u003e\n\u003ch2\u003eC. The Detection Gap\u003c/h2\u003e\n\u003cp\u003eDespite statistically positive differentials, single-frame CFAR detection is not achievable at practical false alarm rates. We evaluate CFAR detection by computing the test statistic (cell-under-test power divided by estimated noise power from surrounding training cells) for 80 independent trials at SNR = 20 dB under UMi/PCA-1.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTABLE VI:\u0026nbsp;\u003c/strong\u003eCFAR DETECTION PERFORMANCE (SNR = 20 dB, UMi, PCA-1)\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"624\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 156px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eTarget Gain\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 156px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eH1/H0 Ratio\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 156px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePd @ Pfa=0.01\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 156px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePd @ Pfa=0.05\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e\u0026minus;10 dB\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.97\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.025\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e\u0026minus;15 dB\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e1.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.013\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.050\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e\u0026minus;20 dB\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e1.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 156px;\"\u003e\n \u003cp\u003e0.037\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eThe H1/H0 ratios near unity indicate that per-frame variance dominates the target-induced power shift. The mean differential of +0.89 dB for \u0026minus;10 dB targets has a standard deviation of \u0026plusmn;1.40 dB, yielding a signal-to-variability ratio of approximately 0.6. For reliable single-frame detection (Pd \u0026gt; 0.9 at Pfa = 0.01), this ratio must exceed approximately 3.0, implying a performance gap of approximately 7 dB (14 dB in power).\u003c/p\u003e\n\u003ch2\u003eD. Communication Performance\u003c/h2\u003e\n\u003cp\u003eThe K\u0026ndash;R iterative receiver, which performs 3 iterations of residual-based symbol refinement (each iteration computes the residual, adds it back to the equalized signal scaled by the inverse channel estimate, and re-decides), provides a consistent communication symbol error rate (SER) improvement of +0.1 to +0.2 dB across all tested SNR values from 0 to 30 dB. This confirms that the R-step sensing extraction does not degrade communication performance\u0026mdash;the two outputs (decoded bits and range-Doppler map) are complementary.\u003c/p\u003e"},{"header":"V. Artifact Suppression Investigation","content":"\u003cp\u003eRecognizing that the raw PCA-1 approach achieves only modest differentials (+0.5 to +2.5 dB) and fails for CFAR detection, we investigated multiple artifact suppression strategies to determine whether additional processing could improve target visibility.\u003c/p\u003e\n\u003ch2\u003eA. Template Subtraction\u003c/h2\u003e\n\u003cp\u003eWe constructed an artifact template by averaging the residual RDM over 100 target-free trials, then subtracted this template (with adaptive scaling) from target-bearing RDMs. This approach failed: the artifact varies per noise realization and per data symbol pattern, making a static template inaccurate. Template subtraction typically made the differential worse (\u0026minus;10 to \u0026minus;100 dB), not better, by introducing new artifacts where the template did not match the current realization.\u003c/p\u003e\n\u003ch2\u003eB. Pilot-Domain Residual\u003c/h2\u003e\n\u003cp\u003eWe also investigated constructing the residual from pilot observations only (Y_p/X_p \u0026minus; Ĥ at pilot positions), avoiding data decision errors entirely. While pilot-domain PCA-1 achieved positive differential at strong target gains (+2.50 dB at \u0026minus;5 dB), it was outperformed by data-domain PCA-1 at moderate and weak gains. The data domain provides 4\u0026times; more observations (256 vs. 64 subcarriers), enabling more accurate SVD-based separation despite the additional decision noise.\u003c/p\u003e\n\u003ch2\u003eC. Channel Robustness\u003c/h2\u003e\n\u003cp\u003eWe tested PCA-1 across four alternative communication channel configurations (strong NLOS, weak NLOS, many-path, and high-Doppler) using the original target set. Results show that PCA-1 gives positive differential only for the default UMi channel (+0.67 dB). The strong NLOS channel gives \u0026minus;0.85 dB, weak NLOS gives \u0026minus;0.35 dB, and many-path gives \u0026minus;0.60 dB. This confirms that PCA-1 is not a universal solution\u0026mdash;it is the correct answer specifically for UMi-like channels where the artifact is cleanly rank-1 and well-separated from the target subspace.\u003c/p\u003e"},{"header":"VI. Discussion","content":"\u003ch2\u003eA. Signal Existence vs. Detectability\u003c/h2\u003e\n\u003cp\u003eOur results reveal a fundamental gap between two distinct questions. The existence question\u0026mdash;\u0026ldquo;Does target-dependent structure persist in the MMSE residual?\u0026rdquo;\u0026mdash;is answered affirmatively: PCA-based artifact removal consistently exposes statistically significant target-dependent power under UMi conditions, across diverse target configurations including multiple simultaneous targets. The detectability question\u0026mdash;\u0026ldquo;Can a single OFDM frame reliably detect targets at practical false alarm rates?\u0026rdquo;\u0026mdash;is answered negatively at the tested parameters, with H1/H0 ratios near unity.\u003c/p\u003e\n\u003cp\u003eThis distinction is important for the ISAC research community. Many ISAC papers implicitly assume that signal existence implies detectability. Our results quantify the gap: approximately 7 dB of additional processing gain is needed to bridge from the current signal-to-variability ratio of 0.6 to the threshold of 3.0 required for reliable detection.\u003c/p\u003e\n\u003ch2\u003eB. Bridging the Gap: Three Mechanisms\u003c/h2\u003e\n\u003cp\u003e\u003cstrong\u003eMIMO beamforming:\u0026nbsp;\u003c/strong\u003eWith Nt transmit and Nr receive antennas, the residual gains Nt \u0026middot; Nr spatial diversity. An 8\u0026times;2 array configuration would provide approximately 12 dB additional gain on the residual, well exceeding the 7 dB gap. This is the most promising path to practical deployment.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eBandwidth increase:\u0026nbsp;\u003c/strong\u003eExpanding from 30.72 MHz to 400 MHz (the maximum 3GPP FR2 component carrier bandwidth) improves range resolution from 4.88 m to 0.375 m. This physically separates target range bins from artifact regions, reducing subspace overlap and enabling cleaner separation.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMulti-frame accumulation:\u0026nbsp;\u003c/strong\u003eCoherent integration over M frames provides approximately 10\u0026middot;log10(M) dB processing gain. Integrating 50 frames (\u0026asymp;17 dB gain) would comfortably exceed the detection threshold, at the cost of requiring target stationarity over the integration window.\u003c/p\u003e\n\u003ch2\u003eC. Why the Artifact Is Low-Rank\u003c/h2\u003e\n\u003cp\u003eThe pilot-based channel estimation pipeline with frequency-domain interpolation produces estimation errors that are smooth functions of subcarrier index. The interpolation kernel acts as a low-pass filter on the estimation error, suppressing high-frequency error components and concentrating energy in the first few spatial frequencies\u0026mdash;which correspond to the first few singular values. This smoothness is independent of the communication channel itself, which explains the universality of the low-rank finding across all seven tested channel models.\u003c/p\u003e\n\u003ch2\u003eD. Why Separability Is Channel-Dependent\u003c/h2\u003e\n\u003cp\u003eAlthough the artifact is universally low-rank, target energy may project onto the same singular vectors when: (i) target delays are similar to communication path delays, producing similar frequency-domain oscillation patterns; (ii) the communication channel itself has rich multipath structure that spreads estimation error across multiple singular vectors, overlapping with the target subspace; or (iii) target Doppler shifts are near zero, aligning with the artifact\u0026rsquo;s dominant DC component.\u003c/p\u003e\n\u003ch2\u003eE. Comparison with LISAC\u003c/h2\u003e\n\u003cp\u003eThe LISAC framework [11] and our work represent complementary perspectives on MMSE residuals in ISAC. LISAC treats the residual as an error signal and trains a neural network to correct communication symbol estimates. We treat the residual as an information carrier and use SVD to characterize its structure for sensing. LISAC has no sensing output; we produce a range-Doppler map. LISAC requires training data and neural network inference; we use a single SVD operation. The two approaches could potentially be combined: LISAC\u0026rsquo;s neural network could be augmented with a sensing output head trained on the residual\u0026rsquo;s target-dependent subspace.\u003c/p\u003e\n\u003ch2\u003eF. Limitations\u003c/h2\u003e\n\u003cp\u003eSeveral limitations should be noted. First, the differentials (+0.5 to +2.5 dB) are small, representing a proof-of-concept rather than a detection-ready system. Second, the channel robustness is limited\u0026mdash;PCA-1 works well only for UMi-like channels. Third, all results are simulation-based with simplified multipath models; full 3GPP TR 38.901 stochastic channel models and hardware-in-the-loop validation would strengthen the findings. Fourth, the SISO configuration provides no spatial diversity; MIMO is essential for practical performance. Fifth, the number of Monte Carlo trials (15\u0026ndash;80 per condition) should be increased to 500+ for publication-quality statistical confidence.\u003c/p\u003e"},{"header":"VII. Conclusion","content":"\u003cp\u003eWe have established three findings regarding the structure of MMSE communication residuals in OFDM ISAC systems.\u003c/p\u003e \u003cp\u003eFirst, the residual artifact from pilot-based channel estimation exhibits a universally low-rank structure across all tested channel conditions, with σ₁/σ₂ ratios from 1.5 to 4.6. The first singular value captures 28\u0026ndash;45% of total residual energy. This structural finding is independent of the communication channel and represents a new characterization of MMSE residuals.\u003c/p\u003e \u003cp\u003eSecond, target-dependent structure can be exposed via SVD-based artifact removal, but the optimal removal rank is channel-specific\u0026mdash;ranging from rank-1 for UMi channels to rank-5 or no removal for rich multipath environments. This channel dependence arises from the alignment between artifact and target subspaces in the SVD domain.\u003c/p\u003e \u003cp\u003eThird, while this target structure is statistically detectable across multiple trials (positive differential in 6 of 9 target scenarios, monotonically increasing with SNR), the per-frame variance prevents single-frame CFAR detection at practical false alarm rates. We quantify this gap precisely: approximately 7 dB of additional processing gain is needed, achievable through MIMO beamforming, wider bandwidth, or multi-frame accumulation.\u003c/p\u003e \u003cp\u003eThese findings define the feasibility boundary for residual-based ISAC sensing: the signal exists, the gap to detection is measurable and specific, and the mechanisms to bridge it are identified. This characterization informs the design of future K\u0026ndash;R receivers that combine model-based artifact removal with additional gain mechanisms to achieve practical sensing from communication residuals without waveform modification or dedicated sensing resources.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eF. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, \u0026ldquo;Joint radar and communication design: Applications, state-of-the-art, and the road ahead,\u0026rdquo; IEEE Trans. Commun., vol. 68, no. 6, pp. 3834\u0026ndash;3862, Jun. 2020.\u003c/li\u003e\n\u003cli\u003eGPP, \u0026ldquo;Study on channel model for frequencies from 0.5 to 100 GHz,\u0026rdquo; 3GPP TR 38.901, v17.0.0, Mar. 2022.\u003c/li\u003e\n\u003cli\u003eGPP, \u0026ldquo;Study on integrated sensing and communication (Release 19),\u0026rdquo; 3GPP TR 22.837, v19.2.0, Jun. 2024.\u003c/li\u003e\n\u003cli\u003eGPP RAN #108, \u0026ldquo;New SID: Study on Integrated Sensing And Communication (ISAC) for NR,\u0026rdquo; RP-251861, Jun. 2025.\u003c/li\u003e\n\u003cli\u003eGPP, \u0026ldquo;Study on channel modelling for ISAC for NR,\u0026rdquo; 3GPP TR 38.765, Release 19, 2025.\u003c/li\u003e\n\u003cli\u003eETSI ISAC ISG, \u0026ldquo;ISAC channel modelling \u0026mdash; Perspectives from ETSI,\u0026rdquo; arXiv:2505.10275, May 2025.\u003c/li\u003e\n\u003cli\u003eSamsung Research, \u0026ldquo;Integrated sensing and communication (ISAC): New monetization opportunities for 5G and beyond,\u0026rdquo; Tech. Rep., Dec. 2025.\u003c/li\u003e\n\u003cli\u003eHuawei, \u0026ldquo;Integrated sensing and communication \u0026mdash; From concept to practice,\u0026rdquo; Tech. Rep., 2025.\u003c/li\u003e\n\u003cli\u003eJ. A. Zhang, F. Liu, C. Masouros, R. W. Heath, Z. Feng, L. Zheng, and A. Petropulu, \u0026ldquo;An overview of signal processing techniques for joint communication and radar sensing,\u0026rdquo; IEEE J. Sel. Topics Signal Process., vol. 15, no. 6, pp. 1295\u0026ndash;1315, Nov. 2021.\u003c/li\u003e\n\u003cli\u003eV. Koivunen, M. F. Keskin, H. Wymeersch, M. Valkama, and N. Gonz\u0026aacute;lez-Prelcic, \u0026ldquo;Multicarrier ISAC: Advances in waveform design, signal processing, and learning under nonidealities,\u0026rdquo; IEEE Signal Process. Mag., vol. 41, no. 5, pp. 17\u0026ndash;30, Sep. 2024.\u003c/li\u003e\n\u003cli\u003eC. Bian, Y. Zhang, M. Hua, K. Meng, and D. G\u0026uuml;nd\u0026uuml;z, \u0026ldquo;LISAC: Learned coded waveform design for ISAC with OFDM,\u0026rdquo; arXiv:2410.10711, Jan. 2025.\u003c/li\u003e\n\u003cli\u003eM. Temiz et al., \u0026ldquo;Bridging the gap via data-aided sensing: Can bistatic ISAC converge to genie performance?\u0026rdquo; arXiv:2505.01280, May 2025.\u003c/li\u003e\n\u003cli\u003eY. Xiong, F. Liu, Y. Cui, W. Yuan, T. X. Han, and G. Caire, \u0026ldquo;On the fundamental tradeoff of integrated sensing and communications under Gaussian channels,\u0026rdquo; IEEE Trans. Inf. Theory, vol. 69, no. 9, pp. 5723\u0026ndash;5751, Sep. 2023.\u003c/li\u003e\n\u003cli\u003eF. Liu and C. Masouros, \u0026ldquo;A tutorial on joint radar and communication transmission for vehicular networks,\u0026rdquo; IEEE Commun. Lett., vol. 25, no. 2, pp. 322\u0026ndash;336, Feb. 2021.\u003c/li\u003e\n\u003cli\u003eZ. Wei, H. Qu, W. Jiang, et al., \u0026ldquo;Integrated sensing and communication: Enabling techniques, applications, tools and data sets, standardization, and future directions,\u0026rdquo; IEEE Internet Things J., vol. 10, no. 13, pp. 11621\u0026ndash;11666, Jul. 2023.\u003c/li\u003e\n\u003cli\u003eC. Sturm and W. Wiesbeck, \u0026ldquo;Waveform design and signal processing aspects for fusion of wireless communications and radar sensing,\u0026rdquo; Proc. IEEE, vol. 99, no. 7, pp. 1236\u0026ndash;1259, Jul. 2011.\u003c/li\u003e\n\u003cli\u003eF. Liu, Y.-F. Liu, A. Li, C. Masouros, and Y. C. Eldar, \u0026ldquo;Cram\u0026eacute;r\u0026ndash;Rao bound optimization for joint radar-communication beamforming,\u0026rdquo; IEEE Trans. Signal Process., vol. 70, pp. 240\u0026ndash;253, Jan. 2022.\u003c/li\u003e\n\u003cli\u003eH. Hua, T. X. Han, and J. Xu, \u0026ldquo;MIMO integrated sensing and communication: CRB-rate tradeoff,\u0026rdquo; IEEE Trans. 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Turunen, et al., \u0026ldquo;Full-duplex OFDM radar with LTE and 5G NR waveforms: Challenges, solutions, and measurements,\u0026rdquo; IEEE Trans. Microw. Theory Techn., vol. 67, no. 10, pp. 4042\u0026ndash;4054, Oct. 2019.\u003c/li\u003e\n\u003cli\u003eGPP, \u0026ldquo;Release 20 \u0026mdash; 5G Advanced,\u0026rdquo; Work Plan SP-260360, presented at TSGs#111, Mar. 2026.\u003c/li\u003e\n\u003cli\u003eS. Mura, D. Tagliaferri, M. Mizmizi, U. Spagnolini, and A. Petropulu, \u0026ldquo;Optimized waveform design for OFDM-based ISAC systems,\u0026rdquo; IEEE Trans. Wireless Commun., vol. 24, no. 6, pp. 5241\u0026ndash;5257, Jun. 2025.\u003c/li\u003e\n\u003cli\u003eR. Bomfin and M. Chafii, \u0026ldquo;On the performance analysis of zero-padding OFDM for monostatic ISAC systems,\u0026rdquo; IEEE Trans. Commun., vol. 72, no. 10, pp. 6304\u0026ndash;6319, Oct. 2024.\u003c/li\u003e\n\u003cli\u003eZ. Wei, H. Qu, and W. Jiang, \u0026ldquo;Iterative signal processing for integrated sensing and communication systems,\u0026rdquo; IEEE Trans. Green Commun. Netw., vol. 7, no. 1, pp. 401\u0026ndash;412, Mar. 2023.\u003c/li\u003e\n\u003cli\u003eB. Li, W. Yuan, F. Liu, N. Wu, and S. Jin, \u0026ldquo;OTFS-based ISAC: How delay Doppler channel estimation assists environment sensing?\u0026rdquo; IEEE Wireless Commun. Lett., vol. 13, no. 10, pp. 2843\u0026ndash;2847, Oct. 2024.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Integrated sensing and communication (ISAC), OFDM, MMSE equalization, singular value decomposition, residual analysis, low-rank structure, 6G, feasibility characterization","lastPublishedDoi":"10.21203/rs.3.rs-9299929/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9299929/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe 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.\u003c/p\u003e \u003cp\u003eFirst, 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\u0026ndash;45% of total residual energy, originating from the structured interpolation error inherent in pilot-based estimation.\u003c/p\u003e \u003cp\u003eSecond, the separability between this artifact and target-dependent structure is channel-specific: rank-1 SVD removal yields statistically significant positive differential power (+\u0026thinsp;0.5 to +\u0026thinsp;2.5 dB) at target locations under urban micro (UMi) propagation, but the optimal removal rank varies across channel conditions\u0026mdash;ranging from PCA-0 (no removal) for rich multipath to PCA-5 for weak non-line-of-sight environments.\u003c/p\u003e \u003cp\u003eThird, 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.\u003c/p\u003e \u003cp\u003eThese findings establish the feasibility boundary for residual-based ISAC sensing and identify the quantified performance gap\u0026mdash;measurable and specific\u0026mdash;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.\u003c/p\u003e","manuscriptTitle":"K–R Residual Analysis for OFDM ISAC: Low-Rank Structure and Sensing Feasibility","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-03 06:44:48","doi":"10.21203/rs.3.rs-9299929/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"dafb1023-9678-48dc-92fe-bcfd0983ee2d","owner":[],"postedDate":"April 3rd, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":65597626,"name":"Cell Communication and Signaling"}],"tags":[],"updatedAt":"2026-04-03T06:44:49+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-03 06:44:48","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9299929","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9299929","identity":"rs-9299929","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}