Energy Efficiency Maximization for Integrated Sensing and Communication in IoT C-RAN

I. INTRODUCTION

• Prior uplink ISAC work commonly kept all RRUs active for sensing; this paper jointly optimizes sensing-RRU activation, user association, and power for EE under an accuracy constraint.Its model-based algorithm decomposes the problem into association, RRU activation, and power-allocation subproblems; ISAC-GNN learns the joint policy using semi-supervised training.Source: §I · pp. 2–3

II. SYSTEM MODEL AND PROBLEM FORMULATION

• A partially centralized uplink C-RAN decodes each UE’s payload locally at one serving RRU and centralizes network coordination and multistatic sensing fusion.RRUs forward decoded payloads and sensing observations to the cloud over ideal, fully synchronized fronthaul links.Source: §II · p. 3, A. Communication Subsystem · p. 4

B. Radar Sensing Subsystem

• Passive multistatic sensing reuses uplink signals, and the CRLB localization bound depends on UE power and sensing-RRU activation.The system is considered in tracking mode, with target location and RCS known and continuously estimated over time.Source: §II · p. 3, B. Radar Sensing Subsystem · p. 4

A. Communication Subsystem

• Each UE transmits to one selected RRU with allocated power; the received uplink signal includes channel gain, the UE signal, and additive noise.The selected RRU decodes that UE’s data before forwarding it to the cloud.Source: A. Communication Subsystem · pp. 3–4

C. Problem Formulation

• The objective maximizes energy efficiency across all UEs and RRUs subject to resource limits and the desired localization performance.The formulation includes UE and RRU power consumption and a maximum acceptable localization error κ.Source: C. Problem Formulation · p. 5
• The CRLB constraint enforces target sensing accuracy, while other constraints cap UE transmit power, require at least one sensing RRU, and assign each UE to one RRU.The CRLB constraint is approximated and relaxed to obtain a tractable formulation.Source: C. Problem Formulation · p. 5
• The resulting problem remains difficult because its objective is non-convex.Source: C. Problem Formulation · p. 5

A. Initialization of Power Allocation p and RRU Activation 𝝉

• The association procedure initializes every UE at maximum transmit power and assumes all RRUs are active for sensing.Source: A. Initialization of Power Allocation… · p. 5

B. User Association

• The association algorithm solves the optimal association separately for each UE and returns the resulting association vector.Source: B. User Association · pp. 5–6
• With power and RRU activation fixed, total power consumption is constant across user associations, so association reduces to maximizing total achievable rate.Source: B. User Association · pp. 5–6
• Each UE’s achievable rate depends only on its own association variable, allowing its optimal RRU association to be selected independently of other UEs’ associations.Source: B. User Association · p. 6

C. RRU Sensing Activation

• With UE powers and associations fixed, RRU activation minimizes the number of active RRUs while satisfying the sensing constraint.The decision-tree procedure starts with all RRUs active and tests deactivations against the constraint.Source: C. RRU Sensing Activation · p. 6
• The tree deactivates the feasible RRU with the minimum estimated sensing-constraint impact and stops when no further deactivation is feasible or one RRU remains.Source: C. RRU Sensing Activation · p. 6
• The activation algorithm has O(K^2) complexity in its longest run, compared with O(2^K) for exhaustive search.Source: C. RRU Sensing Activation · p. 6

D. Fixed-Point Power Allocation

• Power allocation tackles the nonlinear fractional EE problem with a log transformation and alternates between optimizing transformed EE and updating its transformation factors.The method then uses Dinkelbach iterations and a convex reformulation for the power subproblem.Source: D. Fixed-Point Power Allocation · pp. 7–8
• The fixed-point power update converges to a solution of the transformed problem when power stays within its bounds, while the outer iterative procedure monotonically increases the objective and converges.Source: D. Fixed-Point Power Allocation · pp. 7–8

E. Alternative Algorithm

• Each iteration updates association, RRU activation, and power allocation in sequence using the latest preceding updates, stopping when EE improvement falls below a threshold.Source: E. Alternative Algorithm · p. 9
• Under feasible initial power and RRU activation, the Alternative Algorithm’s energy-efficiency value is non-decreasing across iterations and converges.Source: E. Alternative Algorithm · p. 9
• For multiple targets, Schur-complement LMIs preserve convex feasibility of the power-allocation subproblem, so the Alternative Algorithm applies without modification.Source: E. Alternative Algorithm · p. 9

F. Complexity Analysis

• User association has O(KN^2) complexity, RRU activation has O(K^2) complexity, and fixed-point power allocation depends on the iteration counts of its component procedures.Source: F. Complexity Analysis · p. 9
• The overall model-based algorithm’s complexity also depends on the number of outer iterations required for convergence.Source: F. Complexity Analysis · p. 10
• Actual runtime varies with network scale and convergence thresholds across system settings.Source: F. Complexity Analysis · p. 10

• The framework models the ISAC system as a heterogeneous graph and uses a GNN to learn its joint resource-management policy.The section introduces the graph representation, GNN architecture, and its design.Source: §IV · p. 10

A. Graph Modeling the ISAC System

• The bipartite ISCG connects every UE to all RRUs, encoding UE and RRU features and UE–RRU channel-gain and target-path RCS attributes.The graph representation captures permutation equivariance: reordering UE or RRU nodes correspondingly adjusts the solution.Source: A. Graph Modeling the ISAC System · p. 10

B. Graph Neural Networks for Integrated Sensing and Communication Networks

• ISAC-GNN exploits permutation equivariance through an architecture with an initializing layer, repeated ISAC-GNN layers, and an output-processing layer.These are the three components identified for the proposed architecture.Source: B. Graph Neural Networks for… · p. 10

1) Initializing Layer:

• The initializing layer prepares the graph's node and edge features for processing by the ISAC-GNN layers.The text introduces initial feature representations for UE nodes, RRU nodes, and their connecting edges.Source: 1) Initializing Layer · p. 10, A. Graph Modeling the ISAC System · p. 10

2) ISAC-GNN Layers:

• ISAC-GNN layers update node and edge representations using MLPs, whose fixed output dimensions support generalization across system sizes.The paper states that node-wise processing is independent of network size and that the MLP output dimensions remain fixed as UE and RRU counts change.Source: 2) ISAC-GNN Layers · p. 10

3) Output Processing Layer:

• The output layer derives UE power from its final embedding, thresholds RRU activation, and assigns each UE to the RRU with the highest scheduling probability.The sigmoid-based power output satisfies the power limit, while the association rule selects one RRU per UE.Source: 3) Output Processing Layer · pp. 10–11

C. Loss Function

• Semi-supervised training initializes ISAC-GNN with model-based labels, then fine-tunes it unsupervised to maximize system energy efficiency.The training combines a supervised loss for continuous power and discrete activation and association variables with an unsupervised objective for system EE.Source: C. Loss Function · p. 11

D. Complexity Analysis

• The section decomposes the GNN's computational cost into forward-pass and backpropagation components.It introduces the forward-pass cost steps as the basis for the complexity analysis.Source: D. Complexity Analysis · p. 11, 1) Forward Pass · p. 11

1) Forward Pass:

• The forward-pass analysis accounts for message generation, message passing, node updates, and edge updates, with MLP costs scaling with node or edge counts.Under constant embedding size, the stated MLP costs include O(|V|F^2 + |V|) for node processing and O(|E|F^2 + |E|) for edge processing.Source: 1) Forward Pass · p. 11

2) Backward Pass:

• Once trained, ISAC-GNN has computational complexity linear in the numbers of UEs and RRUs, unlike the model-based algorithm's nested iterative procedures.The paper presents this as enabling fast deployment in large-scale ISAC systems.Source: 2) Backward Pass · pp. 11–12

A. Performance of the ISAC-GNN and Distributed Computing Optimization

• ISAC-GNN and DBCA are equivalent in the problems they can solve when they have the same number of layers and communication rounds.The proposition states this correspondence in both directions.Source: A. Performance of the ISAC-GNN and… · p. 12

B. Performance of the ISAC-GNN in energy efficiency maximization problem.

• With a bounded maximum number of neighbors and fixed iterations, Algorithm 5 is equivalent to a DBCA, for which an ISAC-GNN can achieve comparable performance.The stated comparison is for the same training and testing problem sizes.Source: B. Performance of the ISAC-GNN in… · p. 12

A. Simulation Setups

• The simulations target meter-level localization accuracy, use Rayleigh channels, and evaluate one ISAC-GNN model across test configurations without retraining.The selected accuracy threshold is κ=1 m^2; the model-based evaluation uses 100,000 realizations, and learning-based configurations use 2,000 training and 500 testing samples each.Source: A. Simulation Setups · pp. 12–13

B. Model-based Approach Evaluation

• The model-based evaluation combines proposed power allocation, RRU activation, and user association, comparing the resulting scheme with subproblem benchmarks and a decoupled method.The benchmark combinations include equal versus proposed power allocation, nearest versus proposed user association, and full versus proposed RRU activation.Source: B. Model-based Approach Evaluation · p. 13
• About 5.9×10^8 kbits/Joule is the peak EE achieved by Algorithm 5 at K=8 and N=24, where it outperforms all benchmarks.The same comparison reports approximately 6% higher EE from proposed RRU activation than the P-Near-R scheme; about 45% of realizations exceed 5.9×10^8 kbits/Joule for the full scheme.Source: Figure 4 · p. 13, B. Model-based Approach Evaluation · p. 13, Figure 5 · p. 14
• EE rises with more RRUs but falls as UE count grows; at N=27, the benchmark-based method achieves 41% of Algorithm 5's EE.For K=3 and K=9, the fully benchmark-based method reaches 35% and 42% of the proposed approach's EE, respectively.Source: B. Model-based Approach Evaluation · pp. 13–14, Figure 6 · p. 14, Figure 7 · p. 14

C. Learning-based Approach Evaluation

• Semi-supervised training achieves the highest final performance with moderate variability, combining stable supervised initialization with reward optimization.The comparison includes supervised and unsupervised training and Algorithm 5.Source: C. Learning-based Approach Evaluation · p. 14
• On average, ISAC-GNN provides 4% better EE than Algorithm 5, with inference time stabilizing at about 20 ms versus up to 67 times higher runtime for Algorithm 5.The comparison tests one trained model across the listed UE and RRU scenarios.Source: C. Learning-based Approach Evaluation · p. 15, Table IV · p. 15
• At N=27, ISAC-GNN uses about 2.7 W versus 3.5 W for Algorithm 5, a 23% reduction, while achieved spectral efficiencies are nearly identical.The figure plots total system power and spectral efficiency against the number of UEs.Source: C. Learning-based Approach Evaluation · p. 15, Figure 9 · p. 15
• When scaling from (K, N)=(40, 80) to (100, 200), ISAC-GNN improves EE by up to 7.5% and reduces runtime by 99.9% versus Algorithm 5.Algorithm 5 runtime rises above 700 s, while ISAC-GNN inference time remains nearly constant.Source: Table V · p. 15, C. Learning-based Approach Evaluation · p. 16
• At error factor 1−φ^2=0.09, ISAC-GNN maintains EE of 3.5 × 10^8 bits/Joule, higher than all model-based methods.ISAC-GNN was trained on perfect-CSI data; the comparison tests imperfect CSI.Source: C. Learning-based Approach Evaluation · p. 16, Figure 10 · p. 16
• The authors identify imperfect-CSI deployment as an area for further improvement, suggesting noisy-CSI training data or online refinement.Proposed refinements include statistical error models or lightweight model-based updates.Source: C. Learning-based Approach Evaluation · p. 16

VII. CONCLUSIONS

• The conclusion reports an average EE gain of 5% and up to 64-times-faster computation for ISAC-GNN over the model-based approach.It also states that one trained model generalizes across system configurations with different UE and RRU counts.Source: §VII · p. 16

APPENDIX A PROOF OF LEMMA 1

• Algorithm 3's objective value increases across iterations and converges because it is bounded above by the limited power budget.The proof uses the lower-bound relation and optimality of the subproblem solution.Source: Appendix A · pp. 16–17

APPENDIX B PROOF OF LEMMA 2

• The power-allocation iteration converges to a unique fixed point when the reference interference function satisfies positivity, monotonicity, and scalability.The appendix verifies these three properties for the function used in the fixed-point equation.Source: Appendix B · p. 17

APPENDIX C PROOF OF LEMMA 3

• Algorithm 5's objective is non-decreasing at each iteration and bounded, so its convergence is guaranteed.The proof attributes boundedness to constrained variables and a nonvanishing denominator due to circuit and RRU power.Source: Appendix C · p. 17

APPENDIX D PROOF OF PROPOSITION 1

• The DBCA update process is expressed through message generation, neighbor aggregation, and state updates repeated for fixed rounds or until convergence.The appendix reformulates ISAC-GNN updates in a general layer form for the equivalence proof.Source: Appendix D · pp. 17–18

A. ISAC-GNN as a Generalization of DBCA

• ISAC-GNN can reproduce DBCA updates by induction over communication rounds, beginning from shared initial features.The proof relates the DBCA update to the ISAC-GNN node update.Source: A. ISAC-GNN as a Generalization of DBCA · p. 18

B. DBCA Reformulated as ISAC-GNN

• Conversely, the appendix proves by induction that DBCA can be viewed as an ISAC-GNN, with matching initial node features and corresponding update choices.The proof constructs the ISAC-GNN update at each step from the preceding DBCA state.Source: B. DBCA Reformulated as ISAC-GNN · p. 18

APPENDIX E PROOF OF LEMMA 4

• Algorithm 5's user-association, RRU-activation, and power-allocation solvers form an agent-based procedure equivalent to at most 3T DBCA iterations.Each iteration cycles through edge-agent, UE-agent, and RRU-agent updates.Source: Appendix E · p. 19