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Integration of a GNN and U-Net for Hybrid Beamforming in mm-Wave m-MIMO SystemsGurpreet Kaur1,a, Gurmeet Kaur2,b 1Indian Institute of Information Technology Una, Una, India 2Punjabi University, Patiala, India agurpreetkaur@iiitu.ac.in bgurmeetkece@pbi.ac.in Gurpreet Kaur Indian Institute of Information Technology Una, Una, India gurpreetkaur@iiitu.ac.in | Gurmeet Kaur Punjabi University, Patiala, India gurmeetkece@pbi.ac.in |
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Abstract—
Millimeter-Wave (mm-Wave) based Massive Multiple-Input Multiple-Output massive (m-MIMO) systems promise high data rates and high spectral efficiency in next-generation i.e., beyond fifth generation (B5G) wireless networks. Moreover, the fully-digital precoding technique helps further to achieve the highest spectral efficiency in mm-Wave massive MIMO systems but face significant challenges such as hardware complexity and cost. Practically, these systems are infeasible due to complexity and cost of using radio frequency (RF) chain with each antenna in such systems. Hybrid beamforming emerges as a practical solution by combining analog and digital precoding, yet its performance hinges on accurate and low-complexity beamformer design under sparse and dynamic channel conditions. This paper proposes a novel deep learning approach that synergizes Graph Neural Networks (GNNs) and U-Net architectures to efficiently learn optimal hybrid beamforming strategies from channel state information (CSI). For generalizing the system in various configurations and distributions, the GNN is used to capture the spatial and topological relationships between antennas and users. Simultaneously, for extracting the various spatial features for enhanced precoder rebuilding, the U-Net is used to process the structured CSI representations. The proposed approach is trained using a loss function for optimizing the spectral efficiency (SE) and bit error rate (BER). Simulation results show that the integrated GNN and U-Net model achieves superior performance to traditional methods, with significantly reduced computational overhead during inference. The SE percentage improvement with proposed approach is varying from 9–54% as compared to some exiting techniques. Moreover, the proposed approach is able to achieve approximately 92% of the fully digital precoding performance.
Keywords—
Millimeter-Wave (mm-Wave)
Massive Multiple-Input Multiple-Output (m-MIMO)
Hybrid Beamforming
Graph Neural Network (GNN)
U-Net
I. Introduction
For attaining the highest spectral efficiency with reduced hardware complexity in practical millimeter-Wave (mm-Wave) massive multiple-input multiple-output (m-MIMO) systems, the Hybrid beamforming which combines the low-dimensional digital precoding with high-dimensional analog beamforming is used [1], [2]. Due to the prohibitive cost and power consumption for one RF chain per antenna element of fully digital precoding technique makes it infeasible in practice at mmWave frequencies. Ayach et al. proposed mm-Wave MIMO systems with spatially sparse precoding [3]. Alkhateeb et al. is also used the spatial sparse precoding as a base and introduced hybrid precoding and channel estimation via compressive sensing [4]. In [5], various designs for hybrid beamforming including (i) fully-connected design which delivers high flexibility (ii) partially-connected design which is lower in cost, have been explored. However, the main drawbacks faced by these designs is increased hardware cost with high complexity and reduced beamforming gain in fully-connected and partially-connected design respectively [5]. For the reduction of complexity and power consumption, the analog processing techniques with lens antenna arrays and switch-based networks was proposed in [6], [7]. However, it also faced limitations such as hindered system performance with less flexibility. Therefore, Park et al. proposed hybrid precoding using dynamic subarrays to address the above-mentioned issues in wideband mm-Wave MIMO systems [8]. Moreover, the multi-user hybrid precoding with analog beam codebook was also studied in [9], [10]. Sohrabi F. et al. proposed a hybrid beamforming design with large-scale arrays of antennas for orthogonal frequency division multiplexing access (OFDM)-based systems [11].
Other than this, various machine learning techniques have been merged with hybrid beamforming techniques for predicting the beamforming vectors, hybrid end-to-end precoding, estimating the channel, and enabling the efficient beam prediction with limited pilot overhead [12], [13], [14], [15]. Ahmet M. E. et al. presented a beamforming design by using convolutional neural network (CNN) that utilizes the covariance matrices in place of present CSI and attained comparatively better performance in terms of sum rate due to significant reduction in overhead [16]. In addition to this, Zhao X. introduced a WMMSE based design for solving the Hybrid beamforming optimization problem consistently with main focus on maximizing the spectral efficiency [17]. For effectively estimating the frequency-selective channels in mm-Wave range, a deep learning (DL) technique is used in [18]with high improvement in normalized mean squared error (NMSE) at low SNR conditions. Furthermore, Morsali et al., presented a practical DL based hybrid technique with the help of unified neural networks for achieving the optimum performance with restricted number of RF chains [19]. Other than this, Banerjee B. et al., introduced a CGAN (conditional generative adversarial network) -based hybrid beamforming for mm-Wave massive MIMO systems for the improvement in the spectral efficiency [20]. Afterwards, Ramineni et al., presented an analysis for the comparison of different hybrid beamforming techniques and found that when the hybrid techniques are integrated with deep learning techniques the received performance is quite comparable to performance of fully digital precoding technique, and requires significant fewer number of RF chains[21].
It has been noticed in the literature that deep hybrid beamforming has become main area of research in the field of mm-Wave m-MIMO systems with key areas hardware cost, power consumption, and spectral efficiency (SE). Therefore, the main focus of this paper would be as follows:
The development of RF beamformer by using the Graph Neural Networks (GNNs) and U-Net architectures for hybrid beamforming.
The evaluation of improvement in performance in terms of SE and BER.
The calculation of computational complexity.
II. System Model
In the work, we considered the single cell downlink for mm-Wave massive MIMO system for serving
antennas of user equipment (UE) by using
antennas of base station (BS). The RF chains in hybrid beamformers follows
and
for BS and UE respectively. The BS transmits the data in
number of subcarriers using OFDM and on each subcarrier the downlink
data streams, with
. The baseband (BB) precoder
and RF beamformer
are included In
subcarrier of hybrid beamformer
.
Further, the RF beamformer is made up of analog phase shifters with unit modulus
entries. Therefore,
is the final matrix of hybrid beamforming. In the similar manner, the final beamforming matric for UE is
at
,
, and
.
The data symbol vector i.e.,
, for
th subcarrier is having
and
, here
is the identity matrix (
). The BS’s
th subcarrier transmitted signal is represented as:
1
The channel
effected by additive white Gaussian noise (AWGN)
uniformly distributed over all the subcarriers is used for passing the
th subcarrier of transmitted signal. The UE’s
th subcarrier’s received signal after post processing is represented as:
2
The kth subcarrier’s achievable SE (
is represented as:
3
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It has been noticed in[20] that hybrid beamforming based on CGAN is feasible for all the structures i.e., partially connection, adaptive connection, and full connection. Our proposed approach is also suitable for all the structures by just providing a proper dataset for training through integrated GNN and U-Net model. However, in most of the already existing techniques, the RF beamformer is in partially connected structure i.e., connection of each phase shifter of RF chain with only available antennas subset [11], [17], [22]. In this paper, we have used partially connected structure to reduce computational complexity and for the comparison of performance with already existing techniques. The block diagram for hybrid beamforming in partially-connected architecture for mm-Wave m-MIMO System is shown in Fig. 1.The
th subcarrier channel gain matrix using geometric frequency selective model is represented as:
[4]
Here, path loss (
) is same for all the paths and clusters.
is the number of clusters and
is the number of paths per cluster. The gain (complex in nature), angle of arrival (AoA) and departure (AoD) are denoted by
,
, and
. For the response of transmitter and receiver the array vectors is denoted by
respectively. In this work, we have taken perfect CSI at BS in consideration. The main aim of this work is to develop GNN and U-Net integration based hybrid beamforming for improving the SE mentioned in Eq. (3). For achieving the performance near fully digital precoding technique, we have designed the hybrid beamforming matrix instead of focusing solely on transmitter. Therefore, the problem for optimization is represented as:
,
subject to
(5)
Here, the Frobenius norm is denoted by
. For avoiding the problems while training the proposed algorithm, any individual matrices explicit constraints are not imposed except
unit modulus constraint.
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Fig. 3
The block diagram for hybrid beamforming in partially-connected architecture for mm-Wave m-MIMO System
III.
A GNN and U-Net Integration Approach
A.
A GNN and U-NET integraion Overview
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In partially connected architecture or limited RF-chain architectures, for modelling the wireless networks in the form of graph to realize the spatial relationship between users or antennas the GNN is found very effective [23], [24]. On the other hand, U-Net is suitable for digital beamforming precoding to extract and reconstruct the sptially structured features from sparse CSI inputs [25], [26]. In this paper, an integration of GNN and U-net architectures for hybrid beamforming in mm-Wave m-MIMO systems is proposed to address the limitations of conventional methods.
B.
Proposed Integration
In this paper, the phase shifters are utilized for designing the RF beamformer for transmitting the beams in different directions using the phase settings properly. The best direction for transmitting the downlink is decided by values of matrices such as spectral efficiency or sum rate. The fine tuning of
is the main aim of GNN. The total no. of quantization bits used by phase shifters is the main factor for testing the available directions for a tuned beamformer. In this work, we have used 8-bit phase shifters for total of 28 = 256. The
(initial RF beamforming) is selected on the basis of spectral efficiency (maximum value) from DFT codebook by searching different beams (e.g., by 16 beams for 4-bit resolution). At this step, for the calculation of spectral efficiency the BB precoder (based on weighted minimum mean square error (WMMSE)) is used on the temporary basis. After that, the RF beamforming is fine-tuned using GNN.
For addressing the optimization problem mentioned in Eq. (5), we constructed graph
which is an undirected graph with
sets of nodes (equal to number of antenna elements) and
sets of edges. These nodes are used to frame a fully connected graph for each subcarrier with
(edge feature). The initial features are extracted from only known parameter i.e., channel state information as presented below:
6
The proposed GNN based RF beamformer consists of total of
layers. The output feature vector
update of
th node at layer
is represented as follows:
7
Where
is the current GNN layer,
is the indexing of node whose feature vector is being updated,
indexing of the neighbouring node of
node from graph
,
is number of edges of node
and
,
is the ReLU nonlinear activation function,
is the weight matrix. The final output (
) from GNN is obtained through normalization of graph
with
. The loss function from analog beamforming is calculated as:
8
The main goal of GNN is to decrease the value of
. At the end of the training, the outputs are produced for the next stage.
As mentioned previously, the main aim of our proposed design of hybrid beamformer is to combine
and
for getting the results close to
. For designing the BB precoder, we have used an equivalent channel
and used this as the input to U-net. Here,
is the effective BB channel after analog precoding. The output of U-Net is
. The U-net is used in this work to map the reduced dimension channel with the digital precoder and to compensate the interference from
. Therefore, the final output from Hybrid precoder is calculated as follows:
9
After that we have normalized the value of
as;
From this normalised value of
, we have calculated the value of spectral efficiency as:
10
Where
is the average total transmit power.
The complexity (per layer) of GNN for analog beamforming is
and complexity of U-Net for digital beamforming is
. The overall complexity for integration of GNN and U-Net is
.
is number of antennas or (number of graph nodes),
is the feature dimension per node,
is the number of edges,
is the number of subcarriers. The proposed approach achieves real-time hybrid beamforming with approximately 1–5 milliseconds of inference latency making it appropriate for deployment in B5G mm-Wave MIMO systems.
IV. Results And Discussions
In this section, the simulation results for calculating the performance of our proposed design for mm-wave m-MIMO systems in terms of SE and BER is presented in Fig. 2 and Fig. 3. For the comparison purpose, we considered only the cases of RF beamformer in Partially-connected structure. The spacing between the antennas of ULA in BS is
with
RF chains and
antenna elements. And, the spacing between the antennas of ULA in UE is
with
RF chains and
antenna elements, means
streams per subcarrier are transmitted by BS. The operating frequency of system is 28GHz with each subcarrier bandwidth of 32MHz at
=32. The five number of clusters (
) with ten paths per cluster (
) are used for the simulation. The distribution of each path complex gain
is
. The AoA and AoD i.e.,
and
are following the laplace distribution and mean of each cluster is following uniform distribution from 0 to
. In this work, the normalized value of
is obtained by dividing it with
.The training datasets for GNN and U-net consisting of 6000 samples of
from 200 UEs i.e., 30 samples for each UE location. The realizations of AoA and AoD for each location is remaining fixed but it is changing for different locations in the proposed method. The GNN and U-net has been trained with 30 epochs in each phase separately. The testing dataset consisting of 2000 samples of
from 100 locations i.e., 20 samples for each location. The performance results are obtained by taking the average of 30 independent runs of training and testing for same datasets.
Figure 2 presents the Average spectral efficiency (bits/s/Hz) v/s SNR (dB) with proposed hybrid beamforming technique. For comparison purpose, the performance of existing hybrid beamforming techniques i.e., HBF[11], MMSE [17], WMMSE [17], and CGAN[20] with same constraints such as phase shifters with infinite-resolution and unit power is also presented in Fig. 2. For these existing techniques, the condition for stopping the iterative process is two iterations relative difference (
less than
. In this plot, we have used the conventional precoding with SVD for fully digital precoding as in [3]. It has been noted from the plot that our proposed a GNN and U-Net Integration approach based hybrid beamforming is performing better than the already existing (considered) techniques consistently over different values of SNR (dB) due to improved generalization and adaptivity. The analysis in terms of average spectral efficiency at different values of SNR (dB) is shown in Table 1. In Table 1, the approximate performance percentage achieved with different techniques as compared to fully digital precoding performance is presented. The spectral efficiency percentage improvement with proposed approach at -18dB SNR is varying from 9–54% as compared to exiting techniques. Moreover, the proposed approach is able to achieve approximately 92% of the fully digital precoding performance.
Fig. 2
Average spectral efficiency (bits/s/Hz) v/s SNR (dB) with proposed hybrid beamforming technique in comparison to existing hybrid beamforming techniques (HBF[11], MMSE [17], WMMSE [17], and CGAN [20])
SNR (dB) | Average Spectral Efficiency | |||||
|---|---|---|---|---|---|---|
HBF | MMSE | WMMSE | GAN | Proposed | ||
-18 | 33.34 | 49.34 | 53.33 | 66.67 | 73.34 | |
-12 | 42.30 | 47.69 | 53.84 | 65.38 | 78.46 | |
-6 | 47.61 | 52.38 | 57.14 | 69.04 | 75.23 | |
0 | 56.14 | 61.40 | 66.67 | 80.70 | 87.71 | |
6 | 63.51 | 68.11 | 72.97 | 87.29 | 91.89 | |
Fig. 3
shows the BER v/s SNR (dB) with proposed hybrid beamforming technique and other existing techniques HBF, MMSE, WMMSE, and CGAN with same constraints such as phase shifters with infinite-resolution and unit power. It has been observed in Fig. 3 that the performance with proposed technique is higher than other techniques. In this technique, we have used lower learning rate to reduce the higher validation error with larger learning rate. This proposed technique gives lower values of BER due to fine-tuned beamforming and better channel adaption. Also, in case of wideband mm-Wave systems the problem of overfitting is avoided by using the U-net in combination with GNN.
Figure 3. BER v/s SNR (dB) with proposed hybrid beamforming technique in comparison to existing hybrid beamforming techniques (HBF[11], MMSE [17], WMMSE [17], and CGAN [20]) |
V. Conclusion
In this paper, an approach for hybrid beamforming by integrating the GNN and U-net in mm-Wave m-MIMO systems is proposed. Under realistic hardware constraints, this proposed approach would be able to capture the local and even global characteristics to improve the beamforming performance by using the GNN’s relational modelling and U-Net’s spatial feature extraction capability. In the simulation results and discussions section, it has been demonstrated that the proposed approach outperforms existing hybrid beamforming techniques in terms of SE and BER. The future scope of the work is to design the framework for dynamic beam tracking and deploy the large-scale networks in real-time.
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Funding
This research did not receive any specific grant from any funding agency.
VI. Conflict of Interest
The authors declare that they have no conflict of interest.
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Author Contribution
Gurpreet Kaur proposed the idea, simulated the idea in MATLAB software, and wrote the draft of paper. Gurmeet Kaur supervised the entire work and finalized the writing of paper.
References
1.
Rangan, S., Rappaport, T. S., & Erkip, E. (2014). Millimeter-wave cellular wireless networks: Potentials and challenges, Proceedings of the IEEE, vol. 102, no. 3, pp. 366–385. 10.1109/JPROC.2014.2299397
2.
Heath, R. W., Gonzalez-Prelcic, N., Rangan, S., Roh, W., & Sayeed, A. M. An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems. Apr 01 2016 Institute of Electrical and Electronics Engineers Inc 10.1109/JSTSP.2016.2523924
3.
Ayach, O. E., Rajagopal, S., Abu-Surra, S., Pi, Z., & Heath, R. W. (2014). Spatially sparse precoding in millimeter wave MIMO systems. IEEE Trans Wirel Commun, 13(3), 1499–1513. 10.1109/TWC.2014.011714.130846
4.
Alkhateeb, A., El Ayach, O., Leus, G., & Heath, R. W. (Oct. 2014). Channel estimation and hybrid precoding for millimeter wave cellular systems. IEEE Journal on Selected Topics in Signal Processing, 8(5), 831–846. 10.1109/JSTSP.2014.2334278
5.
Lin, C., & Li, G. Y. (Jul. 2016). Energy-Efficient Design of Indoor mmWave and Sub-THz Systems with Antenna Arrays. IEEE Trans Wirel Commun, 15, 4660–4672. 10.1109/TWC.2016.2543733
6.
Zeng, Y., & Zhang, R. (Apr. 2016). Millimeter wave MIMO with lens antenna array: A new path division multiplexing paradigm. IEEE Transactions on Communications, 64(4), 1557–1571. 10.1109/TCOMM.2016.2533490
7.
Han, P. Y. (2015). Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave 5G. IEEE Communications Magazine, pp. 186–194.
8.
Park, S., Alkhateeb, A., & Heath, R. W. (May 2017). Dynamic Subarrays for Hybrid Precoding in Wideband mmWave MIMO Systems. IEEE Trans Wirel Commun, 16(5), 2907–2920. 10.1109/TWC.2017.2671869
9.
Li, Q. C., Niu, H., Papathanassiou, A. T., & Wu, G. (2014). 5G network capacity: Key elements and technologies. IEEE Vehicular Technology Magazine, 9(1), 71–78. 10.1109/MVT.2013.2295070
10.
Ayach, O. E., Heath, R. W., Abu-Surra, S., Rajagopal, S., Pi, Z., & THE CAPACITY OPTIMALITY OF BEAM STEERING IN LARGE MILLIMETER WAVE MIMO SYSTEMS. (2012)., in IEEE 13th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), IEEE.
11.
Sohrabi, F., & Yu, W. (Jul. 2017). Hybrid Analog and Digital Beamforming for mmWave OFDM Large-Scale Antenna Arrays. in IEEE Journal on Selected Areas in Communications (pp. 1432–1443). Institute of Electrical and Electronics Engineers Inc.. 10.1109/JSAC.2017.2698958
12.
Gao, X., Dai, L., Han, S., Chih-Lin, I., & Heath, R. W. (Apr. 2016). Energy-Efficient Hybrid Analog and Digital Precoding for MmWave MIMO Systems with Large Antenna Arrays. IEEE Journal on Selected Areas in Communications, 34(4), 998–1009. 10.1109/JSAC.2016.2549418
13.
Huang, H., Song, Y., Yang, J., Gui, G., & Adachi, F. (2019). Deep-Learning-Based Millimeter-Wave Massive MIMO for Hybrid Precoding, IEEE Trans Veh Technol, vol. 68, no. 3, pp. 3027–3032, Mar. 10.1109/TVT.2019.2893928
14.
Ali, A., González-Prelcic, N., & Heath, R. W. (Feb. 2018). Millimeter Wave Beam-Selection Using Out-of-Band Spatial Information. in IEEE Transactions on Wireless Communications (pp. 1038–1052). Institute of Electrical and Electronics Engineers Inc.. 10.1109/TWC.2017.2773532
15.
Guo, Z., & Zou, W. (2022). Channel Estimation Based on Matrix Completion for mmWave Massive MIMO System. Lecture Notes in Electrical Engineering, 878 LNEE, 834–841. 10.1007/978-981-19-0390-8_104
16.
Elbir, A. M. (Oct. 2020). A Deep Learning Framework for Hybrid Beamforming without Instantaneous CSI Feedback. Ieee Transactions On Vehicular Technology, 69(10), 11743–11755. 10.1109/TVT.2020.3017652
17.
Zhao, X., Lin, T., Zhu, Y., & Zhang, J. (2021). Partially-Connected Hybrid Beamforming for Spectral Efficiency Maximization via a Weighted MMSE Equivalence, IEEE Trans Wirel Commun, vol. 20, no. 12, pp. 8218–8232, Dec. 10.1109/TWC.2021.3091524
18.
Abdallah, A., Celik, A., Mansour, M. M., & Eltawil, A. M. (2022). Deep Learning-Based Frequency-Selective Channel Estimation for Hybrid mmWave MIMO Systems, IEEE Trans Wirel Commun, vol. 21, no. 6, pp. 3804–3821, Jun. 10.1109/TWC.2021.3124202
19.
Morsali, A., Haghighat, A., & Champagne, B. (2021). Deep Learning Framework for Hybrid Analog-Digital Signal Processing in mmWave Massive-MIMO Systems.
20.
Banerjee, B., Elliot, R. C., Krzymien, W. A., & Medra, M. (2024). Hybrid Beamforming for mmWave Massive MIMO Systems Using Conditional Generative Adversarial Networks. Ieee Transactions On Vehicular Technology, 73(10), 15803–15808.
21.
Ramineni, P., Pasula, P., & Panchareddy, T. (Feb. 2025). Performance Analysis of Optimal and HybridBeamforming Techniques and Spectral EfficiencyEnhancement for 5G Millimeter-Wave Systems. International Journal of Wireless and Microwave Technologies, 15(1), 51–65. 10.5815/IJWMT.2025.01.04
22.
Soleimani, M., Elliott, R. C., Krzymień, W. A., Melzer, J., & Mousavi, P. (2020). Hybrid Beamforming for mmWave Massive MIMO Systems Employing DFT-Assisted User Clustering, IEEE Trans Veh Technol, vol. 69, no. 10, pp. 11646–11658, Oct. 10.1109/TVT.2020.3015787
23.
Linfu, Z., Zhiwen, P., & El-Hajjar, M. Graph Neural Network Aided Beamforming for Holographic Millimeter Wave MIMO Systems.
24.
Li, B., & Vu, M. (2024). Graph Neural Network Based Hybrid Beamforming Design in Wideband Terahertz MIMO-OFDM Systems, in IEEE International Symposium on Phased Array Systems and Technology, Institute of Electrical and Electronics Engineers Inc., 10.1109/ARRAY58370.2024.10880428
25.
Ren, X. (2021). A causal u-net based neural beamforming network for real-time multi-channel speech enhancement, in Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH, International Speech Communication Association, pp. 341–345. 10.21437/Interspeech.2021-1457
26.
Lloria, D., Roger, S., León, G., Badía, J. M., Botella-Mascarell, C., & Belloch, J. A. (Jan. 2025). Optimizing Millimeter Wave MIMO Channel Estimation Through GPU-Based Edge Artificial Intelligence. Journal of Supercomputing, 81(1). 10.1007/s11227-024-06795-2