Neural Network-Assisted CLEAN for Channel Modeling in Low-SNR Regimes

# Neural Network-Assisted CLEAN for Channel Modeling in Low-SNR Regimes
Source: [https://arxiv.org/html/2607.27450](https://arxiv.org/html/2607.27450)
###### Abstract

Accurate multipath parameter estimation is critical for modern wireless communication systems, particularly in challenging low\-SNR environments\. Traditional Maximum Likelihood Estimation algorithms, such as CLEAN, provide high\-resolution parameter extraction but suffer from prohibitive computational complexity due to exhaustive grid search\. Conversely, purely data\-driven deep learning approaches lack physical grounding and struggle to generalize across variable multipath densities and off\-grid parameters\. To address these limitations, this paper proposes Neural Network\-Assisted CLEAN \(NN\-CLEAN\), a hybrid framework that embeds a multi\-head residual network directly into the iterative CLEAN extraction loop\. By replacing the exhaustive grid search with rapid, parallelizable forward passes while delegating residual subtraction to exact mathematical models, NN\-CLEAN isolates physical multipath parameters without accumulating non\-physical errors\. Extensive Monte Carlo simulations demonstrate that NN\-CLEAN achieves estimation accuracy exceeding9696% at55dB SNR, matching the traditional Grid\-Search CLEAN \(GS\-CLEAN\) baseline, while providing a massive reduction in computational complexity and substantially outperforming subspace methods and standalone one\-shot neural networks\. Crucially, NN\-CLEAN exhibits a near\-flat scaling in execution runtime and memory consumption as batch sizes increase\. This highly efficient parallelization establishes NN\-CLEAN as a robust, real\-time solution for channel estimation in MIMO systems\.

## IIntroduction

Accurate channel sounding and Direction of Arrival \(DOA\) estimation are fundamental to modern wireless systems, particularly for emerging applications such as Integrated Sensing and Communication \(ISAC\) and advanced spatial multiplexing\. While these technologies offer transformative potential, they face severe practical constraints in complex, dynamic electromagnetic environments where signals are heavily influenced by severe noise and multipath interference\. Consequently, the practical utility of these systems hinges on ensuring robust parameter extraction, specifically within the low Signal\-to\-Noise Ratio \(SNR\) regime\[[8](https://arxiv.org/html/2607.27450#bib.bib17),[13](https://arxiv.org/html/2607.27450#bib.bib18)\]\.

Conventional high\-resolution parameter estimation relies heavily on model\-driven signal processing algorithms\. Subspace\-based methods, such as MUSIC\[[14](https://arxiv.org/html/2607.27450#bib.bib7),[17](https://arxiv.org/html/2607.27450#bib.bib8)\]and ESPRIT\[[12](https://arxiv.org/html/2607.27450#bib.bib5),[18](https://arxiv.org/html/2607.27450#bib.bib6)\], offer high accuracy but are fundamentally limited by the requirement for uncorrelated and non\-coherent sources\[[6](https://arxiv.org/html/2607.27450#bib.bib19)\]\. Furthermore, their performance degrades sharply in low\-SNR environments due to the subspace swap phenomenon, where noise eigenvectors become indistinguishable from signal eigenvectors, leading to catastrophic estimation failures\[[19](https://arxiv.org/html/2607.27450#bib.bib20)\]\.

In contrast, Maximum Likelihood Estimation \(MLE\) methods, such as CLEAN\[[7](https://arxiv.org/html/2607.27450#bib.bib3),[16](https://arxiv.org/html/2607.27450#bib.bib1)\], bypass these coherence restrictions and offer improved robustness against source correlation\[[22](https://arxiv.org/html/2607.27450#bib.bib21)\]\. However, these algorithms typically require exhaustive grid searches across the spatial spectrum, leading to exorbitant computational overhead and memory usage that fail to scale efficiently\[[11](https://arxiv.org/html/2607.27450#bib.bib22)\]\. Moreover, even MLE methods struggle under low\-SNR conditions, as heavy noise increases the probability of the algorithm converging to local optima or dominant sidelobes rather than the true global maximum\[[1](https://arxiv.org/html/2607.27450#bib.bib23)\]\.

Recent studies have increasingly applied Deep Learning \(DL\) to address these computational bottlenecks by treating DOA estimation as a data\-driven regression or classification task\[[9](https://arxiv.org/html/2607.27450#bib.bib10),[5](https://arxiv.org/html/2607.27450#bib.bib11),[20](https://arxiv.org/html/2607.27450#bib.bib12),[3](https://arxiv.org/html/2607.27450#bib.bib13),[23](https://arxiv.org/html/2607.27450#bib.bib24),[21](https://arxiv.org/html/2607.27450#bib.bib25)\]\. While pure DL models offer rapid inference, their reliability is inherently constrained by their training assumptions\. Most existing architectures simplify the estimation problem by assuming incoming signals align perfectly with a predefined discrete spatial grid\. Consequently, when deployed in continuous real\-world environments, they suffer from severe performance degradation due togrid mismatch, where off\-grid signals lead to significant estimation bias and spectral leakage\[[23](https://arxiv.org/html/2607.27450#bib.bib24),[10](https://arxiv.org/html/2607.27450#bib.bib2)\]\. Furthermore, these one\-shot architectures struggle in dynamic environments where the number of multipath components \(MPCs\), denoted asNMPCN\_\{\\text\{MPC\}\}, fluctuates\. Because their output layers have fixed dimensions, they cannot naturally adapt to variable MPC densities\[[21](https://arxiv.org/html/2607.27450#bib.bib25)\]—a fundamentalOut\-of\-Distribution \(OOD\)failure\. Without a mathematically rigorous physical model to anchor their predictions, purely data\-driven approaches remain highly sensitive to noise\. This lack of physical grounding frequently results in the identification of spurious signal peaks in low\-SNR regimes or a complete failure to resolve weak MPCs that deviate from the training distribution\.

These limitations underscore a fundamental gap: traditional mathematical estimators suffer from prohibitive computational costs and susceptibility to noise\-induced local optima, while purely data\-driven DL methods lack the physical robustness required for OOD scenarios, such as variableNMPCN\_\{\\text\{MPC\}\}and continuous off\-grid parameters\. To bridge this gap, we propose Neural Network\-Assisted CLEAN \(NN\-CLEAN\), a hybrid framework that modernizes the traditional CLEAN algorithm by embedding a highly efficient residual network directly into its iterative extraction loop\. Rather than predicting all MPCs simultaneously, NN\-CLEAN utilizes a rapid, parallelizable forward pass solely to isolate the spatial parameters of the dominant path at each iteration\. Crucially, it explicitly addresses the physical limitations of pure DL architectures by delegating the subsequent complex path gain calculation and residual subtraction to exact mathematical models governed by theoretical array manifolds and wave propagation physics\. Because the extracted signal is evaluated and subtracted using true physical constraints, the iterative loop prevents the network from hallucinating spurious peaks or accumulating non\-physical errors\. By combining the pattern\-recognition speed of deep learning with the mathematically grounded structure of MLE, NN\-CLEAN ensures robust generalization to off\-grid MPCs and variableNMPCN\_\{\\text\{MPC\}\}\. This physically anchored framework provides a highly scalable solution that outperforms both traditional subspace methods and standalone one\-shot neural networks\.

The main contributions of this paper are summarized as follows:

1. 1\.NN\-Assisted MLE Framework for Low\-SNR Robustness:By leveraging a NN to guide the iterative parameter extraction, the proposed framework successfully bypasses the spurious sidelobe artifacts and local optima that frequently trap exhaustive grid\-search algorithms\. Furthermore, by delegating the residual subtraction to exact mathematical models, NN\-CLEAN curtails the error propagation inherent to pure DL methods, yielding better estimation accuracy in challenging low\-SNR regimes\.
2. 2\.High Scalability and Fast Inference:NN\-CLEAN resolves the severe computational bottlenecks inherent to multi\-dimensional grid searches\. By isolating spatial parameters through highly parallelizable forward passes, the framework maintains a remarkably flat computational runtime and memory footprint even as batch sizes scale\. This amortized efficiency establishes NN\-CLEAN as an ideal, real\-time solution for demanding, large\-scale deployments such as ISAC and centralized baseband processing\.
3. 3\.Robust Generalization to OOD and Off\-Grid Scenarios:Unlike standalone one\-shot DL models whose reliability is strictly bounded by their training assumptions, NN\-CLEAN explicitly generalizes to Out\-of\-Distribution scenarios\. Because the NN operates locally within a dynamic, iterative loop, the framework seamlessly resolves an unknown and variable number of multipath components \(NMPCN\_\{\\text\{MPC\}\}\), maintaining high\-fidelity estimation even for continuous, strictly off\-grid spatial parameters\.

![Refer to caption](https://arxiv.org/html/2607.27450v1/x1.png)Figure 1:Channel Model Visualization
## IISystem Description and CLEAN Algorithm

### II\-ASystem Model

Consider a quasi\-static double\-directional radio channel between transmitting \(Tx\) and receiving \(Rx\) Uniform Linear Arrays \(ULAs\), with the propagation geometry modeled as in Fig\.[1](https://arxiv.org/html/2607.27450#S1.F1)\. The channel comprises a line\-of\-sight \(LOS\) path and multiple scattered paths, each forming a distinct MPC\. The channel transfer function matrix𝑯​\(fk\)\\bm\{H\}\(f\_\{k\}\), evaluated at frequencyfkf\_\{k\}, is modeled as the superposition of these discrete MPCs\.

Based on the framework in\[[7](https://arxiv.org/html/2607.27450#bib.bib3)\], letbT​\(ϕ\)b\_\{T\}\(\\phi\)andbR​\(ϕ\)b\_\{R\}\(\\phi\)represent theNT×1N\_\{T\}\\times 1and1×NR1\\times N\_\{R\}calibrated, vertically polarized beam patterns of the transmit and receive arrays, respectively, for a given azimuth angleϕ\\phi\. TheNT×NRN\_\{T\}\\times N\_\{R\}channel matrix𝑯​\(fk\)\\bm\{H\}\(f\_\{k\}\)is formulated as:

𝑯​\(fk\)=∑l=1NMPCαl​bT​\(ϕT,l\)​bR​\(ϕR,l\)​exp⁡\(−j​2​π​fk​dlc\)\+𝑵​\(fk\),\\bm\{H\}\(f\_\{k\}\)=\\sum\_\{l=1\}^\{N\_\{\\text\{MPC\}\}\}\\alpha\_\{l\}b\_\{T\}\(\\phi\_\{T,l\}\)b\_\{R\}\(\\phi\_\{R,l\}\)\\exp\\left\(\-j2\\pi f\_\{k\}\\frac\{d\_\{l\}\}\{c\}\\right\)\+\\bm\{N\}\(f\_\{k\}\),\(1\)whereϕT,l\\phi\_\{T,l\}andϕR,l\\phi\_\{R,l\}denote the Azimuth of Departure \(AoD\) and Azimuth of Arrival \(AoA\) for thelthl^\{\\text\{th\}\}MPC, respectively\. The variablesdld\_\{l\}andαl\\alpha\_\{l\}represent the physical travel distance and complex path gain of thelthl^\{\\text\{th\}\}path, whileccis the speed of light\. Finally,𝑵​\(fk\)\\bm\{N\}\(f\_\{k\}\)denotes the additive white Gaussian noise \(AWGN\) matrix\.

### II\-BMPC Estimation based on the CLEAN Algorithm

Jointly estimating the parameters of all MPCs simultaneously requires a multi\-dimensional estimation over a discrete parameter space, which is computationally intractable\. To avoid this complexity, the CLEAN algorithm\[[4](https://arxiv.org/html/2607.27450#bib.bib4)\]is employed as an iterative greedy matching pursuit technique that decouples the problem by extracting one dominant MPC at a time\.

During theithi^\{\\text\{th\}\}iteration, CLEAN isolates the dominant MPC through a discrete search over a three\-dimensional parameter spaceΘ=ΘAoA×ΘAoD×Θdist\\Theta=\\Theta\_\{\\text\{AoA\}\}\\times\\Theta\_\{\\text\{AoD\}\}\\times\\Theta\_\{\\text\{dist\}\}\. This finite grid is constructed by quantizing the azimuth dimensions intoNaN\_\{a\}angular bins and the propagation distance intoNdN\_\{d\}distance bins\. The optimal parameter tuple\{d^i,ϕ^T,i,ϕ^R,i\}∈Θ^\\\{\\hat\{d\}\_\{i\},\\hat\{\\phi\}\_\{T,i\},\\hat\{\\phi\}\_\{R,i\}\\\}\\in\\hat\{\\Theta\}is then extracted by maximizing the squared spatial\-distance correlation:

\{d^i,ϕ^T,i,ϕ^R,i\}\\displaystyle\\\{\\hat\{d\}\_\{i\},\\hat\{\\phi\}\_\{T,i\},\\hat\{\\phi\}\_\{R,i\}\\\}=arg⁡maxd,ϕT,ϕR\|∑k=1Nfexp⁡\(−j​2​π​fk​dc\)\\displaystyle=\\arg\\max\_\{d,\\phi\_\{T\},\\phi\_\{R\}\}\\left\|\\sum\_\{k=1\}^\{N\_\{f\}\}\\exp\\left\(\-j2\\pi f\_\{k\}\\frac\{d\}\{c\}\\right\)\\right\.\(2\)×bR\(ϕR\)\(𝑯rem\(i\)\(fk\)\)†bT\(ϕT\)\|2,\\displaystyle\\quad\\times\\left\.b\_\{R\}\(\\phi\_\{R\}\)\\left\(\\bm\{H\}\_\{\\text\{rem\}\}^\{\(i\)\}\(f\_\{k\}\)\\right\)^\{\\dagger\}b\_\{T\}\(\\phi\_\{T\}\)\\right\|^\{2\},where𝑯rem\(i\)​\(fk\)\\bm\{H\}\_\{\\text\{rem\}\}^\{\(i\)\}\(f\_\{k\}\)denotes the residual channel transfer matrix at iterationii, initialized as𝑯rem\(1\)​\(fk\)=𝑯​\(fk\)\\bm\{H\}\_\{\\text\{rem\}\}^\{\(1\)\}\(f\_\{k\}\)=\\bm\{H\}\(f\_\{k\}\)\.

Once identified, the complex path gainα^i\\hat\{\\alpha\}\_\{i\}of this dominant MPC is computed via a least squares projection onto the residual channel:

α^i=∑k=1Nfexp⁡\(j​2​π​fk​d^ic\)​bT†​\(ϕ^T,i\)​𝑯rem\(i\)​\(fk\)​bR†​\(ϕ^R,i\)Nf​‖bT​\(ϕ^T,i\)‖2​‖bR​\(ϕ^R,i\)‖2\.\\hat\{\\alpha\}\_\{i\}=\\frac\{\\sum\_\{k=1\}^\{N\_\{f\}\}\\exp\\left\(j2\\pi f\_\{k\}\\frac\{\\hat\{d\}\_\{i\}\}\{c\}\\right\)b\_\{T\}^\{\\dagger\}\(\\hat\{\\phi\}\_\{T,i\}\)\\bm\{H\}\_\{\\text\{rem\}\}^\{\(i\)\}\(f\_\{k\}\)b\_\{R\}^\{\\dagger\}\(\\hat\{\\phi\}\_\{R,i\}\)\}\{N\_\{f\}\\left\\\|b\_\{T\}\(\\hat\{\\phi\}\_\{T,i\}\)\\right\\\|^\{2\}\\left\\\|b\_\{R\}\(\\hat\{\\phi\}\_\{R,i\}\)\\right\\\|^\{2\}\}\.\(3\)
The contribution of this dominant MPC is then synthesized and subtracted from the current residual transfer function to form the updated residual for the next iteration:

𝑯rem\(i\+1\)​\(fk\)\\displaystyle\\bm\{H\}\_\{\\text\{rem\}\}^\{\(i\+1\)\}\(f\_\{k\}\)=𝑯rem\(i\)​\(fk\)\\displaystyle=\\bm\{H\}\_\{\\text\{rem\}\}^\{\(i\)\}\(f\_\{k\}\)\(4\)−α^i​bT​\(ϕ^T,i\)​bR​\(ϕ^R,i\)​exp⁡\(−j​2​π​fk​d^ic\)\.\\displaystyle\\quad\-\\hat\{\\alpha\}\_\{i\}b\_\{T\}\(\\hat\{\\phi\}\_\{T,i\}\)b\_\{R\}\(\\hat\{\\phi\}\_\{R,i\}\)\\exp\\left\(\-j2\\pi f\_\{k\}\\frac\{\\hat\{d\}\_\{i\}\}\{c\}\\right\)\.
This iterative extraction and subtraction process repeats until the absolute path gain of the newly extracted MPC falls below a pre\-determined energy threshold or a maximum number of paths is reached\.

Algorithm Bottlenecks:The computational efficacy of traditional Grid\-Search CLEAN \(GS\-CLEAN\) is fundamentally constrained by its reliance on exhaustive parameter evaluation\. For each candidate grid point, evaluating the spatial\-distance correlation in \([2](https://arxiv.org/html/2607.27450#S2.E2)\) requires𝒪​\(Nf​NT​NR\)\\mathcal\{O\}\(N\_\{f\}N\_\{T\}N\_\{R\}\)operations\. Quantizing our three\-parameter search space \(d,ϕT,ϕRd,\\phi\_\{T\},\\phi\_\{R\}\) intoGGbins per dimension inflates the per\-iteration complexity to𝒪​\(G3​Nf​NT​NR\)\\mathcal\{O\}\(G^\{3\}N\_\{f\}N\_\{T\}N\_\{R\}\), while incurring substantial memory overhead for precomputed array manifold dictionaries\. This burden is further compounded by the curse of dimensionality; extending the framework to fully 3D spatial environments \(incorporating elevation angles\) or high\-mobility scenarios \(including Doppler shifts\) expands the search domain to five or more dimensions, driving complexity to an intractable𝒪​\(G5​Nf​NT​NR\)\\mathcal\{O\}\(G^\{5\}N\_\{f\}N\_\{T\}N\_\{R\}\)\. This cubic\-to\-quintic scaling renders exhaustive grid\-search paradigms fundamentally impractical for real\-time, low\-latency applications, thereby necessitating a shift toward efficient, learning\-based estimation frameworks\.

## IIIProposed NN\-CLEAN Algorithm

To overcome the severe computational bottlenecks of multi\-dimensional grid searches, we propose NN\-CLEAN\. As illustrated in Fig\.[2](https://arxiv.org/html/2607.27450#S3.F2), this hybrid framework embeds a NN directly into the CLEAN iterative extraction loop, replacing exhaustive parameter evaluations with a rapid, direct inference\. During theithi^\{\\text\{th\}\}iteration, the complex residual channel matrix𝑯rem\(i\)​\(fk\)\\bm\{H\}\_\{\\text\{rem\}\}^\{\(i\)\}\(f\_\{k\}\)is decoupled into its real and imaginary components to form the input tensor\. A single forward pass maps this observation directly to the dominant MPC, yielding the parameter tupleΘ^i=\{d^i,ϕ^T,i,ϕ^R,i\}\\hat\{\\Theta\}\_\{i\}=\\\{\\hat\{d\}\_\{i\},\\hat\{\\phi\}\_\{T,i\},\\hat\{\\phi\}\_\{R,i\}\\\}\. This streamlined approach maintains high\-fidelity extraction while incurring only a fraction of the traditional memory and computational overhead\.

![Refer to caption](https://arxiv.org/html/2607.27450v1/pictures/NN_clean_overview.jpg)Figure 2:NN\-assisted CLEAN System OverviewAs depicted in Fig\.[3](https://arxiv.org/html/2607.27450#S3.F3), the NN is structured as a multi\-head residual classification architecture that evaluates the channel against pre\-defined parameter grids\. A cascaded backbone of MaxPooling and Residual Network \(ResNet\) efficiently compresses the frequency domain while preserving critical spatial\-frequency features\. The extracted latent features are then routed to independent fully connected \(FC\) heads, which output raw logits representing the spatial probability distributions for the AoA and AoD\. Because the physical distance of a multipath component is inherently coupled with its spatial trajectory, an FC gating mechanism explicitly conditions the final distance head on these resolved angular probabilities to enforce structural consistency\. Finally, anarg⁡max\\arg\\maxoperation is applied to the output logits to select the discrete grid indices with the highest probabilities, mapping them directly to the estimated physical parameters ofΘ^i\\hat\{\\Theta\}\_\{i\}\.

![Refer to caption](https://arxiv.org/html/2607.27450v1/pictures/NN_structure.png)Figure 3:NN Structure Overview
## IVSimulation Results

### IV\-ASimulation Setup

#### IV\-A1Channel Model Setup

The proposed system is implemented and evaluated with a synthesized4×44\\times 4MIMO communication system operating acrossNf=512N\_\{f\}=512uniformly spaced subcarriers in the22–33GHz range\.

Both the TX and RX employ ULAs with half\-wavelength antenna spacing evaluated at a center frequency offc=2\.5f\_\{c\}=2\.5GHz\. To isolate algorithmic performance from hardware imperfections while aligning with the co\-polarized system model, we assume ideal vertically polarized omnidirectional elements \(e\.g\., ideal dipoles\) with negligible mutual coupling\. Consequently, the array’s amplitude response is uniform across the azimuth plane, and its spatial response is governed entirely by frequency\-dependent geometric phase differences\. For anNN\-element array, the theoretical far\-field steering vector at spatial angleϕ\\phiand subcarrierfkf\_\{k\}is defined as:

b​\(ϕ,fk\)=1N​\[1,e−j​π​fkfc​sin⁡\(ϕ\),…,e−j​π​\(N−1\)​fkfc​sin⁡\(ϕ\)\]T\.b\(\\phi,f\_\{k\}\)=\\frac\{1\}\{\\sqrt\{N\}\}\\left\[1,e^\{\-j\\pi\\frac\{f\_\{k\}\}\{f\_\{c\}\}\\sin\(\\phi\)\},\\dots,e^\{\-j\\pi\(N\-1\)\\frac\{f\_\{k\}\}\{f\_\{c\}\}\\sin\(\\phi\)\}\\right\]^\{T\}\.\(5\)
For the grid\-based evaluations and the generation of synthetic data, the parameter spaceΘ\\Thetais discretized as follows:

- •AoD \(ϕT\\phi\_\{T\}\) and AoA \(ϕR\\phi\_\{R\}\):Ranging from−90∘\-90^\{\\circ\}to90∘90^\{\\circ\}with a1∘1^\{\\circ\}resolution \(Na=181N\_\{a\}=181bins\)\.
- •Distance \(dd\):Ranging from55m to1515m with a0\.050\.05m resolution \(Nd=201N\_\{d\}=201bins\)\.

#### IV\-A2Training Setup

The training procedure for the proposed NN\-assisted CLEAN framework is detailed in Algorithm[1](https://arxiv.org/html/2607.27450#alg1)\. The network is optimized using the Adam optimizer \(learning rate10−310^\{\-3\}, batch size128128\)\. To ensure robust generalization, training samples are dynamically synthesized on the fly\. In each training step, a channel comprisingNMPC=4N\_\{\\text\{MPC\}\}=4MPCs is generated, with ground\-truth parameters \(ϕR\\phi\_\{R\},ϕT\\phi\_\{T\},dd\) drawn strictly from their respectiveon\-griddiscrete sets\. The complex path gains undergo simulated Rayleigh fading \(α∼𝒞​𝒩​\(0,2\)\\alpha\\sim\\mathcal\{CN\}\(0,2\)\), and additive white Gaussian noise \(AWGN\) is injected at anSNR∈\[−5,10\]\\text\{SNR\}\\in\[\-5,10\]dB\.

To capture the topological correlation between adjacent discrete bins, we apply Gaussian Label Smoothing \(GLS\)\[[15](https://arxiv.org/html/2607.27450#bib.bib26)\]to the ground\-truth labels\. For a true on\-grid parameter indexμ\\mu, the smoothed target probabilitytjt\_\{j\}at grid indexjjis formulated as a normalized Gaussian distribution:

tj=exp⁡\(−12​\(j−μσ\)2\)∑kexp⁡\(−12​\(k−μσ\)2\),t\_\{j\}=\\frac\{\\exp\\left\(\-\\frac\{1\}\{2\}\\left\(\\frac\{j\-\\mu\}\{\\sigma\}\\right\)^\{2\}\\right\)\}\{\\sum\_\{k\}\\exp\\left\(\-\\frac\{1\}\{2\}\\left\(\\frac\{k\-\\mu\}\{\\sigma\}\\right\)^\{2\}\\right\)\},\(6\)where the spread parameter is empirically set toσ=2\.0\\sigma=2\.0\.

Because the network extracts multipath components sequentially, a distinct loss is computed at each extraction iterationii\. The iteration loss is defined as the weighted Categorical Cross\-Entropy \(CCE\) between the network’s logits \(𝒚^\(i\)\\hat\{\\bm\{y\}\}^\{\(i\)\}\) and the GLS targets \(𝒕\(i\)\\bm\{t\}^\{\(i\)\}\):

ℒiter\(i\)=CCE​\(𝒕ϕR\(i\),𝒚^ϕR\(i\)\)\+CCE​\(𝒕ϕT\(i\),𝒚^ϕT\(i\)\)\+3⋅CCE​\(𝒕d\(i\),𝒚^d\(i\)\),\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(i\)\}=\\text\{CCE\}\\left\(\\bm\{t\}\_\{\\phi\_\{R\}\}^\{\(i\)\},\\hat\{\\bm\{y\}\}\_\{\\phi\_\{R\}\}^\{\(i\)\}\\right\)\+\\text\{CCE\}\\left\(\\bm\{t\}\_\{\\phi\_\{T\}\}^\{\(i\)\},\\hat\{\\bm\{y\}\}\_\{\\phi\_\{T\}\}^\{\(i\)\}\\right\)\+3\\cdot\\text\{CCE\}\\left\(\\bm\{t\}\_\{d\}^\{\(i\)\},\\hat\{\\bm\{y\}\}\_\{d\}^\{\(i\)\}\\right\),\(7\)A weight of33is applied exclusively to the distance loss\. Because precise distance estimation is critical for achieving accurate phase alignment during residual cancellation, this penalty ensures the network prioritizes resolving the distance dimension\.

Algorithm 1Training Procedures for NN\-assisted CLEAN1:Initialize:Model weights, Adam optimizer \(

lr=10−3\\text\{lr\}=10^\{\-3\}\)

2:foreach training step

t=1t=1to

1,000,0001,000,000do

3:Sample

SNR∼𝒰​\(−5,10\)\\text\{SNR\}\\sim\\mathcal\{U\}\(\-5,10\)dB

4:Generate

NMPC=4N\_\{\\text\{MPC\}\}=4MPCs:

\{ϕR,l,ϕT,l,dl\}l=1NMPC∈Θ\\\{\\phi\_\{R,l\},\\phi\_\{T,l\},d\_\{l\}\\\}\_\{l=1\}^\{N\_\{\\text\{MPC\}\}\}\\in\\Theta
5:Sample complex path gains

α∼𝒞​𝒩​\(0,2\)\\alpha\\sim\\mathcal\{CN\}\(0,2\)
6:Synthesize channel

𝑯\\bm\{H\}and add AWGN

7:Generate GLS targets

𝒕ϕR\(i\),𝒕ϕT\(i\),𝒕d\(i\)\\bm\{t\}\_\{\\phi\_\{R\}\}^\{\(i\)\},\\bm\{t\}\_\{\\phi\_\{T\}\}^\{\(i\)\},\\bm\{t\}\_\{d\}^\{\(i\)\}using \([6](https://arxiv.org/html/2607.27450#S4.E6)\)

8:if

t≤500,000t\\leq 500,000then

9:// Phase 1 \(0–50%\): Dominant path detection

10:Extract 1st path only; compute

ℒfinal=ℒiter\(1\)\\mathcal\{L\}\_\{\\text\{final\}\}=\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(1\)\}
11:elseif

t≤750,000t\\leq 750,000then

12:// Phase 2 \(50–75%\): Teacher forcing

13:Extract

NMPCN\_\{\\text\{MPC\}\}paths iteratively usingground\-truthsubtraction;

14:Compute

ℒfinal=1NMPC​∑i=1NMPCℒiter\(i\)\\mathcal\{L\}\_\{\\text\{final\}\}=\\frac\{1\}\{N\_\{\\text\{MPC\}\}\}\\sum\_\{i=1\}^\{N\_\{\\text\{MPC\}\}\}\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(i\)\}
15:else

16:// Phase 3 \(75–100%\): Autoregressive inference

17:Extract

NMPCN\_\{\\text\{MPC\}\}paths iteratively usingpredictedsubtraction;

18:Compute

ℒfinal=1NMPC​∑i=1NMPCℒiter\(i\)\\mathcal\{L\}\_\{\\text\{final\}\}=\\frac\{1\}\{N\_\{\\text\{MPC\}\}\}\\sum\_\{i=1\}^\{N\_\{\\text\{MPC\}\}\}\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(i\)\}
19:endif

20:Backpropagate

ℒfinal\\mathcal\{L\}\_\{\\text\{final\}\}and update model weights

21:endfor

Finally, to mitigate error propagation inherent in iterative subtraction, the network is trained over1,000,0001,000,000steps using a three\-phase curriculum\. The final backpropagated loss,ℒfinal\\mathcal\{L\}\_\{\\text\{final\}\}, adapts dynamically:

- •Phase 1 \(0%–50%\):Focuses solely on dominant path detection \(ℒfinal=ℒiter\(1\)\\mathcal\{L\}\_\{\\text\{final\}\}=\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(1\)\}\), learning fundamental feature representations without subtraction noise\.
- •Phase 2 \(50%–75%\):Extracts allNMPCN\_\{\\text\{MPC\}\}paths iteratively\. Perfect residual subtraction \(teacher forcing\) is employed using ground\-truth parameters to compute the subtracted interference, yielding an averaged lossℒfinal=1NMPC​∑i=1NMPCℒiter\(i\)\\mathcal\{L\}\_\{\\text\{final\}\}=\\frac\{1\}\{N\_\{\\text\{MPC\}\}\}\\sum\_\{i=1\}^\{N\_\{\\text\{MPC\}\}\}\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(i\)\}\.
- •Phase 3 \(75%–100%\):Full autoregressive inference\. The network subtracts its own predicted components to learn resilience against accumulated errors, utilizing the same averaged lossℒfinal=1NMPC​∑i=1NMPCℒiter\(i\)\\mathcal\{L\}\_\{\\text\{final\}\}=\\frac\{1\}\{N\_\{\\text\{MPC\}\}\}\\sum\_\{i=1\}^\{N\_\{\\text\{MPC\}\}\}\\mathcal\{L\}\_\{\\text\{iter\}\}^\{\(i\)\}\.

![Refer to caption](https://arxiv.org/html/2607.27450v1/pictures/accuracy_comparison_2.png)Figure 4:Estimation Accuracy Comparison Between Different Methods![Refer to caption](https://arxiv.org/html/2607.27450v1/x2.png)\(a\)Spatial MCDηang\\eta\_\{\\text\{ang\}\}
![Refer to caption](https://arxiv.org/html/2607.27450v1/pictures/dist.png)\(b\)Distance MCDηd\\eta\_\{\\text\{d\}\}
![Refer to caption](https://arxiv.org/html/2607.27450v1/x3.png)\(c\)Energy Captured Ratio

Figure 5:CDF comparisons for spatial MCD, distance MCD, and energy captured ratio across different SNR \(NMPC=2N\_\{\\text\{MPC\}\}=2\)\.![Refer to caption](https://arxiv.org/html/2607.27450v1/x4.png)Figure 6:Runtime and Memory Comparison Between NN\-CLEAN and GS\-CLEAN

### IV\-BResults and Analysis

The proposed NN\-CLEAN framework is evaluated via Monte Carlo simulations, comprising10,00010,000independent channel realizations per test configuration\. To rigorously assess generalization against real\-world propagation complexities, the testing data is synthesized entirelyoff\-grid\. Drawing the ground\-truth parameters \(ϕR,ϕT,d\\phi\_\{R\},\\phi\_\{T\},d\) from continuous uniform distributions—rather than discrete training bins—intentionally introduces realistic grid mismatch and spectral leakage to challenge the framework’s robustness\.

Under these stringent conditions, Fig\.[4](https://arxiv.org/html/2607.27450#S4.F4)evaluates the multipath resolvability of NN\-CLEAN against four established baselines: traditional GS\-CLEAN\[[7](https://arxiv.org/html/2607.27450#bib.bib3)\], 3D\-MUSIC\[[17](https://arxiv.org/html/2607.27450#bib.bib8)\], 3D\-ESPRIT\[[18](https://arxiv.org/html/2607.27450#bib.bib6)\], and a standalone NN one\-shot approach\. Performance is benchmarked across varying MPC densities \(NMPC∈\{1,2,3,4\}N\_\{\\text\{MPC\}\}\\in\\\{1,2,3,4\\\}\) at an SNR of 5 dB\. Due to the spatial subspace limitations inherent to the simulated antenna configuration, evaluations for the subspace methods are strictly capped atNMPC=3N\_\{\\text\{MPC\}\}=3\. Estimation accuracy is defined as the ratio of correctly identified MPCs to the total number of simulated paths\. Specifically, an extracted MPC is considered a correct match if its predicted AoA, AoD, and distance simultaneously fall within a±2\\pm 2grid index tolerance of the ground truth\. To explicitly isolate the efficacy of the proposed iterative framework, the one\-shot baseline utilizes the identical NN architecture as NN\-CLEAN but is trained to regress allNMPCN\_\{\\text\{MPC\}\}components simultaneously in a single forward pass, assuming ideal heuristic pairing\.

As shown in Fig\.[4](https://arxiv.org/html/2607.27450#S4.F4), the empirical results demonstrate that NN\-CLEAN achieves peak estimation accuracy in sparse multipath scenarios \(NMPC∈\{1,2\}N\_\{\\text\{MPC\}\}\\in\\\{1,2\\\}\), successfully bypassing the severe side\-lobe artifacts that frequently limit traditional subspace and grid\-search algorithms\. As multipath density increases, both CLEAN\-based approaches exhibit a linear decay in accuracy\. Although GS\-CLEAN degrades at a slightly slower rate—eventually surpassing NN\-CLEAN in dense environments \(NMPC∈\{3,4\}N\_\{\\text\{MPC\}\}\\in\\\{3,4\\\}\) due to cumulative error propagation within the NN’s sequential extraction loop—the proposed framework maintains a substantial performance margin over the subspace methods\. Crucially, NN\-CLEAN consistently outperforms the standalone one\-shot baseline across all tested densities\. The severe degradation of the one\-shot model explicitly highlights that embedding the NN within the iterative CLEAN loop is vital for mitigating accuracy loss asNMPCN\_\{\\text\{MPC\}\}grows\. Ultimately, because the dominant 1–2 paths capture the overwhelming majority of channel energy in practical deployments, NN\-CLEAN’s superior resolvability in these primary scenarios establishes it as a highly favorable framework for real\-time systems\.

Beyond resolving the existence of MPCs, it is critical to evaluate the absolute parameter precision of the identified paths\. Fig\.[5](https://arxiv.org/html/2607.27450#S4.F5)evaluates the estimation precision for aNMPC=2N\_\{\\text\{MPC\}\}=2scenario across a sweep ofSNR∈\[−5,10\]\\text\{SNR\}\\in\[\-5,10\]dB\. Performance is explicitly quantified via three metrics: the Cumulative Distribution Function \(CDF\) of the Root Mean Square Error \(RMSE\) for the angular Multipath Component Distance \(MCD\)\[[2](https://arxiv.org/html/2607.27450#bib.bib15)\]\(ηang\\eta\_\{\\text\{ang\}\}\), the distance MCD RMSE \(ηd\\eta\_\{d\}\), and the overall energy captured ratio \(ϵ\\epsilon\)\.

To prevent angular wrap\-around discontinuities, the angular MCD between a ground\-truth path and its estimate is formulated as:

ηang=12​\|ej​ϕR−ej​ϕ^R\|2\+\|ej​ϕT−ej​ϕ^T\|2\.\\eta\_\{\\text\{ang\}\}=\\frac\{1\}\{2\}\\sqrt\{\\left\|e^\{j\\phi\_\{R\}\}\-e^\{j\\hat\{\\phi\}\_\{R\}\}\\right\|^\{2\}\+\\left\|e^\{j\\phi\_\{T\}\}\-e^\{j\\hat\{\\phi\}\_\{T\}\}\\right\|^\{2\}\}\.\(8\)Correspondingly, the normalized distance MCD is defined as:

ηd=\(\|d−d^\|Δ​dmax⋅dstdΔ​dmax\)2\.\\eta\_\{d\}=\\sqrt\{\\left\(\\frac\{\|d\-\\hat\{d\}\|\}\{\\Delta d\_\{\\text\{max\}\}\}\\cdot\\frac\{d\_\{\\text\{std\}\}\}\{\\Delta d\_\{\\text\{max\}\}\}\\right\)^\{2\}\}\.\(9\)The overall angular and distance RMSEs are then computed from these individual path errors across all successfully matched components\. Finally, the fidelity of the overall channel reconstruction is measured by the energy captured ratioϵ=‖∑i=1NMPC𝑯^i‖2/‖𝑯‖2\\epsilon=\\left\\\|\\sum\_\{i=1\}^\{N\_\{\\text\{MPC\}\}\}\\hat\{\\bm\{H\}\}\_\{i\}\\right\\\|^\{2\}\\big/\\\|\\bm\{H\}\\\|^\{2\}, where𝑯^i\\hat\{\\bm\{H\}\}\_\{i\}represents the channel matrix reconstructed from theii\-th estimated MPC\.

As depicted in Fig\.[5](https://arxiv.org/html/2607.27450#S4.F5), the empirical CDFs for the angular MCD \(Fig\.[5](https://arxiv.org/html/2607.27450#S4.F5)a\), distance MCD \(Fig\.[5](https://arxiv.org/html/2607.27450#S4.F5)b\), and energy captured ratio \(Fig\.[5](https://arxiv.org/html/2607.27450#S4.F5)c\) demonstrate a shared trend: estimation fidelity predictably degrades as the SNR decreases\. Specifically, the absolute error scales remain exceptionally small for both algorithms\. Although the exhaustive GS\-CLEAN method yields strictly superior precision across all evaluated metrics, the performance gap is remarkably narrow\. The curves for NN\-CLEAN closely align with the grid\-search baseline, maintaining nearly identical distributions despite the tight scale of the plots\. This confirms that while the NN sacrifices a marginal degree of precision due to the discretization of its output layer, it still provides highly accurate parameter estimations and reliable signal reconstruction\.

Ultimately, this marginal precision trade\-off is justified by a massive reduction in computational complexity\. To isolate the fundamental per\-iteration processing cost, Fig\. 6 evaluates execution runtime and memory consumption forNMPC=1N\_\{\\text\{MPC\}\}=1across a sweep of batch sizes\. Both NN\-CLEAN and GS\-CLEAN are GPU\-parallelized and evaluated on an NVIDIA GeForce RTX 4070 GPU with 12 GB of memory, with peak GPU memory recorded using TensorFlow statistics after warm\-up\. Here, the batch size denotes the number of independent channel realizations processed concurrently by a centralized processor, such as observations from multiple users or vehicles in an ISAC or multi\-user MIMO system\. The results reveal a stark divergence in scalability: at a minimal workload \(Batch=1\\text\{Batch\}=1\), both algorithms exhibit comparable execution times due to the fixed overhead of NN kernel initialization\. However, even at this baseline, NN\-CLEAN demonstrates superior architectural efficiency, consuming approximately half the memory of GS\-CLEAN\. As the batch size increases, the computational burden of GS\-CLEAN escalates linearly, driven by the repetitive requirement to allocate high\-resolution cost volumes and perform exhaustive 3D searches for every channel realization\. In contrast, NN\-CLEAN leverages the massive parallelization capabilities of the GPU to process multiple realizations in a single forward pass, maintaining a near\-flat trend for both runtime and memory footprint\. This significant amortized efficiency establishes NN\-CLEAN as a highly scalable solution for high\-throughput, real\-time channel estimation in MIMO systems\.

## VConclusion

In this paper, we introduced NN\-CLEAN, a novel hybrid framework that resolves the critical computational bottlenecks of the traditional CLEAN algorithm\. By embedding a multi\-head residual neural network directly into the iterative CLEAN extraction loop, the proposed method replaces exhaustive multi\-dimensional grid searches with highly efficient, parallelizable inference\. Evaluated under stringent off\-grid conditions across varying multipath component \(NMPCN\_\{\\text\{MPC\}\}\) counts and SNR regimes, NN\-CLEAN demonstrated superior resolvability compared to traditional subspace methods and standalone one\-shot learning models\. While the discretization of the network’s output layer results in a marginal sacrifice in absolute precision compared to the exhaustive GS\-CLEAN baseline, NN\-CLEAN maintains exceptionally tight alignment in both parameter accuracy and signal reconstruction fidelity\. Most importantly, this minor precision trade\-off yields profound computational benefits\. Unlike traditional methods whose processing and memory demands scale linearly with batch size, NN\-CLEAN leverages GPU parallelization to achieve near\-constant runtime and memory footprint\. Ultimately, by anchoring the iterative subtraction of predicted MPCs to exact wave propagation physics, NN\-CLEAN provides a highly robust, scalable, and physically grounded solution for real\-time channel estimation in next\-generation MIMO and ISAC systems\.

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