A Physics-Guided Receiver for WiFi that Learns Hardware Nonlinearities without Training Data

A Physics-Guided Receiver for WiFi that Learns Hardware Nonlinearities without Training Data | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article A Physics-Guided Receiver for WiFi that Learns Hardware Nonlinearities without Training Data Ramakrishna Pasupuleti This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9192715/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract Modern WiFi systems operating under IEEE 802.11ax (WiFi 6) and 802.11be (WiFi 7) suffer from hardware nonlinearities — including power amplifier saturation, ADC quantisation, and OFDM clipping — that conventional linear receivers treat as white Gaussian noise, leaving structured, learnable residuals uncorrected. We propose the K-R receiver, a physics-guided two-step framework that combines model-based MMSE equalisation with a closed-form nonlinear residual corrector trained per packet using existing preamble pilots, requiring no offline dataset and no backpropagation. The 12-feature residual corrector includes a cubic amplitude term physically grounded in the standard Rapp PA model, enabling the framework to exploit model-consistent nonlinear structure rather than learning arbitrary corrections. Under IEEE 802.11ax-compliant simulations across 12 independent scenarios, the proposed receiver achieves consistent spectral efficiency gains of approximately + 0.40–0.43 bps/Hz over MMSE, remains stable under hardware mismatch conditions where deep-learning baselines degrade, and achieves over 60× higher energy efficiency than offline-trained neural receivers. An analytical MSE decomposition reveals that the channel-error correction gain dominates the quantisation gain at low SNR — precisely at the coverage edge, where improvements matter most. These results suggest that physics-guided online learning is a practical and principled alternative to offline-trained neural receivers for real-world WiFi deployment. Physical sciences/Engineering Physical sciences/Mathematics and computing Physical sciences/Physics WiFi 6/7 IEEE 802.11ax power amplifier nonlinearity Rapp model K-R receiver residual learning closed-form training coverage edge MIMO energy efficiency Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction The WiFi nonlinearity problem The global WiFi ecosystem is undergoing a fundamental transition. IEEE 802.11ax (WiFi 6), ratified in 2021, and its successor IEEE 802.11be (WiFi 7), under active development, support modulation orders of 1024-QAM and 4096-QAM respectively — an eight-fold increase in constellation density compared to the 802.11n generation.¹ This escalation in spectral ambition places unprecedented demands on the linearity of receiver hardware. A WiFi 6 access point operating at 1024-QAM can tolerate less than 0.5 dB of in-band nonlinear distortion before symbol error rates become unacceptable; at 4096-QAM, this margin shrinks further. Yet real-world WiFi hardware exhibits three fundamental nonlinear impairments that are difficult to eliminate. First, power amplifier (PA) saturation: the solid-state PAs used in access points and user devices follow the standard Rapp model¹, exhibiting amplitude-dependent gain compression that becomes severe when the OFDM signal's high peak-to-average power ratio (PAPR) drives the PA beyond its linear operating region. Second, ADC quantisation: low-cost WiFi chips and IoT edge devices increasingly adopt 4–6 bit ADCs to reduce power consumption², introducing structured granularity noise with non-Gaussian statistics. Third, OFDM clipping: transmitters deliberately apply hard amplitude limiting to protect hardware from PAPR spikes³, introducing in-band distortion proportional to clipping severity. These three impairments coexist in every real WiFi deployment — yet they are jointly absent from nearly all published receiver designs. Why current methods fail The standard WiFi receiver applies frequency-domain MMSE equalisation to the received OFDM signal. MMSE is optimal under the assumption of additive white Gaussian noise, which is a reasonable approximation when the receiver's hardware impairments are small. Under realistic PA saturation, ADC quantisation, and clipping, however, the residual after MMSE is not Gaussian — it is structured, amplitude-dependent, and physically predictable from the hardware model. MMSE leaves this structured residual entirely uncorrected, treating it as irreducible noise. The resulting performance ceiling is increasingly apparent at 256-QAM and above. Deep learning receivers, including OAMP-Net⁴, DetNet⁵, and deep convolutional neural network equalisers⁶, have attracted significant attention as nonlinear receiver solutions. These methods can, in principle, learn arbitrary nonlinear input-output mappings from labelled training data. In practice, however, they share a structural weakness that makes them poorly suited to commercial WiFi deployment: they require offline training on datasets representing the target channel and hardware conditions. In WiFi, the channel changes every packet, the PA operating point drifts with temperature and load, and no two access points have identical hardware characteristics. An offline-trained model calibrated to one set of conditions exhibits distribution mismatch — and therefore performance collapse — when those conditions change. This paper provides direct empirical evidence of this collapse and demonstrates a principled alternative. The K-R approach: bridging model-based and data-driven design We propose the K-R receiver for IEEE 802.11ax/be, building on our earlier contributions in mmWave 5G⁷ and LiFi⁸ to address WiFi-specific hardware nonlinearities. The K-R framework is motivated by a simple observation: after MMSE equalisation, the residual error in a PA-impaired OFDM system is not random — it is dominated by the third-order distortion term of the Rapp model, which generates a residual component proportional to |r_q|³, where r_q is the MMSE output. If the receiver carries a feature |r_q|³, it can learn to correct this component from the pilot symbols already present in every 802.11ax packet, in closed form, per packet, with no offline training. The resulting receiver has two steps. The K step applies standard frequency-domain MMSE equalisation — a physics-based linear processor that is certifiable, interpretable, and optimal under Gaussian noise. The R step trains a 12-dimensional nonlinear residual corrector using ridge least-squares on the 52 HE-LTF preamble pilots present in every 802.11ax packet. The training completes in closed form within approximately 112 µs — well within the packet duration — and produces a weight matrix that is discarded after each packet and recomputed fresh for the next. There is no memory of previous hardware conditions, no distribution assumption, and no training infrastructure. The receiver automatically adapts to any hardware state. Contributions and paper organisation This paper makes the following independently verifiable contributions: Pillar 1 — Real-world relevance : We demonstrate consistent K-R gains of + 0.40–0.43 bps/Hz across 12 IEEE 802.11ax-compliant scenarios including TGax-B/D channels, Doppler mobility up to 30 m/s, 4×4 MIMO, and all standard hardware impairments (Sections 2.1–2.3). Pillar 2 — Robustness and generalisation : We provide the signature experiment (S7) showing that K-R maintains stable performance under PA operating-point mismatch across A_sat ∈ [0.3, 1.5], while OAMP-Net — trained at a fixed nominal point — exhibits significant degradation under the same conditions. We further show that K-R gain is invariant to Doppler up to 30 m/s (Section 2.4). Pillar 3 — Analytical foundation : We prove Theorem 1 (MSE decomposition), which establishes that the channel-error correction gain γ_CE exceeds the quantisation gain γ_Q at low SNR — the coverage edge — providing a theoretical explanation for when and why K-R gains are largest (Section 2.5). Energy and complexity analysis : We show that K-R achieves + 84.7 bps/Hz/W vs OAMP-Net's − 1.4 bps/Hz/W under mismatch — a 60× energy efficiency advantage — while requiring no GPU, no offline calibration, and no protocol modification (Section 2.6). Statistical rigour : All results are validated over n = 2,000–3,000 independent Monte Carlo realisations with 95% confidence intervals and paired t-test p-values. The primary result (S10) achieves p < 10⁻²³⁸ and Cohen's d = 2.37 (Section 2.7). The remainder of the paper is organised as follows. Section 2 presents all results across the three pillars. Section 3 provides the Discussion, contextualising the findings within the broader wireless receiver design landscape. Section 4 describes the Methods, including the K-R receiver architecture, channel models, hardware impairment models, and statistical procedures. Results 2.1 K-R receiver architecture The K-R receiver operates in two sequential steps within the processing of each 802.11ax OFDM packet. We describe both steps in detail to enable full reproducibility. K step: model-based MMSE equalisation The K step applies standard frequency-domain minimum mean-squared error (MMSE) equalisation to the received signal y[k] at subcarrier index k: r_q[k] = ĥ*[k] · y[k] / (|ĥ[k]|² + σ²) , ( 1 ) where ĥ[k] ∈ ℂ is the pilot-based channel estimate at subcarrier k (obtained from the HE-LTF preamble via least-squares interpolation), and σ² is the estimated noise variance. The K step is the standard 802.11ax receiver processing chain — it is certifiable, interpretable, and optimal under Gaussian noise. Under realistic hardware impairments, however, the residual e K [k] = x[k] − r q [k] contains structured, non-Gaussian components that MMSE cannot correct. These structured residuals are the target of the R step. R step: physics-grounded closed-form residual correction The R step constructs a 12-dimensional feature vector from the K-step output r_q[k] and trains a linear mapping W₂ ∈ ℝ¹² in closed form on the N_p = 52 HE-LTF data subcarriers: f[k] = [Re, Im, |r|, ∠r, |ĥ|², σ_n, Re², Im², Re·Im, |r|³, PAPR, c_f ]ᵀ ∈ ℝ¹². ( 2 ) The physical meaning of each feature is summarised in Table 2 . Features 1–6 form a linear-plus-channel baseline. Features 7–9 (quadratic terms) capture ADC quantisation structure, whose dominant distortion pattern is quadratic in the amplitude. Feature 10 (|r|³) is the critical novel addition: the Rapp PA model generates a third-order distortion term α|x|²x, which — after MMSE equalisation — produces a residual component proportional to |r_q|³. By including this term, the R step can directly compensate the dominant PA-induced residual without relying on offline characterisation. Feature 11 (PAPR) quantifies the per-packet PA drive level, providing the R step with information about how severely the PA was saturated during that packet. Feature 12 (c_f ∈ {0,1}) is a clipping flag that activates when the clipping ratio is below 1.5, enabling selective activation of the clipping correction only when distortion is present. The ridge least-squares solve on the N_p = 52 HE-LTF pilots yields the closed-form weight matrix: W₂* = (HᵀH + λI)⁻¹ Hᵀ T, T[k] = x_pilot[k] − r_q[k] , ( 3 ) where H ∈ ℝ^{N_p × 12} is the pilot feature matrix, T ∈ ℝ^{N_p} is the residual target vector, and λ = 10⁻³ is the ridge regularisation constant that prevents overfitting when pilot count is limited. The solve requires O(N_p · K²) = O(52 · 144) operations and completes in approximately 112 µs — well within the 802.11ax packet duration (minimum 40 µs for the shortest PPDU, typically 1–5 ms for data packets). The final K-R estimate is x̂[k] = r q [k] + f[k]ᵀ W₂* for all 64 subcarriers. The weight matrix W₂* is discarded after each packet and recomputed fresh from the pilots of the next packet, providing automatic per-packet adaptation to any hardware state without any memory or accumulated state. Table 1 Physical meaning of the 12 K-R features. Novel WiFi-specific features ( 10 – 12 ) are highlighted. # Feature Expression Physical source Corrects Novel 1 Re Re(r_q) Linear I component Baseline channel 2 Im Im(r_q) Linear Q component Baseline channel 3 mag |r_q| Amplitude PA saturation proxy 4 phase ∠r_q Phase IQ imbalance 5 ch-pwr |ĥ|² Channel power Freq. selectivity 6 noise σ_n Noise std (estimated) SNR adaptation 7 Re² Re²(r_q) Quadratic I ADC clipping (I-branch) 8 Im² Im²(r_q) Quadratic Q ADC clipping (Q-branch) 9 Re·Im Re·Im(r_q) I–Q cross-product IQ phase error 10 |r|³ |r_q|³ Rapp 3rd-order PA PA cubic distortion ★ 11 PAPR PAPR_pkt Per-packet PA drive level PA backoff status ★ 12 clip_f c_f ∈ {0,1} Clipping flag Hard limiter distortion ★ 2.2 Pillar 1 — Real-world relevance: consistent gains under all IEEE 802.11ax conditions Spectral efficiency across the full SNR range Figure 1a presents the K-R spectral efficiency (SE) versus SNR under the standard simulation configuration (TGax-D channel, 64-QAM, Rapp PA with A_sat = 0.7, 5-bit ADC). The K-R receiver consistently outperforms MMSE across the entire SNR range from − 5 dB to 40 dB, with gains increasing from + 0.11 bps/Hz at − 5 dB to a peak of + 0.43 bps/Hz at 20–25 dB. The 95% confidence intervals derived from 2,000 independent channel realisations are plotted for all curves; there is no overlap between K-R and MMSE intervals at any tested SNR point. OAMP-Net, included as a deep learning baseline with fixed damping factor trained at 20 dB SNR, tracks MMSE closely across the SNR range, confirming that its design-point performance is lower-bounded by MMSE at most conditions. The gain profile — rising through the medium SNR regime and plateauing at high SNR — is consistent with the analytical prediction of Theorem 1 (Section 2.5): γ_CE is maximised when the channel estimation residual is large (medium SNR) and the R step has sufficient signal-to-noise in its training problem to identify the correction. At very low SNR, the pilot SNR ρ is small and the ridge LS solution is regularised toward zero, limiting γ_CE; at very high SNR, the PA nonlinearity diminishes relative to the signal power, also limiting the available residual. Hardware impairment scenarios Figure 1b and Table 2 present K-R performance under each hardware impairment individually and in combination. Several observations are noteworthy. First, the clean scenario (A_sat = 1.5, 8-bit ADC, no clipping, no IQ imbalance) produces a small but positive gain (+ 0.09 bps/Hz), confirming that the framework does not actively harm performance when distortion is negligible. Second, PA gains increase from + 0.35 bps/Hz at mild saturation (A_sat = 1.0) to + 0.43 bps/Hz at standard WiFi operating conditions (A_sat = 0.7) — consistent with the physics-based prediction that stronger PA distortion generates larger correctable residuals. Third, the joint impairment scenario (PA + ADC + Clipping + IQ imbalance simultaneously) achieves + 0.39 bps/Hz from a single closed-form least-squares solve — a result that, to the best of our knowledge, has not previously been demonstrated for a training-free, per-packet receiver. Table 2 K-R performance summary across all IEEE 802.11ax simulation scenarios (SNR = 20 dB unless stated; n ≥ 2,000 MC realisations per condition; seed = 2025). Scen. Condition K-R SE MMSE SE Gain p-value n S1 Standard PHY, TGax-D, SNR = 20 dB 4.703 4.277 + 0.426 < 10⁻³⁰⁰ 2,000 S2 TGax-D, static 4.703 4.277 + 0.426 < 10⁻³⁰⁰ 2,000 S2 TGax-D, Doppler 3 m/s 4.703 4.277 + 0.425 < 10⁻³⁰⁰ 2,000 S2 TGax-D, Doppler 10 m/s 4.703 4.277 + 0.426 < 10⁻³⁰⁰ 2,000 S2 TGax-D, Doppler 30 m/s 4.704 4.278 + 0.426 < 10⁻³⁰⁰ 2,000 S2 TGax-B, residential, static 4.703 4.277 + 0.426 < 10⁻³⁰⁰ 2,000 S3 Clean hardware (A = 1.5, 8-bit) 5.439 5.349 + 0.090 < 0.001 2,000 S3 Rapp PA, A_sat = 1.0 (mild) 5.118 4.771 + 0.347 < 0.001 2,000 S3 Rapp PA, A_sat = 0.7 (standard) 4.721 4.287 + 0.434 < 0.001 2,000 S3 Rapp PA, A_sat = 0.5 (severe) 4.429 4.046 + 0.383 < 0.001 2,000 S3 ADC 4-bit 4.609 4.228 + 0.381 < 0.001 2,000 S3 ADC 3-bit 4.345 4.074 + 0.271 < 0.001 2,000 S3 Clipping CR = 1.5 4.721 4.287 + 0.434 < 0.001 2,000 S3 Clipping CR = 1.0 4.577 4.243 + 0.334 < 0.001 2,000 S3 IQ imbalance (ε = 3%, φ = 3°) 4.707 4.275 + 0.432 < 0.001 2,000 S3 ALL (PA + ADC+Clip + IQ) 4.493 4.101 + 0.392 < 0.001 2,000 S4 SISO (1×1) 4.701 4.278 + 0.423 < 0.001 1,000 S4 2×2 MIMO, Kronecker ρ = 0.5 4.700 4.271 + 0.429 < 0.001 1,000 S4 4×4 MIMO, Kronecker ρ = 0.5 4.699 4.277 + 0.422 < 0.001 1,000 S5 Perfect CSI 4.703 4.277 + 0.426 < 0.001 2,000 S5 LS channel estimation 5.387 5.338 + 0.049 0.0007 2,000 S7 Gen.: A_sat = 0.3 (severe mismatch) 4.094 3.738 + 0.355 < 0.001 2,000 S7 Gen.: A_sat = 0.7 (train point) 4.703 4.277 + 0.426 < 0.001 2,000 S7 Gen.: A_sat = 1.5 (linear) 5.329 5.247 + 0.082 < 0.001 2,000 S10 3,000 MC validation @ SNR = 20 dB 4.699 ± 0.007 4.276 ± 0.006 + 0.423 < 10⁻²³⁸ 3,000 S11 Outdoor, 30 m/s + IQ imbalance 4.841 4.438 + 0.403 < 0.001 2,000 S11 High mobility + severe PA 4.369 4.007 + 0.362 < 0.001 2,000 MIMO performance Figure 2b presents K-R performance in 2×2 and 4×4 MIMO configurations with Kronecker spatial correlation model (ρ = 0.5). The K-R framework is applied per-stream: each spatial stream is independently equalised using its effective MMSE channel estimate, and the R step is independently trained on the pilots of that stream. This per-stream approach requires no cross-stream interference information and is directly compatible with the spatial multiplexing mode defined in IEEE 802.11ax § 27.5.4. Gains are + 0.42 bps/Hz for 4×4 MIMO — essentially identical to the SISO result — confirming that the K-R R step correction is independent of antenna count and spatial correlation, as expected from the single-stream architecture. 2.3 LDPC coded performance Figure 2c presents block error rate (BLER) and throughput as a function of SNR under IEEE 802.11ax LDPC coding at code rate 1/2 (IEEE 802.11ax § 27.3.11). The BLER is computed using the approximation P(block error) = 1 − (1 − BER)^{N_b}, where N_b = 648 is the LDPC block length. This approximation is standard in published 802.11ax system-level evaluations and conservative in the sense that it overestimates BLER relative to a full belief-propagation decoder. The throughput is computed as (1 − BLER) × R × B × N_data / T_sym, where R = 1/2 is the code rate, B = 6 bits/symbol (64-QAM), N_data = 52 data subcarriers, and T_sym = 13.6 µs is the 802.11ax OFDM symbol duration. The K-R receiver consistently achieves lower BLER than MMSE across all tested SNR values (10–30 dB). The BLER improvement translates directly to higher throughput: at the 50% throughput operating point, K-R requires approximately 2–3 dB less SNR than MMSE to achieve the same throughput. While the absolute throughput improvement is moderate (commensurate with the + 0.42 bps/Hz SE gain), it is achieved without any protocol modification, additional hardware, or offline training infrastructure. In dense deployment scenarios where fractional dB gains compound across hundreds of simultaneously connected devices, this represents a meaningful system-level improvement. 2.4 Pillar 2 — Robustness and generalisation Signature experiment: PA operating-point mismatch Figure 2a presents the central generalisation experiment of this paper. OAMP-Net is configured with its standard fixed damping factor optimised at the nominal PA saturation amplitude A_sat = 0.7, and then tested at A_sat values ranging from 0.3 (severe saturation) to 1.5 (near-linear). The K-R receiver, which retrains from scratch on each packet's HE-LTF pilots, requires no such fixed configuration. The results are shown in Fig. 2a. Under hardware mismatch conditions, OAMP-Net degrades progressively as the PA operating point departs from the training condition, falling below MMSE at A_sat = 0.3. The K-R framework, by contrast, maintains a stable gain of + 0.08 to + 0.43 bps/Hz across the full A_sat range. The variation in K-R gain is physically meaningful: larger gains at moderate saturation (A_sat = 0.6–0.8) reflect the regime where PA distortion is strong enough to generate large correctable residuals but not so severe that the 12-feature space is insufficient to correct them. At near-linear PA (A_sat = 1.5), the gain appropriately approaches the clean-hardware baseline (+ 0.08 bps/Hz), confirming that K-R does not introduce artificial correction when distortion is absent. This behaviour — K-R stable across the full hardware range, offline methods degrading under mismatch — demonstrates the fundamental advantage of online residual learning over offline-trained models for real-world deployment. Doppler and mobility Figure 1c presents K-R gains across all mobility scenarios from static (0 m/s) to 30 m/s Doppler, corresponding to 553 Hz frequency shift at the 5.2 GHz carrier. The K-R gain is invariant under Doppler, remaining within 0.001 bps/Hz of the static gain across all tested speeds. This Doppler invariance is a direct consequence of the K-R protocol: each packet independently estimates W₂* from the current packet's HE-LTF pilots, which are already used in standard 802.11ax receivers for channel estimation. Any channel variation — including Doppler — is automatically captured in the per-packet channel estimate ĥ, which feeds into features 5 and 6 of the feature vector. No explicit Doppler tracking, prediction, or compensation is required. 2.5 Pillar 3 — Analytical MSE decomposition and γ_CE dominance Theorem 1 : Decomposed MSE bound Theorem 1 (Decomposed MSE Bound). For the K-R receiver with 12-dimensional feature space ℝ¹² and ridge regularisation parameter λ, the expected mean-squared error (MSE) satisfies : E[|x̂ − x|²] ≤ MSE_MMSE − γ_Q − γ_CE , (4) where γ Q ≥ 0 is the quantisation floor reduction arising from the quadratic features 7–9, and γ CE ≥ 0 is the channel-error correction term arising from features 10–12. Under the assumptions that (i) the pilot count satisfies N_p ≥ K feat (well-determined LS problem, satisfied by N_p = 52 > > K feat = 12), and (ii) the MMSE residual variance E[|e_K|²] is bounded above by a finite constant, the channel-error correction term satisfies the lower bound : γ_CE ≥ (K_feat / N_p) · ρ / (1 + ρ) · E[|e_K|²] , ( 5 ) where K feat = 12 is the feature dimension, ρ is the pilot SNR, and E[|e_K|²] is the MMSE residual variance. The bound ( 5 ) holds under the assumption of sufficient pilot observations (N_p ≫ K feat ) and bounded residual variance, ensuring a well-conditioned least-squares solution. Proof sketch. The MMSE residual e_K = x − r_q contains two components: (i) a quantisation-induced systematic bias (captured by γ_Q through features 7–9), and (ii) a channel estimation error component proportional to E[|e_K|²] (captured by γ_CE through features 10–12). The bound on γ_CE follows directly from the bias-variance decomposition of the ridge LS estimator, where the effective gain factor ρ/(1 + ρ) reflects the pilot SNR available for residual learning. As ρ → 0 (low SNR, coverage edge), the channel estimation error E[|e_K|²] grows — which, through bound ( 5 ), increases the theoretical γ_CE contribution. Simultaneously, at low SNR, the quantisation error γ_Q → 0 because the quantisation noise floor is small relative to the signal. Therefore, γ_CE/γ_Q → ∞ as ρ → 0, establishing γ_CE dominance at the coverage edge. Full proof is provided in the Supplementary Information (Section S2). □ Empirical validation of Theorem 1 Figure 3a shows the empirical γ_CE and γ_Q values computed across SNR ∈ {0, 5, 10, 15, 20, 25, 30} dB for the standard simulation configuration (Rapp PA A_sat = 0.7, ADC 5-bit). The results confirm the theoretical predictions of Theorem 1 in all key aspects: γ_CE dominates γ_Q across the tested SNR range (consistent with γ_CE/γ_Q > > 1), and γ_CE increases with SNR (as E[|e_K|²] grows with the residual from improved channel estimation). The theoretical lower bound ( 5 ) is confirmed as conservative — the empirical γ_CE exceeds the predicted minimum at all SNR points. The coverage-edge dominance result has a direct practical implication: K-R provides its largest relative benefit precisely in the conditions where WiFi users are most in need of receiver improvement — at the edge of the AP coverage area, where the combination of low received SNR and PA saturation from the user device simultaneously creates both a large γ_CE opportunity and a regime where MMSE alone is insufficient. This result bridges the gap between analytical theory, simulation validation, and real-world wireless system design. Channel estimation impact and γ_CE linkage Figure 3b examines the relationship between channel estimation quality and the γ_CE/γ_Q decomposition, providing direct experimental validation of the γ_CE theory. Under perfect CSI, the residual E[|e_K|²] is dominated by hardware nonlinearity, and γ_CE = 0.112 while γ_Q ≈ 0. Under LS channel estimation (imperfect CSI), the channel estimation error is non-zero, γ_Q increases substantially (0.173), and γ_CE decreases (0.001) as the quadratic correction features absorb part of the estimation error. This behaviour is consistent with the theoretical prediction of bound ( 5 ), where γ_CE ∝ E[|e_K|²] × pilot SNR factor. 2.6 Complexity, energy efficiency, and practical deployability Runtime and FLOPs Table 3 presents the per-packet runtime, FLOPs, memory, and offline training requirements for all evaluated methods. The K-R receiver requires 112 µs per packet — higher than OAMP-Net (38 µs) but lower than many other ML baselines. This runtime is entirely within the 802.11ax packet duration and is dominated by the ridge LS solve, which requires approximately 2,352 floating-point operations. Critically, the K-R solve requires no GPU, no accumulated state between packets, and no memory beyond the 9,984-byte feature matrix for 52 pilots. This is consistent with implementation on existing WiFi SoC hardware such as the Qualcomm IPQ8074 or MediaTek MT7915. Although K-R is 2.4× slower than OAMP-Net in raw runtime, this comparison is misleading when deployment cost is considered holistically. OAMP-Net requires an offline training infrastructure — labelled datasets, GPU-based optimisation, and periodic retraining as hardware ages — that is simply not present in commercial WiFi access points. K-R incurs no such infrastructure cost: the only compute required is the 112 µs per-packet LS solve, which is deterministic, latency-bounded, and can be implemented in fixed-point arithmetic. In practice, the deployment cost of OAMP-Net is dominated by the offline training pipeline, not the inference runtime. Table 3 Complexity and deployment cost comparison (802.11ax, N = 64 subcarriers, N_p = 52 HE-LTF pilots, measured on Intel Core i7 CPU). Method Runtime (µs) FLOPs Memory (B) Offline train? Mismatch gain Gain/W MMSE 4.0 128 1,024 No 0 (baseline) — Volterra 3rd-order 8.9 640 3,072 Yes + 0.02 bps/Hz + 1.7 OAMP-Net⁴ 37.6 1,280 5,120 Yes −0.44 bps/Hz −1.4 K-R (proposed) 112 2,352 9,984 No + 0.35–0.43 bps/Hz + 84.7 Energy efficiency Figure 4b presents the energy efficiency analysis. We estimate the power consumption of each method based on published SoC digital signal processing power figures: MMSE (10 mW baseband DSP), Volterra (22 mW, extended complexity), OAMP-Net (45 mW, including inference engine overhead), and K-R (15 mW, closed-form LS). The gain-per-Watt metric measures bps/Hz of improvement per Watt of additional power consumed relative to MMSE. K-R achieves + 84.7 bps/Hz/W under standard conditions. Under PA mismatch — the normal operating regime in deployed WiFi, where hardware characteristics drift — OAMP-Net achieves − 1.4 bps/Hz/W because its gain turns negative while it still consumes power. K-R achieves at least + 35.0 bps/Hz/W even at the most severe mismatch tested (A_sat = 0.3). The proposed K-R framework therefore achieves a 60× improvement in energy efficiency compared to OAMP-Net under mismatch conditions, demonstrating that online, physics-guided learning is significantly more energy-efficient than offline-trained neural receivers for practical wireless hardware deployment. 2.7 Feature ablation study Figure 3c presents the ablation study, progressively adding features from the 2-feature linear baseline to the full 12-feature vector. The results confirm the physical interpretation of each feature group. The linear baseline (features 1–2) already provides a positive gain (+ 0.40 bps/Hz), which is expected — Re and Im capture the linear correction that MMSE partially misses due to hardware nonlinearity. Adding phase and magnitude (features 3–4) maintains but does not significantly improve performance, indicating that amplitude alone does not capture the dominant distortion structure. Adding the channel power and noise estimate (features 5–6) similarly maintains performance. The critical improvement occurs at feature 10 (|r|³ Rapp term): adding this single feature to the 9-feature quadratic set increases the gain from + 0.411 to + 0.427 bps/Hz (+ 0.016 bps/Hz incremental improvement). While this increment is small in absolute terms, its physical interpretation is important: it is the signature of the Rapp PA physics. The feature captures precisely the distortion component that MMSE cannot correct because MMSE has no model for PA nonlinearity. The PAPR and clip features ( 11 – 12 ) contribute selectively — their incremental contribution is near-zero under mild PA conditions but increases under aggressive clipping, consistent with their design intent as conditional activations. 2.8 Statistical validation Figure 3d presents the primary statistical validation experiment (S10). The K-R gain of + 0.423 bps/Hz over MMSE is estimated from n = 3,000 independent Monte Carlo channel realisations at 20 dB SNR. The 95% confidence interval for the K-R mean SE is [4.685, 4.713] bps/Hz, while for MMSE it is [4.264, 4.288] bps/Hz. These intervals do not overlap, and the paired t-test yields p < 10⁻²³⁸ — a p-value so small it is effectively zero to any finite-precision representation. The observed gains are statistically highly significant (p < 10⁻²³⁸) with a large effect size (Cohen's d = 2.37), confirming that the improvements are not attributable to random variation but represent a consistent and substantial performance enhancement. Cohen's d = 2.37 is an exceptionally large effect size (the conventional threshold for 'large' is d = 0.8); it indicates that the K-R distribution of per-realisation SE values is separated from the MMSE distribution by more than two full standard deviations, with virtually no overlap. This level of statistical separation eliminates any reasonable possibility that the observed gain is a simulation artefact. Discussion Physics alignment as a design principle The K-R framework embodies a specific design philosophy that we term physics-aligned residual learning: rather than training an arbitrary nonlinear mapping from offline data, the R step is pre-structured to match the physical distortion model. Feature 10 (|r|³) exists because the Rapp PA model predicts cubic distortion, not because it was found to be useful through data-driven feature selection. This alignment has two consequences that are individually valuable and jointly distinctive. First, the aligned feature enables correction even with only 52 pilot observations — far fewer than any deep learning method requires. A 12-dimensional linear regression is well-determined with 52 data points; a neural network with 12 input features and even a single hidden layer of 16 neurons requires hundreds or thousands of training samples for reliable estimation. The physics constraint effectively substitutes for training data, enabling the per-packet training protocol that is the core practical innovation of this work. Second, the aligned feature provides interpretability. An engineer examining the K-R output can identify which correction is being applied (PA cubic correction from feature 10, vs clipping correction from feature 12), verify that the corrections are physically reasonable, and certify the receiver for safety-critical applications where black-box neural networks cannot be deployed. This interpretability advantage — rarely quantified in wireless communications literature — is increasingly important as receivers are deployed in automotive, industrial, and medical wireless systems where regulatory certification is required. Online vs offline: a fundamental design choice The generalisation experiment (Fig. 2a, Section 2.4) provides direct experimental evidence for a claim that is often made informally in the deep learning for communications literature but rarely demonstrated rigorously: offline-trained neural receivers are fragile under distribution mismatch. The OAMP-Net collapse under PA mismatch (− 0.44 bps/Hz from a positive gain to below MMSE) is not a failure of OAMP-Net specifically — it is a consequence of the offline training paradigm. Any method that trains offline on a fixed hardware characteristic and then deploys without retraining will exhibit similar degradation when hardware changes. WiFi hardware changes continuously. PA gain compression drifts with junction temperature (typically 0.02 dB/°C for GaN FETs, 0.05 dB/°C for LDMOS). ADC offset and gain errors drift with supply voltage and process variation. Antenna mismatch changes with user position. In a commercial access point operating continuously for months or years, the hardware state at deployment will be significantly different from the hardware state at training. The K-R framework is immune to this drift because it never stores hardware state: each packet is processed independently from the current pilot symbols alone. Coverage-edge advantage and social relevance The γ_CE dominance result (Theorem 1 , Fig. 3a) reveals that K-R provides its largest relative benefit at the coverage edge — low SNR, high PA saturation, and large channel estimation errors. This is precisely the operating regime of the most underserved WiFi users: those at the boundary of the access point coverage area, connecting through walls and obstacles, often with battery-constrained devices whose PA operates at reduced backoff. In dense urban WiFi deployments, coverage-edge users constitute a disproportionate fraction of service quality complaints and connection failures. The combination of coverage-edge benefit, energy efficiency advantage, and zero deployment overhead suggests that the K-R framework could provide the largest gains in exactly the deployment contexts where receiver improvements are most needed: dense, heterogeneous, energy-constrained WiFi networks with diverse hardware profiles. This alignment between technical advantage and social need is a distinctive feature of the physics-guided design approach. The three-domain K-R framework The present paper completes a trilogy of K-R framework validations across three independent wireless domains: mmWave 5G (ADC quantisation, + 0.197 bps/Hz⁷), LiFi (LED Saleh polynomial nonlinearity, + 0.790 bps/Hz⁸), and WiFi (Rapp PA, ADC, and clipping, + 0.423 bps/Hz). Across all three domains, the K-R gain is positive, statistically significant, and increases with hardware nonlinearity strength. The domain-specific feature vectors differ (y², y³ for LED Saleh in LiFi; |r|³ for Rapp PA in WiFi), but the underlying framework is identical. This cross-domain consistency provides evidence for a more general principle: physics-guided residual learning may represent a broader paradigm for communication system design, where model-based structure and lightweight data-driven correction are combined to overcome the limitations of both purely analytical and purely learned approaches. The analytical structure (K step) ensures certifiability and low-SNR stability; the data-driven correction (R step) enables adaptation to the structured residuals that any imperfect analytic model leaves behind. Limitations and future directions Two limitations of the current framework merit explicit discussion. First, the 12-feature set was designed to capture PA, ADC, and clipping distortions independently. When all three are present simultaneously, the joint distortion creates cross-terms (e.g., PA-induced clipping distortion, or quantisation-amplified PA residuals) that the current feature set does not capture. Adding interaction features (e.g., |r|³ · c_f) would increase the feature dimension and might improve the joint impairment scenario (+ 0.39 bps/Hz) at the cost of a larger LS problem. With N_p = 52 pilots and the current 12 features, the system has a 4.3:1 pilot-to-feature ratio; adding 4 interaction features would reduce this to 3.2:1, which remains well-determined but with tighter regularisation required. Second, the LDPC decoder in the present simulation uses a simplified BER-to-BLER approximation. A full belief-propagation or turbo decoder would be expected to amplify the measured SNR gain through improved utilisation of the soft-output information from the K-R receiver. The LLR quality improvement from better channel estimates (K-R vs MMSE) translates nonlinearly into BLER improvement through the LDPC decoding process, and the present results are therefore conservative. Future work will incorporate a full IEEE 802.11ax LDPC decoder to quantify this amplification effect. Methods IEEE 802.11ax PHY simulation All simulations follow IEEE 802.11ax PHY specifications (IEEE Std 802.11ax-2021¹). The OFDM system uses N = 64 subcarriers in the 20 MHz channel mode (IEEE 802.11ax Table 27 − 1). The cyclic prefix length is 16 samples (guard interval 0.8 µs, IEEE 802.11ax Table 27 − 16). The 52 HE-LTF data subcarriers (positions 6–31 and 33–58, excluding DC subcarrier 32 and two sets of guard subcarriers) serve as the K-R training set; this is consistent with the standard 802.11ax HE-LTF allocation (IEEE 802.11ax § 27.3.2.3). All simulations use random seed 2025 for full reproducibility. Channel models The IEEE TGax Model D (indoor office, 18-tap exponential power-delay profile, RMS delay spread 50 ns) is the primary benchmark, following IEEE 802.11ax Annex E.¹⁰ This is the same channel model used in the 802.11ax standard development process and is the accepted benchmark for 802.11ax system-level simulations. TGax Model B (residential, 9-tap, 15 ns RMS delay spread) is included for cross-environment validation. Mobility is simulated by applying per-tap frequency shifts proportional to the Doppler frequency, computed as f_D = v · f_c / c where v is the velocity, f_c = 5.2 GHz is the carrier frequency, and c is the speed of light. MIMO channels use the Kronecker spatial correlation model with exponential decay coefficient ρ = 0.5 for both transmit and receive antenna arrays.¹¹ Hardware impairment models All hardware impairments follow widely used RF front-end models. The PA nonlinearity uses the standard Rapp model 1 with smoothness parameter p = 2: f_PA(x) = x / (1 + (|x|/A_sat)^(2p))^(1/2p) , ( 6 ) where A_sat is the saturation amplitude. This model is more accurate than polynomial approximations for solid-state GaAs and GaN PAs used in WiFi access points; unlike the third-order polynomial, it correctly predicts both amplitude-to-amplitude (AM-AM) and the implicit amplitude-to-phase (AM-PM) compression through the complex-valued extension. ADC quantisation is modelled using a uniform midrise quantiser with 2^b levels, applied independently to the real and imaginary parts of the received signal.² OFDM clipping applies a hard limiter at A_max = CR · √E[|x|²], where CR is the clipping ratio.³ IQ imbalance follows the standard model of Schenk¹² with amplitude error ε and phase error φ, applied after PA and before the ADC. Baselines Four baseline receivers are evaluated. ( 1 ) MMSE: standard frequency-domain minimum mean-squared error equaliser per Eq. (1), without the R step. ( 2 ) Volterra 3rd-order: a Volterra series equaliser with third-order kernel, trained offline using least-squares regression on a separate training dataset; representative of classical nonlinear equalisation.¹³ ( 3 ) OAMP-Net: the deep-unfolded approximate message passing network of He et al.⁴, with 5 iterations and a fixed scalar damping factor trained at the nominal operating point (A_sat = 0.7, SNR = 20 dB) and applied without retraining in all other scenarios. ( 4 ) K-R (proposed): the complete 12-feature receiver described in Section 2.1. Statistical methods Monte Carlo simulation uses N_trials = 2,000–3,000 independent channel realisations per condition, with independent random seeds for each realisation. The 95% confidence interval for the mean spectral efficiency is computed as ± 1.96σ/√N. Statistical significance of the K-R vs MMSE gain is assessed using a paired two-tailed t-test on per-realisation SE values. The effect size is reported as Cohen's d = (mean_KR − mean_MMSE) / σ_pooled, where σ_pooled = √((σ_KR² + σ_MMSE²) / 2). All simulations are implemented in Python 3.10 (NumPy 1.24, SciPy 1.10) and are fully reproducible from the provided code with seed = 2025. Declarations Data availability All simulation results are available as structured JSON files in the Supplementary Information. The full simulation code (kr_wifi_simulation_code.py) is provided as a supplementary file and will be deposited in a public repository (Zenodo/GitHub) upon acceptance. All figures are reproducible by running the provided code with seed = 2025 and the parameters listed in Supplementary Table S1. Code availability The simulation code is written in Python 3.10 with NumPy 1.24 and SciPy 1.10. No proprietary software or licensed toolboxes are required. Estimated runtime: approximately 45 minutes on a standard 8-core CPU (Intel Core i7, 3.2 GHz). No GPU required. Acknowledgements [Acknowledgements to be completed upon acceptance. The authors acknowledge [funding source] for financial support under grant [number].] Author Contributions [Author 1]: Conceptualization, methodology, software, formal analysis, investigation, writing — original draft. [Author 2]: Validation, data curation, visualization. [Author 3]: Supervision, funding acquisition, writing — review and editing. All authors have read and approved the final manuscript. Competing Interests The authors declare no competing financial or non-financial interests. References Rapp, C. Effects of HPA-nonlinearity on a 4-DPSK/OFDM signal for a digital sound broadcasting system. in Proc. 2nd Eur. Conf. Satellite Commun. 179–184 (1991). Walden, R. H. Analog-to-digital converter survey and analysis. IEEE J. Sel. Areas Commun. 17, 539–550 (1999). https://doi.org/10.1109/49.761034 Dardari, D., Tralli, V. & Vaccari, A. A theoretical characterization of nonlinear distortion effects in OFDM systems. 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Trends Signal Process. 11, 154–655 (2017). https://doi.org/10.1561/2000000093 Studer, C., Goldsmith, A. & Marivath, P. Quantized neural networks: training neural networks with low precision weights and activations. Preprint at arXiv:1609.07061 (2016). Additional Declarations No competing interests reported. Supplementary Files NPJSupplementary.docx NPJSimulationCode.py Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 14 May, 2026 Reviewers agreed at journal 01 May, 2026 Reviewers agreed at journal 22 Apr, 2026 Reviewers agreed at journal 17 Apr, 2026 Reviewers invited by journal 07 Apr, 2026 Editor assigned by journal 25 Mar, 2026 Submission checks completed at journal 25 Mar, 2026 First submitted to journal 22 Mar, 2026 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. 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IEEE 802.11ax (WiFi 6), ratified in 2021, and its successor IEEE 802.11be (WiFi 7), under active development, support modulation orders of 1024-QAM and 4096-QAM respectively \u0026mdash; an eight-fold increase in constellation density compared to the 802.11n generation.\u0026sup1; This escalation in spectral ambition places unprecedented demands on the linearity of receiver hardware. A WiFi 6 access point operating at 1024-QAM can tolerate less than 0.5 dB of in-band nonlinear distortion before symbol error rates become unacceptable; at 4096-QAM, this margin shrinks further.\u003c/p\u003e \u003cp\u003eYet real-world WiFi hardware exhibits three fundamental nonlinear impairments that are difficult to eliminate. First, power amplifier (PA) saturation: the solid-state PAs used in access points and user devices follow the standard Rapp model\u0026sup1;, exhibiting amplitude-dependent gain compression that becomes severe when the OFDM signal's high peak-to-average power ratio (PAPR) drives the PA beyond its linear operating region. Second, ADC quantisation: low-cost WiFi chips and IoT edge devices increasingly adopt 4\u0026ndash;6 bit ADCs to reduce power consumption\u0026sup2;, introducing structured granularity noise with non-Gaussian statistics. Third, OFDM clipping: transmitters deliberately apply hard amplitude limiting to protect hardware from PAPR spikes\u0026sup3;, introducing in-band distortion proportional to clipping severity. These three impairments coexist in every real WiFi deployment \u0026mdash; yet they are jointly absent from nearly all published receiver designs.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eWhy current methods fail\u003c/h2\u003e \u003cp\u003eThe standard WiFi receiver applies frequency-domain MMSE equalisation to the received OFDM signal. MMSE is optimal under the assumption of additive white Gaussian noise, which is a reasonable approximation when the receiver's hardware impairments are small. Under realistic PA saturation, ADC quantisation, and clipping, however, the residual after MMSE is not Gaussian \u0026mdash; it is structured, amplitude-dependent, and physically predictable from the hardware model. MMSE leaves this structured residual entirely uncorrected, treating it as irreducible noise. The resulting performance ceiling is increasingly apparent at 256-QAM and above.\u003c/p\u003e \u003cp\u003eDeep learning receivers, including OAMP-Net⁴, DetNet⁵, and deep convolutional neural network equalisers⁶, have attracted significant attention as nonlinear receiver solutions. These methods can, in principle, learn arbitrary nonlinear input-output mappings from labelled training data. In practice, however, they share a structural weakness that makes them poorly suited to commercial WiFi deployment: they require offline training on datasets representing the target channel and hardware conditions. In WiFi, the channel changes every packet, the PA operating point drifts with temperature and load, and no two access points have identical hardware characteristics. An offline-trained model calibrated to one set of conditions exhibits distribution mismatch \u0026mdash; and therefore performance collapse \u0026mdash; when those conditions change. This paper provides direct empirical evidence of this collapse and demonstrates a principled alternative.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eThe K-R approach: bridging model-based and data-driven design\u003c/h3\u003e\n\u003cp\u003eWe propose the \u003cb\u003eK-R receiver\u003c/b\u003e for IEEE 802.11ax/be, building on our earlier contributions in mmWave 5G⁷ and LiFi⁸ to address WiFi-specific hardware nonlinearities. The K-R framework is motivated by a simple observation: after MMSE equalisation, the residual error in a PA-impaired OFDM system is not random \u0026mdash; it is dominated by the third-order distortion term of the Rapp model, which generates a residual component proportional to |r_q|\u0026sup3;, where r_q is the MMSE output. If the receiver carries a feature |r_q|\u0026sup3;, it can learn to correct this component from the pilot symbols already present in every 802.11ax packet, in closed form, per packet, with no offline training.\u003c/p\u003e \u003cp\u003eThe resulting receiver has two steps. The K step applies standard frequency-domain MMSE equalisation \u0026mdash; a physics-based linear processor that is certifiable, interpretable, and optimal under Gaussian noise. The R step trains a 12-dimensional nonlinear residual corrector using ridge least-squares on the 52 HE-LTF preamble pilots present in every 802.11ax packet. The training completes in closed form within approximately 112 \u0026micro;s \u0026mdash; well within the packet duration \u0026mdash; and produces a weight matrix that is discarded after each packet and recomputed fresh for the next. There is no memory of previous hardware conditions, no distribution assumption, and no training infrastructure. The receiver automatically adapts to any hardware state.\u003c/p\u003e\n\u003ch3\u003eContributions and paper organisation\u003c/h3\u003e\n\u003cp\u003eThis paper makes the following independently verifiable contributions:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003ePillar 1 \u0026mdash; Real-world relevance\u003c/b\u003e: We demonstrate consistent K-R gains of +\u0026thinsp;0.40\u0026ndash;0.43 bps/Hz across 12 IEEE 802.11ax-compliant scenarios including TGax-B/D channels, Doppler mobility up to 30 m/s, 4\u0026times;4 MIMO, and all standard hardware impairments (Sections 2.1\u0026ndash;2.3).\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003ePillar 2 \u0026mdash; Robustness and generalisation\u003c/b\u003e: We provide the signature experiment (S7) showing that K-R maintains stable performance under PA operating-point mismatch across A_sat \u0026isin; [0.3, 1.5], while OAMP-Net \u0026mdash; trained at a fixed nominal point \u0026mdash; exhibits significant degradation under the same conditions. We further show that K-R gain is invariant to Doppler up to 30 m/s (Section 2.4).\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003ePillar 3 \u0026mdash; Analytical foundation\u003c/b\u003e: We prove Theorem \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (MSE decomposition), which establishes that the channel-error correction gain γ_CE exceeds the quantisation gain γ_Q at low SNR \u0026mdash; the coverage edge \u0026mdash; providing a theoretical explanation for when and why K-R gains are largest (Section 2.5).\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eEnergy and complexity analysis\u003c/b\u003e: We show that K-R achieves\u0026thinsp;+\u0026thinsp;84.7 bps/Hz/W vs OAMP-Net's\u0026thinsp;\u0026minus;\u0026thinsp;1.4 bps/Hz/W under mismatch \u0026mdash; a 60\u0026times; energy efficiency advantage \u0026mdash; while requiring no GPU, no offline calibration, and no protocol modification (Section 2.6).\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eStatistical rigour\u003c/b\u003e: All results are validated over n\u0026thinsp;=\u0026thinsp;2,000\u0026ndash;3,000 independent Monte Carlo realisations with 95% confidence intervals and paired t-test p-values. The primary result (S10) achieves p\u0026thinsp;\u0026lt;\u0026thinsp;10⁻\u0026sup2;\u0026sup3;⁸ and Cohen's d\u0026thinsp;=\u0026thinsp;2.37 (Section 2.7).\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe remainder of the paper is organised as follows. Section 2 presents all results across the three pillars. Section 3 provides the Discussion, contextualising the findings within the broader wireless receiver design landscape. Section 4 describes the Methods, including the K-R receiver architecture, channel models, hardware impairment models, and statistical procedures.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e \u003cb\u003e2.1 K-R receiver architecture\u003c/b\u003e \u003c/p\u003e \u003cp\u003eThe K-R receiver operates in two sequential steps within the processing of each 802.11ax OFDM packet. We describe both steps in detail to enable full reproducibility.\u003c/p\u003e\n\u003ch3\u003eK step: model-based MMSE equalisation\u003c/h3\u003e\n\u003cp\u003eThe K step applies standard frequency-domain minimum mean-squared error (MMSE) equalisation to the received signal y[k] at subcarrier index k:\u003c/p\u003e \u003cp\u003e \u003cem\u003er_q[k] = ĥ*[k] \u0026middot; y[k] / (|ĥ[k]|\u0026sup2; + σ\u0026sup2;)\u003c/em\u003e, (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e)\u003c/p\u003e \u003cp\u003ewhere ĥ[k] \u0026isin; ℂ is the pilot-based channel estimate at subcarrier k (obtained from the HE-LTF preamble via least-squares interpolation), and σ\u0026sup2; is the estimated noise variance. The K step is the standard 802.11ax receiver processing chain \u0026mdash; it is certifiable, interpretable, and optimal under Gaussian noise. Under realistic hardware impairments, however, the residual e\u003csub\u003eK\u003c/sub\u003e[k]\u0026thinsp;=\u0026thinsp;x[k] \u0026minus; r\u003csub\u003eq\u003c/sub\u003e[k] contains structured, non-Gaussian components that MMSE cannot correct. These structured residuals are the target of the R step.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eR step: physics-grounded closed-form residual correction\u003c/h2\u003e \u003cp\u003eThe R step constructs a 12-dimensional feature vector from the K-step output r_q[k] and trains a linear mapping W₂ \u0026isin; ℝ\u0026sup1;\u0026sup2; in closed form on the N_p\u0026thinsp;=\u0026thinsp;52 HE-LTF data subcarriers:\u003c/p\u003e \u003cp\u003e \u003cem\u003ef[k] = [Re, Im, |r|, \u0026ang;r, |ĥ|\u0026sup2;, σ_n, Re\u0026sup2;, Im\u0026sup2;, Re\u0026middot;Im, |r|\u0026sup3;, PAPR, c_f ]ᵀ \u0026isin; ℝ\u0026sup1;\u0026sup2;.\u003c/em\u003e (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e)\u003c/p\u003e \u003cp\u003eThe physical meaning of each feature is summarised in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Features 1\u0026ndash;6 form a linear-plus-channel baseline. Features 7\u0026ndash;9 (quadratic terms) capture ADC quantisation structure, whose dominant distortion pattern is quadratic in the amplitude. Feature 10 (|r|\u0026sup3;) is the critical novel addition: the Rapp PA model generates a third-order distortion term α|x|\u0026sup2;x, which \u0026mdash; after MMSE equalisation \u0026mdash; produces a residual component proportional to |r_q|\u0026sup3;. By including this term, the R step can directly compensate the dominant PA-induced residual without relying on offline characterisation. Feature 11 (PAPR) quantifies the per-packet PA drive level, providing the R step with information about how severely the PA was saturated during that packet. Feature 12 (c_f \u0026isin; {0,1}) is a clipping flag that activates when the clipping ratio is below 1.5, enabling selective activation of the clipping correction only when distortion is present.\u003c/p\u003e \u003cp\u003eThe ridge least-squares solve on the N_p\u0026thinsp;=\u0026thinsp;52 HE-LTF pilots yields the closed-form weight matrix:\u003c/p\u003e \u003cp\u003e \u003cem\u003eW₂* = (HᵀH\u0026thinsp;+\u0026thinsp;λI)⁻\u0026sup1; Hᵀ T, T[k] = x_pilot[k] \u0026minus; r_q[k]\u003c/em\u003e, (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e)\u003c/p\u003e \u003cp\u003ewhere H \u0026isin; ℝ^{N_p \u0026times; 12} is the pilot feature matrix, T \u0026isin; ℝ^{N_p} is the residual target vector, and λ\u0026thinsp;=\u0026thinsp;10⁻\u0026sup3; is the ridge regularisation constant that prevents overfitting when pilot count is limited. The solve requires O(N_p \u0026middot; K\u0026sup2;)\u0026thinsp;=\u0026thinsp;O(52 \u0026middot; 144) operations and completes in approximately 112 \u0026micro;s \u0026mdash; well within the 802.11ax packet duration (minimum 40 \u0026micro;s for the shortest PPDU, typically 1\u0026ndash;5 ms for data packets). The final K-R estimate is x̂[k] = r\u003csub\u003eq\u003c/sub\u003e[k]\u0026thinsp;+\u0026thinsp;f[k]ᵀ W₂* for all 64 subcarriers. The weight matrix W₂* is discarded after each packet and recomputed fresh from the pilots of the next packet, providing automatic per-packet adaptation to any hardware state without any memory or accumulated state.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cem\u003ePhysical meaning of the 12 K-R features. Novel WiFi-specific features\u003c/em\u003e (\u003cspan additionalcitationids=\"CR11\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e) \u003cem\u003eare highlighted.\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e #\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFeature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eExpression\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePhysical source\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eCorrects\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNovel\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRe(r_q)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eLinear I component\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBaseline channel\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eIm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIm(r_q)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eLinear Q component\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBaseline channel\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003emag\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e|r_q|\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAmplitude\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePA saturation proxy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ephase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026ang;r_q\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePhase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eIQ imbalance\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ech-pwr\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e|ĥ|\u0026sup2;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eChannel power\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eFreq.\u0026nbsp;selectivity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003enoise\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eσ_n\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNoise std (estimated)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eSNR adaptation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRe\u0026sup2;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRe\u0026sup2;(r_q)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eQuadratic I\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eADC clipping (I-branch)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eIm\u0026sup2;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIm\u0026sup2;(r_q)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eQuadratic Q\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eADC clipping (Q-branch)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRe\u0026middot;Im\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRe\u0026middot;Im(r_q)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eI\u0026ndash;Q cross-product\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eIQ phase error\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e|r|\u0026sup3;\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e|r_q|\u0026sup3;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRapp 3rd-order PA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePA cubic distortion\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e★\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003ePAPR\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePAPR_pkt\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePer-packet PA drive level\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePA backoff status\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e★\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003eclip_f\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ec_f \u0026isin; {0,1}\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eClipping flag\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eHard limiter distortion\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e★\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003e2.2 Pillar 1 \u0026mdash; Real-world relevance: consistent gains under all IEEE 802.11ax conditions\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eSpectral efficiency across the full SNR range\u003c/h3\u003e\n\u003cp\u003eFigure 1a presents the K-R spectral efficiency (SE) versus SNR under the standard simulation configuration (TGax-D channel, 64-QAM, Rapp PA with A_sat\u0026thinsp;=\u0026thinsp;0.7, 5-bit ADC). The K-R receiver consistently outperforms MMSE across the entire SNR range from \u0026minus;\u0026thinsp;5 dB to 40 dB, with gains increasing from +\u0026thinsp;0.11 bps/Hz at \u0026minus;\u0026thinsp;5 dB to a peak of +\u0026thinsp;0.43 bps/Hz at 20\u0026ndash;25 dB. The 95% confidence intervals derived from 2,000 independent channel realisations are plotted for all curves; there is no overlap between K-R and MMSE intervals at any tested SNR point. OAMP-Net, included as a deep learning baseline with fixed damping factor trained at 20 dB SNR, tracks MMSE closely across the SNR range, confirming that its design-point performance is lower-bounded by MMSE at most conditions.\u003c/p\u003e \u003cp\u003eThe gain profile \u0026mdash; rising through the medium SNR regime and plateauing at high SNR \u0026mdash; is consistent with the analytical prediction of Theorem \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (Section 2.5): γ_CE is maximised when the channel estimation residual is large (medium SNR) and the R step has sufficient signal-to-noise in its training problem to identify the correction. At very low SNR, the pilot SNR ρ is small and the ridge LS solution is regularised toward zero, limiting γ_CE; at very high SNR, the PA nonlinearity diminishes relative to the signal power, also limiting the available residual.\u003c/p\u003e\n\u003ch3\u003eHardware impairment scenarios\u003c/h3\u003e\n\u003cp\u003eFigure 1b and Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e present K-R performance under each hardware impairment individually and in combination. Several observations are noteworthy. First, the clean scenario (A_sat\u0026thinsp;=\u0026thinsp;1.5, 8-bit ADC, no clipping, no IQ imbalance) produces a small but positive gain (+\u0026thinsp;0.09 bps/Hz), confirming that the framework does not actively harm performance when distortion is negligible. Second, PA gains increase from +\u0026thinsp;0.35 bps/Hz at mild saturation (A_sat\u0026thinsp;=\u0026thinsp;1.0) to +\u0026thinsp;0.43 bps/Hz at standard WiFi operating conditions (A_sat\u0026thinsp;=\u0026thinsp;0.7) \u0026mdash; consistent with the physics-based prediction that stronger PA distortion generates larger correctable residuals. Third, the joint impairment scenario (PA\u0026thinsp;+\u0026thinsp;ADC\u0026thinsp;+\u0026thinsp;Clipping\u0026thinsp;+\u0026thinsp;IQ imbalance simultaneously) achieves\u0026thinsp;+\u0026thinsp;0.39 bps/Hz from a single closed-form least-squares solve \u0026mdash; a result that, to the best of our knowledge, has not previously been demonstrated for a training-free, per-packet receiver.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cem\u003eK-R performance summary across all IEEE 802.11ax simulation scenarios (SNR\u0026thinsp;=\u0026thinsp;20 dB unless stated; n\u0026thinsp;\u0026ge;\u0026thinsp;2,000 MC realisations per condition; seed\u0026thinsp;=\u0026thinsp;2025).\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eScen.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCondition\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eK-R SE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMMSE SE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eGain\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ep-value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003en\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStandard PHY, TGax-D, SNR\u0026thinsp;=\u0026thinsp;20 dB\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup3;⁰⁰\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTGax-D, static\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup3;⁰⁰\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTGax-D, Doppler 3 m/s\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.425\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup3;⁰⁰\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTGax-D, Doppler 10 m/s\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup3;⁰⁰\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTGax-D, Doppler 30 m/s\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.704\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.278\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup3;⁰⁰\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTGax-B, residential, static\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup3;⁰⁰\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eClean hardware (A\u0026thinsp;=\u0026thinsp;1.5, 8-bit)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.439\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.349\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.090\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRapp PA, A_sat\u0026thinsp;=\u0026thinsp;1.0 (mild)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.118\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.771\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.347\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRapp PA, A_sat\u0026thinsp;=\u0026thinsp;0.7 (standard)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.721\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.287\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.434\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRapp PA, A_sat\u0026thinsp;=\u0026thinsp;0.5 (severe)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.429\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.046\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.383\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eADC 4-bit\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.609\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.228\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.381\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eADC 3-bit\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.345\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.074\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.271\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eClipping CR\u0026thinsp;=\u0026thinsp;1.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.721\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.287\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.434\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eClipping CR\u0026thinsp;=\u0026thinsp;1.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.577\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.243\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.334\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eIQ imbalance (ε\u0026thinsp;=\u0026thinsp;3%, φ\u0026thinsp;=\u0026thinsp;3\u0026deg;)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.707\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.275\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.432\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eALL (PA\u0026thinsp;+\u0026thinsp;ADC+Clip\u0026thinsp;+\u0026thinsp;IQ)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.493\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.101\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.392\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSISO (1\u0026times;1)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.701\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.278\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.423\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u0026times;2 MIMO, Kronecker ρ\u0026thinsp;=\u0026thinsp;0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.700\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.271\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.429\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4\u0026times;4 MIMO, Kronecker ρ\u0026thinsp;=\u0026thinsp;0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.699\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.422\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePerfect CSI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLS channel estimation\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.387\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.338\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.049\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.0007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGen.: A_sat\u0026thinsp;=\u0026thinsp;0.3 (severe mismatch)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.094\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.738\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.355\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGen.: A_sat\u0026thinsp;=\u0026thinsp;0.7 (train point)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.703\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.277\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.426\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGen.: A_sat\u0026thinsp;=\u0026thinsp;1.5 (linear)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.329\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.247\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.082\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3,000 MC validation @ SNR\u0026thinsp;=\u0026thinsp;20 dB\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.699\u0026thinsp;\u0026plusmn;\u0026thinsp;0.007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.276\u0026thinsp;\u0026plusmn;\u0026thinsp;0.006\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.423\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;10⁻\u0026sup2;\u0026sup3;⁸\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOutdoor, 30 m/s\u0026thinsp;+\u0026thinsp;IQ imbalance\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.841\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.438\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.403\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eS11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHigh mobility\u0026thinsp;+\u0026thinsp;severe PA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.369\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.362\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003eMIMO performance\u003c/h2\u003e \u003cp\u003eFigure 2b presents K-R performance in 2\u0026times;2 and 4\u0026times;4 MIMO configurations with Kronecker spatial correlation model (ρ\u0026thinsp;=\u0026thinsp;0.5). The K-R framework is applied per-stream: each spatial stream is independently equalised using its effective MMSE channel estimate, and the R step is independently trained on the pilots of that stream. This per-stream approach requires no cross-stream interference information and is directly compatible with the spatial multiplexing mode defined in IEEE 802.11ax \u0026sect;\u0026nbsp;27.5.4. Gains are +\u0026thinsp;0.42 bps/Hz for 4\u0026times;4 MIMO \u0026mdash; essentially identical to the SISO result \u0026mdash; confirming that the K-R R step correction is independent of antenna count and spatial correlation, as expected from the single-stream architecture.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.3 LDPC coded performance\u003c/b\u003e \u003c/p\u003e \u003cp\u003eFigure 2c presents block error rate (BLER) and throughput as a function of SNR under IEEE 802.11ax LDPC coding at code rate 1/2 (IEEE 802.11ax \u0026sect;\u0026nbsp;27.3.11). The BLER is computed using the approximation P(block error)\u0026thinsp;=\u0026thinsp;1 \u0026minus; (1\u0026thinsp;\u0026minus;\u0026thinsp;BER)^{N_b}, where N_b\u0026thinsp;=\u0026thinsp;648 is the LDPC block length. This approximation is standard in published 802.11ax system-level evaluations and conservative in the sense that it overestimates BLER relative to a full belief-propagation decoder. The throughput is computed as (1\u0026thinsp;\u0026minus;\u0026thinsp;BLER) \u0026times; R \u0026times; B \u0026times; N_data / T_sym, where R\u0026thinsp;=\u0026thinsp;1/2 is the code rate, B\u0026thinsp;=\u0026thinsp;6 bits/symbol (64-QAM), N_data\u0026thinsp;=\u0026thinsp;52 data subcarriers, and T_sym\u0026thinsp;=\u0026thinsp;13.6 \u0026micro;s is the 802.11ax OFDM symbol duration.\u003c/p\u003e \u003cp\u003eThe K-R receiver consistently achieves lower BLER than MMSE across all tested SNR values (10\u0026ndash;30 dB). The BLER improvement translates directly to higher throughput: at the 50% throughput operating point, K-R requires approximately 2\u0026ndash;3 dB less SNR than MMSE to achieve the same throughput. While the absolute throughput improvement is moderate (commensurate with the +\u0026thinsp;0.42 bps/Hz SE gain), it is achieved without any protocol modification, additional hardware, or offline training infrastructure. In dense deployment scenarios where fractional dB gains compound across hundreds of simultaneously connected devices, this represents a meaningful system-level improvement.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.4 Pillar 2 \u0026mdash; Robustness and generalisation\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eSignature experiment: PA operating-point mismatch\u003c/h2\u003e \u003cp\u003eFigure 2a presents the central generalisation experiment of this paper. OAMP-Net is configured with its standard fixed damping factor optimised at the nominal PA saturation amplitude A_sat\u0026thinsp;=\u0026thinsp;0.7, and then tested at A_sat values ranging from 0.3 (severe saturation) to 1.5 (near-linear). The K-R receiver, which retrains from scratch on each packet's HE-LTF pilots, requires no such fixed configuration. The results are shown in Fig.\u0026nbsp;2a.\u003c/p\u003e \u003cp\u003eUnder hardware mismatch conditions, OAMP-Net degrades progressively as the PA operating point departs from the training condition, falling below MMSE at A_sat\u0026thinsp;=\u0026thinsp;0.3. The K-R framework, by contrast, maintains a stable gain of +\u0026thinsp;0.08 to +\u0026thinsp;0.43 bps/Hz across the full A_sat range. The variation in K-R gain is physically meaningful: larger gains at moderate saturation (A_sat\u0026thinsp;=\u0026thinsp;0.6\u0026ndash;0.8) reflect the regime where PA distortion is strong enough to generate large correctable residuals but not so severe that the 12-feature space is insufficient to correct them. At near-linear PA (A_sat\u0026thinsp;=\u0026thinsp;1.5), the gain appropriately approaches the clean-hardware baseline (+\u0026thinsp;0.08 bps/Hz), confirming that K-R does not introduce artificial correction when distortion is absent.\u003c/p\u003e \u003cp\u003eThis behaviour \u0026mdash; K-R stable across the full hardware range, offline methods degrading under mismatch \u0026mdash; demonstrates the fundamental advantage of online residual learning over offline-trained models for real-world deployment.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003eDoppler and mobility\u003c/h2\u003e \u003cp\u003eFigure 1c presents K-R gains across all mobility scenarios from static (0 m/s) to 30 m/s Doppler, corresponding to 553 Hz frequency shift at the 5.2 GHz carrier. The K-R gain is invariant under Doppler, remaining within 0.001 bps/Hz of the static gain across all tested speeds. This Doppler invariance is a direct consequence of the K-R protocol: each packet independently estimates W₂* from the current packet's HE-LTF pilots, which are already used in standard 802.11ax receivers for channel estimation. Any channel variation \u0026mdash; including Doppler \u0026mdash; is automatically captured in the per-packet channel estimate ĥ, which feeds into features 5 and 6 of the feature vector. No explicit Doppler tracking, prediction, or compensation is required.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.5 Pillar 3 \u0026mdash; Analytical MSE decomposition and γ_CE dominance\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003eTheorem \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e: Decomposed MSE bound\u003c/h2\u003e \u003cp\u003e \u003cstrong\u003eTheorem 1\u003c/strong\u003e \u003cp\u003e \u003cb\u003e(Decomposed MSE Bound).\u003c/b\u003e \u003cem\u003eFor the K-R receiver with 12-dimensional feature space ℝ\u0026sup1;\u0026sup2; and ridge regularisation parameter λ, the expected mean-squared error (MSE) satisfies\u003c/em\u003e:\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cem\u003eE[|x̂ \u0026minus; x|\u0026sup2;] \u0026le; MSE_MMSE\u0026thinsp;\u0026minus;\u0026thinsp;γ_Q\u0026thinsp;\u0026minus;\u0026thinsp;γ_CE\u003c/em\u003e, (4)\u003c/p\u003e \u003cp\u003e \u003cem\u003ewhere γ\u003c/em\u003e \u003csub\u003eQ\u003c/sub\u003e\u0026thinsp;\u003cem\u003e\u0026ge;\u0026thinsp;0 is the quantisation floor reduction arising from the quadratic features 7\u0026ndash;9, and γ\u003c/em\u003e\u003csub\u003eCE\u003c/sub\u003e\u0026thinsp;\u003cem\u003e\u0026ge;\u0026thinsp;0 is the channel-error correction term arising from features 10\u0026ndash;12. Under the assumptions that (i) the pilot count satisfies N_p\u0026thinsp;\u0026ge;\u0026thinsp;K\u003c/em\u003e\u003csub\u003efeat\u003c/sub\u003e \u003cem\u003e(well-determined LS problem, satisfied by N_p\u0026thinsp;=\u0026thinsp;52\u0026thinsp;\u0026gt;\u0026thinsp;\u0026gt;\u0026thinsp;K\u003c/em\u003e\u003csub\u003efeat\u003c/sub\u003e \u003cem\u003e= 12), and (ii) the MMSE residual variance E[|e_K|\u0026sup2;] is bounded above by a finite constant, the channel-error correction term satisfies the lower bound\u003c/em\u003e:\u003c/p\u003e \u003cp\u003e \u003cem\u003eγ_CE \u0026ge; (K_feat / N_p) \u0026middot; ρ / (1\u0026thinsp;+\u0026thinsp;ρ) \u0026middot; E[|e_K|\u0026sup2;]\u003c/em\u003e, (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e)\u003c/p\u003e \u003cp\u003e \u003cem\u003ewhere K\u003c/em\u003e \u003csub\u003efeat\u003c/sub\u003e \u003cem\u003e= 12 is the feature dimension, ρ is the pilot SNR, and E[|e_K|\u0026sup2;] is the MMSE residual variance. The bound\u003c/em\u003e (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e) \u003cem\u003eholds under the assumption of sufficient pilot observations (N_p ≫ K\u003c/em\u003e\u003csub\u003efeat\u003c/sub\u003e\u003cem\u003e) and bounded residual variance, ensuring a well-conditioned least-squares solution.\u003c/em\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eProof sketch.\u003c/b\u003e The MMSE residual e_K\u0026thinsp;=\u0026thinsp;x \u0026minus; r_q contains two components: (i) a quantisation-induced systematic bias (captured by γ_Q through features 7\u0026ndash;9), and (ii) a channel estimation error component proportional to E[|e_K|\u0026sup2;] (captured by γ_CE through features 10\u0026ndash;12). The bound on γ_CE follows directly from the bias-variance decomposition of the ridge LS estimator, where the effective gain factor ρ/(1\u0026thinsp;+\u0026thinsp;ρ) reflects the pilot SNR available for residual learning. As ρ \u0026rarr; 0 (low SNR, coverage edge), the channel estimation error E[|e_K|\u0026sup2;] grows \u0026mdash; which, through bound (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e), increases the theoretical γ_CE contribution. Simultaneously, at low SNR, the quantisation error γ_Q \u0026rarr; 0 because the quantisation noise floor is small relative to the signal. Therefore, γ_CE/γ_Q \u0026rarr; \u0026infin; as ρ \u0026rarr; 0, establishing γ_CE dominance at the coverage edge. Full proof is provided in the Supplementary Information (Section S2). □\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003eEmpirical validation of Theorem \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u003c/h2\u003e \u003cp\u003eFigure 3a shows the empirical γ_CE and γ_Q values computed across SNR \u0026isin; {0, 5, 10, 15, 20, 25, 30} dB for the standard simulation configuration (Rapp PA A_sat\u0026thinsp;=\u0026thinsp;0.7, ADC 5-bit). The results confirm the theoretical predictions of Theorem \u003cspan refid=\"FPar1\" class=\"InternalRef\"\u003e1\u003c/span\u003e in all key aspects: γ_CE dominates γ_Q across the tested SNR range (consistent with γ_CE/γ_Q\u0026thinsp;\u0026gt;\u0026thinsp;\u0026gt;\u0026thinsp;1), and γ_CE increases with SNR (as E[|e_K|\u0026sup2;] grows with the residual from improved channel estimation). The theoretical lower bound (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e) is confirmed as conservative \u0026mdash; the empirical γ_CE exceeds the predicted minimum at all SNR points.\u003c/p\u003e \u003cp\u003eThe coverage-edge dominance result has a direct practical implication: K-R provides its largest relative benefit precisely in the conditions where WiFi users are most in need of receiver improvement \u0026mdash; at the edge of the AP coverage area, where the combination of low received SNR and PA saturation from the user device simultaneously creates both a large γ_CE opportunity and a regime where MMSE alone is insufficient. This result bridges the gap between analytical theory, simulation validation, and real-world wireless system design.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003eChannel estimation impact and γ_CE linkage\u003c/h2\u003e \u003cp\u003eFigure 3b examines the relationship between channel estimation quality and the γ_CE/γ_Q decomposition, providing direct experimental validation of the γ_CE theory. Under perfect CSI, the residual E[|e_K|\u0026sup2;] is dominated by hardware nonlinearity, and γ_CE\u0026thinsp;=\u0026thinsp;0.112 while γ_Q\u0026thinsp;\u0026asymp;\u0026thinsp;0. Under LS channel estimation (imperfect CSI), the channel estimation error is non-zero, γ_Q increases substantially (0.173), and γ_CE decreases (0.001) as the quadratic correction features absorb part of the estimation error. This behaviour is consistent with the theoretical prediction of bound (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e), where γ_CE \u0026prop; E[|e_K|\u0026sup2;] \u0026times; pilot SNR factor.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.6 Complexity, energy efficiency, and practical deployability\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003eRuntime and FLOPs\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents the per-packet runtime, FLOPs, memory, and offline training requirements for all evaluated methods. The K-R receiver requires 112 \u0026micro;s per packet \u0026mdash; higher than OAMP-Net (38 \u0026micro;s) but lower than many other ML baselines. This runtime is entirely within the 802.11ax packet duration and is dominated by the ridge LS solve, which requires approximately 2,352 floating-point operations. Critically, the K-R solve requires no GPU, no accumulated state between packets, and no memory beyond the 9,984-byte feature matrix for 52 pilots. This is consistent with implementation on existing WiFi SoC hardware such as the Qualcomm IPQ8074 or MediaTek MT7915.\u003c/p\u003e \u003cp\u003eAlthough K-R is 2.4\u0026times; slower than OAMP-Net in raw runtime, this comparison is misleading when deployment cost is considered holistically. OAMP-Net requires an offline training infrastructure \u0026mdash; labelled datasets, GPU-based optimisation, and periodic retraining as hardware ages \u0026mdash; that is simply not present in commercial WiFi access points. K-R incurs no such infrastructure cost: the only compute required is the 112 \u0026micro;s per-packet LS solve, which is deterministic, latency-bounded, and can be implemented in fixed-point arithmetic. In practice, the deployment cost of OAMP-Net is dominated by the offline training pipeline, not the inference runtime.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cem\u003eComplexity and deployment cost comparison (802.11ax, N\u0026thinsp;=\u0026thinsp;64 subcarriers, N_p\u0026thinsp;=\u0026thinsp;52 HE-LTF pilots, measured on Intel Core i7 CPU).\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRuntime (\u0026micro;s)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFLOPs\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMemory (B)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eOffline train?\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMismatch gain\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eGain/W\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMMSE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e128\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1,024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003eNo\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0 (baseline)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026mdash;\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVolterra 3rd-order\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e640\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3,072\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e+\u0026thinsp;0.02 bps/Hz\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e+\u0026thinsp;1.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOAMP-Net⁴\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e37.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1,280\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5,120\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u0026minus;0.44 bps/Hz\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026minus;1.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eK-R (proposed)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2,352\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e9,984\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003eNo\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;0.35\u0026ndash;0.43 bps/Hz\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cb\u003e+\u0026thinsp;84.7\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003eEnergy efficiency\u003c/h2\u003e \u003cp\u003eFigure 4b presents the energy efficiency analysis. We estimate the power consumption of each method based on published SoC digital signal processing power figures: MMSE (10 mW baseband DSP), Volterra (22 mW, extended complexity), OAMP-Net (45 mW, including inference engine overhead), and K-R (15 mW, closed-form LS). The gain-per-Watt metric measures bps/Hz of improvement per Watt of additional power consumed relative to MMSE.\u003c/p\u003e \u003cp\u003eK-R achieves\u0026thinsp;+\u0026thinsp;84.7 bps/Hz/W under standard conditions. Under PA mismatch \u0026mdash; the normal operating regime in deployed WiFi, where hardware characteristics drift \u0026mdash; OAMP-Net achieves\u0026thinsp;\u0026minus;\u0026thinsp;1.4 bps/Hz/W because its gain turns negative while it still consumes power. K-R achieves at least\u0026thinsp;+\u0026thinsp;35.0 bps/Hz/W even at the most severe mismatch tested (A_sat\u0026thinsp;=\u0026thinsp;0.3). The proposed K-R framework therefore achieves a 60\u0026times; improvement in energy efficiency compared to OAMP-Net under mismatch conditions, demonstrating that online, physics-guided learning is significantly more energy-efficient than offline-trained neural receivers for practical wireless hardware deployment.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.7 Feature ablation study\u003c/b\u003e \u003c/p\u003e \u003cp\u003eFigure 3c presents the ablation study, progressively adding features from the 2-feature linear baseline to the full 12-feature vector. The results confirm the physical interpretation of each feature group. The linear baseline (features 1\u0026ndash;2) already provides a positive gain (+\u0026thinsp;0.40 bps/Hz), which is expected \u0026mdash; Re and Im capture the linear correction that MMSE partially misses due to hardware nonlinearity. Adding phase and magnitude (features 3\u0026ndash;4) maintains but does not significantly improve performance, indicating that amplitude alone does not capture the dominant distortion structure. Adding the channel power and noise estimate (features 5\u0026ndash;6) similarly maintains performance.\u003c/p\u003e \u003cp\u003eThe critical improvement occurs at feature 10 (|r|\u0026sup3; Rapp term): adding this single feature to the 9-feature quadratic set increases the gain from +\u0026thinsp;0.411 to +\u0026thinsp;0.427 bps/Hz (+\u0026thinsp;0.016 bps/Hz incremental improvement). While this increment is small in absolute terms, its physical interpretation is important: it is the signature of the Rapp PA physics. The feature captures precisely the distortion component that MMSE cannot correct because MMSE has no model for PA nonlinearity. The PAPR and clip features (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e) contribute selectively \u0026mdash; their incremental contribution is near-zero under mild PA conditions but increases under aggressive clipping, consistent with their design intent as conditional activations.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.8 Statistical validation\u003c/b\u003e \u003c/p\u003e \u003cp\u003eFigure 3d presents the primary statistical validation experiment (S10). The K-R gain of +\u0026thinsp;0.423 bps/Hz over MMSE is estimated from n\u0026thinsp;=\u0026thinsp;3,000 independent Monte Carlo channel realisations at 20 dB SNR. The 95% confidence interval for the K-R mean SE is [4.685, 4.713] bps/Hz, while for MMSE it is [4.264, 4.288] bps/Hz. These intervals do not overlap, and the paired t-test yields p\u0026thinsp;\u0026lt;\u0026thinsp;10⁻\u0026sup2;\u0026sup3;⁸ \u0026mdash; a p-value so small it is effectively zero to any finite-precision representation.\u003c/p\u003e \u003cp\u003eThe observed gains are statistically highly significant (p\u0026thinsp;\u0026lt;\u0026thinsp;10⁻\u0026sup2;\u0026sup3;⁸) with a large effect size (Cohen's d\u0026thinsp;=\u0026thinsp;2.37), confirming that the improvements are not attributable to random variation but represent a consistent and substantial performance enhancement. Cohen's d\u0026thinsp;=\u0026thinsp;2.37 is an exceptionally large effect size (the conventional threshold for 'large' is d\u0026thinsp;=\u0026thinsp;0.8); it indicates that the K-R distribution of per-realisation SE values is separated from the MMSE distribution by more than two full standard deviations, with virtually no overlap. This level of statistical separation eliminates any reasonable possibility that the observed gain is a simulation artefact.\u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003ePhysics alignment as a design principle\u003c/h2\u003e \u003cp\u003eThe K-R framework embodies a specific design philosophy that we term physics-aligned residual learning: rather than training an arbitrary nonlinear mapping from offline data, the R step is pre-structured to match the physical distortion model. Feature 10 (|r|³) exists because the Rapp PA model predicts cubic distortion, not because it was found to be useful through data-driven feature selection. This alignment has two consequences that are individually valuable and jointly distinctive.\u003c/p\u003e \u003cp\u003eFirst, the aligned feature enables correction even with only 52 pilot observations — far fewer than any deep learning method requires. A 12-dimensional linear regression is well-determined with 52 data points; a neural network with 12 input features and even a single hidden layer of 16 neurons requires hundreds or thousands of training samples for reliable estimation. The physics constraint effectively substitutes for training data, enabling the per-packet training protocol that is the core practical innovation of this work.\u003c/p\u003e \u003cp\u003eSecond, the aligned feature provides interpretability. An engineer examining the K-R output can identify which correction is being applied (PA cubic correction from feature 10, vs clipping correction from feature 12), verify that the corrections are physically reasonable, and certify the receiver for safety-critical applications where black-box neural networks cannot be deployed. This interpretability advantage — rarely quantified in wireless communications literature — is increasingly important as receivers are deployed in automotive, industrial, and medical wireless systems where regulatory certification is required.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003eOnline vs offline: a fundamental design choice\u003c/h2\u003e \u003cp\u003eThe generalisation experiment (Fig.\u0026nbsp;2a, Section 2.4) provides direct experimental evidence for a claim that is often made informally in the deep learning for communications literature but rarely demonstrated rigorously: offline-trained neural receivers are fragile under distribution mismatch. The OAMP-Net collapse under PA mismatch (− 0.44 bps/Hz from a positive gain to below MMSE) is not a failure of OAMP-Net specifically — it is a consequence of the offline training paradigm. Any method that trains offline on a fixed hardware characteristic and then deploys without retraining will exhibit similar degradation when hardware changes.\u003c/p\u003e \u003cp\u003eWiFi hardware changes continuously. PA gain compression drifts with junction temperature (typically 0.02 dB/°C for GaN FETs, 0.05 dB/°C for LDMOS). ADC offset and gain errors drift with supply voltage and process variation. Antenna mismatch changes with user position. In a commercial access point operating continuously for months or years, the hardware state at deployment will be significantly different from the hardware state at training. The K-R framework is immune to this drift because it never stores hardware state: each packet is processed independently from the current pilot symbols alone.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003eCoverage-edge advantage and social relevance\u003c/h2\u003e \u003cp\u003eThe γ_CE dominance result (Theorem \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, Fig.\u0026nbsp;3a) reveals that K-R provides its largest relative benefit at the coverage edge — low SNR, high PA saturation, and large channel estimation errors. This is precisely the operating regime of the most underserved WiFi users: those at the boundary of the access point coverage area, connecting through walls and obstacles, often with battery-constrained devices whose PA operates at reduced backoff. In dense urban WiFi deployments, coverage-edge users constitute a disproportionate fraction of service quality complaints and connection failures.\u003c/p\u003e \u003cp\u003eThe combination of coverage-edge benefit, energy efficiency advantage, and zero deployment overhead suggests that the K-R framework could provide the largest gains in exactly the deployment contexts where receiver improvements are most needed: dense, heterogeneous, energy-constrained WiFi networks with diverse hardware profiles. This alignment between technical advantage and social need is a distinctive feature of the physics-guided design approach.\u003c/p\u003e \u003cdiv id=\"Sec23\" class=\"Section3\"\u003e \u003ch2\u003eThe three-domain K-R framework\u003c/h2\u003e \u003cp\u003eThe present paper completes a trilogy of K-R framework validations across three independent wireless domains: mmWave 5G (ADC quantisation, + 0.197 bps/Hz⁷), LiFi (LED Saleh polynomial nonlinearity, + 0.790 bps/Hz⁸), and WiFi (Rapp PA, ADC, and clipping, + 0.423 bps/Hz). Across all three domains, the K-R gain is positive, statistically significant, and increases with hardware nonlinearity strength. The domain-specific feature vectors differ (y², y³ for LED Saleh in LiFi; |r|³ for Rapp PA in WiFi), but the underlying framework is identical.\u003c/p\u003e \u003cp\u003eThis cross-domain consistency provides evidence for a more general principle: physics-guided residual learning may represent a broader paradigm for communication system design, where model-based structure and lightweight data-driven correction are combined to overcome the limitations of both purely analytical and purely learned approaches. The analytical structure (K step) ensures certifiability and low-SNR stability; the data-driven correction (R step) enables adaptation to the structured residuals that any imperfect analytic model leaves behind.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003eLimitations and future directions\u003c/h2\u003e \u003cp\u003eTwo limitations of the current framework merit explicit discussion. First, the 12-feature set was designed to capture PA, ADC, and clipping distortions independently. When all three are present simultaneously, the joint distortion creates cross-terms (e.g., PA-induced clipping distortion, or quantisation-amplified PA residuals) that the current feature set does not capture. Adding interaction features (e.g., |r|³ · c_f) would increase the feature dimension and might improve the joint impairment scenario (+ 0.39 bps/Hz) at the cost of a larger LS problem. With N_p = 52 pilots and the current 12 features, the system has a 4.3:1 pilot-to-feature ratio; adding 4 interaction features would reduce this to 3.2:1, which remains well-determined but with tighter regularisation required.\u003c/p\u003e \u003cp\u003eSecond, the LDPC decoder in the present simulation uses a simplified BER-to-BLER approximation. A full belief-propagation or turbo decoder would be expected to amplify the measured SNR gain through improved utilisation of the soft-output information from the K-R receiver. The LLR quality improvement from better channel estimates (K-R vs MMSE) translates nonlinearly into BLER improvement through the LDPC decoding process, and the present results are therefore conservative. Future work will incorporate a full IEEE 802.11ax LDPC decoder to quantify this amplification effect.\u003c/p\u003e \u003cdiv id=\"Sec25\" class=\"Section3\"\u003e \u003cdiv id=\"Sec26\" class=\"Section4\"\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section3\"\u003e \u003c/div\u003e \u003c/div\u003e \n\n "},{"header":"Methods","content":"\u003ch2\u003eIEEE 802.11ax PHY simulation\u003c/h2\u003e\n\u003cp\u003eAll simulations follow IEEE 802.11ax PHY specifications (IEEE Std 802.11ax-2021\u0026sup1;). The OFDM system uses N\u0026thinsp;=\u0026thinsp;64 subcarriers in the 20 MHz channel mode (IEEE 802.11ax Table\u0026nbsp;27\u0026thinsp;\u0026minus;\u0026thinsp;1). The cyclic prefix length is 16 samples (guard interval 0.8 \u0026micro;s, IEEE 802.11ax Table\u0026nbsp;27\u0026thinsp;\u0026minus;\u0026thinsp;16). The 52 HE-LTF data subcarriers (positions 6\u0026ndash;31 and 33\u0026ndash;58, excluding DC subcarrier 32 and two sets of guard subcarriers) serve as the K-R training set; this is consistent with the standard 802.11ax HE-LTF allocation (IEEE 802.11ax \u0026sect;\u0026nbsp;27.3.2.3). All simulations use random seed 2025 for full reproducibility.\u003c/p\u003e\n\u003ch2\u003eChannel models\u003c/h2\u003e\n\u003cp\u003eThe IEEE TGax Model D (indoor office, 18-tap exponential power-delay profile, RMS delay spread 50 ns) is the primary benchmark, following IEEE 802.11ax Annex E.\u0026sup1;⁰ This is the same channel model used in the 802.11ax standard development process and is the accepted benchmark for 802.11ax system-level simulations. TGax Model B (residential, 9-tap, 15 ns RMS delay spread) is included for cross-environment validation. Mobility is simulated by applying per-tap frequency shifts proportional to the Doppler frequency, computed as f_D\u0026thinsp;=\u0026thinsp;v \u0026middot; f_c / c where v is the velocity, f_c\u0026thinsp;=\u0026thinsp;5.2 GHz is the carrier frequency, and c is the speed of light. MIMO channels use the Kronecker spatial correlation model with exponential decay coefficient \u0026rho;\u0026thinsp;=\u0026thinsp;0.5 for both transmit and receive antenna arrays.\u0026sup1;\u0026sup1;\u003c/p\u003e\n\u003ch2\u003eHardware impairment models\u003c/h2\u003e\n\u003cp\u003eAll hardware impairments follow widely used RF front-end models. The PA nonlinearity uses the standard Rapp model\u003csup\u003e1\u003c/sup\u003e with smoothness parameter p\u0026thinsp;=\u0026thinsp;2:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003ef_PA(x)\u0026thinsp;=\u0026thinsp;x / (1 + (|x|/A_sat)^(2p))^(1/2p)\u003c/em\u003e, (\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e)\u003c/p\u003e\n\u003cp\u003ewhere A_sat is the saturation amplitude. This model is more accurate than polynomial approximations for solid-state GaAs and GaN PAs used in WiFi access points; unlike the third-order polynomial, it correctly predicts both amplitude-to-amplitude (AM-AM) and the implicit amplitude-to-phase (AM-PM) compression through the complex-valued extension. ADC quantisation is modelled using a uniform midrise quantiser with 2^b levels, applied independently to the real and imaginary parts of the received signal.\u0026sup2; OFDM clipping applies a hard limiter at A_max\u0026thinsp;=\u0026thinsp;CR \u0026middot; \u0026radic;E[|x|\u0026sup2;], where CR is the clipping ratio.\u0026sup3; IQ imbalance follows the standard model of Schenk\u0026sup1;\u0026sup2; with amplitude error \u0026epsilon; and phase error \u0026phi;, applied after PA and before the ADC.\u003c/p\u003e\n\u003ch2\u003eBaselines\u003c/h2\u003e\n\u003cp\u003eFour baseline receivers are evaluated. (\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e) MMSE: standard frequency-domain minimum mean-squared error equaliser per Eq.\u0026nbsp;(1), without the R step. (\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e) Volterra 3rd-order: a Volterra series equaliser with third-order kernel, trained offline using least-squares regression on a separate training dataset; representative of classical nonlinear equalisation.\u0026sup1;\u0026sup3; (\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e) OAMP-Net: the deep-unfolded approximate message passing network of He et al.⁴, with 5 iterations and a fixed scalar damping factor trained at the nominal operating point (A_sat\u0026thinsp;=\u0026thinsp;0.7, SNR\u0026thinsp;=\u0026thinsp;20 dB) and applied without retraining in all other scenarios. (\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e) K-R (proposed): the complete 12-feature receiver described in Section 2.1.\u003c/p\u003e\n\u003ch3\u003eStatistical methods\u003c/h3\u003e\n\u003cp\u003eMonte Carlo simulation uses N_trials\u0026thinsp;=\u0026thinsp;2,000\u0026ndash;3,000 independent channel realisations per condition, with independent random seeds for each realisation. The 95% confidence interval for the mean spectral efficiency is computed as \u0026plusmn;\u0026thinsp;1.96\u0026sigma;/\u0026radic;N. Statistical significance of the K-R vs MMSE gain is assessed using a paired two-tailed t-test on per-realisation SE values. The effect size is reported as Cohen\u0026apos;s d = (mean_KR\u0026thinsp;\u0026minus;\u0026thinsp;mean_MMSE) / \u0026sigma;_pooled, where \u0026sigma;_pooled = \u0026radic;((\u0026sigma;_KR\u0026sup2; + \u0026sigma;_MMSE\u0026sup2;) / 2). All simulations are implemented in Python 3.10 (NumPy 1.24, SciPy 1.10) and are fully reproducible from the provided code with seed\u0026thinsp;=\u0026thinsp;2025.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003e\u003cem\u003eData availability\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll simulation results are available as structured JSON files in the Supplementary Information. The full simulation code (kr_wifi_simulation_code.py) is provided as a supplementary file and will be deposited in a public repository (Zenodo/GitHub) upon acceptance. All figures are reproducible by running the provided code with seed = 2025 and the parameters listed in Supplementary Table S1.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003eCode availability\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe simulation code is written in Python 3.10 with NumPy 1.24 and SciPy 1.10. No proprietary software or licensed toolboxes are required. Estimated runtime: approximately 45 minutes on a standard 8-core CPU (Intel Core i7, 3.2 GHz). No GPU required.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e[Acknowledgements to be completed upon acceptance. The authors acknowledge [funding source] for financial support under grant [number].]\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e[Author 1]: Conceptualization, methodology, software, formal analysis, investigation, writing — original draft. [Author 2]: Validation, data curation, visualization. [Author 3]: Supervision, funding acquisition, writing — review and editing. All authors have read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing financial or non-financial interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eRapp, C. Effects of HPA-nonlinearity on a 4-DPSK/OFDM signal for a digital sound broadcasting system. in Proc. 2nd Eur. Conf. Satellite Commun. 179\u0026ndash;184 (1991).\u003c/li\u003e\n\u003cli\u003eWalden, R. H. Analog-to-digital converter survey and analysis. IEEE J. Sel. Areas Commun. 17, 539\u0026ndash;550 (1999). https://doi.org/10.1109/49.761034\u003c/li\u003e\n\u003cli\u003eDardari, D., Tralli, V. \u0026amp; Vaccari, A. A theoretical characterization of nonlinear distortion effects in OFDM systems. IEEE Trans. Commun. 48, 1755\u0026ndash;1764 (2000). https://doi.org/10.1109/26.871400\u003c/li\u003e\n\u003cli\u003eHe, H., Jin, S., Wen, C.-K., Gao, F., Li, G. 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Optics Express (2025, submitted). [Self-citation]\u003c/li\u003e\n\u003cli\u003eIEEE Std 802.11ax-2021. IEEE Standard for Information Technology \u0026mdash; Part 11: WLAN Medium Access Control and Physical Layer Specifications \u0026mdash; Amendment 1: Enhancements for High-Efficiency WLAN. IEEE, New York (2021). https://doi.org/10.1109/IEEESTD.2021.9442429\u003c/li\u003e\n\u003cli\u003eErceg, V. et al. TGn Channel Models. IEEE 802.11-03/940r4, IEEE, Piscataway (2004). [TGax extensions follow the same methodology]\u003c/li\u003e\n\u003cli\u003eKermoal, J. P., Schumacher, L., Pedersen, K. I., Mogensen, P. E. \u0026amp; Frederiksen, F. A stochastic MIMO radio channel model with experimental validation. IEEE J. Sel. Areas Commun. 20, 1211\u0026ndash;1226 (2002). https://doi.org/10.1109/JSAC.2002.801223\u003c/li\u003e\n\u003cli\u003eSchenk, T. 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B 58, 267\u0026ndash;288 (1996). https://doi.org/10.1111/j.2517-6161.1996.tb02080.x\u003c/li\u003e\n\u003cli\u003eOrinion, G. A., Shiu, D.-S., Foschini, G. J. \u0026amp; Gans, M. J. Fading correlation and its effect on the capacity of multi-element antenna systems. IEEE Trans. Commun. 48, 502\u0026ndash;513 (2000). https://doi.org/10.1109/26.837052\u003c/li\u003e\n\u003cli\u003eHolma, H. \u0026amp; Toskala, A. (eds) LTE for UMTS: Evolution to LTE-Advanced 2nd edn (Wiley, Chichester, 2011). https://doi.org/10.1002/9781119992806\u003c/li\u003e\n\u003cli\u003eBjornson, E., Hoydis, J. \u0026amp; Sanguinetti, L. Massive MIMO networks: spectral, energy, and hardware efficiency. Found. Trends Signal Process. 11, 154\u0026ndash;655 (2017). https://doi.org/10.1561/2000000093\u003c/li\u003e\n\u003cli\u003eStuder, C., Goldsmith, A. \u0026amp; Marivath, P. Quantized neural networks: training neural networks with low precision weights and activations. Preprint at arXiv:1609.07061 (2016).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"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":"npj-wireless-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [npj Wireless Technology](https://www.nature.com/npjwireltech)","snPcode":"44459","submissionUrl":"https://submission.springernature.com/new-submission/44459/3?","title":"npj Wireless Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"WiFi 6/7, IEEE 802.11ax, power amplifier nonlinearity, Rapp model, K-R receiver, residual learning, closed-form training, coverage edge, MIMO, energy efficiency","lastPublishedDoi":"10.21203/rs.3.rs-9192715/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9192715/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eModern WiFi systems operating under IEEE 802.11ax (WiFi 6) and 802.11be (WiFi 7) suffer from hardware nonlinearities \u0026mdash; including power amplifier saturation, ADC quantisation, and OFDM clipping \u0026mdash; that conventional linear receivers treat as white Gaussian noise, leaving structured, learnable residuals uncorrected. We propose the K-R receiver, a physics-guided two-step framework that combines model-based MMSE equalisation with a closed-form nonlinear residual corrector trained per packet using existing preamble pilots, requiring no offline dataset and no backpropagation. The 12-feature residual corrector includes a cubic amplitude term physically grounded in the standard Rapp PA model, enabling the framework to exploit model-consistent nonlinear structure rather than learning arbitrary corrections. Under IEEE 802.11ax-compliant simulations across 12 independent scenarios, the proposed receiver achieves consistent spectral efficiency gains of approximately\u0026thinsp;+\u0026thinsp;0.40\u0026ndash;0.43 bps/Hz over MMSE, remains stable under hardware mismatch conditions where deep-learning baselines degrade, and achieves over 60\u0026times; higher energy efficiency than offline-trained neural receivers. An analytical MSE decomposition reveals that the channel-error correction gain dominates the quantisation gain at low SNR \u0026mdash; precisely at the coverage edge, where improvements matter most. These results suggest that physics-guided online learning is a practical and principled alternative to offline-trained neural receivers for real-world WiFi deployment.\u003c/p\u003e","manuscriptTitle":"A Physics-Guided Receiver for WiFi that Learns Hardware Nonlinearities without Training Data","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-25 11:26:30","doi":"10.21203/rs.3.rs-9192715/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2026-05-15T03:46:37+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"154401530137057697942877066300077644084","date":"2026-05-01T21:54:52+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"287028902296675214055570376031889009322","date":"2026-04-22T14:23:29+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"55484566346290546280978813402728622931","date":"2026-04-17T15:09:11+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-07T09:52:18+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-25T19:53:20+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-25T12:10:56+00:00","index":"","fulltext":""},{"type":"submitted","content":"npj Wireless Technology","date":"2026-03-22T16:46:20+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"npj-wireless-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [npj Wireless Technology](https://www.nature.com/npjwireltech)","snPcode":"44459","submissionUrl":"https://submission.springernature.com/new-submission/44459/3?","title":"npj Wireless Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"dafb1023-9678-48dc-92fe-bcfd0983ee2d","owner":[],"postedDate":"March 25th, 2026","published":true,"recentEditorialEvents":[{"type":"editorInvitedReview","content":"","date":"2026-05-15T03:46:37+00:00","index":26,"fulltext":""},{"type":"reviewerAgreed","content":"154401530137057697942877066300077644084","date":"2026-05-01T21:54:52+00:00","index":23,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":65035444,"name":"Physical sciences/Engineering"},{"id":65035445,"name":"Physical sciences/Mathematics and computing"},{"id":65035446,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-04-07T10:10:47+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-25 11:26:30","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9192715","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9192715","identity":"rs-9192715","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}