Can graph neural network-based detection mitigate the impact of hardware imperfections?

Can graph neural network-based detection mitigate the impact of hardware imperfections? Lamprini Mitsiou∗, Stylianos Trevlakis∗, Argiris Tsiolas†, Dimitrios J. Vergados†, Angelos Michalas†, and Alexandros-Apostolos A. Boulogeorgos† ∗Research & Development Department, InnoCube P.C., Thessaloniki 55535, Greece. E-mails: [email protected], [email protected]. †Department of Electrical and Computer Engineering, University of Western Macedonia, Kozani 50100, Greece. E-mail: [email protected], [email protected], [email protected], [email protected]. Abstract—Until recently, researchers used machine learning methods to compensate for hardware imperfections at the symbol level, indicating that optimum radio-frequency transceiver performance is possible. Nevertheless, such approaches neglect the error correcting codes used in wireless networks, which inspires machine learning (ML)-approaches that learn and minimise hardware imperfections at the bit level. In the present work, we evaluate a graph neural network (GNN)-based intelligent detector’s in-phase and quadrature imbalance (IQI) mitigation capabilities. We focus on a high-frequency, high-directional wireless system where IQI affects both the transmitter (TX) and the receiver (RX). The TX uses a GNN-based decoder, whilst the RX uses a linear error correcting algorithm. The bit error rate (BER) is computed using appropriate Monte Carlo simulations to quantify performance. Finally, the outcomes are compared to both traditional systems using conventional detectors and wireless systems using belief propagation based detectors. Due to the utilization of graph neural networks, the proposed algorithm is highly scalable with few training parameters and is able to adapt to various code parameters. Index Terms—Belief propagation, bit error rate, graph neural networks, hardware imperfection mitigation, in-phase and quadrature imbalance, machine learning. I. INTRODUCTION As the wireless world search for unexploited resources in higher frequency bands, like millimeter wave and terahertz, new challenges are identifies and call for novel solutions [1]– [5]. One of the most important challenges is dealing with the impact of transceiver hardware imperfections. As discussed in [6]–[12], hardware imperfections, such as local oscillators’ phase noise, amplifier’s non-linearity and especially up and down-converters in-phase and quadrature imbalance (IQI), significantly limit the reliability of high-frequency wireless systems. Note that, as described [13], the hardware imperfection of wireless systems in the higher frequency range cannot be completely avoided, even with new technological solutions supported by integrated microwave photonics. Motivated by this, several researchers have presented hardware imperfections mitigation solutions [14]–[18]. In particular, in [14], the authors reported a widely linear IQI calibration structure that estimates the IQI parameters using either secondorder statistics or least-square-based model fitting. The authors of [15] used higher order statistics-based approaches in order to estimate the amplifier non-linearity in the presence of IQI and documented an IQI parameter maximum-likelihood estimation approach. The aforementioned approaches are twostep processes that are usually energy consuming. To counterbalance this, the authors of [16] presented a realvalued time-delay neural network that is used as an one-step mitigation process; thus, simplifies the compensation process. In [17], a shortcut real-valued time-delay neural network for compensating IQI and amplifiers non-linearity was introduced. Finally, in [18], a neural network-based digital predistortion as a solution to countermeasure the impact of cross-talk, amplifier non-linearity, IQI, and direct current offset was presented. In other words, the authors of [15], [17], [18] aimed to employ machine learning methodologies in order to compensate the impact of hardware imperfections at a symbol-level, proving that with such approaches the ideal radio-frequency (RF) transceiver performance are reachable. However, following such approach, it is impossible to exploit the characteristics of the error correction codes that are employed in nowadays wireless systems. This observation motivates the design of machine learning (ML)-approaches that learn and de-emphasize the impact of hardware imperfections in a bit level. These approaches should be scalable and have a relatively low-number of training parameters in order to be adaptive in different code parameters. Motivated by this as well as the close relation between the Tanner graphs, which can be used to represent codes, and the graph neural networks (GNNs), in this paper, we assess the IQI mitigation capabilities of an intelligent detector that employs GNN. In particular, we consider a high-frequency and high-directional wireless system in which both the transmitter (TX) and the receiver (RX) suffer from IQI. A linear error correction code is employed by the TX, while a GNN-based decoder is used by the RX. To quantify the performance, the bit error rate (BER) is derived through respective Monte Carlo simulations. The results are benchmarked against conventional systems that employ traditional detectors, as well as wireless systems that use belief propagation (BF) based detectors. The organization of the rest of the paper is as follows: The system model is described in Section II. Section III reports the operation and training procedures of the intelligent detectors. This is the accepted manuscript version of the paper prior to IEEE formatting and copyediting. © 2025 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses. Results and related discussions are documented in Section IV. Finally, the conclusions and main message of this contribution is summarized in Section V. Notations: In what follows, [·||·]stands for the concatenation operator. Moreover, ⊕represents the message aggregation function, which, for this contribution is the mean value. The logarithm to base 2is denoted as log2(·). The sum of xifor i∈[1, N]is represented as PN i=1 xi, while QN i=1 xiis the product of xifor i∈[1, N]. Finally, Pr (E)is the probability of the event E. II. SYSTEM MODEL As illustrated in Fig. 1, we consider a high-directional wireless system that consists of a TX and a RX. Both the TX and RX employ analog beamforming and their beams are assumed to be perfectly aligned. The TX consists of a bit source that outputs a tuple of b bits, i.e., the codeword, which is the input of a (N, K, L)linear encoder, where Nis the codeword length, while Kand Lare respectively the number of ones in each column and row of the parity check matrix, P. The linear encoder uses zero-padding in order to be able to support odd codeword lengths and outputs a bit tuple c. Let Lbe the function that describes the operation of the linear encoder, then c=L{b}.(1) The output of the linear encoder is in turn inputted in a quadrature amplitude modulation (QAM) mapper. Let M{·} be the function that models the operation of the QAM mapper. Then, the output of the QAM mapper can be described as x=M{c}.(2) The symbol vector, x, is forwarded to the up-converter. We assume that the up-converter suffers from in-phase and quadrature imbalance. As a consequence, the baseband equivalent signal at the up-converter’s output can be expressed as [19] s=Kt 1x+Kt 2x∗,(3) where Kt 1and Kt 2are the IQI coefficients that, based on [20], can be written as [21] Kt 1=1 + gtexp (j θt) 2(4) and Kt 2=1 + gtexp (j θt) 2,(5) with gtand θtdenoting the IQI-infused amplitude and phase mismatched, respectively. Notice that Kt 1= 1 −Kt 2∗.(6) Moreover, the TX image rejection ratio (IRR) can be obtained as It=|Kt 1|2 |Kt 2|2.(7) The up-converter is followed by the analog beamformer. The baseband equivalent at the output of the TX can be written as sb=u s,(8) where ustands for the TX beamforming vector. The baseband equivalent signal at the output of the RX beamformer can be expressed as rb=v H sb+n,(9) where vand Hstands for the RX beam-vector and the channel matrix respectively, while nis an additive white Gaussian noise vector. Each element of nis modeled as an zero-mean complex Gaussian process of variance No. Additionally, E[ninj] = 0, for i6=j. With the aid of (8), (9) can be rewritten as rb=v H u s +n.(10) As reported in [22], since the TX and RX beams are perfectly aligned, v H u =h, (11) where his a scalar that represents the channel coefficient. As a consequence, (10) yields rb=hs+n.(12) Notice that the impact of multi-path fading is respectively low. Thus, the channel coefficient models only the deterministic path-gain. The output of the RX beamformer is connected to a downconverter that suffers from IQI. The down-converter introduces IQI; as a result, the baseband equivalent signal at the output of the down-converter can be expressed as [20] r=Kr 1rb+Kr 2r∗ b,(13) where Kr 1=1 + grexp (−j θr) 2(14) and Kr 2=1−grexp (j θr) 2.(15) In (14) and (15), grand θrare respectively the RX amplitude and phase mismatches. The RX IRR can be written as Ir=|Kr 1|2 |Kr 2|2.(16) From (12), (13) can be expressed as r=Kr 1(hs+n) + Kr 2(hs+n)∗(17) or r=Kr 1hs+Kr 2hs∗+Kr 1n+Kr 2n∗.(18) Bit source bLinear encoder cQAM mapper xUp converter Analog beamformer ssb Transmitter Channel Receiver Analog beamformer rDown converter rb QAM demapper Zero-padding remover l Linear deencoder Fig. 1: System model. By applying (3) to (18), the baseband equivalent received signal at the output of the down-converter can be expressed as r=Kr 1hKt 1x+Kt 2x∗+Kr 2hKt 1x+Kt 2x∗∗ +Kr 1n+Kr 2n∗,(19) or equivalently r=Kr 1Kt 1+Kr 2Kt 2hx+Kr 1Kt 2+Kr 2Kt 1hx∗ +Kr 1n+Kr 2n∗.(20) Thus, the received signal-to-distortion-plus-noise-ratio (SDNR) is given by γ=|Kr 1Kt 1+Kr 2Kt 2|2ρ |Kr 1Kt 2+Kr 2Kt 1|2ρ+|Kr 1|2+|Kr 2|2,(21) where ρstands for the signal-to-noise-ratio (SNR) of the ideal wireless system, i.e., the one that does not suffer from IQI, and can be expressed as ρ=h2Px No .(22) In (22), Pxstands for the average transmission power. The output of the down-converter is inserted to the QAM demapper, which returns the log-likelihood ratios (LLRs) of the received signals. Let D{·} represent the QAM demapper’s operation. Then, the LLRs at the output of the QAM demapper can be written as l=D{r},(23) where the k−th value of lcan be obtained as lk,m = log2 Pr (ck= 1 |rm) Pr (ck= 0 |rm).(24) Note that rmstands for the m−th value of mthat carriers the ckbit. The LLR vector is inputted in the zero-padding remover that outputs only the LLR elements of that corresponds to the coded message. In turn, the output of the zero-pading remover is inserted in the decoder that provides an estimation of the transmitted codeword. III. INTELLIGENT DETECTORS This section focuses on presenting intelligent detectors. Specifically, Section III-A presents a BP-based detector, while Section III-B reports a graph neural network-based detector. A. Belief propagation Let GP= (V ∪ C,E)with V,Cand Erespectively stand for the variable nodes (VNs), check nodes (CNs), edge nodes (ENs) of the Tanner graph GP. Note that each row of Pstands for a CN and each column for a VN. The Tanner graph can be seen as a deep neural network, in which the input layer receives the LLRs. The nodes in the hidden layers represent processing nodes. Each processing node is connected with a number of edges of the Tanner graph. As a consequence, each hidden layer consists of Enodes. The number of nodes is equal to the size of E. The output layer has Nprocessing elements and its responsibility is to provide an estimation of the transmitted codeword. If the number of iterations is set to L, then the number of hidden layer are 2L. The processing element pkof the hidden layer kis associated with the VN, vk, and CN, ck, outputs [23] tk,ek=(lvk+Pe′ k−1,ck−16=cktk−1,e′,for kodd 2 tan−1Qe′′ ,vk−16=vktanh tk−1,e′′ 2,for keven (25) where e′= (vk, ck−1),e′′ = (vk−1, ck)and lvkis the self LLR message of vk. The k−th node of the output layer reports ok=lv2L+X e′ t2L,e′.(26) B. Graph neural network Similar to the BP approach, we consider a Tanner graph Gg= (Vg∪ Fg,Eg), where Vgis the set of the VNs. Each VN stands for a specific element of c.Fgrepresents the set of the CNs, and Egis the set of ENs. If Pi,j = 1, then vg,i is connected to fg,j , where Pi,j is the i, j element of P,vg,i and fg,j are the iand jelements of the sets Vg, and Fg, respectively. To denote the set of all VNs that are connected to the vg,i, we use Vg(vg,i). Similarly, the set of all the CNs that are associated with fg,j is represented by Fg(fg,j). To train the graph neural network, we use a function for the update of the edge messages and another one that update the nodes. Let mv,i,j be the updated message from vito vj, then mv,i,j =gmhwvg,i ||wfg,j ||gmvg,i ,fg,j i,amv,f ,(27) where wvg,i and wfg,j are vectors computed by each node for the VN vg,i and the CN fg,j, respectively. Moreover, amv,f stands for the trainable parameters. Finally, gm(·)represents the parametrized function. Let mu,j,i be the updated message value from the CN ujto the VN vj. Thus, it can be obtain as mf,i,j =gnhwfg,j ||wvg,i ||gmfg,j ,vg,i i,amf,v ,(28) To evaluate the updated value of wvg,i , we apply w ′ vg,i =gnwvg,i || ⊕vi∈Vgmv,i,j||gvg,i av,(29) where avstands for the trainable parameters of the VN. Following a similar approach, the CN values can be updated as w ′ fg,j =gnwfg,i || ⊕fi∈Cgmf,i,j||gfg,i af,(30) where avand afare the VN and CN parameters, respectively. The training process consists of two phases: i) initialization, and ii) iterative optimization. In this paper, we employ the Gloron uniform initalizer [24] to find the initial values of the trainable parameters, and the Adam optimizer to find their (sub)optimal values. As a loss function, the binary crossentropy is applied. IV. RESULTS & DISCUSSION This section presents Monte Carlo simulations that reveal the effectiveness of the ML-based detection approaches in the mitigating the impact of IQI and benchmarks GNN against BP and conventional approaches. The following scenario is considered. A wireless system that operates in the 120 GHz band and employs low-density parity check code (LDPC) with code-rate equal to 0.714. The parity check code has 63 ×45 size. The zero-padding adds 1bit if and only if the length of the codeword is odd. A quadrature phase shift keying modulator is used by the TX and the corresponding demodulator by the RX. Both the TX and RX suffer from IQI with phase error equal to 5o. The BF and GNN respectively perform 20 and 8iterations. Figure 2 depicts the BER as a function of the SNR for different IQI levels and coding/decoding schemes. As benchmarks, 23456789 10−5 10−4 10−3 10−2 10−1 SNR (dB) BER Conventional (Ideal RF) BP (Ideal RF) GNN (Ideal RF) Conventional (It=Ir= 20 dB) BP (It=Ir= 20 dB) GNN (It=Ir= 20 dB) Conventional (It=Ir= 30 dB) BP (It=Ir= 30 dB) GNN (It=Ir= 30 dB) Fig. 2: BER vs SNR for different coding schemes and levels of IQI. the cases of conventional detectors and ideal RF front-end are considered. As expected, for given detector and level of IQI, as the SNR increases, the error performance improves. For example, for the ideal RF front-end with conventional detector case, as the SNR increases from 7to 9 dB, the BER decreases for more than one order of magnitude. For the same SNR variation and for the case in which a conventional detector is employed, but both the TX and RX suffer of IQI with IRR equal to 20 dB, the BER decreases for about two orders of magnitude. Moreover, for conventional decoders and a SNR beyond 3 dB, as the level of IQI increases, i.e., as IRR decreases, the error performance degrades. For instance, for a SNR equal to 7 dB dB, as IRR increases from 20 to 30 dB, the BER decreases from 1.17 ×10−3to 7.88 ×10−4. On the other hand, for either BF or GNN-based decoders and a fixed SNR, as the level of IQI increases, an error performance improvement is observed. For example, for BF-based detector and SNR equals 7 dB, the BER decreases from 1.53 ×10−4 to 4.45 ×10−5, as the IRR decreases from 30 to 20 dB. For the same SNR, but GNN-based detector, the BER decreases from 5.15 ×10−5to 3.52 ×10−6, as the IRR decreases from 30 to 20 dB. As explained in [25], this is due to the TX IQI induced diversity order that can be exploited by the intelligent detectors. Additionally, from this figure, we observe that for ideal RF and a given SNR that is beyond 4.5 dB, GNNbased detectors outperforms both BF-based and conventional detectors. On the other hand, in wireless systems in which their transceiver suffer from IQI, for any given SNR, GNNbased detectors outperforms both BF-based and conventional detectors. For instance, for SNR equals 7 dB and both the IRR of the TX and and RX equal to 20 dB, the GNN-based detector achieves a BER that is equal to 3.52 ×10−6, while, for the same SNR and IRR, BF-based detectors achieve a BER that equals 4.65×10−5, and conventional detectors achieve a BER, which is equal to 1.17 ×10−3. Notice that the GNN-based detectors uses 8iterations, while the BF-based one 20. In other words, with less iterations the GNN-based detector achieves better performance than the BF-based detector. V. CONCLUSIONS In this paper, we presented a GNN-based intelligent detector and demonstrated its ability to prevent IQI. Specifically, we consider a two-way wireless system in which the TX and RX are both susceptible to IQI at high frequencies and in opposite directions. 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