# GRACE: Gradient-based XAI Scheme for Channel Estimation in Wireless Communications

Abdul Karim Gizzini, Yahia Medjahdi, Mouna Ben Mabrouk

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Abdul Karim Gizzini, Yahia Medjahdi, Mouna Ben Mabrouk. GRACE: Gradient-based XAI Scheme for Channel Estimation in Wireless Communications. 2024 IEEE International Mediterranean Conference on Communications and Networking (MeditCom), IEEE, Jul 2024, Madrid, France. pp.572-577, 10.1109/MeditCom61057.2024.10621232. hal-04973925

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# GRACE: Gradient-based XAI Scheme for Channel Estimation in Wireless Communications

Abdul Karim Gizzini*, Yahia Medjahdi§, Mouna Ben Mabrouk*

*SogetiLabs Research and Innovation (part of Capgemini), F-92130, Issy Les Moulineaux, France

§ IMT Nord Europe, Institut Mines Télécom, Centre for Digital Systems, F-59653 Villeneuve d'Ascq, France

Email: {abdul.gizzini, mouna.benmabrouk}@sogeti.com, yahia.medjahdi@imt-nord-europe.fr

**Abstract**—The support of artificial intelligence (AI) is a key element in future 6G networks, where the concept of native AI will be introduced. Moreover, AI is widely employed in different critical applications such as autonomous driving and medical diagnosis. In such applications, using AI as black-box models is risky and challenging since the logic behind the decision-making methodology of the AI model is unclear. Consequently, it is crucial to understand and trust the decisions taken by these models to ensure their safe and efficient deployment. Tackling this issue can be achieved by developing explainable AI (XAI) schemes that aim to explain the logic behind the black-box model behavior. One solution to achieve this is to identify the relevant model inputs that are contributing the most towards its decisions. In this context, this paper proposes a gradient-based XAI scheme for channel estimation denoted as GRACE that aims to classify the model inputs using gradient backpropagation. Unlike perturbation-based input filtering, the proposed GRACE filtering strategy is based on the internal model architecture. Hence it provides a more efficient and trustworthy input classification. Performance evaluation in terms of bit error rate (BER) shows that identifying the relevant model inputs by the proposed GRACE leads to a better BER performance in comparison to the recently proposed perturbation-based scheme.

**Index Terms**—6G, AI, XAI, channel estimation, input filtering, perturbation-based, gradient-based

## I. INTRODUCTION

Artificial intelligence (AI) is expected to play an important role in the overall design of future 6G networks aimed to be native AI networks [1]. Indeed, distributed AI will be embedded within the functionality of all layers to support demands for high data rates and low latency-critical applications. According to [2], the AI-enabled intelligent architecture for 6G networks consists of several layers including the intelligent sensing layer. In the latter, accurate environment monitoring is critical for 6G smart applications like autonomous driving [3].

A robust PHY layer design consists in a reliable intelligent sensing layer. However, estimating the wireless channel highly impacts such reliability. Actually, a precisely estimated channel influences the follow-up equalization and decoding operations at the receiver [4]. In this context, channel estimation is challenging among different PHY layer applications due to the doubly selective nature of the channel in mobile applications.

Recently, deep learning (DL) has been employed in the PHY layer of wireless communications [5], [6], including channel estimation [7]–[12] to overcome the limitations of conventional channel estimation schemes. These limitations

lie in the poor generalization ability of conventional schemes and high computational complexity. In addition, conventional channel estimators require a large number of training sequences besides the prior knowledge of the channel and noise statistics as the case in the least square (LS) and linear minimum mean-squared error (LMMSE) schemes, respectively. In contrast, DL-based channel estimation schemes provide a good performance-complexity trade-off, robustness, and generalization in different scenarios.

Among different DL networks, feed-forward neural networks (FNNs) have been widely used in literature as a post-processing unit added to some conventional channel estimation. Even though these FNN-based channel estimation schemes can achieve good performance, but still the logic behind choosing the FNN inputs as well as the used architecture, i.e., the number of hidden layers and neurons as well as the suitable FNN inputs is still unclear. Therefore, the majority of the FNN-based channel estimation schemes consider a huge number of neurons within the employed FNN model [13], resulting in a considerable complexity cost. Moreover, these models take several inputs to perform the channel estimation task such as the received signal, received pilots, and previously estimated channel, among others.

Regardless of the recorded success in providing good performance, FNN-based channel estimation schemes encounter mainly two issues: (1) Identifying the relevant inputs: filtering the most important FNN inputs is crucial for optimizing the performance of the employed FNN model. (2) Trustworthy deployment: The majority of the utilized DL models are black-box models with an ambiguous decision-making methodology. Therefore, researchers and industrial leaders are not able to trust the employment of these models in real-case sensitive applications. Therefore, using explainable models, known as explainable AI (XAI) models, is important for safe and efficient AI deployment. It is worth mentioning that developing XAI schemes ensure the transparency of the employed black-box model, as well as optimizing the model input and architecture accordingly, leading to safe, efficient, and reliable AI employment in real-case scenarios.

The majority of XAI-related works in the literature are surveys and reviews about the guidelines and importance of using XAI in wireless communications [14]. We note that XAI schemes have been employed in different applications such as network optimization [15], open radio access network (O-
RAN) architecture [16], resource allocation [17], and secure IoT [18]. These XAI schemes might be difficult to be adapted to the physical layer applications because in such applications there are no clear discriminative features within the model inputs. To fill this gap, a perturbation-based XAI scheme denoted as XAI-CHEST [19] has been proposed recently. The main objective of XAI-CHEST is to provide the interpretability of the FNN models employed in the channel estimation tasks by identifying the relevant and irrelevant FNN inputs contributing to the model decision-making behavior using an external perturbation-based strategy. However, relying on external interpretability is not enough since the interpretability is provided only by analyzing the impact of inducing noise at the FNN input without considering the internal model architecture.

This paper proposes a new gradient-based XAI scheme for channel estimation denoted as GRACE. The proposed scheme aims to classify the FNN inputs based on the internal gradients obtained by the layer-wise relevance propagation (LRP) XAI technique [20]. Unlike the XAI-CHEST scheme, the proposed input filtering strategy is based on the internal FNN model architecture, leading to more efficient and trustworthy input classification. Simulation results reveal that the proposed GRACE scheme provides a better input filtering strategy than the XAI-CHEST scheme. It should be noted that FNN-based channel estimation schemes are discussed among other DL-based channel estimation schemes since we are targeting low-complex and low-latency DL-based solutions.

The remainder of this paper is organized as follows: Section II presents the system model. Section III illustrates the proposed scheme. Section IV discusses the performance evaluation of the proposed scheme and the XAI-CHEST scheme in terms of bit error rate (BER). Finally, Section V concludes the paper.

## II. SYSTEM MODEL

In this paper, we consider orthogonal frequency-division multiplexing (OFDM) based transmission where the i-th transmitted frequency-domain OFDM symbol $x_i \in \mathbb{C}^{K_{on} \times 1}$ can be expressed as follows:

$$x_i[k] = \begin{cases} x_{i,d}[k], & k \in \mathcal{K}_d \\ x_{i,p}[k], & k \in \mathcal{K}_p \\ 0, & k \in \mathcal{K}_n \end{cases} \quad (1)$$

where $0 \le k \le K-1$ denotes the subcarrier index. We note that $K_{on}$ useful subcarriers are used where $K_{on} = K_p + K_d$. $x_{i,p} \in \mathbb{C}^{K_p \times 1}$ and $x_{i,d} \in \mathbb{C}^{K_d \times 1}$ represent the allocated pilot symbols and the modulated data symbols at a set of subcarriers denoted $K_p$ and $K_d$, respectively. $K_n = K - K_{on}$ denotes the null guard band subcarriers.

The received frequency-domain OFDM symbol denoted as $r_i \in \mathbb{C}^{K_{on} \times 1}$ is expressed as follows:

$$r_i[k] = h_i[k]s_i[k] + e_i[k] + v_i[k], \quad k \in \mathcal{K}_{on}. \quad (2)$$

$h_i \in \mathbb{C}^{K_{on} \times 1}$ refers to the frequency response of the doubly-selective channel at the i-th OFDM symbol and k-th subcarrier. $v_i$ signifies the additive white Gaussian noise (AWGN) of variance $\sigma^2$. $e_i$ refers to the Doppler-induced inter-carrier interference derived in [21] and can be expressed as:

$$e_i[k] = \frac{1}{K} \sum_{q=0}^{K-1} \sum_{q \neq k}^{K-1} h_i[q, n] e^{-j2\pi \frac{n(k-q)}{K}} s_i[q]. \quad (3)$$

## III. PROPOSED GRADIENT-BASED SCHEME

This section presents the proposed GRACE input filtering strategy where the LRP XAI technique is employed to compute the input relevance scores. LRP is an explanation technique applicable to different DL networks. LRP is initially inspired by Taylor decomposition [22] where the objective is to assign a relevance score to each neuron within the model architecture by backward propagating the model prediction, i.e., the FNN-based estimated channel in our case. This propagation is subject to a conservation property following Kirchoff's conservation laws [23], where the relevance of the current layer must be redistributed to the lower layer in equal amounts.

Let $1 \le m \le M$ and $1 \le n \le N$ denote the indices of neurons within two consecutive layers $l$ and $l+1$ of the FNN, where $M$ and $N$ represent the total number of neurons in each layer, respectively. Propagating relevance scores at a given layer onto neurons of the lower layer is achieved by applying the rule:

$$\text{LRP}_{m}^{(l)} = \sum_{n=1}^{N} \frac{z_{mn}}{\sum_{m=1}^{M} z_{mn}} \text{LRP}_{n}^{(l+1)}. \quad (4)$$

Here, $z_{mn}$ denotes the contribution of neuron $m$ in the $l$-th layer to the relevance of neuron $n$ in the $l+1$ layer. In this work, we employ the basic zero rule of LRP. Hence, $z_{mn}$ can be expressed as follows:

$$z_{mn} = a_m w_{mn}, \quad (5)$$

where $a_m$ and $w_{mn}$ denote the activation and weight of the m-th neuron. We note that the $\text{LRP}_m^{(l)}$ can be calculated by applying the following steps [20]:

1) The denominator in (4) is calculated by applying a forward pass on the considered layer, such that:

$$d_n = \sum_{m=1}^{M} a_m w_{mn}, \quad \forall n. \quad (6)$$

2) Element-wise division, such that:

$$q_n = \frac{\text{LRP}_n^{(l+1)}}{d_n}, \quad \forall n. \quad (7)$$

3) The term $\sum_{n=1}^{N} w_{mn}$ within the numerator of (4) is calculated by applying a backward pass such that:
<table><caption>Table I
CHARACTERISTICS OF THE EMPLOYED CHANNEL MODELS FOLLOWING JAKE'S DOPPLER SPECTRUM.</caption><thead><tr><th>Channel model</th><th>Average path gains [dB]</th><th>Path delays [ns]</th></tr></thead><tbody><tr><td>VTV-EX</td><td>[0, 0, 0, -6.3, -6.3, -25.1, -25.1, -22.7, -2.27, -22.7]</td><td>[0, 1, 2, 100, 101, 200, 201, 202, 300, 301, 302]</td></tr><tr><td>VTV-SDWW</td><td>[0, 0, -11.2, -11.2, -19, -21.9, -25.3, -25.3, -24.4,-28, -26.1, -26.1]</td><td>[0, 1, 100, 101, 200, 300, 400, 401, 500, 600, 700, 701]</td></tr></tbody></table>

$$c_m = [\nabla(\sum_{n=1}^{N} d_n(a)q_n)]_m, \forall m. \quad (8)$$

where $a = [a_1, a_2, ..., a_m]$ is the vector of lower-layer activations.

4) The final step is the element-wise product, such that:

$$LRP_m^{(l)} = a_m c_m, \forall m. \quad (9)$$

We note that the FNN forward pass (6) serves to enforce the conservation property expressed as:

$$\sum_{m=1}^{M} \text{LRP}_m = \sum_{n=1}^{N} \text{LRP}_n = \hat{h}_{\Phi_i}'' \quad (10)$$

Where $\hat{h}_{\Phi_i}'' \in \mathbb{R}^{2K_{\text{on}} \times 1}$ denotes the FNN-based estimated channel where a prior conventional estimated denoted as $\Phi$ is applied. The LRP propagation starts by considering the $\hat{h}_{\Phi_i}''$ as the relevance scores of the FNN output layer and terminates once reaching the FNN input denoted as $\hat{h}_{\Phi_i}' \in \mathbb{R}^{2K_{\text{on}} \times 1}$, where the relevance of the FNN inputs $s_{g-\Phi_i}' \in \mathbb{R}^{2K_{\text{on}} \times 1}$ are computed. Hence, the FNN input filtering can be applied using the LRP input relevance scores. It is worth mentioning that the proposed GRACE input filtering strategy computes the relevant scores of the FNN inputs using the backpropagation of the FNN output. Hence, it provides better relevance scores than the perturbation-based input filtering strategy, where the relevance scores $s_{p-\Phi_i}' \in \mathbb{R}^{2K_{\text{on}} \times 1}$ are assigned by an external model without any interaction with the internal architecture of the studied model. We note that a better performance can be achieved by filtering out the irrelevant FNN inputs, i.e., having low relevance scores as we discuss in the next section.

## IV. SIMULATION RESULTS

This section illustrates the performance evaluation of the proposed GRACE scheme in comparison to the XAI-CHEST scheme. The STA-FNN channel estimation scheme [8] is used as a case study, i.e., $\Phi = \text{STA}$. BER performance is evaluated by taking into consideration several subcarrier configurations, i.e., full, relevant, and irrelevant. The performance evaluation is performed according to the frequency selectivity of the channel, where QPSK modulation is employed. We note that the considered channel models [24] as shown in Table I are: (i) Low-frequency selectivity (LFS), where VTV Expressway (VTV-EX) scenario is employed. (ii) High-frequency selectivity (HFS), where VTV Expressway Same Direction with Wall (VTV-SDWW) scenario is considered. In both scenarios,

Doppler frequency $f_d = 1000$ Hz is considered. Moreover, the considered STA-FNN scheme is trained using a 100,000 OFDM symbols dataset, split into 80% training, and 20% testing. ADAM optimizer is used with a learning rate $lr = 0.001$. Moreover, the training data are splited into 128 batches trained for 500 epochs. Simulation parameters are based on the IEEE 802.11p standard [4], where $I = 50$ OFDM symbols, and $K = 64$ subcarriers are used with 156.25 kHz subcarrier spacing. Moreover, the comb pilot allocation is used, hence, $K_p = 4$, $K_d = 48$, $K_n = 12$.

Figure 1 shows the performance evaluation of the proposed GRACE FNN inputs filtering strategy in the considered LFS and HFS channel models employing QPSK modulation order. We note that the LRP relevance scores are calculated using (4) where $\hat{h}_{\text{STA}_i}''$ is back-propagated to obtain the relevance scores of the STA-FNN inputs $s_{g-\text{STA}_i}'$ which correspond to the conventional STA estimated channel. Accordingly, the elements of $\hat{h}_{\text{STA}_i}'$ are assigned with either negative, neutral, or positive relevance indicating their contribution in estimating $\hat{h}_{\text{STA}_i}''$. Starting with the LFS channel model, we can notice from Figure 1(a) that the proposed GRACE scheme divides $\hat{h}_{\text{STA}_i}'$ into 20 negative, 10 neutral, and 22 positive elements including 4 high positive elements. We note that these high positive elements represent the relevance scores of the estimated channel at the pilot subcarriers. This signifies that the estimated channels at the pilot subcarriers are the most relevant STA-FNN inputs. Whereas, the other elements have lower or negative contributions in estimating $\hat{h}_{\text{STA}_i}''$. To further study the impact of the assigned LRP relevance scores, the BER performance is analyzed considering different combinations, as shown in Figure 1(b). We can notice that employing only $\hat{h}_{\text{STA}_i}'$ having positive relevance improves the performance by 5 dB gain in terms of signal-to-noise ratio (SNR) for a BER = $10^{-6}$ compared to giving the full $\hat{h}_{\text{STA}_i}'$ vector<fnref n="1" /> as an input to the STA-FNN model. It is worth mentioning that the same performance can be achieved by employing only the estimated channel at the pilots. Hence, the low positive subcarriers have no impact on improving the STA-FNN model performance. In addition to that, employing the positive and neutral elements leads to a close performance as employing the full $\hat{h}_{\text{STA}_i}'$ vector. Therefore, they negatively impact the performance of the studied STA-FNN model. Moreover, using the negative elements significantly degrades the BER performance, which is the case also when employing the negative and neutral elements.

<footnote n="1">¹The full $\hat{h}_{\text{STA}_i}'$ includes all the subcarriers, i.e., $K_{\text{on}} = 52$.</footnote>
<figure><img src="image_2.png" /><figcaption>(a) LFS channel model: subcarriers vs LRP relevance scores</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(b) LFS channel model: BER performance</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(c) HFS channel model: subcarriers vs LRP relevance scores</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(d) HFS channel model: BER performance</figcaption></figure>

<caption>Figure 1. Performance evaluation of the proposed GRACE FNN inputs filtering strategy employing QPSK modulation.</caption>

Concerning the HFS channel model, we can notice from Figure 1(c) that the number of negative and neutral elements decreases whereas the number of positive elements increases in comparison to the LFS channel model. This is due to the fact that in the HFS channel model, the channel frequency selectivity is higher, thus, the STA-FNN model needs more inputs to accurately estimate the $\hat{h}''_{STA_i}$ channel. Moreover, the gap between the low positive and high positive elements decreases in comparison to the LFS channel model. Figure 1(d) shows the BER performance employing different combinations. We can notice that employing only the pilots performs almost similar to employing the full $\hat{h}''_{STA_i}$ vector, which is also the case where the positive elements are employed. However, employing the $\hat{h}'_{STA_i}$ elements having the highest 15 relevance scores, i.e., the relevance score is greater than 1, outperforms the full case by around 3 dB gain in terms of SNR for a BER = $10^{-4}$. Therefore, we can conclude that employing only the pilots as STA-FNN inputs is enough to improve the

BER performance in the LFS channel model due to the slow variation of the channel. However, as the channel selectivity becomes more challenging, more inputs must be given to the STA-FNN model to improve the BER performance.

Figure 2 illustrates the comparison between the proposed GRACE scheme and the XAI-CHEST scheme. Recall that the XAI-CHEST is an external perturbation-based scheme, where adjustable noise weights $s'_{p-STA_i}$ are assigned as relevance scores to the STA-FNN inputs. This noise is then used to classify the STA-FNN inputs, where subcarriers assigned higher noise are irrelevant and vice-versa. Figures 2(a) and 2(c) show the noise weights and the complement of the normalized LRP scores obtained by the XAI-CHEST and the proposed GRACE schemes, respectively. Hence, lower relevance scores correspond to more relevant STA-FNN inputs. We can notice from Figure 2(a) that the relevance scores distribution is shifted towards one, where the majority of subcarriers are assigned a score equal to one. This signifies that these subcarriers
<figure><img src="image_6.png" /><figcaption>(a) LFS channel model: Relevance distribution</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) LFS channel model: BER performance</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) HFS channel model: Relevance distribution</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(d) HFS channel model: BER performance</figcaption></figure>

<caption>Figure 2. Performance evaluation of the proposed GRACE scheme compared to the XAI-CHEST scheme [19].</caption>

are not important for the decision-making methodology of the studied STA-FNN model. This is because, in the LFS channel model, the channel does not vary rapidly among the subcarriers, thus, few subcarriers are needed to accomplish the channel estimation task by the STA-FNN model. It is worth mentioning that the pilot subcarriers are assigned the lowest score in both schemes which reveals that the STA-FNN model is not able to neglect the estimated channels at the pilots, where considering them is crucial to achieve high estimation accuracy. This is consistent with the channel estimation rules, where the channel estimates at the pilots are very close to the ideal channel estimation. In the HFS channel model, the relevance scores distribution is shifted towards zero as shown in Figure 2(c). This signifies that the STA-FNN model lacks certainty regarding whether the subcarriers can be disregarded or not. However, using the proposed GRACE scheme allows to better select the relevant STA-FNN inputs where $R = 15$ improves the BER performance compared

to the selected relevant inputs by the XAI-CHEST scheme, i.e., $R = 28$, as illustrated in Figure 2(d). Moreover, the proposed GRACE scheme has a high variance in distributing the relevance scores across the $\hat{h}'_{STA_i}$ elements. Hence, the proposed GRACE scheme provides a more efficient input filtering strategy compared to the XAI-CHEST scheme. This is due to the fact that the relevance scores in the XAI-CHEST scheme are obtained by an external model, whereas, the proposed scheme computes the relevance scores using the backpropagation of the STA-FNN estimated channel through its internal architecture. Thus, offering a more trustworthy relevance score assignment in comparison to the XAI-CHEST scheme. However, we note that a performance-threshold optimization is needed to accurately select the most relevant subcarriers in the HFS channel model, but this optimization problem is more feasible using the proposed GRACE scheme due to the high variance in the distribution of the relevance scores.
V. CONCLUSION

Ensuring the transparency and trustworthiness of AI is a
crucial need for its efficient and safe deployment in critical
applications such as autonomous driving. This goal could be
achieved by studying the sensitivity of the model to the input
data and feeding only the relevant ones in the training process.
In this context, this paper proposes the GRACE scheme that
classifies the model input using the LRP XAI technique.
The GRACE scheme provides a better input filtering strategy
than the recently proposed XAI-CHEST schemes since the
relevance scores are calculated considering the internal model
architecture. This would guarantee that the backpropagation
of the model output to the model input serves as the basis
for the input classification. As a future perspective, LRP
rules will be exploited to further optimize the architecture of
the employed model. Therefore, better optimizing the studied
model performance.

ACKNOWLEDGEMENT

This work was partially supported by the framework of the
EcoTrain Project through the French Agency for Ecological
Transition (ADEME) under Grant ADEME-"Digitalisation et
Décarbonation du Transport Ferrovaire".

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