Deep Learning-Assisted Parallel Interference Cancellation for Grant-Free NOMA in Machine-Type Communication

In recent years, to address the challenge we mentioned, a compressed sensing (CS) approach has been popularly used for AD and CE in the coherent scheme [16, 17, 18, 19] and AD and DD in the non-coherent scheme [14, 11, 12]. This approach exploits the sparse activity patterns exhibited by numerous devices. It also provides performance guarantees for AD, CE, and DD in an asymptotic regime where the number of devices and the length of spreading sequences approach infinity while maintaining a fixed ratio between them [11]. Despite these advantages, CS-based techniques have certain limitations and practical issues to address. First, since the length of the spreading sequence is finite due to the limited coherence time, the performance guarantees obtained in the asymptotic regime cannot be applied. This challenge becomes particularly noticeable in critical or low-latency mMTC scenarios, where a short spreading sequence is essential to satisfy the latency requirements [20, 21]. Second, the restricted isometric property (RIP), which is a fundamental requirement for ensuring the accuracy of CS recovery techniques, might be violated when the number of active devices is large [22]. These inherent limitations of the CS approach can result in significant performance degradation for AD, CE, and DD, thereby constraining its broader applicability to diverse grant-free NOMA scenarios.

Successive interference cancellation (SIC) has been recognized as a promising alternative to tackle AD, CE, and DD problems in grant-free NOMA [23, 24, 25]. The main advantage of the SIC lies in its ability to iteratively eliminate interference, thereby enhancing the overall system performance. However, in the absence of accurate knowledge about activity and channel gain, determining the proper cancellation order becomes challenging. This difficulty may result in significant error propagation, leading to a degradation in overall performance [25]. Parallel interference cancellation (PIC) can address the aforementioned problem as this approach does not require a prior knowledge about the channel gain order [26, 27, 28]. Furthermore, the PIC approach can achieve lower latency compared to the SIC approach [28]. Despite these benefits, most PIC techniques to date do not fully account for certain unique characteristics of grant-free NOMA systems, such as non-orthogonal spreading sequences and sparse activity patterns. Neglecting these characteristics can potentially lead to error propagation, thereby limiting the performance gains offered by PIC. Moreover, to the best of our knowledge, none of the existing works have investigated the advantages of a PIC approach for AD, CE, and DD in grant-free NOMA, despite its potential advantages in addressing the limitations of the CS and SIC approaches.

In this paper, we propose a deep learning (DL)-assisted PIC approach for grant-free NOMA systems. Specifically, we develop three PIC frameworks based on the proposed approach: (i) the pilot-only PIC, (ii) the data-aided PIC, and (iii) the non-coherent PIC. The key idea of these frameworks is to eliminate inter-user interference in an iterative and parallel manner inspired by PIC, while jointly optimizing the AD, CE, and DD performances via end-to-end training facilitated by DL. Notably, our frameworks surpass existing techniques in both performance and computational efficiency without requiring prior knowledge of channel gain orders and device activities. The major contributions of this paper can be summarized as follows:

• •

  We present a DL-assisted pilot-only PIC framework for the coherent scheme in grant-free NOMA. The proposed framework comprises multiple stages, each incorporating three modules: CE, IC, and AD modules. Each module performs its specified task, while simultaneously collaborating to enhance the overall performance of the grant-free NOMA system. Furthermore, the proposed framework supports end-to-end training, which can improve overall performance by training complex relationships and dependencies between different modules.

• •

  We extend the pilot-only PIC framework into the data-aided PIC framework. The most prominent feature of this framework is that it utilizes the received data signals as well as the received pilot signals for CE. To this end, in the data-aided framework, both CE and DD modules are included as a part of end-to-end learning, while PIC is applied to both the received pilot and data signals. By incorporating additional information for the CE module, this framework not only improves CE performance but also enhances AD and DD performances compared to the pilot-only PIC framework.

• •

  We extend the pilot-only PIC framework into the non-coherent PIC framework. The distinctive feature of this framework is its ability to simultaneously perform AD and DD, using a single unified module. By doing so, the non-coherent PIC framework effectively addresses the inherent inefficiency of coherent schemes, which necessitate the transmission of pilot signals even when dealing with small data bits.

• •

  Through extensive numerical evaluation, we demonstrate the superior performance of our frameworks compared to traditional techniques. Our simulation results show that, in a scenario with $20$ devices, $18$-length coherence interval, and $2$ data bits, the proposed pilot-only PIC framework achieves more than a $1.25$-fold decrease in both AD and DD errors and a $3.3$dB decrease in CE error compared to the AMP technique. Furthermore, it is demonstrated that, in the same scenario above, the proposed data-aided PIC framework achieves more than a twofold decrease in both AD and DD errors and a $6.9$dB decrease in CE error compared to the AMP technique. Moreover, the proposed pilot-only and data-aided PIC frameworks achieves these improvements with less computational complexities. The effectiveness of the proposed non-coherent PIC framework is also verified when the number of data bits is small. The computational complexities of the proposed frameworks are also analyzed in terms of simulation time.

We consider an uplink grant-free NOMA system comprising $K$ devices and one BS, as illustrated in Fig. 1. Without loss of generality, we assume that both the devices and the BS are equipped with a single antenna. The channel between device $k$ and the BS is modeled as follows [11, 12]:

where $\beta_{k}$ is a large-scale fading coefficient and $h_{k}\sim\mathcal{CN}(0,1)$ is a small-scale fading channel. The channel $g_{k}$ remains constant during the transmission of $\tau$ samples. The device $k$ becomes active with a probability of $\epsilon$ and aims to transmit $J$ data bits $[b_{1},b_{2},\ldots,b_{J}]^{\sf T}\in\{0,1\}^{J}$ to the BS. Under this system, two communication schemes can be considered [10, 11]: (i) coherent scheme and (ii) non-coherent scheme. The overall diagram of these two schemes is illustrated in Fig. 2, with detailed explanations provided below.

• •

  Coherent scheme: In this scheme, each device $k$ has a unique spreading sequence ${\bm{s}}_{k}\in\mathbb{C}^{L}$, known to the BS. All spreading sequences have unit power, i.e., $\|{\bm{s}}_{k}\|_{2}=1,\forall k$, and their lengths are smaller than the number of devices, i.e., $L<K$. When a certain device $k$ is active, it modulates $J$ data bits to $D$ symbols. The $d$th symbol is represented as $x_{k,d}\in\mathcal{C}$, where $\mathcal{C}$ denotes a constellation set, and $d\in\{1,\ldots,D\}$. Subsequently, active device $k$ spreads $d$th data symbol $x_{k,d}\in\mathcal{C}$ using the spreading sequence, generating ${\bm{s}}_{k}x_{k,d}$. During the coherent transmission, each active device $k$ first transmits its spreading sequence ${\bm{s}}_{k}$ as a pilot signal for AD and CE, and then transmits the $D$ sequences $\{{\bm{s}}_{k}x_{k,d}\}_{\forall d}$ as data signals to the BS. Under this transmission strategy, the sequence length $L$ should be less than $\tau/(D+1)$ due to the limited channel coherence time. The received pilot and data signals at the BS are given by

  $\displaystyle{\bm{y}}_{\rm p}$

  $\displaystyle=\sum_{k=1}^{K}\rho{\bm{s}}_{k}a_{k}g_{k}+{\bm{n}}_{\rm p}=\sum_{k=1}^{K}{\bm{s}}_{k}\gamma_{k}+{\bm{n}}_{\rm p},$

  (2)

  and

  $\displaystyle{\bm{y}}_{{\rm d},d}$

  $\displaystyle=\sum_{k=1}^{K}\rho{\bm{s}}_{k}a_{k}g_{k}x_{k,d}+{\bm{n}}_{{\rm d},d}=\sum_{k=1}^{K}{\bm{s}}_{k}\gamma_{k}x_{k,d}+{\bm{n}}_{{\rm d},d},$

  (3)

  respectively, where $\rho$ is the transmission power of the devices, $a_{k}\in\{0,1\}$ is the device activity indicator with $\mathbb{P}[a_{k}=1]=\epsilon$ and $\mathbb{P}[a_{k}=0]=1-\epsilon$, and ${\bm{n}}_{\rm p}$ and ${\bm{n}}_{{\rm d},d}$ represent the additive white Gaussian noise (AWGN) at the BS. After receiving the superimposed signals in (2) and (3), the BS estimates the device activity indicators and the effective channels, subsequently detecting the data symbols $\{x_{k,d}\}_{\forall k,d}$.

• •

  Non-coherent scheme: In this scheme, each device $k$ has a set of spreading sequences $\mathcal{S}_{k}=\{{\bm{s}}_{k,1},{\bm{s}}_{k,2},\ldots,{\bm{s}}_{k,2^{J}}\}$, where each ${\bm{s}}_{k,j}\in\mathbb{C}^{L}$ is known to the BS and has unit power, i.e., $\|{\bm{s}}_{k,j}\|_{2}=1,\forall k,j$. When a device $k$ is active, it chooses one of these sequences based on the $J$ data bits. Then, the active device $k$ transmits this selected spreading sequence to the BS. Under the non-coherent transmission strategy, the sequence length $L$ should be less than $\tau$. The received signal at the BS is given by

  	$\displaystyle{\bm{y}}_{\rm nc}$	$\displaystyle=\sum_{k=1}^{K}\sum_{j=1}^{2^{J}}\rho{\bm{s}}_{k,j}a_{k,j}g_{k}+{\bm{n}}_{\rm nc}$
  		$\displaystyle=\sum_{k=1}^{K}\sum_{j=1}^{2^{J}}{\bm{s}}_{k,j}\gamma_{k,j}+{\bm{n}}_{\rm nc},$		(4)

  where ${\bm{n}}_{\rm nc}$ is the AWGN at the BS, $a_{k,j}\in\{0,1\}$ indicates whether or not $j$th sequence of device $k$ is transmitted, and satisfies

  $\displaystyle\sum_{j=1}^{2^{J}}{a}_{k,j}=\begin{cases}{1,}&{\text{with probability }\epsilon},\\ {0,}&{\text{with probability }1-\epsilon}.\end{cases}$

  (5)

  In this scheme, we use the notation $a_{k}=\sum_{j=1}^{2^{J}}{a}_{k,j}\in\{0,1\}$ to represent the device activity indicator, which aligns with the terminology used in the coherent scheme. After receiving the superimposed signal in (• ‣ II-A), the BS estimates $\{a_{k,j}\}_{\forall k,j}$, and demodulates them to obtain data bits.

In the non-coherent scheme, the BS should take into account $2^{J}K$ sequences, which are far longer than the $K$ sequences in the coherent scheme. Consequently, the non-coherent scheme exhibits expensive computational complexity, particularly when the number $J$ of data bits is large. Despite this computational overhead, the non-coherent scheme offers superior performance in scenarios where $J$ is small, as reported in [11].

Let $\hat{a}_{k}$, $\hat{\gamma}_{k}$, $\hat{b}_{k,j}$ be the estimated activity indicator, effective channel, and $j$th data bit of device $k$, respectively. Then, the performance of AD is evaluated using the false alarm probability $P_{\rm fa}=\mathbb{P}[\hat{a}_{k}=1|a_{k}=0]$ and the miss detection probability $P_{\rm md}=\mathbb{P}[\hat{a}_{k}=0|a_{k}=1]$ [29]. The average error probability is given by

If $P_{\rm fa}$ and $P_{\rm md}$ are equal, one can trivially have that $P_{\rm err}=P_{\rm fa}=P_{\rm md}$. The performance of CE is evaluated by the normalized mean-squared error (NMSE) defined as

The performance of DD is assessed using the bit error rate (BER) defined as

where, if $a_{k}=0$ and $\hat{a}_{k}=1$, $J$ bit errors are considered to occur for device $k$, and vice versa.

A major challenge in realizing grant-free NOMA in Sec. II-A is the limited length of the spreading sequence. This limitation is inevitable due to the finite channel coherence time and scarce radio resources available for data transmission. This challenge becomes even more crucial in scenarios involving critical or low-latency mMTC, where a small length of spreading sequences is necessary to meet stringent latency requirements. Due to this inherent limitation in spreading sequence length, it is impossible to guarantee the optimality of existing AD, CE, and DD techniques based on the CS approach. Another key challenge comes from the lack¹¹1Typically, in grant-free NOMA, attaining the above information may require additional communication overhead between the BS and each device, which can lead to increased resource consumption and potential delays in the communication process. of the knowledge of channel gain orders. This can lead to significant error propagation of the SIC approach for AD, CE, and DD techniques. Although existing PIC-based CE techniques do not require prior knowledge about channel gain orders, they can still suffer from significant error propagation in grant-free NOMA due to non-orthogonal spreading sequences and the absence of accurate knowledge about device activities. To establish grant-free NOMA as a viable solution, it is crucial to devise a comprehensive framework overcoming the aforementioned challenges without compromising the performance of AD, CE, and DD.

The proposed pilot-only PIC framework includes $T$ stages, each consisting of $K$ CE modules and $K$ IC modules. Following these stages, we integrate $K$ AD modules that estimate the activity of each device in a parallel manner. The high-level illustration of the proposed framework is presented in Fig. 3, while detailed explanations of each component of our framework are provided below.

In the grant-free NOMA, the CE and AD problems are categorized as regression and classification problems. These two tasks are closely related, as both tasks rely on the same received signal and have dependencies stemming from activity indicators. To accurately estimate both $\{\tilde{\bm{\gamma}}_{k}\}_{\forall k}$ and $\{a_{k}\}_{\forall k}$, we employ a joint loss function that can optimize the performance of both regression and classification tasks.

Firstly, we use the regression loss function to measure the discrepancy between the estimated and true channels.

where $w_{t}$ is the weight associated with each stage. In the regression loss in (17), we leverage the mean absolute error (MAE) instead of the more commonly used mean squared error (MSE) in regression tasks [30, 31]. Our motivation behind this choice comes from the sparse characteristic of effective channels $\{\tilde{\bm{\gamma}}_{k}\}_{\forall k}$ in grant-free NOMA systems. Specifically, this characteristic implies that while the majority of devices exhibit zero channel gains, a small subset has much higher channel gains, resulting in outliers within the dataset. Traditionally, the MSE loss serves as a prevalent metric used to evaluate the performance of CE [30, 31]. However, in the presence of outliers in effective channels, the MSE loss may prioritize minimizing errors, particularly for these outliers, while potentially neglecting the majority of the data [32]. In contrast, MAE is known to be robust to outliers [32], aligning well with the characteristics of grant-free NOMA scenarios. Therefore, our decision to use MAE enhances the overall performance and robustness of our framework while mitigating error propagation.

Next, we present the classification loss function which measures the difference between the activity scores and the true activity indicators as follows:

This loss function is commonly known as the binary cross-entropy (BCE) loss function and is highly effective for binary label training. By adopting the BCE loss function, we can effectively train $K$ AD modules as well as the entire framework. Combining the loss functions in (17) and (18), we obtain the joint loss function:

where $\lambda\in(0,1)$ is a hyper-parameter determined by the relative importance of CE performance compared to AD performance.

Our proposed framework offers end-to-end training facilitated by the IC module and joint loss function (19), enhancing performance by capturing complex inter-module relationships. Our framework also addresses the vanishing gradient problem by computing MAE loss at each stage $t$, not just the final stage. Additionally, our framework integrates joint training for regression (CE) and classification (AD) tasks. This feature not only streamlines the training pipeline but also reduces the size of the DNN compared to strategies that separate regression and classification tasks.

The proposed data-aided PIC framework includes $T$ uniform stages, each consisting of $K$ CE modules, $K$ IC modules, and $K$ DD modules. A notable distinction between the data-aided PIC framework and the pilot-only PIC framework lies in the inclusion of the DD module and the exclusion of the AD module. The DD module is responsible for estimating both the device activity indicator $a_{k}$ and the data symbols $\{x_{k,d}\}_{\forall d}$. Another notable distinction is the input of the CE module, which includes $\{{\bm{y}}_{{\rm d},d}\}_{\forall d}$ as well as ${\bm{y}}_{\rm p}$ to extract the common channel information between received signals. The high-level illustration of the $t$th stage of the data-aided PIC framework is depicted in Fig. 4, with detailed explanations provided below.

The proposed non-coherent PIC framework includes $T$ uniform stages, each consisting of $K$ CE modules, $K$ IC modules, and $K$ DD modules. Unlike the pilot-only and data-aided frameworks, this framework is specifically designed for the non-coherent scheme introduced in Sec. II-A. The high-level illustration of the $t$th stage of the non-coherent PIC framework is depicted in Fig. 5.

In our simulations, we consider an uplink grant-free NOMA system where $20$ devices are distributed uniformly within a cell, with their locations ranging from $50$ m to $500$ m from the center of the cell²²2In mMTC scenarios with an extremely large number of devices, to efficiently accommodate these devices, we can consider the strategies to form groups for these devices and allocate different communication resources to each group, as discussed in [33]. This strategy is especially effective in critical or low-latency mMTC scenarios where the sequence length $L$ should be sufficiently small [20, 21]. . The devices transmit spreading sequences with a power of $\rho=20$ dBm and a bandwidth of $100$ kHz [34]. For data transmission, the devices modulate the data bits using BPSK. The power spectral density of the AWGN at the BS is $-169$ dBm/Hz, and $\beta_{k}$ follows the path-loss model $-128.1-36.7\log_{10}d_{k}$ dB, where $d_{k}$ is the distance between device $k$ and the BS [19, 12]. We generate each spreading sequence ${\bm{s}}_{k}$ or ${\bm{s}}_{k,j}$ from a Gaussian distribution $\mathcal{CN}({\bm{0}},\frac{1}{L}{\bm{I}})$ and subsequently normalize it to ensure $\|{\bm{s}}_{k}\|_{2}=1$ or $\|{\bm{s}}_{k,j}\|_{2}=1$. It should be noted that, unlike CS algorithms, the proposed frameworks do not impose any specific structure constraints on the spreading sequences.

For training, we first generate training data samples with $\epsilon=0.25$. Then, we perform post-processing on both the training and test datasets to ensure that every data sample contains at least one active device, which is essential for evaluating NMSE in (7). We train our model with a batch size of $1024$ for $100$ epochs using the Adam optimizer in [35] with a learning rate of $0.001$, $\lambda=0.5$, and $w_{t}=1,\forall t$. All the AD, CE, and DD modules in our frameworks consist of two hidden layers, each comprising $64$ neurons and rectified linear unit (ReLU) activation functions.

For comparison, in the coherent scheme, we consider the approximate message passing (AMP) in [18], the least absolute shrinkage and selection operator (LASSO) in [36], and an FCNN. In the LASSO framework, the objective is to minimize the following loss function:

where ${\bm{S}}=[{\bm{s}}_{1},\ldots,{\bm{s}}_{K}]$, and $\nu$ is set to $0.05$. In the AMP and LASSO frameworks, device activity is determined based on whether the estimated channel gain is larger than a predefined threshold. The FCNN framework involves two distinct DNNs, each performing CE and AD tasks. These networks comprise $10$ hidden layers with $2048$ neurons per layer and ReLU activation functions. Other training parameters are the same as those of our framework. In the coherent scheme, all frameworks employ the MRC method in (III-A4) to detect the symbols $\{x_{k,d}\}_{\forall k,d}$. For comparison, in the non-coherent scheme, we modify the AMP and LASSO frameworks to ensure the constraint in (5), similar to [11, 12]. We refer to these modified frameworks as Non-coherent AMP and Non-coherent LASSO, respectively.

In Fig. 6, we compare the false alarm probability $P_{\rm fa}$ and miss detection probability $P_{\rm md}$ of different grant-free NOMA frameworks with the coherent scheme for various values of $\epsilon$. For the proposed frameworks, we set $L=6$, $T=4$ for the pilot-only PIC, and $T=2$ for the data-aided PIC. Their simulation times are comparable but significantly lower than those of the other frameworks. Fig. 6 shows that the proposed pilot-only and data-aided PIC frameworks outperform all existing frameworks regardless of $\epsilon$, even though they were trained with a fixed $\epsilon=0.25$. In particular, the performance gap between the proposed and existing frameworks increases as $\epsilon$ decreases. This result not only highlights the superiority of the proposed frameworks but also demonstrates their effective applicability to diverse grant-free NOMA scenarios. Fig. 6 also shows that the proposed data-aided PIC framework achieves the lowest error rates for all $\epsilon$. This result suggests that the data-aided PIC framework effectively extracts the common channel information shared between the received pilot and data signals. Meanwhile, our frameworks exhibit higher AD accuracy compared to the LASSO framework. This improvement can be attributed to our optimization strategy which relies on the joint loss function in (19), differing from the loss function in (40).

In Fig. 7, we compare the average error probability $P_{\rm err}$, NMSE, and BER of different grant-free NOMA frameworks with the coherent scheme for various values of $\tau$. In this simulation, we set $\epsilon=0.1$, $J=2$, $L=\tau/3$, and determine appropriate thresholds to ensure the condition $P_{\rm err}=P_{\rm fa}=P_{\rm md}$. The BER is then evaluated under this specified condition. Fig. 7 shows that $P_{\rm err}$, NMSE, and BER of all frameworks decrease as $\tau$ increases. Fig. 7 also demonstrates that the proposed pilot-only and data-aided PIC frameworks outperform other frameworks across all performance metrics. The performance gaps between the AMP and proposed frameworks significantly increase as $\tau$ decreases. Under this result, we note that the CS-based AMP technique should satisfy the RIP condition to ensure the recovery performance. Meanwhile, the performance gaps between the FCNN and the proposed frameworks indicate that incorporating communication knowledge from the PIC into DL is more effective than treating DL as a black box. In particular, the NMSE results demonstrate that it is difficult to optimize a naive FCNN without knowledge of the communication.

In Fig. 8, we compare $P_{\rm err}$ and BER of different grant-free NOMA frameworks with the non-coherent scheme for various values of $\epsilon$. In this simulation, we set $\tau=10$, $K=10$, $T=4$, $J=2$, and $\rho=10$ dBm. Fig. 8 shows that the proposed non-coherent PIC framework outperforms the existing frameworks. In particular, the performance gap in terms of both $P_{\rm err}$ and BER between the proposed and existing frameworks increases as $\epsilon$ decreases, even though it was trained with a fixed $\epsilon=0.25$.

In Fig. 9, we compare $P_{\rm err}$ and BER of different grant-free NOMA frameworks with the non-coherent scheme for various values of transmission power $\rho$. In this simulation, we set $\epsilon=0.2$, $\tau=10$, $K=10$, $T=4$, and $J=2$. Fig. 9 shows that the proposed framework achieves the lowest $P_{\rm err}$ and BER across varying $\rho$. This result suggests the robustness of the proposed framework for various values of $\rho$.

In Table I, we compare the coherent and non-coherent schemes using different grant-free NOMA frameworks. In this simulation, we set $\epsilon=0.2$, $\tau=10$, $K=10$, and $\rho=15$ dBm. For the proposed frameworks, we set $T=7$ for the pilot-only PIC, $T=3$ for the data-aided PIC, and $T=4$ for the non-coherent PIC. Their simulation times are comparable but significantly lower than those of the other frameworks. It should be noted that the sequence length is determined differently between the coherent and non-coherent schemes under the fixed $\tau=10$. Specifically, for the coherent scheme, it is calculated as $L=\lfloor\tau/(J+1)\rfloor$, and for the non-coherent scheme, it is set to $L=\tau=10$. Table I shows that the grant-free NOMA frameworks with the non-coherent scheme achieve lower BER compared to those with the coherent scheme. In particular, the proposed non-coherent PIC framework achieves the lowest error rates. This result suggests that the non-coherent scheme is particularly useful for scenarios transmitting a small number of data bits [11].